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UNIVERSITA' DEGLI STUDI DI PADOVA

Sede Amministrativa: Università degli Studi di Padova

Dipartimento di TECNICA E GESTIONE DEI SISTEMI INDUSTRIALI

SCUOLA DI DOTTORATO DI RICERCA IN INGEGNERIA INDUSTRIALE INDIRIZZO PROGETTAZIONE MECCANICA E INGEGNERIA MOTOCICLISTICA CICLO XXI

NOTCH MECHANICS UNDER ELASTIC AND ELASTIC-PLASTIC CONDITIONS

Direttore della Scuola: Ch.mo Prof. Paolo Francesco Bariani Supervisore: Ch.mo Prof. Paolo Lazzarin

Dottorando: Dr. Michele Zappalorto

Ringraziamenti Molte sono le persone che in questi tre anni, in modo diverso, mi sono state vicine e la cui presenza ha reso indimenticabili numerose giornate. Per prima cosa devo ringraziare il Professor Lazzarin, per la fiducia che ha dimostrato nei miei confronti, per il costante supporto con cui mi ha affiancato durante l’attività di ricerca, i preziosi consigli e incoraggiamenti, per l’entusiasmo e la passione che è riuscito a trasmettermi e che sono stati il motore primo di tutti i risultati contenuti in questo lavoro. Lo ringrazio di cuore per le possibilità che mi ha dato e per quelle mi sta dando. Un ringraziamento speciale va al mio caro amico e collega Filippo. Lo ringrazio per tutti i momenti di amicizia e scienza che ci hanno legato e che mi hanno permesso di crescere come uomo e come ricercatore. Ringrazio il Professor Quaresimin, per la fiducia che ha dimostrato nei miei confronti, che non mancherò di ricambiare con impegno e dedizione. Ringrazio mamma e papà, che con i loro sacrifici mi hanno permesso di intraprendere la difficile e affascinante strada della conoscenza. Li ringrazio per aver condiviso con me, non solo le gioie per i piccoli traguardi raggiunti, ma anche i momenti di stress e di tensione. Ringrazio Lisa che ha sempre dimostrato di apprezzare e ricambiare il mio affetto, anche quando non sono stato in grado di manifestarlo come avrei voluto. Un grazie davvero unico va a Silvia, per aver sempre creduto in me, per essermi stata vicina e avermi capito, per aver sempre ricambiato il mio amore, costante e incondizionato. Lo stesso amore che ci sta conducendo verso una vita insieme ricca di gioia per essere l’uno con l’altra. Ringrazio la famiglia di Silvia, Giuliano, Messalina e Matteo, che mi hanno accolto con affetto e mi hanno fatto sempre sentire parte integrante della loro famiglia. Ringrazio di cuore il mio amico Gabriele, che più di tutti forse mi ha sempre incoraggiato a essere me stesso e a non piegarmi alle circostanze esterne. Grazie

per tutti i momenti preziosi e unici di amicizia che ci hanno tenuto uniti in questi anni. Un grazie infine a tutti i giovani ricercatori, dottorandi, tecnici e borsisti che in questi anni ho avuto modo di conoscere e apprezzare: Fabio S., Fabio G., Riccardo, Giorgio K., Mauro, Paolo, Dario, Filippo, Barbara, Marco, Manuel, Nicola.

The mental features discoursed of as the analytical, are, in themselves, but little susceptible of analysis. We appreciate them only in their effects. We know of them, among other things, that they are to their possessor, when inordinately possessed, a source of the liveliest enjoyment. […]. Yet to calculate is not in itself to analyze.[…]. The analytical power should not be confounded with simple ingenuity; for while the analyst is necessarily ingenious, the ingenious man is often remarkably incapable of analysis. E. A. Poe

Ai miei genitori, che con l’esempio mi hanno insegnato l’importanza del sacrificio, ed il valore delle cose che con esso possono essere ottenute A Silvia, che con gioia e amore ha illuminato la mia vita

i

Contents

Contents Abstract

I

Sommario

III

Introduction

V

1. Basic equations of solid mechanics and elasticity

1

1.1. Introduction

1

1.2. Equation of continuum mechanics

1

1.2.1. 1.2.2. 1.2.3. 1.2.4. 1.2.5. 1.2.6. 1.2.7.

Equilibrium equations Traction vector and principal stresses Change of Coordinates: the directioncosine matrix relating two bases Displacement field Strain field Equations in Cylindrical Coordinates (r, θ, z) Equations in Spherical Coordinates (r, θ, φ)

1 2 3 6 6 8 9

1.3. Equation of elasticity

10

1.4. Plane problems of elasticity theory

10

1.4.1. 1.4.2.

The Airy stress function Stress transformation formulae in twodimensional problems Equations in polar coordinates Kolosov equations

13 13 14

2. A review of some up-to-now known solutions for linear and nonlinear problems

15

1.4.3. 1.4.4.

11

2.1. Introduction

15

2.2. Some solution of notch problems under linear elastic conditions using the Airy stress function

15

ii

Contents

2.2.1.

2.2.2.

2.2.3.

2.2.4. 2.2.5.

2.2.6. 2.2.7.

A line force acting on the surface of a half space (Timoshenko and Goodier, 1970) A half space subject to periodic traction on the surface (Timoshenko and Goodier, 1970) Lamé Problem (Timoshenko and Goodier, 1970) Stress field symmetric about an axis (Timoshenko and Goodier, 1970) A circular hole in an infinite sheet under remote shear (Timoshenko and Goodier, 1970) A circular hole in an infinite sheet under remote tension (Kirsch, 1898) Williams’s solution for a sharp Vnotch under mode I and II loadings

15

17 18

19

20 22 24

2.3. Lazzarin and Tovo’s equations for blunt V and U shaped notches under mode I and II

28

2.4. Solutions under elastic-plastic conditions

33

2.4.1.

Solution for mode I loaded cracks and V-shaped notch problems in power hardening materials Slip line fields theory

33 35

3. Linear Elastic Solutions For Notches Under Torsion

37

2.4.2.

3.1. Introduction

37

3.2. Semielliptic circumferential notches under torsion

39

3.2.1. 3.2.2. 3.2.3. 3.2.4. 3.2.5. 3.2.6. 3.2.7. 3.2.8.

Fundamental complex potentials in antiplane elasticity Elliptic coordinate system Formulation of the boundary value problem Boundary conditions for an infinite shaft Solution of a cracked infinite shaft Semicircular circumferential notch in an axisymmetric shaft under torsion Boundary conditions for a finite size shaft Stresses in polar coordinates

39 41 44 44 48 51 52 57

iii

Contents

3.2.9. Strain and displacement fields 3.2.10. Numerical results 3.2.11. Circumferential elliptic notch with an arbitrary orientation angle β 3.2.12. A comparison with the solution by Smith 3.3. Hyperbolic and parabolic notches in round shafts under torsion and uniform antiplane shear loadings 3.3.1.

3.3.2. 3.3.3. 3.3.4. 3.3.5.

3.3.6. 3.3.7. 3.3.8. 3.3.9.

Further comments on the fundamental complex potentials in antiplane elasticity Hyperbolic coordinate system Hyperbolic-parabolic coordinate system A first class of solutions by using the hyperbolic transformation A second class of solutions by using the hyperbolic-parabolic transformation The influence of finite dimensions on stress distributions A comparison with numerical results Limitations and applicability ranges of the two classes of solutions proposed An explicit link between plane and antiplane elasticity problems

57 59

59 62

64

64 65 68 69

77 90 90 93 93

3.4. Averaged strain energy density

94

3.5. Exact solution for a semicircular notch under torsion

97

3.6. J-integral for blunt notches under mode III loadings

101

3.6.1. 3.6.2. 3.6.3.

Introduction to J-integral J-integral for hyperbolic and parabolic notches J-integral for semi-elliptical notches under torsion

101 101 103

3.7. An analytical link between the distributions of different kind of notches under torsion

104

3.8. Elastic Notch Stress Intensity Factors for sharply V-notched rounded bars under torsion

105

iv

Contents

3.8.1.

3.8.2.

3.8.3. 3.8.4.

3.8.5.

3.8.6. 3.8.7. 3.8.8. 3.8.9.

The stress intensity factor for circumferentially-cracked rounded bars under torsion loading Analytical solutions for the notch stress intensity factors due to deep sharp V-notches under torsion Hollow section Notch stress intensity factors for shallow sharp v-notches in solid bars under torsion Notch stress intensity factors for finite sharp v-notches in solid bars under torsion Comparison with the results by Noda and Takase Results of FE analyses and final discussion Appendix A. Kt for hollow sections Appendix B. Expressions for the nondimensional NSIF within the range 1 ≤ R/a ≤ 100

4. NonLinear Solutions For Mode Loaded Notches Under Torsion

105

108 111

112

113 114 116 118

120

III 123

4.1. Introduction

123

4.2. Elastic-plastic stress fields ahead of parabolic notches under antiplane shear loading

126

4.2.1. 4.2.2. 4.2.3. 4.2.4. 4.2.5. 4.2.6. 4.2.7.

Parabolic transformation A material with an elastic-perfectlyplastic behaviour A material with an isotropic hardening A link with Neuber’s solution Some remarks about the equivalent strain energy density criterion Further remarks on the plastic zone shape Final observations

4.3. Plastic stress fields for pointed V-notches under torsion 4.3.1. 4.3.2. 4.3.3.

Basic equations for materials with a nonlinear powerlaw behaviour Linear-elastic stress fields Elastic-plastic fields

126 127 136 144 146 149 154 154 154 161 162

v

Contents

4.3.4.

Link between plastic and elastic NSIFs under small scale yielding conditions 4.3.5. Scale effect 4.3.6. Plastic zone size under small scale yielding 4.3.7. Platic J-integral under mode III 4.3.8. Averaged strain energy density 4.3.9. Further remarks 4.3.10. Comparison to FE analyses and discussion 4.3.11. Final observations 4.4. A reformulated version of the Neuber rule accounting for the influence of the notch opening angle 4.4.1. 4.4.2. 4.4.3.

Basic equations Description of the notch profile Boundary conditions and suitable forms for ψ 4.4.4. Closed form solution 4.4.5. Limit solutions 4.4.6. Some limitations to the solution 4.4.7. Some remarks about the use of notch stress intensity factor and mean strain energy density 4.4.8. Appendix A: A comparison with Neuber for the case 2α=0 4.4.9. Appendix B. A further discussion on the boundary conditions for ψ 4.4.10. Final remarks 4.5. A uniform solution for nonlinear stress and strain distributions at mode-III-loaded sharp and blunt notches 4.5.1. 4.5.2. 4.5.3. 4.5.4. 4.5.5. 4.5.6. 4.5.7. 4.5.8. 4.5.9.

Basic equations Linearization of the problem: general formulation in terms of strains Linearization of the problem: general formulation in terms of stresses Neuber’s conformal transformation Boundary conditions for ψ General forms for the function Ψ: formulation in terms of strains General forms for the function Ψ: formulation in terms of strains Leading order term based solution Leading order solution for ω1=1

163 164 166 168 172 175 177 188

189 189 191 192 193 196 200

200 203 206 208

209 209 210 210 211 212 213 213 214 215

vi

Contents

4.5.10. Leading order based solution for Neuber’s special nonlinear law 4.5.11. Closed form solution for nonlinear elastic materials with a power law 4.5.12. Solution for the Ramberg-Osgood type law 4.5.13. Appendix A. On the conversion of hypergeometric series into polynomial expressions 4.5.14. Appendix B. Limit solution for the Ramberg-Osgood type law

5. Fatigue assessments of welded joints

216 229 231

233 236

239

5.1. Introduction

239

5.2. Fatigue strength of welded joints under uniaxial and multiaxial loading

242

5.2.1. 5.2.2. 5.2.3. 5.2.4. 5.2.5. 5.2.6. 5.2.7.

Analytical preliminaries The N -SIF approach for fatigue strength assessments A synthesis in terms of averaged strain energy density Other data due to Susmel and Tovo Multiaxial fatigue loading of welded joint made of structural steels Multiaxial fatigue loading of welded joint made of aluminium alloys Final observations

242 244 245 248 249 254 256

5.3. Fatigue strength of three dimensional welded joints: a synthesis based on the Strain Energy Density over a control volume

258

6. Some advantages derived from the use of the strain energy density over a control volume

261

6.1. Introduction

261

6.2. A link between strain energy density averaged over a control volume and notch stress intensity factors

266

6.3. 2D models: transverse non-load-carrying fillet welded joints

268

vii

Contents

6.4. Local SED-based approach applied to 3D welded joints

274

6.5. Fatigue strenght of three-dimensional welded joints

277

6.5.1. Data due to Maddox 6.5.2. Data due to Lihavainen and Marquis 6.5.3. Data due to Fricke and Doerk 6.5.4. Data due to Ferreira et al 6.6. Conclusions

277 277 278 279 282

7. Notch effect: a comparison between torsion and tension

285

7.1. Introduction

285

7.2. Extension of Atzori and Lazzarin’s diagram to notches under torsion loadings

286

7.3. Size effect under torsion

288

7.4. Influence of the notch shape on stress distributions

290

7.5. Influence of the notch acuity on stress distribution and intensity

291

7.6. Some expressions for the notch sensitivity under torsion

294

Conclusions

297

Bibliography

301

I

Abstract

Abstract Deliberately created or inadvertently induced, notches and defects unavoidably exist in engineering components. Then, fatigue strength assessments often need linear and nonlinear stresses and strains at the notch root or in the close neighbourhood of it. Whilst there is a large body of work on notch root stresses under tensile loading, in the previous literature there has been relatively little attention paid to torsional loading of prismatic shafts. Nevertheless, the engineering use of torque carrying shafts is extensive and they are susceptible to crack formation at notches and grooves under both static and cyclic conditions. In this work, a comprehensive evaluation of the components of the linear and non-linear stress fields ahead various kind of mode III loaded notches is presented. These expressions are also used to provide closed form solutions for some local parameters such as the averaged strain energy density and Rice J-integral. Finally an assessment of fatigue strength of welded joints subjected to multiaxial loading (combined Mode I and Mode III) as well as to complex threedimensional welded joints based on the local energy is presented.

III

Sommario

Sommario Variazioni geometriche, come fori e intagli, sono comunemente presenti nella maggior parte dei componenti meccanici. Tali discontinuità, causa di una perturbazione della distribuzione di tensione nominale, comportano un aumento locale delle tensioni e delle deformazioni. La conoscenza delle distribuzioni di tensione nelle adiacenze di tali variazioni geometriche è quindi di grande importanza nella valutazione della resistenza a fatica di componenti strutturali. Mentre in letteratura vi sono numerose soluzioni teoriche per componenti piani soggetti a trazione o flessione, relativamente pochi sono i contributi relativi a casi di torsione in travi prismatiche o assialsimmetriche. Tuttavia, gli alberi soggetti a coppia torcente rappresentano un caso di notevole interesse applicativo, essendo potenzialmente interessati da fenomeni di innesco e propagazione di cricche di fatica dovute a effetti di intaglio di diverso tipo. Il lavoro riporta delle soluzioni analitiche in forma chiusa per le distribuzioni di tensione generate da intagli circonferenziali in componenti assialsimmetrici soggetti a torsione, in condizioni lineari elastiche ed elastoplastiche. Tali soluzioni sono inoltre utilizzate per determinare delle espressioni in forma chiusa per alcuni parametri locali, quali la densità di energia di deformazione e il J-integral di Rice, e per discutere dal punto di vista teorico alcuni aspetti peculiari relativi all’effetto d’intaglio in presenza di sollecitazioni torsionali. Viene infine proposta una sintesi di un elevato numero di risultati sperimentali, tratti dalla letteratura, relativi a giunzioni saldate tridimensionali soggetti a fatica monoassiale (trazione o flessione) e multiassiale (Modo I e Modo III combinati) in termini di densità di energia di deformazione.

Introduction

V

Introduction Knowledge of the linear elastic stress fields ahead of notches is essential in the high cycle fatigue assessment of structural components. The most famous analytical contribution to the study of circumferentially blunt notched shafts under torsion is that due to Neuber (1958), who addressed the problem of ‘deep’ and ‘shallow’ notches and was able to determine in both cases the theoretical stress concentration factor Kt. In order to overcome some limitations of the analytical approaches, various numerical techniques have been used to obtain approximate solutions for the stress concentration factor of notched components subjected to torsion, see, amongst others, Rushton (1967), Hamada and Kitagawa (1968), Matthews and Hooke, (1971) and Peterson (1974). Worthy of mention are also some recent contributions due to Noda and Takase who accurately determined, by means of the body force method, the stress concentration factors of blunt V-notches in round bars under torsion loading (Noda and Takase, 2006), as well as the notch stress intensity factors of sharp, zero radius, V-notches (Noda and Takase, 2003). Considering the local stress distributions and not only Kt is essential in dealing with the structural integrity of notched components. Creager and Paris (1967) gave the elastic stress fields in the vicinity of the tip of blunt cracks, or ‘slim’ parabolic notches, under Mode I, II and III loading. The intensities of the fields were expressed in terms of generalised Stress Intensity Factors, later correlated by Glinka to the maximum elastic stress in plane problems (Glinka, 1985). A major difference of a notch under Mode I and Mode II loading compared with a cracked body, is that the bluntness of the notch results in the presence of stress terms proportional to x0.5 and x1.5, x being the distance from the notch tip. Under mode III loading, only the term proportional to x0.5 is present, and the analogy with the crack case is stronger.

VI

Introduction

The elastic stress distribution problem for sharp V-shaped notches in round bars under antiplane shear was solved by Seweryn and Molski (1996), Dunn et al. (1997), Qian and Hasebe (1997), who also dealt with the problem of the singularity at the interface of a bi-material V-notch. Chapter 3 of this work presents a set of closed form solutions for linear elastic stress, strain and displacement fields induced by different kind of circumferential notches in axisymmetric shafts under torsion or uniform antiplane loading. The boundary value problems is formulated according to the complex potential function approach in combination with elliptic, hyperbolic and parabolic coordinate systems. Different classes of solutions are then proposed whose accuracy and range of applicability are discussed in detail taking advantage of a large number of results from FE analyses. The finite size effect on the elastic stress distributions is considered as well. It is also shown that some well-known solutions of linear elastic fracture mechanics and notch mechanics can be seen as special cases of the general solutions reported herein. Finally, the developed analytical frame is used to tie the Mode III Notch Stress Intensity Factor (NSIF) to the maximum shear stress at the notch root, as well as to give closed-form expressions for the strain energy averaged over a given control volume embracing the notch root and for the mode III J-integral when applied to blunt notches. In Chapter 7, the above-mentioned solutions for linear elastic stress fields are also used to discuss some relevant differences existing in the notch effect under Mode I and Mode III loadings. When the notch root radius is zero or small, the stress close to the notch tip becomes very high, exceeding the yield limit and causing a plastic zone of which the size is comparable with that of the process zone controlling fracture mechanisms. In such a case, the knowledge of the role played by local yielding on stress distributions ahead of crack or notch tip is of major importance on the behaviour and the reliability in service of the structural components.

Introduction

VII

In the literature, great attention has been paid to the determination of the elastic-plastic stress and strain fields ahead of cracks and sharp V-notches (i.e. reentrant corners), under mode I, II and III loadings. The asymptotic elastic-plastic stress fields ahead of a crack in a power hardening material under mode I and II were obtained by Hutchinson (1968a,b) and by Rice and Rosengren (1968). In the ‘HRR solution’, the crack tip fields for stresses and strains are given as a function either of a dominant term or of the Jintegral. The role played by the non-singular stress terms and by the specimen geometry on the applicability of the small scale yielding hypothesis was also discussed in detail (Larsson and Carlsson, 1973, Rice, 1974). Afterwards the effects of the higher order terms on the crack tip stress fields under mode I and mode II were analysed (Sharma and Aravas, 1991, Xia et al. ,1993), with extensions to sharp V-notches (Kuang and Xu, 1987, Xia and Wang, 1993) for which also an approximated solution was discussed (Filippi et al., 2002) as well as the mixed mode case in the presence of large opening angles (Lazzarin et al., 2001). In the aforementioned papers the material is modelled according to a power hardening law and the determination of the angular functions is carried out numerically in the ambit of the J2 deformation theory. The numerical solution is generally based on multi-shooting techniques or on special FEM techniques (Zhang and Joseph, 1998, Chen and Ushijima, 2000). Differently from the plane problems, the antiplane problem under yielding can be solved in closed-form by using some elegant mappings (Rice, 1967a), which reduce the non-linear governing equations into a linear equation system and make it possible an analytical solution for cracks and sharp V-notches (Rice, 1967a, Yuan and Yang, 1995, Yang et al., 1996, Wang and Kuang, 1999). By proceeding on parallel tracks, great efforts were done to determine elasticplastic stresses and strains at the root of blunt notches under different loading modes. The obtained degree of accuracy was found to depend on the loading mode and the far-field-applied-load magnitude. With the aim to find a general relation linking stresses and strains, a strip with two symmetric notches under longitudinal shear was analysed by Neuber (1961) who reduced the solution of the

VIII

Introduction

problem to a system of two ordinary differential equations. Neuber was able to obtain a closed-form solution according to which the geometrical mean value of the effective stress and strain concentration factors for any stress-strain law is equal to the elastic (Hoookean) theoretical stress concentration factor. Neuber’s rule is based on the hypothesis that the connection between real stress and nominal stress could be represented by a special function, i.e. the “Leading function” (Neuber, 1961). As a possible alternative to Neuber’s rule, in the presence of small scale yielding, the Equivalent Strain Energy Density (ESED) criterion was formulated by Glinka and co-workers (Molsky and Glinka, 1981, Glinka 1985a-b, Glinka et al., 1988) considering the root of blunt notches under tensile or bending loadings, the torsion loading being excluded. Considering sharp V-notches instead of blunt notches, the point-related ESED criterion was later extended to a finite size volume centred on the sharp V-notch tip (Lazzarin and Zambardi, 2002). In the case of rounded notches a solution providing the stress and strain distributions within the plastic region is not reported in the literature, nor there exists a model providing a transition between fracture mechanics and notch mechanics as a function of the notch root radius value. Chapter 4 deals with these problems. More precisely, the aims of the analyses can be summarised as follows: -

to give a common analytical frame to the analysis of stress and strain fields due to cracks, sharp and rounded notches under antiplane shear loading;

-

to describe analytically the influence of the notch root radius on the extension of the plastic zone ahead of the notch;

-

to provide an explicit link between the elastic Notch Stress Intensity Factors (NSIFs) and the plastic NSIFs;

-

to express analytically the variability of the strain energy density at the notch tip or over a finite size volume surrounding the crack notch tip with respect to the linear elastic case.

Finally, Chapters 5 and 6 address to the fatigue assessment of welded joints considering plane and three-dimensional welded joints under uniaxial or multiaxial loading conditions. The weld toe region is modelled like a sharp, zero radius, V-shaped notch and the mean value of the strain energy density (SED) is

Introduction

IX

evaluated by means of FE analyses over a circular sector just centred on the weld toe. It is also underlined as set of advantages deriving by the use of the SED as a fatigue relevant parameter; in particular as opposed to the direct evaluation of the NSIFs, the mean value of the elastic SED on the control volume can be accurately determined by using relatively coarse meshes. This fact is of major importance for the applicability of the method to components of complex geometry.

1

Basic Equations of solid mechanics and elasticity

1 Basic equations of solid mechanics and elasticity 1.1 Introduction This chapter is aimed to recall the basic equation of solid mechanics and elasticity, which will be basic in the following to develop new solution within this topic.

1.2 Equation of continuum mechanics Most of the problems here discussed are related to continuous, homogeneous and isotropic bodies. With continuous we refer to a body that is continuous at infinitesimal scale; with homogeneous we refer to a body whose material properties do not change abruptly; finally an isotropic body is one whose properties do not vary with direction.

1.2.1 Equilibrium equations In the classical theory of elasticity the effects of inertia and body forces are commonly neglected. If these conditions are guaranteed, the body will be in a state of equilibrium according to the following system of equations: ∂σ xx ∂τ xy ∂τ xz + + =0 ∂x ∂y ∂z ∂τ xy ∂σ yy ∂τ yz + + =0 ∂x ∂y ∂z ∂τ xz ∂τ yz ∂σ zz + + =0 ∂x ∂y ∂z

(1.2.1)

2


Equilibrium also requires that shear stresses be symmetrical in the absence of a body couple, namely:

τij = τ ji

(1.2.2)

This assumption reduces the number of stresses to be determined from nine to six.

1.2.2 Traction vector and principal stresses On the surface of a body, the stresses produce a force per unit of area called traction T, whose components can be expressed in a matrix form:

⎡σ xx τ xy τ xz ⎤ ⎧nx ⎫ ⎢ ⎥⎪ ⎪ T = ⎢τ xy σ yy τ yz ⎥ ⎨n y ⎬ ⎢τ xz τ yz σ zz ⎥ ⎪⎩ nz ⎪⎭ ⎣ ⎦

(1.2.3)

where ni are the components of an outward normal unit vector of the surface. Furthermore it has to be remembered that the axes of the Cartesian coordinate system can always be rotated at any point such that all shear stresses disappear from the surface of the stresses cube. The magnitude of stress at this given point is characterised simply by three normal stresses (σ1, σ2, σ3), which are referred as principal stresses. Principal stresses can be determined by noting that when a plane is the principal plane, the traction on the plane is normal to the plane, namely, the traction vector T must be in the same direction as the unit normal vector n. Let the magnitude of the traction be σ . On the principal plane, the traction vector is in the direction of the normal vector, T = σn . In the matrix notion, we have ⎡ t1 ⎤ ⎡ n1 ⎤ ⎢ t ⎥ = σ⎢n ⎥ . ⎢ 2⎥ ⎢ 2⎥ ⎣⎢ t 3 ⎦⎥ ⎣⎢ n 3 ⎦⎥

(1.2.4)

Here both T and n are vectors, but σ is a scalar representing the magnitude of the principal stress. By noting that the traction vector is the stress matrix times the normal vector, so that

3

Basic Equations of solid mechanics and elasticity ⎡ σ11 ⎢σ ⎢ 21 ⎢⎣σ 31

σ12 σ 22 σ 32

σ13 ⎤ ⎡ n1 ⎤ ⎡ n1 ⎤ ⎥ ⎢ ⎥ σ 23 n 2 = σ ⎢ n 2 ⎥ . ⎥⎢ ⎥ ⎢ ⎥ σ 33 ⎥⎦ ⎢⎣ n 3 ⎥⎦ ⎢⎣ n 3 ⎥⎦

(1.2.5)

This is an eigenvalue problem. When the stress state is known, i.e., the stress matrix is given, it is possible to solve the above eigenvalue problem to determine the eigenvalue σ and the eigenvector n. The eigenvalue σ is the principal stress, and the eigenvector n is the principal direction. Because the stress tensor is a 3 by 3 symmetric matrix, you can always find three real eigenvalues, i.e., principal stresses, σ a ,σ b ,σ c . We distinguish three cases:

a. If the three principal stresses are unequal, the three principal directions are orthogonal (e.g., pure shear state). b. If two principal stresses are equal, but the third is different, the two equal principal stresses can be in any directions in a plane, and the third principal direction is normal to the plane (e.g., pure tensile state). c. If all the three principal stresses are equal, any direction is a principal direction. This stress state is called a hydrostatic state. The magnitude of the maximum shear stress, τmax, that a body sustains at a point is related to the principal stresses by the formula: τ max =

max σα − σβ 2

(1.2.6)

where max σα − σβ is the greatest difference between principal stresses. τ max acts on a plane with the normal vector 45° from the principal directions.

1.2.3 Change of Coordinates: the direction-cosine matrix relating two bases

In the 3D space, let e1, e2 and e3 be an orthonormal basis, namely ei ⋅ e j = δ ij , where δij is the Kronecker delta. The base vectors are ordered to follow the righthand rule. Let e'1 , e'2 , e'3 be a new orthonormal basis, namely eα ⋅ e β = δ αβ .

4

Basic Equations of solid mechanics and elasticity Let the angle between the two vectors ei and e'α be θ iα . Denote the direction

cosine of the two vectors by: liα = ei ⋅ e'α = cosθiα

(1.2.7)

We follow the convention that the first index of liα refers to a coordinate in the old basis, and the second to a coordinate in the new basis. For the two bases, there are a total of 9 direction cosines. We can list liα as a 3 by 3 matrix. By our convention, the rows refer to the old basis, and the columns to the new basis. Note that liα is the component of the vector e'α projected on the vector ei . We can express each new base vector as a linear combination of the three old base vectors

e'α = l1α e1 + l2α e 2 + l3α e3

(1.2.16)

Similarly, we can express the old basis as a linear combination of the new basis

ei = li1e'1 +li 2e'2 +li 3e'3

(1.2.17)

Using the summation convention, we write more concisely as

ei = liα e'α

(1.2.18)

Let now f be a vector. It is a linear combination of the base vectors:

f = f i ei

(1.2.19)

where f1 , f 2 , f 3 are the components of the vector, and are commonly written as a column. Let f '1 , f '2 , f '3 be the components of the vector f in the new basis, namely,

f = f 'α e'α

(1.2.20)

Recall the transformation between the two sets of basis, ei = liα e'α , we write that

f = f iei = f iliα e'α

(1.2.21)

A comparison between the two expressions gives that f 'α = f iliα

(1.2.22)

Thus, the component column in the new basis is the transpose of the direction-cosine matrix times the component column in the old basis.

5


Similarly, we can show that f i = liα f 'α

(1.2.23)

The component column in the old basis is the direction-cosine matrix times the component column in the old basis. If one supposes that the new basis and the old basis differ by an angle θ around the axis e3, the direction cosines are

e1 ⋅ e'1 = cosθ , e1 ⋅ e'2 = − sin θ , e1 ⋅ e'3 = 0, e 2 ⋅ e'1 = sin θ , e 2 ⋅ e'2 = cosθ , e 2 ⋅ e'3 = 0,

(1.2.24)

e3 ⋅ e'1 = 0, e3 ⋅ e'2 = 0, e3 ⋅ e'3 = 1.

e2

e2 '

e1 '

θ

θ

e1

Figure 1.2.1 angle between vectors ei and e'i Consequently, the matrix of the direction cosines is ⎡cosθ [liα ] = ⎢⎢ sin θ ⎢⎣ 0

− sin θ cosθ 0

0⎤ 0⎥⎥ 1⎥⎦

(1.2.25)

The components of a vector transform as ⎡ f '1 ⎤ ⎡ cosθ ⎢ f ' ⎥ = ⎢− sin θ ⎢ 2⎥ ⎢ ⎢⎣ f '3 ⎥⎦ ⎢⎣ 0

sin θ cosθ 0

0⎤ ⎡ f1 ⎤ 0⎥⎥ ⎢⎢ f 2 ⎥⎥ 1⎥⎦ ⎢⎣ f 3 ⎥⎦

(1.2.26)

Thus, f '1 = f1 cosθ + f 2 sin θ f '2 = − f1 sin θ + f 2 cosθ

(1.2.27)

f '3 = f 3

Since {T'} = [σ' ]{n '} = [l iα ]{T} = [l iα ][σ]{n} = [l iα ][σ][l iα ] {n '} , the components of a t

stress state transform as: ⎡ σ'11 ⎢ σ' ⎢ 21 ⎢⎣σ' 31

σ'12 σ' 22 σ' 32

σ'13 ⎤ ⎡ cos θ sin θ 0⎤ ⎡ σ11 σ' 23 ⎥ = ⎢ − sin θ cos θ 0⎥ ⎢σ 21 ⎥ ⎢ ⎥⎢ σ' 33 ⎥⎦ ⎢⎣ 0 0 1⎥⎦ ⎢⎣σ 31

σ12 σ 22 σ 32

σ13 ⎤ ⎡cos θ − sin θ 0⎤ σ 23 ⎥ ⎢ sin θ cos θ 0⎥ ⎥⎢ ⎥ 0 1⎥⎦ σ 33 ⎥⎦ ⎢⎣ 0

6


(1.2.28) Thus,

σ '11 =

σ 11 + σ 22

σ 11 − σ 22

cos 2θ + σ 12 sin 2θ 2 2 σ + σ 22 σ 11 − σ 22 σ '22 = 11 − cos 2θ − σ 12 sin 2θ 2 2 σ − σ 22 sin 2θ + σ 12 cos 2θ σ '12 = − 11 2 σ '13 = σ 13 cosθ + σ 23 sin θ +

(1.2.29)

σ '23 = −σ 13 sin θ + σ 23 cosθ σ '33 = σ 33

1.2.4 Displacement field The displacement is the vector by which a material particle moves relative to its position in the reference configuration. If all the material particles in the body move by the same displacement vector, the body as a whole moves by a rigidbody translation.

If the material particles in the body move relative to one

another, the body deforms. We label each material particle by its coordinates (x, y, z ) in the reference configuration.

At time t, the material particle (x, y, z ) has the displacement

u ( x, y, z , t ) in the x-direction, v( x, y, z , t ) in the y-direction, and w( x, y, z , t ) in the

z-direction.

A function of spatial coordinates is known as a field.

The

displacement field is a time-dependent vector field.

1.2.5 Strain field For a given displacement field, we can calculate the strain field. Consider two material particles in the reference configuration: particle A at (x, y, z ) and particle B at (x + dx, y, z ) . In the reference configuration, the two particles are distance dx apart. At a given time t, the two particles move to new locations. The xcomponent of the displacement of particle A is u ( x, y, z , t ) , and that of particle B is u ( x + dx, y, z , t ) .

Consequently, the distance between the two particles

elongates by u ( x + dx, y, z , t ) − u ( x, y, z , t ) . The axial strain in the x-direction is

7


u ( x, y + dy, z , t ) C dy

u ( x, y , z , t )

B

A dx Figure 1.2.2. Schematic representation of deformation

εx =

u (x + dx, y, z, t ) − u (x, y, z, t ) ∂u = . dx ∂x

(1.2.30)

This is a strain of material particles in the vicinity of (x, y, z ) at time t. The function ε x ( x, y, z , t ) is a component of the strain field of the body. The shear strain is defined as follows.

Consider two lines of material

particles. In the reference configuration, the two lines are perpendicular to each other. The deformation changes the included angle by some amount. This change in the angle defines the shear strain, γ . We now translate this definition into a strain-displacement relation. Consider three material particles A, B, and C. In the reference configuration, their coordinates are A (x, y, z ) , B (x + dx, y, z ) , and C (x, y + dy, z ) .

In the deformed configuration, in the x-direction, particle A

moves by u ( x, y, z , t ) and particle C by u ( x, y + dy, z , t ) .

Consequently, the

deformation rotates line AC about axis z by an angle u (x, y + dy, z, t ) − u (x, y, z, t ) ∂u = . dy ∂y

(1.2.31)

Similarly, the deformation rotates line AB about axis z by an angle v (x + dx, y, z, t ) − v (x, y, z, t ) ∂v = . dx ∂x

(1.2.32)

Consequently, the shear strain in the xy plane is the net change in the included angle:

8


γ xy =

∂u ∂v + . ∂y ∂x

(1.2.33)

For a body in the three-dimensional space, the strain state of a material particle is described by a total of six components. The strains relate to the displacements as ∂u ∂v ∂w , εy = , εz = ∂x ∂y ∂z ∂u ∂v ∂v ∂w = + , γ yz = + , ∂y ∂x ∂z ∂y

εx = γ xy

γ zx

∂w ∂u = + ∂x ∂z

(1.2.34)

Another definition of the shear strain relates the definition here by ε xy = γ xy / 2 . With this new definition, we can write the six strain-displacement

relations neatly as ε ij =

1 ⎛⎜ ∂u i ∂u j ⎞⎟ . + 2 ⎜⎝ ∂x j ∂x i ⎟⎠

(1.2.35)

In general six equation of compatibility must be satisfied in order to obtain a single-valued displacement field: 2 2 ∂ 2ε xx ∂ ε yy ∂ γ xy + − =0 ∂y 2 ∂x 2 ∂x∂y

∂ 2ε xx ∂ 2ε zz ∂ 2γ xz + − =0 ∂z 2 ∂x 2 ∂x∂z ∂ 2ε yy ∂ 2ε zz ∂ 2γ yz + − =0 ∂z 2 ∂y 2 ∂y∂z 2 2 ∂ 2ε xx ∂ ε xy ∂ ε yz ∂ 2ε zx = − + =0 ∂y∂z ∂x∂z ∂x 2 ∂y∂x

∂ 2ε yy ∂x∂z

=

∂ 2ε xy ∂y∂z

+

∂ 2ε yz ∂y∂x

−

(1.2.36)

∂ 2ε zx =0 ∂y 2

∂ ε xy ∂ 2ε yz ∂ 2ε zx ∂ ε zz =− + + =0 ∂x∂y ∂z 2 ∂x∂z ∂y∂z 2

2

1.2.6 Equations in Cylindrical Coordinates (r, θ, z) Suppose u, v, w are the displacement components in the radial, circumferential and axial directions, respectively. Equations of Equilibrium are:

9

Basic Equations of solid mechanics and elasticity ∂σ r 1 ∂σ rθ ∂σ rz σ r − σ θ + + + =0 ∂r r ∂θ ∂z r ∂σ rθ 1 ∂σ θ ∂σ θz 2σ rθ + + + =0 r ∂θ r ∂r ∂z ∂σ rz 1 ∂σ θz ∂σ z σ rz + + + =0 ∂r r ∂θ r ∂z

∂u ∂r 1 ⎛ ∂v ⎞ εθ = ⎜ + u ⎟ r ⎝ ∂θ ⎠ ∂w εz = ∂z εr =

1 ⎛ ∂v 1 ∂w ⎞ ε θz = ⎜ + ⎟ 2 ⎝ ∂z r ∂θ ⎠ 1 ⎛ ∂w ∂u ⎞ ε zr = ⎜ + ⎟ 2 ⎝ ∂r ∂z ⎠ 1 ⎛ 1 ∂u ∂v v ⎞ ε rθ = ⎜ + − ⎟ 2 ⎝ r ∂θ ∂r r ⎠

(1.2.37)

(1.2.38)

1.2.7 Equations in Spherical Coordinates (r, θ, φ) Suppose r is the radial distance, θ is measured from the positive z-axis to a radius and φ is measured round the z-axis in a right-handed sense. u, v, w are the displacements components in the r, θ, φ directions, respectively. ∂σ r 1 ∂σ rθ 1 ∂σ rφ 2σ r − σ θ − σ φ − σ rθ cot θ + + + =0 ∂r r ∂θ r sin θ ∂φ r ∂σ rθ 1 ∂σθ 1 ∂σθφ (σ θ − σ φ ) cot θ + 3σ rθ + + + =0 r ∂θ r sin θ ∂φ r ∂r ∂σ rφ 1 ∂σθφ 1 ∂σφ 3σ rφ + 2σθrφ cot θ + + =0 + r ∂θ r sin θ ∂φ r ∂r

(1.2.39)

∂u ∂r 1 ⎛ ∂v ⎞ εθ = ⎜ + u ⎟ r ⎝ ∂θ ⎠

εr =

εφ = ε θφ

⎞ 1 ⎛ ∂w ⎜⎜ + u sin θ + ν cos θ ⎟⎟ r sin θ ⎝ ∂φ ⎠

1 ⎛ ∂w 1 ∂v 1 ∂ν ⎞ ⎟ = ⎜⎜ − w cot θ + + 2r ⎝ ∂θ sin θ ∂z r ∂φ ⎟⎠

1 ⎛ ∂w w 1 ∂u ⎞ ⎟ − + ε rφ = ⎜⎜ 2 ⎝ ∂r r r sin θ ∂φ ⎟⎠ 1 ⎛ 1 ∂u ∂ν ν ⎞ ε rθ = ⎜ + − ⎟ 2 ⎝ r ∂θ ∂r r ⎠

(1.2.40)

10


1.3 Equation of elasticity The following stress-strain relationships hold true for linear elasticity: 1 [σxx − ν(σyy + σzz )] E 1 ε yy = [σ yy − ν(σ xx + σzz )] E 1 εzz = [σzz − ν( σ yy + σ xx )] E

γ xz =

ε xx =

γ yz = γ xy =

τ xz G τ yz G τ xy

(1.3.1)

G

1.4 Plane problems of elasticity theory The plane problems of elasticity are generally designated as plane strain and plane stress problems. Plane strain conditions are typically met by thick plates that are loaded in the plane. The loading is invariant along the z-direction.

Consequently, the

displacement field takes the form u (x , y ) , v (x, y ) , w = 0 . From the strain displacement relations, we find that only the three in-plane strains are nonzero: ε xx (x, y ), ε yy (x, y ), γ xy (x, y ) . The three out-of-plane strains vanish:

ε zz = γ xz = γ yz = 0 so that, the stress-strain relations imply that

τ xz = τ yz = 0 .

From ε zz = 0 and ε zz = (σ zz − νσ xx − νσ yy ) , we obtain that:

σ zz = ν(σ xx + σ yy )

(1.4.1)

Differently, plane stress conditions are met by thin plates loaded in the plane, so that it seems reasonable to guess that the stress field in the sheet only has nonzero components in its plane: σ xx ,σ yy ,τ xy , and the components out of the plane vanish: σ zz = τ xz = τ yz = 0 .

(1.4.2)

For both cases the stress field in the plane is described by three functions

σ xx ( x, y ), σ yy ( x, y ), τ xy ( x, y ) . Then, in light of the simplified stress states the equilibrium equations reduce to: ∂σ xx ∂τ xy + = 0, ∂x ∂y

∂τ xy ∂x

+

∂σ yy ∂y

=0

while compatibility of strains requires:

(1.4.3)

11

Basic Equations of solid mechanics and elasticity 2 2 ∂ 2 ε xx ∂ ε yy ∂ γ xy = + ∂x∂y ∂x 2 ∂y 2

(1.4.4)

The stress-strain relationships for plane stress are then:

ε xx =

σ xx

ε yy = γ xy

E

σ yy

−ν

σ yy

−ν

E

σ xx

E E 2(1 + ν ) τ xy = E

ε zz = −

ν

E

(σ

xx

(1.4.5)

+ σ yy )

γ xz = γ yz = 0 While for plane strain one has:

ε xx =

2 1 (σ xx −νσ yy −νσ zz ) = 1 −ν ⎛⎜σ xx − ν σ yy ⎞⎟ E E ⎝ 1 −ν ⎠

2 1 (σ yy −νσ xx −νσ zz ) = 1 −ν ⎛⎜σ yy − ν σ xx ⎞⎟ E E ⎝ 1 −ν ⎠ 2(1 +ν ) τ xy = E

ε yy = γ xy

(1.4.6)

1.4.1 The Airy stress function Consider two functions f ( x, y ) and g ( x, y ) satisfying ∂f ∂g = ∂x ∂y

(1.4.7)

Cauchy-Schwarz conditions assures that there exists a function A( x, y ) , such that: f =

∂A ∂A , g= . ∂y ∂x

(1.4.8)

We now apply the above theorem to the equilibrium equations. From:

∂σ xx ∂τ xy =0 + ∂y ∂x

(1.4.9)

we deduce that there exists a function A( x, y ) , such that σ xx =

∂A ∂A , τ xy = − ∂y ∂x

(1.4.10)

12


From: ∂τ xy

+

∂x

∂σ yy ∂y

=0

(1.4.11)

we deduce that that there exists a function B( x, y ) , such that σ yy =

∂B ∂B , τ xy = − ∂x ∂y

(1.4.12)

Finally, from the properties ∂A ∂B = ∂x ∂y

(1.4.13)

we deduce that that there exists a function φ ( x, y ) , such that: A=

∂φ ∂φ , B= ∂y ∂x

(1.4.14)

The function φ ( x, y ) is known as the Airy (1862) stress function. The three components of the stress field can now be represented by the stress function: σ xx =

∂ 2φ ∂ 2φ ∂ 2φ , , τ = − σ = yy xy ∂y∂x ∂y 2 ∂y 2

(1.4.15)

Using the stress-strain relations, we can also express the three components of strain field in terms of the Airy stress function:

ε xx =

1 ⎛ ∂ 2φ ∂ 2φ ⎞ ⎜⎜ 2 −ν 2 ⎟⎟ E ⎝ ∂y ∂x ⎠

ε yy =

1 ⎛ ∂ 2φ ∂ 2φ ⎞ ⎜⎜ 2 −ν 2 ⎟⎟ E ⎝ ∂x ∂y ⎠

γ xy = −

(1.4.16)

2(1 + ν ) ∂ 2φ E ∂x∂y

Inserting the expressions of the strains in terms of φ ( x, y ) into the compatibility equation, and we obtain that ∂ 4φ ∂ 4φ ∂ 4φ +2 2 2 + 4 =0 ∂y ∂x ∂y ∂x 4

(1.4.17)

This equations can also be written as ⎛ ∂2 ∂ 2 ⎞⎛ ∂ 2 φ ∂ 2 φ ⎞ ⎜⎜ 2 + 2 ⎟⎟⎜⎜ 2 + 2 ⎟⎟ = 0 ∂y ⎠⎝ ∂x ∂y ⎠ ⎝ ∂x

This is the so-called biharmonic equation.

(1.4.18)

13


Thus, a procedure to solve a plane problem of elasticity is to solve for φ ( x, y ) from the above partial differential equation, and then calculate stresses and strains. After the strains are obtained, the displacement field can be obtained by integrating the strain-displacement relations.

1.4.2 Stress transformation formulae in two-dimensional problems In plane problems, the formulae for stress transformation reported in the previous section becomes:

σ 11 + σ 22

σ 11 − σ 22

cos 2θ + σ 12 sin 2θ 2 2 σ + σ 22 σ 11 − σ 22 σ '22 = 11 − cos 2θ − σ 12 sin 2θ 2 2 σ − σ 22 σ '12 = − 11 sin 2θ + σ 12 cos 2θ 2

σ '11 =

+

(1.4.20)

Then, if we represent the state of stress in a polar, (r ,θ ) , coordinate system, the components of the stress state are σ rr ,σ θθ ,τ rθ can be determined as:

σ xx + σ yy

σ xx − σ yy

cos 2θ + τ xy sin 2θ 2 2 σ + σ yy σ xx − σ yy σ θθ = xx − cos 2θ − τ xy sin 2θ 2 2 σ − σ yy τ rθ = − xx sin 2θ + τ xy cos 2θ 2

σ rr =

+

(1.4.21)

1.4.3 Equations in polar coordinates The Airy stress function is a function of the polar coordinates, φ (r ,θ ) . The stresses are expressed in terms of the Airy stress function: ∂ 2φ 1 ∂φ + 2 2 r ∂θ r ∂r 2 ∂φ σ θθ = 2 ∂r ∂ ⎛ ∂φ ⎞ τ rθ = − ⎜ ⎟ ∂r ⎝ r∂θ ⎠

σ rr =

The biharmonic equation is:

(1.4.22)

14


⎛ ∂2 ∂ ∂ 2 ⎞⎛ ∂ 2φ ∂φ ∂ 2φ ⎞ ⎜⎜ 2 + + 2 2 ⎟⎟⎜⎜ 2 + + 2 2 ⎟⎟ = 0 r∂r r ∂θ ⎠⎝ ∂r r∂r r ∂θ ⎠ ⎝ ∂r

(1.4.23)

The stress-strain relations in polar coordinates are similar to those in the rectangular coordinates:

ε rr = ε θθ = γ rθ

σ rr E

σ θθ

−ν

σ θθ

−ν

σ rr

E

E E 2(1 +ν ) = τ rθ E

(1.4.24)

The strain-displacement relations are ∂ur ∂r u ∂u . = r+ θ r r∂θ ∂u u ∂u = r + θ − θ r∂θ ∂r r

ε rr = ε θθ γ rθ

(1.4.25)

1.4.4 Kolosov equations The Kolosov formulation of plane problems of elasticity follows: σ xx + σ yy = 4 Re ψ' ( z )

σ xx − σ yy − 2iτzy = 2[z ψ ' ' ( z ) + χ ' ' ( z )]

(1.4.26)

where, as is well known, ψ and χ are two holomorphic functions to be determined.

15

A review of some up-to-now known solutions for linear and nonlinear problems

2 A review of some up-to-now known solutions for linear and nonlinear problems 2.1

Introduction

The aim of this chapter is to provide an overview of some classical solutions to notch problems under linear and nonlinear conditions and to discuss the methods of solution for these problems, which will be useful for the analytical developments carried out in the subsequent chapters.

2.2

Some solution under linear elastic conditions using the Airy stress function

2.2.1

A line force acting on the surface of a half space (Timoshenko and Goodier, 1970)

Consider a half space of an elastic material subjected to a line force on its surface. Let P be the force per unit length. The half space is represented by the condition x > 0 , and the force points in the direction of x. This problem has no length scale.

Linearity and dimensional considerations requires that the stress field take the form

σ ij (r ,θ ) =

P g ij (θ ) r

(2.2.1)

where g ij (θ ) are dimensionless functions of θ . We guess that the stress function takes the form

φ (r ) = rPf (θ )

(2.2.2)

16


where f (θ ) is a dimensionless function of θ .

Inserting this form into the

biharmonic equation, we obtain an ODE for f (θ ) :

f +2

d2 f d4 f + =0 dθ 2 dθ 4

(2.2.3)

The general solution is

φ (r ,θ ) = rP( A sin θ + B cos θ + Cθ sin θ + Dθ cos θ )

(2.2.4)

Observe that r sin θ = y and r cos θ = x do not contribute to any stress, so we

drop these two terms. By the symmetry of the problem, we look for stress field symmetric about θ = 0 , so that we will drop the term θ cos θ . Consequently, the stress function takes the form

φ (r ,θ ) = rPCθ sin θ

(2.2.5)

We can calculate the components of the stress field:

σ rr =

2CP cos θ , σ θθ = τ rθ = 0 r

(2.2.6)

This field satisfies the traction boundary conditions, σ θθ = τ rθ = 0 at θ = 0 and θ = π . To determine C, we require that the resultant force acting on a cylindrical surface of radius r balance the line force P. On each element rdθ of the surface, the radial stress provides a vertical component of force σ rr cos θrdθ . The force balance of the half cylinder requires that π /2

P+

∫π σ

rr

cos θrdθ = 0

(2.2.7)

− /2

Integrating, we obtain that C = −1 / π . The stress components in the x-y coordinates are then: 2P cos 4 θ πx 2P σ yy = − sin 2 θ cos 2 θ πx 2P τ xy = − sin θ cos3 θ πx

σ xx = −

(2.2.8)

17


2.2.2 A half space subject to periodic traction on the surface (Timoshenko and Goodier, 1970)

An elastic material occupies a half space, x > 0 . On the surface of the material, x = 0 , the traction vector is prescribed

σ xx (0, y, z ) = σ 0 cos ky, τ xy (0, y, z ) = τ xz (0, y, z ) = 0

(2.2.9)

The material clearly deforms under the plane strain conditions.

It is

reasonable to guess that the Airy function should take the form φ(x , y ) = f (x ) cos ky

(2.2.10)

The biharmonic equation becomes 2 d 4f 2 d f − 2 k + k 4f = 0 dx 4 dx 2

(2.2.11)

This is a homogenous ordinary differential equation with constant coefficients. A solution is of the form f (x ) = e αx

(2.2.12)

Insert this form into the ordinary differential equation and we obtain that

(α

2

− k2 ) = 0 . 2

The algebraic equation has double roots of α = − k , and double roots of

α = + k . Consequently, the general solution is of the form f (x ) = Ae kx + Be − kx + Cxe kx + Dxe − kx

(2.2.13)

where A, B, C and D are constants of integration. We expect that the stress field vanishes as x → +∞ , so that the stress function should be of the form f (x ) = Be − kx + Dxe − kx

(2.2.14)

We next determine the constants B and D by using the boundary conditions. The stress fields are σ xx

∂ 2φ = 2 = − (Be − kx + Dxe − kx )k 2 cos ky ∂y

∂ 2φ ⎛ D ⎞ τ xy = − = ⎜ − Be − kx + e − kx − Dxe − kx ⎟ k 2 sin ky ∂x∂y ⎝ k ⎠ 2 D ∂ φ ⎛ ⎞ σ yy = 2 = ⎜ Be − kx − 2 e − kx + Dxe − kx ⎟ k 2 cos ky k ∂x ⎝ ⎠

(2.2.15)

18

A review of some up-to-now known solutions for linear and nonlinear problems Recall the boundary conditions

σ xx (0, y ) = σ 0 cos ky, τ xy (0, y ) = 0 .

(2.2.16)

We find that B = −σ 0 / k 2 , D = −σ 0 / k

(2.2.17)

The stress field inside the material is σ xx = σ 0 (1 + kx )e − kx cos ky τ xy = −σ 0 kxe − kx sin ky

(2.2.18)

σ yy = σ 0 (1 − kx )e − kx cos ky

The stress field decays exponentially.

2.2.3 Lamé Problem (Timoshenko and Goodier, 1970)

Consider a spherical cavity in a large body, under remote hydrostatic tension. The symmetry of the problem makes spherical coordinate system convenient. For the solution we can follow the simple approach reported hereafter. Since the two equal hoop stresses σ θ , σ φ are equal by symmetry, we can use equilibrium equations to express σθ in terms of σ r (see chapter 1):

σθ = σ r +

r dσ r 2 dr

(2.2.19)

Then, we can use the Hooke’s law to express both strains in terms of σ r :

εr =

1⎡ (1 − 2ν )σ r − νr dσ r ⎤⎥ ⎢ E⎣ dr ⎦

1⎡ r dσ r ⎤ ε θ = ⎢(1 − 2ν )σ r + (1 − ν ) 2 dr ⎥⎦ E⎣

(2.2.20)

Then, since ε r = ∂u / ∂r and ε θ = u / r , we can write the radial strain as a function of the hoop strain:

ε r = d (rε θ ) / dr

(2.2.21)

Finally, rearranging, one obtains the following Equations for σr: d 2σ r 4 dσ r + =0 dr 2 r dr

(2.2.22)


19

This is an equidimensional equation, whose solution takes the form σ r = r m . Substituting σ r = r m into Eq. (2.2.22), one finds two roots, m = 0 and m = -3. Then:

B r3 B σθ = A − 3 2r

σr = A+

(2.2.23)

The boundary conditions are

− Prescribed remote stress: σ r = S

as r = ∞ .

− Traction-free at the cavity surface: σ r = 0

as

r=a

Upon determining the two constants A and B, we obtain the stress distribution

⎡

3 ⎛a⎞ ⎤ ⎝ r ⎠ ⎥⎦

⎡

3 1⎛a⎞ ⎤ 2 ⎝ r ⎠ ⎥⎦

σ r = S ⎢1 − ⎜ ⎟ ⎥, σ θ = S ⎢1 + ⎜ ⎟ ⎥ ⎢⎣

⎢⎣

(2.2.24)

Then, in this case, the stress concentration factor is 3/2.

2.2.4 Stress field symmetric about an axis (Timoshenko and Goodier, 1970)

In the presence of a geometric symmetric about an axis, the stress field reasonably depends only on the radial distance r, so that φ = φ(r ) . Then, the biharmonic equation simplifies as follows: ⎛ d 2 1 d ⎞ ⎛ d 2 φ 1 dφ ⎞ ⎜⎜ 2 + ⎟⎜ ⎟=0 + r dr ⎟⎠⎜⎝ dr 2 r dr ⎟⎠ ⎝ dr

(2.2.25)

This is an equi-dimensional equation, whose solutions can be sought in the form φ = r m . Inserting φ = r m into the biharmonic equation, we obtain the algebraic equation m 2 (m − 2 )2 = 0 .Consequently, the general solution is: φ(r ) = A log r + Br 2 log r + Cr 2 + D . where A, B, C and D are constants to determine. The components of the stress field are then:

(2.2.26)

20

A review of some up-to-now known solutions for linear and nonlinear problems 1 ∂φ A ∂ 2φ + = + B(1 + 2 log r ) + 2C 2 2 r ∂θ r ∂r r 2 A ∂ 2φ σ θθ = 2 = − 2 + B(3 + 2 log r ) + 2C r ∂r ∂ ⎛ ∂φ ⎞ τ rθ = − ⎜ ⎟=0 ∂r ⎝ r∂θ ⎠

σ rr =

(2.2.27)

By using now again the expression ε r = d (rε θ ) / dr in combination with Hooke’s law: Eε r = [σ r − νσ θ ]

Eε θ = [σ θ − νσ r ]

(2.2.28)

One obtains the following equation:

∂σ ⎞ ⎛ ∂σ σ r (1 + ν) − σθ (1 + ν) − r⎜ r − ν θ ⎟ = 0 ∂r ⎠ ⎝ ∂r

(2.2.29)

Substituting into Eq. (2.2.29) the expressions for stresses, Eqs. (2.2.27), one obtains B=0. Then the complete stress field can be re-written in the well-known formulation: C2 r2 C = C1 − 22 r

σ rr = C1 + σ θθ

(2.2.30)

From these equation it is easy to show that, in the case of a hole of radius a in an infinite sheet subject to a remote biaxial stress S, the stress field in the sheet is ⎡

2 ⎛a⎞ ⎤ ⎢⎣ ⎝ r ⎠ ⎥⎦ ⎡ ⎛ a ⎞2 ⎤ = S ⎢1 + ⎜ ⎟ ⎥ ⎣⎢ ⎝ r ⎠ ⎦⎥

σ rr = S ⎢1 − ⎜ ⎟ ⎥ σ θθ

(2.2.31)

2.2.5 A circular hole in an infinite sheet under remote shear (Timoshenko and Goodier, 1970)

Remote from the hole, the sheet is in a state of pure shear:

τ xy = S , σ xx = σ yy = 0

(2.2.32)

The remote state of stress can also be expressed in polar coordinates:

A review of some up-to-now known solutions for linear and nonlinear problems σ rr = S sin 2θ σ θθ = − S sin 2θ τ rθ = S cos 2θ

21

(2.2.33)

∂ ⎛ ∂φ ⎞ ∂ 2φ ∂ 2φ 1 ∂φ + = , and τ rθ = − ⎜ σ ⎟, θθ 2 2 2 ∂r ⎝ r∂θ ⎠ r ∂θ r ∂r ∂r we guess that the stress function must be in the form Remembering that σ rr =

φ (r ,θ ) = f (r )sin 2θ

(2.2.34)

Under this circumstances, the biharmonic equation can be re-rewritten in the following form: ⎛ d2 d 4 ⎞⎛ ∂ 2 f 4f ⎞ ∂f ⎜⎜ 2 + − 2 ⎟⎟ = 0 − 2 ⎟⎟⎜⎜ 2 + rdr r ⎠⎝ ∂r r∂r r ⎠ ⎝ dr

(2.2.35)

Eq. (2.2.35) is an Euler equation, whose solution are in the form f (r ) = r m ; inserting into the bi-harmonic equation, the following equation can be obtained for m:

((m − 2)

2

)(

)

− 4 m2 − 4 = 0

(2.2.36)

whose roots are 2, -2, 0, 4. Consequently, the stress function results: ⎛ ⎝

φ (r ,θ ) = ⎜ Ar 2 + Br 4 +

C ⎞ + D ⎟ sin 2θ 2 r ⎠

(2.2.37)

The stress components inside the sheet are ∂ 2φ 1 ∂φ 6C 4 D ⎞ ⎛ + = −⎜ 2 A + 4 + 2 ⎟ sin 2θ 2 2 r ∂θ r ∂r r r ⎠ ⎝ 2 ∂φ ⎛ 6C ⎞ = 2 = ⎜ 2 A + 12 Br 2 + 4 ⎟ sin 2θ r ⎠ ∂r ⎝

σ rr = σ θθ

τ rθ = −

(2.2.38)

∂ ⎛ ∂φ ⎞ ⎛ 6C 2 D ⎞ 2 ⎜ ⎟ = ⎜ − 2 A − 6 Br + 4 + 2 ⎟ cos 2θ ∂r ⎝ r∂θ ⎠ ⎝ r r ⎠

The following boundary conditions can now be imposed: −

Remote from the hole, namely, r → ∞ , σ rr = S sin 2θ , τ rθ = S cos 2θ ; these conditions results in A = −S / 2 and B = 0 ;

−

On the surface of the hole, namely, r = a, σ rr = 0, τ rθ = 0 , giving D = Sa 2 and C = − Sa 4 / 2 . The stress field inside the sheet is then:

22

A review of some up-to-now known solutions for linear and nonlinear problems 4 2 ⎡ ⎛a⎞ ⎛a⎞ ⎤ σ rr = S ⎢1 + 3⎜ ⎟ − 4⎜ ⎟ ⎥ sin 2θ ⎝r⎠ ⎝ r ⎠ ⎦⎥ ⎣⎢

⎡

4 ⎛a⎞ ⎤ ⎝ r ⎠ ⎥⎦

σ θθ = − S ⎢1 + 3⎜ ⎟ ⎥ sin 2θ ⎢⎣

τ rθ

(2.2.39)

4 2 ⎡ ⎛a⎞ ⎛a⎞ ⎤ = S ⎢1 − 3⎜ ⎟ + 2⎜ ⎟ ⎥ cos 2θ ⎝r⎠ ⎝ r ⎠ ⎦⎥ ⎣⎢

2.2.6 A circular hole in an infinite sheet under remote tension (Kirsch, 1898)

Remote from the hole, the sheet is in a state of pure tension, namely τ xy = σ yy = 0 and σ xx = S

Figure 2.2.1. A circular hole in an infinite sheet under remote tension

The remote state of stress can also be expressed in polar coordinates: S (1 + cos 2θ ) 2 S σ θθ = (1 − cos 2θ ) 2 S τ rθ = − sin 2θ 2

σ rr =

(2.2.40)

It is then evident that the stress field can be considered as the superimposition of two distinct stress fields: a.

One symmetric about an axis, which does not depend on the polar coordinate θ, and whose solution as been already determined in 2.2.3; and takes the form:


23

2 S ⎡ ⎛a⎞ ⎤ σ rr = ⎢1 − ⎜ ⎟ ⎥ 2 ⎣⎢ ⎝ r ⎠ ⎦⎥

σ θθ =

S ⎡ ⎛a⎞ ⎢1 + ⎜ ⎟ 2 ⎢⎣ ⎝ r ⎠

2

⎤ ⎥ ⎥⎦

(2.2.41)

τ rθ = 0 b.

One not symmetric, which does depend on the polar coordinate θ. In this case, we may guess that the stress function must be in the form φ(r, θ) = f (r ) cos 2θ . Under this circumstances, the biharmonic equation can

be re-rewritten again as ⎛ d2 d ∂f 4 ⎞⎛ ∂ 2 f 4f ⎜⎜ 2 + − 2 ⎟⎟⎜⎜ 2 + − 2 rdr r ⎠⎝ ∂r r∂r r ⎝ dr

⎞ ⎟⎟ = 0 ⎠

(2.2.42)

so that the solution is (see the previous section): C ⎛ ⎞ φ(r, θ) = ⎜ Ar 2 + Br 4 + 2 + D ⎟ cos 2θ r ⎝ ⎠

(2.2.43)

Then: 1 ∂φ 6C 4 D ⎞ ∂ 2φ ⎛ + = −⎜ 2 A + 4 + 2 ⎟ cos 2θ 2 2 r ∂θ r ∂r r r ⎠ ⎝ 6C ⎞ ∂ 2φ ⎛ = 2 = ⎜ 2 A + 12 Br 2 + 4 ⎟ cos 2θ ∂r r ⎠ ⎝

σ rr = σ θθ

τ rθ = −

(2.2.44)

6C 2 D ⎞ ∂ ⎛ ∂φ ⎞ ⎛ 2 ⎜ ⎟ = ⎜ 2 A + 6 Br − 4 − 2 ⎟ sin 2θ ∂r ⎝ r∂θ ⎠ ⎝ r r ⎠

The four-needed boundary conditions are: −

Remote from the hole, namely, r → ∞ , σ rr =

S S cos 2θ, τ rθ = − sin 2θ , 2 2

giving A = −S / 4, B = 0 . −

On the surface of the hole, namely, r = a, σ rr = 0, τ rθ = 0 , giving D = Sa 2 / 2 and C = −Sa 4 / 4 . Then stress fields take the form:

24

A review of some up-to-now known solutions for linear and nonlinear problems 4 2 S⎡ ⎛a⎞ ⎛a⎞ ⎤ σ rr = ⎢1 + 3⎜ ⎟ − 4⎜ ⎟ ⎥ cos 2θ 2 ⎣⎢ ⎝r⎠ ⎝ r ⎠ ⎦⎥

S⎡ 2 ⎢⎣

4 ⎛a⎞ ⎤ ⎝ r ⎠ ⎥⎦

σ θθ = − ⎢1 + 3⎜ ⎟ ⎥ cos 2θ τ rθ

(2.2.45)

4 2 S⎡ ⎛a⎞ ⎛a⎞ ⎤ = − ⎢1 − 3⎜ ⎟ + 2⎜ ⎟ ⎥ sin 2θ 2 ⎣⎢ ⎝r⎠ ⎝ r ⎠ ⎦⎥

Adding the solution for the two sub-problems, we can obtain the complete solution. 2 ⎤ S ⎡ ⎛a⎞ ⎤ + − cos 2 1 θ ⎜ ⎟ ⎢ ⎥ ⎥ 2 ⎢⎣ ⎝ r ⎠ ⎥⎦ ⎥⎦ 4 2 S⎡ S ⎡ ⎛a⎞ ⎤ ⎛a⎞ ⎤ = − ⎢1 + 3⎜ ⎟ ⎥ cos 2θ + ⎢1 + ⎜ ⎟ ⎥ 2 ⎣⎢ 2 ⎣⎢ ⎝ r ⎠ ⎦⎥ ⎝ r ⎠ ⎦⎥

S⎡ ⎛a⎞ ⎛a⎞ ⎢1 + 3⎜ ⎟ − 4⎜ ⎟ 2 ⎢⎣ ⎝r⎠ ⎝r⎠ 4

σ rr = σ θθ

S⎡ 2 ⎢⎣

⎛a⎞ ⎝r⎠

4

2

(2.2.46)

2 ⎛a⎞ ⎤ ⎝ r ⎠ ⎥⎦

τ rθ = − ⎢1 − 3⎜ ⎟ + 2⎜ ⎟ ⎥ sin 2θ Note that: −

When θ=π/2 and r=a, σθθ=3S, so that the stress concentration factor is 3;

−

When θ=0, and r=a, σθθ=- S, and hole edge is under compression.

2.2.7 Williams’s solution for a sharp V-notch under mode I and II loadings

Williams considered the problem of stress distribution resulting from re-entrant corners writing the Airy stress function by using separation of variables (Williams 1952): φ(r, θ) = r λ +1 ⋅ f (θ)

(2.2.47)

Since:

σ rr =

2 ⎤ ∂ 2φ 1 ∂φ λ −1 ⎡ d f (θ ) r + = + (λ + 1) ⋅ f (θ )⎥ ⎢ 2 2 2 r ∂θ r ∂r ⎣ dθ ⎦

σ θθ =

∂ 2φ ⎡ ⎤ = r λ −1 ⎢λ ⋅ (λ + 1) ⋅ f (θ )⎥ 2 ∂r ⎣ ⎦

τ rθ = − then:

df (θ ) ⎤ ∂ ⎛ ∂φ ⎞ λ −1 ⎡ ⎜ ⎟ = r ⎢− λ ⋅ dθ ⎥⎦ ∂r ⎝ r∂θ ⎠ ⎣

(2.2.48)

25

A review of some up-to-now known solutions for linear and nonlinear problems ⎛ ∂ 2φ ∂φ ∂ 2φ ⎞ ⎜⎜ 2 + + 2 2 ⎟⎟ = σ rr + σθθ = r∂r r ∂θ ⎠ ⎝ ∂r r

λ −1

(2.2.49)

⎡ ∂ 2f ( θ, λ ) ⎤ λ −1 ⎡ ⎤ ⎢ ∂θ2 + (λ + 1) ⋅ f ( θ)⎥ + r ⎢λ ⋅ (λ + 1) ⋅ f ( θ)⎥ ⎦ ⎣ ⎣ ⎦

Substituting in the bi-harmonic equations, the following equation results for f: ∂ 4f ∂θ

4

[

+ (λ − 1)2 + (λ + 1)2

]∂∂θf + [(λ − 1) ⋅ (λ + 1) ]f = 0 2

2

2

(2.2.50)

2

y σθθ r 2α

σrθ

σrr

θ x

γ Figure 2.2.2. Polar coordinate system at the tip of a sharp V-notch.

It can be shown that solutions for f satisfying Eq. (2.2.50) are in the form: f(θ) = b1 sin(λ + 1)θ + b 2 cos(λ + 1)θ + b3 sin(λ − 1)θ + b 4 cos(λ − 1)θ

(2.2.51)

Boundary conditions for the free egde notch takes the form:

σ θθ (±γ ) = τ rθ (±γ ) = 0

(2.2.52)

Boundary conditions can be also rewritten in terms of f:: f (± γ, λ ) =

∂f (± γ, λ ) =0 ∂θ

(2.2.53)

Or, more explicitly, substituting f: ± b1 sin(λ + 1)γ + b2 cos(λ + 1)γ ± b3 sin(λ −1)γ + b4 cos(λ −1)γ = 0 + b1 (λ + 1) ⋅ cos(λ + 1)γ ± b2 (λ + 1) ⋅ sin(λ + 1)γ +

(2.2.54)

+ b3 (λ −1) ⋅ cos(λ −1)γ ± b4 (λ −1) ⋅ sin(λ −1)γ = 0

By dividing the terms due to the symmetric mode (Mode I) and the skewsymmetric one (Mode II), respectively:

26

A review of some up-to-now known solutions for linear and nonlinear problems ⎧⎪+ b 2 cos(λ1 + 1) γ + b 4 cos(λ1 − 1) γ = 0 ⎨ ⎪⎩+ b 2 (λ1 + 1) ⋅ sin (λ1 + 1) γ + b 4 (λ1 − 1) ⋅ sin (λ1 − 1) γ = 0

(2.2.55)

⎧⎪+ b1sin (λ 2 + 1) γ + b 3sin (λ 2 − 1) γ = 0 ⎨ ⎪⎩+ b1 (λ 2 + 1) ⋅ cos(λ 2 + 1) γ + b 3 (λ 2 − 1) ⋅ cos(λ 2 − 1) γ = 0

(2.2.56)

The eigenvalues of the problem can be determined thanks to the RouchèCapelli condition on the determinant, providing the following two equations: (λ1 − 1) ⋅ sin(λ1 − 1) γ ⋅ cos(λ1 + 1) γ +

(2.2.57)

− (λ1 + 1) ⋅ cos(λ1 − 1) γ ⋅ sin(λ1 + 1) γ = 0

(λ 2 + 1) ⋅ sin(λ 2 − 1) γ ⋅ cos(λ 2 + 1) γ +

(2.2.58)

− (λ 2 − 1) ⋅ cos(λ 2 − 1) γ ⋅ sin(λ 2 + 1) γ = 0

which can also be rewritten in the following synthetic form: sin 2λ1γ + λ1 ⋅ sin 2 γ = 0

(2.2.59)

sin 2λ 2 γ − λ 2 ⋅ sin 2 γ = 0

(2.2.60)

For Mode I and Mode II, respectively. Solutions of Eqs. (2.2.59) and (2.2.60)

(1-λi)

are plotted in Figure 2.2.3.

0.50

(1-λ1)

mode I mode II

0.45 0.40 0.35 0.30 0.25 0.20

(1-λ2)

2α

0.15 0.10 0.05 0.00 0

20

40

60

80

100

120

140

160

2α [°]

180

Figure 2.2.3. Plotting of William’s eigeinvalues.

We are interested in the smallest but positive root of the previous equations, since the strain energy density close to the tip must remain finite.

27


By considering again the system of equation we can determine the remaining parameters: b 2 = −b 4

(λ1 − 1) sin (λ1 − 1) γ (λ − 1) = b4 1 ⋅ χ1 ⋅ (λ1 + 1) sin (λ1 + 1) γ (λ1 + 1)

(2.2.61)

where χ1 satisfies: χ1 =

sin (λ1 − 1) γ sin (1 − λ1 ) γ =− sin (λ1 + 1) γ sin (1 + λ1 ) γ

(2.2.62)

Further: b1 = −b 3

sin (λ 2 − 1) γ = −b 3 ⋅ χ 2 sin (λ 2 + 1) γ

(2.2.63)

where χ2 =

sin (λ 2 − 1) γ sin (1 − λ 2 ) γ =− sin (λ 2 + 1) γ sin (1 + λ 2 ) γ

(2.2.64)

Finally, expressions for stress distributions takes the following form:

σ θθ (r ,θ ) = λ1r λ −1b4 [(1 + λ1 ) cos(1 − λ1 )θ + χ1 (1 − λ1 ) cos(1 + λ1 )θ ] 1

σ rr (r ,θ ) = λ1r λ −1b4 [(3 − λ1 ) cos(1 − λ1 )θ − χ1 (1 − λ1 ) cos(1 + λ1 )θ ] 1

(2.2.65)

τ rθ (r ,θ ) = λ1r λ −1b4 [(1 − λ1 ) sin(1 − λ1 )θ + χ1 (1 − λ1 ) sin(1 + λ1 )θ ] 1

For mode I loadings, and:

σ θθ (r ,θ ) = λ2 r λ −1b3 [(1 + λ2 ) sin(1 − λ2 )θ + χ 2 (1 + λ2 ) sin(1 + λ2 )θ ] 2

σ rr (r ,θ ) = λ2 r λ −1b3 [(3 − λ2 ) sin(1 − λ2 )θ − χ 2 (1 + λ2 ) sin(1 + λ2 )θ ] (2.2.66) 2

τ rθ (r ,θ ) = λ2 r λ −1b3 [(1 − λ2 ) cos(1 − λ2 )θ + χ 2 (1 + λ2 ) cos(1 + λ2 )θ ] 2

for mode II. Stress fields are defined as a a function of two unknown parameters which can be linked to the geometry and the far fields applied load through the notch stress intensity factors defined by Gross and Meldenson (1972):

[ 2π lim [r

] (r, θ = 0)]

K1N = 2π lim r1−λ1 σ θθ (r, θ = 0) r →0

K 2N = So that:

r →0

1−λ 2

σ rθ

(2.2.67) (2.2.68)

28


σ θθ (r, θ) =

1 r λ1 −1 ⋅ K1N [(1 + λ1 ) cos(1 − λ1 )θ + χ1 (1 − λ1 ) cos(1 + λ1 )θ] 2π (1 + λ1 ) + χ1 (1 − λ1 )

σ rr (r, θ) =

1 r λ1 −1 ⋅ K1N [(3 − λ1 ) cos(1 − λ1 )θ − χ1 (1 − λ1 ) cos(1 + λ1 )θ] 2π (1 + λ1 ) + χ1 (1 − λ1 )

1 r λ1 −1 ⋅ K1N [(1 − λ1 ) sin(1 − λ1 )θ + χ1 (1 − λ1 ) sin(1 + λ1 )θ] τ rθ (r, θ) = 2π (1 + λ1 ) + χ1 (1 − λ1 ) (2.2.69) For mode I and: σ θθ (r, θ) =

r λ 2 −1 ⋅ K 2N 1 [(1 + λ 2 )sin (1 − λ 2 )θ + χ 2 (1 + λ 2 )sin (1 + λ 2 )θ] 2π (1 − λ 2 ) + χ 2 (1 + λ 2 )

r λ 2 −1 ⋅ K 2N 1 [(3 − λ 2 )sin (1 − λ 2 )θ − χ 2 (1 + λ 2 )sin (1 + λ 2 )θ] σ rr (r, θ) = 2π (1 − λ 2 ) + χ 2 (1 + λ 2 ) σ rθ (r, θ) =

r λ 2 −1 ⋅ K 2N 1 [(1 − λ 2 ) cos(1 − λ 2 )θ + χ 2 (1 + λ 2 ) cos(1 + λ 2 )θ] 2π (1 − λ 2 ) + χ 2 (1 + λ 2 ) (2.2.70)

For Mode II, respectively.

2.3

Lazzarin and Tovo’s equations for blunt V and U shaped notches under Mode I and II

With the aim to determine stress components in the proximity of an open notch (which becomes a V-crack when the notch radius is equal to zero) under plane stress-strain conditions, Lazzarin and Tovo (1996) used Kolosoff-Muskhelishvili's method, adopting the following analytical functions: ϕ( z ) = a z

λ

ψ( z ) = b z + c z λ

μ

(2.3.1)

where coefficients a, b, c are complex, exponents λ and μ are real, with λ > μ by hypothesis.

29


σϑ

σr

τr ϑ

2α

r

ρ

r0 Figure 2.3.1. Coordinate system and symbols used for the stress field components (figure from Lazzarin and Tovo, 1996).

Polar components of stresses result (Figure 2.3.1):

σ ϑ = λr λ −1 [a1 (1 + λ ) cos(1 − λ )ϑ + a2 (1 + λ )sin (1 − λ )ϑ + b1 cos(1 + λ )ϑ − b2 sin (1 + λ )ϑ ] + + μr μ −1 [c1 cos(1 + μ )ϑ − c2 sin (1 + μ )ϑ ]

σ r = λr λ −1 [a1 (3 − λ ) cos(1 − λ )ϑ + a2 (3 − λ )sin (1 − λ )ϑ − b1 cos(1 + λ )ϑ + b2 sin (1 + λ )ϑ ] + + μr μ −1 [− c1 cos(1 + μ )ϑ + c2 sin (1 + μ )ϑ ]

τ rϑ = λr λ −1 [a1 (1 − λ )sin (1 − λ )ϑ − a2 (1 − λ ) cos(1 − λ )ϑ + b1 sin (1 + λ )ϑ + b2 cos(1 + λ )ϑ ] + + μr μ −1 [c1 sin (1 + μ )ϑ + c2 cos(1 + μ )ϑ ]

(2.3.2) In order to impose the boundary conditions Lazzarin and Tovo (1996) used Neuber’s curvilinear coordinates u and v. The analytical link between the Cartesian coordinates the auxiliary curvilinear coordinates is: x + iy = re iϑ = z = w q = ( u + iv ) q

(2.3.3)

The parametric curve corresponding in the curvilinear system to the u=0 condition, describes in the (x,y) system a sharp angle equal to 2α where: 2α = π( 2 − q )

(2.3.4)

Moreover the curve u = u0, corresponding to the free edge of a generic smooth notch with notch root radius equal to: ρ=

q ⋅ u q0 q ⋅ r0 = ( q − 1) ( q − 1)

Being β = −

(2.3.5)

ϑ the angle included between the principal directions of the polar q

and the curvilinear systems, the components of the stress field in the (u,v)

30


curvilinear system can be now derived from (2.3.2) by imposing the local rotation of the reference system previously defined (Lazzarin and Tovo, 1996): 1 1 2ϑ 2ϑ −τ rϑ sin 2 2 q q 1 1 2ϑ 2ϑ σ v = (σ r + σ ϑ )− (σ r − σ ϑ ) cos +τ rϑ sin q q 2 2 1 2ϑ 2ϑ τ uv = (σ r − σ ϑ )sin +τ rϑ cos q q 2

σ u = (σ r + σ ϑ )+ (σ r − σ ϑ ) cos

(2.3.6)

The following boundary conditions have been imposed along the free edge of the notch:

(σ u )u =u (τ uv )u =u

0

0

=0 (2.3.7)

=0

Lazzarin and Tovo noted that: −

condition (2.3.7) should be satisfied sufficiently far away from the tip, where terms related to exponent "μ" are meaningless;

−

the stress field should be correct in the close neighbourhood of the tip. Hence the first two terms of the series expansions along the free edge, related to σu and τuv stresses, have to be zero. Under these considerations, general boundary conditions (2.3.7) become for σu (Lazzarin and Tovo, 1996):

( σu ) u= u

=0 ⇒ 0

v >> u 0

0

r →∞

ϑ →±

( σu ) uv==0u = ( σr ) ϑr==u0

0

q

(

1−λ

)

lim r σϑ = 0

(2.3.8)

qπ 2

=0

(2.3.9)

⎛ ∂σu ⎞ ⎛ ∂σr ⎞ ⎛ ∂σr ⎞ 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 = ⇒ − = =0 τ ( ) r ϑ r u = ⎝ ∂v ⎠uv==0u ⎝ ∂ϑ ⎠ϑr==u0 q ⎝ ∂ϑ ⎠ϑr==u0 ϑ= 0 0

q

0

q

0

q

(2.3.10)

0

Other three boundary conditions can be stated, in a similar fashion, in τuv. Eq. (2.3.8) in σu and the analogous one in τuv give the same conditions proposed by Williams (1952) for V-cracks and reported in the previous section. Eq. (2.3.8) gives n-solutions or Eigenvalues later characterized by a second index n. The problem being linear, the general solution can be given as a linear combination of all particular solutions; however, the stress field defined by the


31

eigenvalue with the lowest modulus represents the prevailing component and it will be later considered in the analyses. The values of the constants, associated to each Eigenvalue, are given by: b1n = −a1n (1 + λ1n )

cos(1 − λ1n )qπ / 2 = cos(1 + λ1n )qπ / 2

sin(1 − λ1n )qπ / 2 = −a1n (1 − λ1n ) = a1n (1 − λ1n )χ1n sin(1 + λ1n )qπ / 2 b 2n = a 2 n (1 + λ 2n )

sin(1 − λ 2 n )qπ / 2 = sin(1 + λ 2n )qπ / 2

cos(1 − λ 2 n )qπ / 2 = a 2 n (1 − λ 2n ) = −a 2 n (1 + λ 2n )χ 2n cos(1 + λ 2n )qπ / 2

(2.3.11)

(2.3.12)

where χik coefficients are: χi k = −

( ) sin (1 + λ )qπ / 2 sin 1 − λ i k qπ / 2

(2.3.13)

ik

The remaining simplified boundary conditions can be used to determine the other values of constants and parameters. Using boundary conditions, four expressions can be derived for the real constants c1n and c2n: 1 ⎡ ⎤ μ1n r0μ −λ ⎢ 1 − (1 + μ )⎥ c1n = 1n ⎥ ⎢q ⎣ ⎦

(3 − λ1n ) − χ1n (1 − λ1n ) ⎡ ⎤ ⎢ (1 + λ1n )⎤ + χ (1 − λ )⎡(1 + λ ) − 1 ⎤ ⎥⎥ = λ1n a1n ⎡ ⎢ ⎢(1 − λ1n )2 − 1n 1n ⎢ 1n q ⎥⎦ q ⎥⎦ ⎥⎦ ⎢⎣ ⎣ ⎣

(2.3.14)

⎡ ⎤ 1 ⎤ (1 − λ2 n ) + χ2 n (1 + λ2 n ) μ −λ ⎡ μ 2 n r0 ⎢ c2 n = λ 2 n a 2 n ⎢ 2 ⎥ (2.3.15) ⎥ ⎣ −(1 + μ 2 n ) ⎦ ⎢⎣ ( 3 − λ 2 n )(1 − λ 2 n ) − χ 2 n (1 + λ 2 n ) ⎥⎦

[

]

By imposing that systems (2.3.14) and (2.3.15) have nontrivial solutions, the values of the parameter μ and of the real and imaginary components of the variable c can be obtained for mode I and mode II:

⎧⎡ ⎡ (1 + λ1n ) ⎤ 2 1 ⎤⎫ ⎪ ⎢(1 − λ1n ) − ⎥ + χ (1 − λ1n )⎢(1 + λ1n ) − ⎥ ⎪ q ⎦ 1n q ⎦⎪ ⎣ 1 ⎪⎣ ⎬ −1 μ1n = − ⎨ q ⎪ ( 3 − λ1n ) − χ1n (1 − λ1n ) ⎪ ⎪ ⎪ ⎩ ⎭

(2.3.16)

32

A review of some up-to-now known solutions for linear and nonlinear problems μ2n

c1n =

[

]

⎧⎪ ( 3 − λ )(1 − λ ) − χ (1 + λ ) 2 ⎫⎪ 2n 2n 2n 2n ⎬ −1 = −⎨ (1 − λ2 n ) + χ2 n (1 + λ2 n ) ⎪⎩ ⎪⎭

λ1n λ r μ1n 0

1 n −μ1 n

c2 n =

[

a1n ( 3 − λ1n ) − χ1n (1 − λ1n )

λ2n λ r μ2n 0

2 n −μ 2 n

]

[

a 2 n (1 − λ 2 n ) + χ 2 n (1 + λ 2 n )

(2.3.17)

(2.3.18)

]

(2.3.19)

Stress field components in the neighborhood of the notch tips are defined except for a constant value. Afterwards the second index of the Eigenvalues and of the associate constants will be omitted for the sake of simplicity of notation, since only the more interesting solution, referred to index 1, will be considered. Stress components for mode I (tension) are then given by:

⎧⎡⎧ (1 + λ1 ) cos(1 − λ1 )ϑ ⎫ ⎧ cos(1 + λ1 )ϑ⎫⎤ ⎧σϑ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪⎥ ⎪ ⎪ ⎢ λ −1 ⎨ σ r ⎬ = λ1r 1 a1 ⎨⎢⎨(3 − λ1 ) cos(1 − λ1 )ϑ⎬ + χ1 (1 − λ1 )⎨− cos(1 + λ1 )ϑ⎬⎥ ⎪ sin (1 + λ )ϑ ⎪⎥ ⎪τ ⎪ ⎪⎢⎪ (1 − λ )sin (1 − λ )ϑ ⎪ 1 1 ⎭ 1 ⎭⎦ ⎩ ⎩ rϑ ⎭ ⎩⎣ ⎩ ⎛r⎞ + ⎜⎜ ⎟⎟ ⎝ r0 ⎠

μ1 −λ1

⎧ cos(1 + μ1 )ϑ ⎫⎫ [(3 − λ1 ) − χ1(1 − λ1 )]⎪⎨− cos(1 + μ1 )ϑ⎪⎬⎪⎬ ⎪ sin (1 + μ )ϑ ⎪⎪ 1 ⎩ ⎭⎭

(2.3.20) For mode II (shear) by: ⎧⎡⎧ (1 + λ 2 )sin (1 − λ 2 )ϑ ⎫ ⎧ sin (1 + λ 2 )ϑ ⎫⎤ ⎧ σϑ ⎫ ⎪ ⎪ ⎪⎥ ⎪ ⎪ λ 2 −1 ⎪⎢ ⎪ a 2 ⎨⎢⎨(3 − λ 2 )sin (1 − λ 2 )ϑ⎬ + χ 2 (1 + λ 2 )⎨− sin (1 + λ 2 )ϑ ⎬⎥ + ⎨ σr ⎬ = λ 2r ⎪ cos(1 + λ )ϑ⎪⎥ ⎪τ ⎪ ⎪⎢⎪(1 − λ ) cos(1 − λ )ϑ⎪ 2 2 ⎭ 2 ⎭⎦ ⎩ ⎩ rϑ ⎭ ⎩⎣ ⎩ ⎛r⎞ − ⎜⎜ ⎟⎟ ⎝ r0 ⎠

μ 2 −λ 2

⎧ sin (1 + μ 2 )ϑ ⎫⎫ [(1 − λ 2 ) + χ 2 (1 + λ 2 )]⎪⎨− sin (1 + μ 2 )ϑ⎪⎬⎪⎬ ⎪ cos(1 + μ )ϑ ⎪⎪ 2 ⎩ ⎭⎭

(2.3.21) In the presence of a traction loading, the maximum stress at the notch tip can be derived by imposing r=r0 and ϑ=0, so that: a1 =

σ max,tr 4 λ1 r0λ −1

(2.3.22)

According to LEFM, stress components can also be linked to a field parameter, in such a way that the parameter will be equal to stress intensity factors

33


KI and KII as soon as the notch degenerates into a crack. (ρ=0, 2α=0); then such a parameter should be representative of the only main stress term, related to exponent λ. Eqs. (2.3.20) and (2.3.21) can be particularized by using definitions given by Gross and Mendelson (1972) for open cracks: K1 = 2π lim ( σϑ ) ϑ=0 r

K 2 = 2π lim ( τ rϑ ) ϑ=0 r

1−λ1

1−λ 2

r →0

r →0

(2.3.24)

Applying definitions (2.3.24) as long as the component in μ related to the fillet radius is equal to zero (so that also c1,2=0), the real and imaginary parts of the complex constant a are: a1 =

K1

λ1 2π [(1 + λ1 ) + χ1 (1 − λ1 )]

a2 = −

(2.3.25)

K2

λ2 2π [(1 − λ2 ) + χ 2 (1 + λ2 )]

On the basis of these last expressions, in presence of a tensile loading, the maximum stress at the notch tip can be linked to the field parameter by means of the notch radius (represented by the parameter r0) and the opening angle (explicitly according to q, implicitly according to λ). Thus: σmax,tr =

2.4

[

4 K1r0 λ−1

2π (1 + λ1 ) + χ1 (1 − λ1 )

]

(2.3.26)

Solutions under elastic-plastic conditions

2.4.1 Solution for mode I loaded cracks and sharp V notches in power hardening materials

Let us assume that the material obeys the Ramberg-Osgood law, according to which the uniaxial tensile strain ε is related to the uniaxial stress σ as: ε=

σ ⎛σ⎞ +⎜ ⎟ E ⎝K⎠

n

1≤n≤∞

(2.4.1)

The generalized stress-strain relation is (Hutchinson, 1968a,b): 1 − 2ν σ kk 3 α ⎛ σe ⎞ ⎜ ⎟ δij + ε ij = (1 + ν) + E E 3 2 E ⎜⎝ σ o ⎟⎠ s ij

n −1

s ij

(2.4.2)

34


where the summation convention is used for repeated indices. The quantities σe , σ0, Sij are the effective Von Mises stress, the yield stress and the deviatoric stresses, respectively, while α is a material constant, ν is the Poisson coefficient and δij the Kronecker delta. In more detail: 1

⎡3 ⎤2 σ e = ⎢ sij ⋅ sij ⎥ ⎣2 ⎦ 1 sij = σ ij − σ kk δ ij 3 E α = n σ on −1 K

(2.4.3)

In non dimensional terms, (2.4.2) can be written as follows: εij

= (1 + ν )

αε o

sij α

+

1 − 2ν σ kk 3 δij + σ en − 1sij 3 α 2

(2.4.4)

where ε0 = σ0 / E. In the immediate vicinity of the notch tip, the contribution to the singular stress and strain field by the elastic term in (2.4.2) is small when compared to the contribution of the plastic term, as shown by Hutchinson (1968a,b). Therefore, it is sufficient for the asymptotic analysis to use the following simplified expression: ε ij =

3 n −1 σ e s ij 2

(2.4.5)

Stresses can be expressed as follows (Yuan and Lin, 1994, Lazzarin et al., 2001): σij(r,θ) σo

~ (0) (θ) = As r − s σ ij

(2.4.6)

where A is a parameter to determine. The von Mises stress and the deviatoric components of the stresses can also be given as: σ (r ,θ ) σ

= As r − s σ~ ( 0) (θ ) e

σ

= As r − s ~ s ( 0) (θ ) ij

e

o s (r , θ ) ij o

(2.4.7)

35

A review of some up-to-now known solutions for linear and nonlinear problems where: 1

~ (0) (θ) = ⎛⎜ 3 ~s (0) ⋅ ~s (0) ⎞⎟ 2 σ e ij ⎠ ⎝ 2 ij

(2.4.8)

By introducing the equilibrium and compatibility conditions: ε

∂ ( r σ ) ∂σ r + rθ − σ = 0 θθ ∂r ∂θ ∂ ( r σ ) ∂σ rθ + θθ + σ = 0 rθ ∂r ∂θ

rr

=

∂u

r ∂r 1 ∂u

θ + 1u r ∂θ r r 1 ⎛ 1 ∂u r ∂u θ 1 ⎞⎟ ε = ⎜ + − u rθ 2 ⎜ r ∂θ r θ ⎟⎠ ∂r ⎝ ε

θθ

=

(2.4.9)

the governing equations result in a homogeneous systems of differential equations. ~ ( 0) − σ ~ ( 0) + σ ~ ( 0) = 0 ⎧(−s + 1)σ rr θθ rθ,θ ⎪ ( 0) (0) ~ ~ ⎪σ θθ,θ + (−s + 2)σ rθ = 0 ⎪ 3 ~ (0) n −1~ ( 0) ⎪(−sn + 1)~ u r(0) − σ srr = 0 e ⎨ 2 ⎪~ (0) ~ (0) 3 ~ ( 0) n −1~ (0) sθθ = 0 ⎪ u r + u θ ,θ − 2 σ e ⎪1 3 ~ (0) n −1~ (0) ⎪ (~ σ rθ = 0 u r(,0θ) − sn~ u θ( 0) ) − σ e 2 ⎩2

where, as usual, the symbol f,θ

means

(2.4.10)

∂f . By imposing under plane strain ∂θ

~ ( 0 ) = 1 (σ ~ (0) + σ ~ (0) ) and by using for the displacement hypothesis the condition σ zz rr θθ 2

~ u r the third equation, the system has three stress components σrr, σθθ, σrθ and one

displacement component uθ as unknown eigenfunctions. Boundary conditions for a symmetric loading mode (Mode I) are: uθ (r ,0) = 0

τ rθ (r ,0) = 0 σ θθ (r , γ ) = 0 τ rθ (r , γ ) = 0

(2.4.11)

2.4.2 Slip line fields theory

The slip line field theory simplifies the governing equations proposed in the previous chapter as due to due the J2 deformation theory by considering a rigid

36


plastic material, so that the von Mises stress is kept constant and equal to the yield stress σY, once plastic deformation occurs. We will consider only the solution of slip lines in its original intent, that is under the hypothesis of plane strain. Slip lines are defined as trajectories of maximum shear stress. Assumes now yielding occurs, so that the maximum shear stress reaches the yield value k = σ Y / 3 . Stresses in polar coordinates can be written as function of the shear and normal stress expressed in the plane of maximum shear: σ xx = σ − k sin 2χ σ yy = σ + k sin 2χ

(2.4.12)

τ xy = k cos 2χ By inserting Eqs. (2.4.12) into Equilibrium equations one obtains: ∂σ

⎛ ∂χ ∂χ ⎞ − 2k ⎜⎜ cos 2χ + sin 2χ ⎟⎟ = 0 ∂y ⎠ ∂x ∂x ⎝ ∂σ ⎛ ∂χ ∂χ ⎞ − 2k ⎜⎜ sin 2χ − cos 2χ ⎟⎟ = 0 ∂y ∂x ∂y ⎠ ⎝

(2.4.13)

One can convert Eq. (2.4.13) in the α and β coordinates, being them the orthogonal coordinates in the plane of maximum shear by noting that: ∂ ∂ ∂x ∂ ∂y ∂ ∂ = + = cos χ + sin χ ∂s α ∂x ∂s α ∂y ∂s α ∂x ∂y ∂ ∂ ∂x ∂ ∂y ∂ ∂ + = cos χ − sin χ = ∂sβ ∂x ∂sβ ∂y ∂sβ ∂y ∂x

(2.4.14)

Thanks to Eqs. (2.4.14), Eqs. (2.4.13) can be rewritten as: ∂σ (σ − 2kχ ) = 0 ∂s α ∂σ (σ + 2kχ ) = 0 ∂sβ

(2.4.15)

which results in Hencky’s equations: σ − 2kχ = const along α slips lines σ + 2kχ = const along β slips lines

(2.4.16)

Linear Elastic Solutions For Notches Under Torsion

37

3 Linear Elastic Solutions For Notches Under Torsion 3.1 Introduction Knowledge of the linear elastic stress fields ahead of notches is essential in the high cycle fatigue assessment of structural components. The most famous analytical contribution to the study of circumferentially blunt notched shafts under torsion is that due to Neuber (1958), who addressed the problem of ‘deep’ and ‘shallow’ notches and was able to determine in both cases the theoretical stress concentration factor Kt. Thanks to Neuber’s solution, a number of plots for Kt related to common geometries under torsion were drawn (Peterson, 1974). Before Neuber, other authors had focused their attention to antiplane stress problems dealing with keyway grooved or longitudinally cracked cylindrical shafts (Filon, 1900, Shepherd, 1932, Wigglesworth and Stevenson ,1939, Wigglesworth, 1939). Some analytical tools reported in these papers will be reconsidered herein. In order to overcome some limitations of the analytical approaches, various numerical techniques have been used to obtain approximate solutions for the stress concentration factor of notched components subjected to torsion, see, amongst others, Rushton (1967), Hamada and Kitagawa (1968), Matthews and Hooke, (1971) and Peterson (1974). Worthy of mention are also some recent contributions due to Noda and Takase who accurately determined, by means of the body force method, the stress concentration factors of blunt V-notches in round bars under torsion loading (Noda and Takase, 2006), as well as the notch stress intensity factors of sharp, zero radius, V-notches (Noda and Takase, 2003). Considering the local stress distributions and not only Kt is essential in dealing with the structural integrity of notched components. Creager and Paris (1967) gave the elastic stress fields in the vicinity of the tip of blunt cracks, or

38


‘slim’ parabolic notches, under Mode I, II and III loading. The intensities of the fields were expressed in terms of generalised Stress Intensity Factors, later correlated by Glinka to the maximum elastic stress in plane problems (Glinka, 1985). A major difference of a notch under Mode I and Mode II loading compared with a cracked body, is that the bluntness of the notch results in the presence of stress terms proportional to x0.5 and x1.5, x being the distance from the notch tip. Under mode III loading, only the term proportional to x0.5 is present, and the analogy with the crack case is stronger. The stress distribution problem for sharp V-shaped notches in round bars under antiplane shear was solved by Seweryn and Molski (1996), Dunn et al. (1997), Qian and Hasebe (1997), who also dealt with the problem of the singularity at the interface of a bi-material V-notch. In this chapter a set of closed form solutions for stress, strain and displacement fields induced by semi-elliptic, hyperbolic and parabolic circumferential notches in axisymmetric shafts under torsion or uniform antiplane loading is provided. The boundary value problems is formulated according to the complex potential function approach, and using, elliptic, hyperbolic and parabolic coordinate systems. Different classes of solutions are proposed, whose accuracy and range of applicability is discussed in detail taking advantage of a large number of results from FE analyses. The finite size effect on the elastic stress distributions is considered as well. It is also shown that some well-known solutions of linear elastic fracture mechanics and notch mechanics can be seen as special cases of the general solutions reported herein. Finally, the developed analytical frame is used to tie the Mode III Notch Stress Intensity Factor (NSIF) to the maximum shear stress at the notch root, as well as to give closed-form expressions for the strain energy averaged over a given control volume embracing the notch root, as well as for the Mode III Jintegral. Finally, taking advantage of the analytical frame, the last section of the chapter is focused on the determination of analytical expressions for the Mode III Notch Stress Intensity Factors for circumferentially-sharply-notched rounded bards

39


under torsion loading, just starting from the theoretical stress distributions of the corresponding notch problem.

3.2 Semielliptic circumferential notches under torsion 3.2.1 Fundamental complex potentials in antiplane elasticity Consider an axisymmetric body weakened by a circumferential notch of a generic shape, made of an isotropic and homogenous material obeying the theory of the linear elastic deformations. Consider also a Cartesian reference system (x, y, z) having the origin at an appropriate distance from the notch tip. Suppose now that the body is loaded by a remote shear stress τ∞, resulting only in displacements w in the z direction, normal to the plane of the notch characterized by the x and y axes, as shown in Figure 3.2.1.

y

τ τyz τxz

τzx τzy x

z τ

Figure 3.2.1. Axi-symmetric body weakened by a circumferential notch and subjected to antiplane shear stresses. As a direct consequence of the displacement field, all the components of the strain tensor are zero, except for the sliding components in the xz and yz planes (Timoshenko and Goodier, 1970):

γ xz = Since

∂w ∂w and γ yz = . ∂x ∂y

(3.2.1)

40


γ xz =

τ xz G and

γ yz =

τ yz G

(3.2.2)

in the absence of body forces, the equilibrium condition in the z direction results in ∂τ xz ∂τ yz + =0 ∂x ∂y .

(3.2.3)

Consequently ∂2w ∂2w =0 + ∂x 2 ∂y 2

(3.2.4)

and then w is an harmonic function. Following the theory of complex numbers, a real function that is a solution of Laplace’s equation can be always written as the real part of an analytical (holomorphic) function H(z) (Timoshenko and Goodier, 1970). This means that the expression

H ( z ) + H ( z ) = 2Gw

(3.2.5)

is valid. Differentiating Eq.(3.2.5) with respect to x and y, one obtains

2G

∂w = H ' ( z) + H ' ( z ) ∂x

(3.2.6)

2G

∂w = i (H ' ( z ) − H ' ( z ) ) ∂y .

(3.2.7)

and

Multiplying Eq.(3.2.7) by i and subtracting from Eq.(3.2.6), the result is 2G

∂w ∂w − i 2G = 2H ' ( z) ∂x ∂y

(3.2.8)

Finally, taking advantage of Eqs.(3.2.1) and (3.2.2)

τ xz − iτ yz = H ' ( z ) .

(3.2.9)

Eq. (3.2.9) can also be re-written in polar coordinates by using the following relationships

τ xz = τ rz cos ϕ − τ ϕz sin ϕ

(3.2.10)

τ yz = τ rz sin ϕ + τ ϕz cos ϕ

(3.2.11)

then

41

Linear Elastic Solutions For Notches Under Torsion τ xz − iτ yz = τ rz cos ϕ − τ ϕz sin ϕ − i (τ rz sin ϕ + τ ϕz cos ϕ ) =

τ rz (cos ϕ − i sin ϕ ) − τ ϕz (i cos ϕ + sin ϕ )

(3.2.12)

Multiplying Eq. (3.2.12) throughout by –1, one obtains − τ xz + iτ yz = i 2τ rz (cos ϕ − i sin ϕ )

− iτ ϕz (i 2 cos ϕ + i sin ϕ ) = −τ rz (cos ϕ − i sin ϕ ) +

iτ ϕz (cos ϕ − i sin ϕ ) = (cos ϕ − i sin ϕ )(iτ ϕz − τ rz )

(3.2.13)

and finally

τ rz − iτ ϕz = eiϕ H ' ( z ) .

(3.2.14)

We can also observe that, using the property of shear stresses

τ ij = τ ji ,

(3.2.15)

Eqs. (3.2.9) and (3.2.14) can be re-written in the form

τ zx − iτ zy = H ' ( z )

(3.2.16)

τ zr − iτ zϕ = eiϕ H ' ( z ) .

(3.2.17)

It is also worth noting that the only hypothesis used about the complex function H(z) concerns its holomorphism. Nothing has been said about the form of H(z), which will vary from case to case depending on the relevant boundary conditions.

3.2.2

Elliptic coordinate system

We shall use here an orthogonal curvilinear coordinate system (Timoshenko and Goodier, 1970, Inglis, 1913, Stevenson, 1945 Gao, 1996) generated by the transformation

z = c cosh ζ ,

(3.2.18)

where c is a constant and z = x + iy and ζ = ξ + iη are complex variables in the physical and the transformed planes, respectively. In terms of the components x and y, Eq.(3.2.18) becomes x = c cosh ξ cosη

y = c sinh ξ sin η

(3.2.19)

and so cosη =

x ; c cosh ξ

sin η =

y . c sinh ξ

(3.2.20a,b)

42

Linear Elastic Solutions For Notches Under Torsion Elimination of η requires x2 y2 + 2 = 1. c 2 cosh 2 ξ c sinh 2 ξ

(3.2.21)

Different values of ξ result in a family of ellipses, all characterised by the same foci (see figure 3.2.2):

(

)

x = ± c 2 ⋅ cosh 2 ξ − sinh 2 ξ = ±c .

(3.2.22) y

y

r η x

(a)

x

(b)

Figure 3.2.2 (a) Family of ellipses with the same foci ; (b) parametric drawing of the ellipsis When varying η, for ξ = ξ 0 , Eq. (3.2.21) describes a particular ellipse of the family. Consider an elliptic profile of semi-axes a and b, having the centre located at the origin of the coordinate system and the major axis along the x direction. According to Eq. (3.2.21), the semi-axes are ⎧a = c cosh ξ 0 ⎨ ⎩ b = c sinh ξ 0 .

(3.2.23)

Moreover ⎧c = a2 − b2 ⎪ ⎛a⎞ ⎨ ⎪ξ 0 = arccos h⎜⎝ c ⎟⎠ ⎩ .

(3.2.24)

By inverting Eq. (3.2.18) for positive values of x and y, one obtains 2 ⎡z ⎤ ⎛z⎞ ⎛z⎞ ⎢ ζ = ξ + iη = arccos h⎜ ⎟ = ln + ⎜ ⎟ − 1⎥ ⎢c ⎥ ⎝c⎠ ⎝c⎠ ⎣ ⎦

and

(3.2.25)

43


2 ⎧ ⎡z ⎤ ⎫⎪ ⎪ ⎢ ⎛z⎞ ξ = Re⎨ln + ⎜ ⎟ − 1⎥ ⎬ . ⎥⎪ ⎝c⎠ ⎪⎩ ⎢⎣ c ⎦⎭

(3.2.26)

Eq. (3.2.25) can be solved by observing that z 2 − c2 = =4

(x − y − c ) + i(2 xy) , (x − y − c ) + 4 x y ⎛⎜ cos β2 + i sin β2 ⎞⎟ = A⎛⎜ cos β2 + i sin β2 ⎞⎟ 2

2

2

2

2 2

2

2

2

⎝

⎠

⎝

⎠

(3.2.27) where

(

)

2

A = 4 x2 − y2 − c2 + 4x2 y2

(3.2.28)

and ⎧ ⎞ ⎛ 2 xy ⎟ ⎪ arctan⎜⎜ 2 2 2 ⎟ x − y −c ⎠ ⎪ ⎝ β =⎨ ⎞ 2 xy ⎪arctan⎛⎜ ⎟+π 2 2 2 ⎟ ⎜ ⎪⎩ ⎝ x − y −c ⎠

x2 − y2 − c2 > 0

if

if x 2 − y 2 − c 2 < 0

.

(3.2.29)

The terms on the right hand side of Eq. (3.2.25) can be written in the form 2 2 ⎡z ⎤ ⎛z ⎞ ⎛z⎞ ⎛z⎞ ln ⎢ + ⎜ ⎟ − 1⎥ = ln z + z 2 − c 2 − ln c + iph⎜ + ⎜ ⎟ − 1 ⎟ ⎜c ⎟ ⎢c ⎥ ⎝c⎠ ⎝c⎠ ⎣ ⎦ ⎝ ⎠

(3.2.30)

Finally, Eq. (3.2.26) turns out to be ⎛ ⎝

ξ = ln z + z 2 − c 2 − ln c = ln ⎜ x + A cos 2

2

β⎞

β⎞ ⎛ ⎟ + i⎜ y + A sin ⎟ − ln c 2⎠ ⎝ 2⎠

β⎞ ⎛ β⎞ ⎛ = ln ⎜ x + A cos ⎟ + ⎜ y + A sin ⎟ − ln c 2⎠ ⎝ 2⎠ ⎝

. (3.2.31)

It is now possible to determine η by using one of Eqs. (3.2.19). The values of

⎡ π π⎤ interest for η are within the range ⎢− ; ⎥ , which is the domain for which the ⎣ 2 2⎦ sine function can be inverted. It is then more convenient to determine η as ⎛

y ⎞ ⎟⎟ ⎝ c sinh ξ ⎠

η = arcsin⎜⎜

(3.2.32)

44


3.2.3 Formulation of the boundary value problem Consider a semi-elliptic circumferential notch in an axisymmetric shaft under torsion, as shown in Figure 3.2.3.

Mt a

Mt

2R

Figure 3.2.3 Notched shaft under torsion loading. The solution will be obtained using the complex potentials method and the elliptic coordinate system already described. The form of the function H(z) was chosen to be H ( z ) = Ac cosh ζ + Bc sinh ζ

(3.2.33)

which coincides with one of the two potentials already used in the solution of the plane problem of an infinite elastic plate with an elliptic hole under uniaxial or biaxial tension (Timoshenko and Goodier, 1970, Stevenson, 1945, Gao, 1996). So, by using Eq. (3.2.16) and remembering that H ' ( z) =

∂z = c sinh ζ , one can write ∂ζ

∂H ( z ) ∂ζ ⋅ = ( A1 + iA2 ) + ( B1 + iB2 ) coth ζ = τ zx − iτ zy ∂ζ ∂z

(3.2.34)

The elliptic transformation allow us to completely describe the notch profile by the curve ξ = ξ 0 .

3.2.4 Boundary conditions for an infinite shaft Consider now a coordinate system (x,y,z) having the origin located at centre of the

45

Linear Elastic Solutions For Notches Under Torsion notch, see Figure 3.2.4.

τzr

τzy y

τzx

ξ0

z

y

τzϕ

r ϕ

x

z

x

(b)

(a)

Figure 3.2.4. Cartesian (a) and polar (b) coordinate systems The boundary conditions that allow us to determine the complex parameters A and B in the simplest way are

1. At z → ∞ , τ zy = τ and τ zx = 0 ; 2. On the notch flank ( ξ = ξ 0 ), τ zξ = 0 . When ( ξ = ξ 0 ) and (η =

π 2

) then

τ zy = τ zξ ; 3. When η =

π 2

, τ zx = 0 , to satisfy the global boundary conditions.

One should also note that when ζ → ∞ , then the function coth ζ = 1 . Consequently H ' ( z ) = ( A1 + iA2 ) + ( B1 + iB2 ) = τ zx − iτ zy

(3.2.35)

H ' ( z ) , from Eq. (3.2.34), can be written in a different way. By using a

convenient form for the function (Timoshenko and Goodier, 1970):

H ' ( z) = ( A1 + iA2 ) + ( B1 + iB2 )

sinh 2ξ − i sin 2η = cosh 2ξ − cos 2η

⎛ ⎞ sinh 2ξ sin 2η + B2 ⎜⎜ A1 + B1 ⎟+ cosh 2ξ − cos 2η cosh 2ξ − cos 2η ⎟⎠ ⎝ ⎛ ⎞ sinh 2ξ sin 2η + i⎜⎜ A2 + B2 − B1 ⎟ cosh 2ξ − cos 2η cosh 2ξ − cos 2η ⎟⎠ ⎝

(3.2.36)

46

Linear Elastic Solutions For Notches Under Torsion Re-arranging sinh 2ξ sin 2η + B2 cosh 2ξ − cos 2η cosh 2ξ − cos 2η sinh 2ξ sin 2η τ zy = − A2 − B2 + B1 cosh 2ξ − cos 2η cosh 2ξ − cos 2η

τ zx = A1 + B 1

(3.2.37)

Remembering that sinh 2ξ 0 =

2ab a 2 + b2 and cosh 2 ξ = , 0 a2 − b2 a2 − b2

(3.2.38)

the boundary conditions for the shear stress components are

τ zy η = π = − A2 − B2 2

ξ =ξ 0

b sinh 2ξ 0 = − A2 − B2 = 0 a cosh 2ξ 0 + 1

(3.2.39)

and

τ zx η = π = A1 + B1 2

sinh 2ξ cosh 2ξ + 1

= 0.

(3.2.40)

In conclusion, the system is ⎧ A2 + B2 = −τ ⎪A +B =0 1 ⎪ 1 ⎪ b , ⎨ A 2 = − B2 a ⎪ ⎪ A + B sinh 2ξ = 0 1 ⎪⎩ 1 cosh 2ξ + 1

(3.2.41)

so that

τb ⎧ ⎪ A2 = a − b ⎪⎪ A = 0 . ⎨ 1 τa ⎪ B2 = − a−b ⎪ ⎪⎩ B1 = 0

(3.2.42)

Eq. (3.2.37) combined with Eq. (3.2.42) gives ⎧ ⎞ τ ⎛ sinh 2ξ − b ⎟⎟ ⎜⎜ a ⎪⎪τ zy = a − b ⎝ cosh 2ξ − cos 2η ⎠. ⎨ τ a η sin 2 ⎪τ zx = − ⎪⎩ a − b cosh 2ξ − cos 2η

Therefore, at the notch tip, the shear stress component becomes

(3.2.43)

47


τ zy

η =0 ξ =ξ 0

=

τ ⎛

⎞ sinh 2ξ 0 τ ⎛ a ⎞ ⎛ a⎞ ⎜⎜ a − b ⎟⎟ = ⎜ a − b ⎟ = τ ⎜1 + ⎟ a − b ⎝ cosh 2ξ 0 − 1 ⎠ ⎝ b⎠ ⎠ a −b⎝ b

(3.2.44)

and theoretical stress concentration factor is then Kt = 1 +

a b

(3.2.45)

This result agrees with Neuber’s solution for a shallow circumferential notch (Neuber, 1958). For a semicircular notch (a=b), Eq. (3.2.45) gives K t = 2 . The solution found, which is exact only in the case of an infinite diameter shaft, can be also applied to finite sized shafts, at least when the a/R ratio is less than 0.05. In doing so, the error in τmax was found to be less than 10 %. Rg

c cosη (coshξ−coshξ0) a−c cosηcoshξ0

Figure 3.2.5. Relationships to determine the coordinates of the point on the notched component

The linear decrease of the nominal shear stress can be taken into account by means of some simple geometrical considerations, see Figure 3.2.5. These allow us to correct the stress fields given by Eqs. (3.2.43) in the following way ⎧ τ ⎛ ⎞⎫ sinh 2ξ − b ⎟⎟⎪ ⎜⎜ a ⎪ ⎧τ zy ⎫ ⎪ a − b ⎝ cosh 2ξ − cos 2η ⎠⎪⎬⎛⎜1 − c cosη (cosh ξ − cosh ξ 0 ) ⎞⎟ , ⎨ ⎬=⎨ ⎜ sin 2η R + (a − c cosη cosh ξ 0 ) ⎟⎠ ⎩τ zx ⎭ ⎪ − τa ⎪⎝ ⎪⎩ a − b cosh 2ξ − cos 2η ⎪⎭

(3.2.46) where R is the radius of the net section. Figures 3.2.6, 3.2.7 and 3.2.8 show the results of a finite element analysis carried out on a finite diameter shaft with R=200 mm, weakened by a semi-elliptic

48


circumferential notch with a=1 mm and b=0.5 mm.

3.2.5 Solution of a cracked infinite shaft The crack case can be treated as limit condition of a semi-elliptic notch with bÆ0. In this case b = ξ 0 = 0 and c = a , so that

cosh ζ =

z ; sinh ζ = a

z 2 − a2 . a

(3.2.47)

3

τzy / τ

Mt

a=1 R=200 a/b=2

2

a

2R

Mt

1

Eq. (3.2.46) FEM

0

1

10

100

x [mm]

Figure 3.2.6. Plot of the stress component τ zy along the x-direction. The stress component is normalised with respect to the nominal stress.

3

Eq. (3.2.46) Eq. (3.2.46) FEM FEM

τzy / τ τzj / τ

2

Mt

a=1 R=200 a/b=2

a

τzx / τ 2R

Mt

1

0

0

10

20

30

40

50

60

70

80

90

η [degrees]

Figure 3.2.7. Plots of the stress components τ zy and τ zx along the notch flank.

Both stresses normalised with respect to the nominal stress.

49


τzy / τ

1.2

τzy / τ

τzj / τ

1.0

a=1 R=200 a/b=2

0.8 0.6 Eq. (3.2.46) FEM, a'=4, a'/b'=2 FEM, a'=60, a'/b'=2

0.4

τzx/ τ

0.2 0

0

10

20

30

40 50 η [degrees]

60

70

80

90

Figure 3.2.8. Stress fields plotted along two elliptical paths centred on the origin of the (x,y) system; a’=4 mm and a’=60 mm, whereas a’/b’=2; Stresses normalised with respect to the nominal stress.

Substitution of these values into Eqs. (3.2.42) results in the following values for Ai and Bi ⎧ A2 = 0 ⎪ A =0 ⎪ 1 ⎨ ⎪ B2 = −τ ⎪⎩ B1 = 0

(3.2.48)

Substitution into Eq. (3.2.34) gives H ' ( z ) = ( A1 + iA2 ) + ( B1 + iB2 ) coth ζ = = iB2

cosh ζ = −iτ sinh ζ

z z −a 2

2

(3.2.49)

= τ zx − iτ zy

Along the crack axis (y=0), shear stress components are x ⎧ ⎪τ zy = τ ⎨ x2 − a2 , ⎪⎩τ zx = 0

(3.2.50)

Then the relationship for the Mode III stress intensity factor KIII is K III = lim 2π ( x − a)τ zy x→a

= lim 2π ( x − a) x→a

y =0

=

τa ( x − a)( x + a)

= τ πa

.

(3.2.51)

Furthermore, multiplying Eq. (3.2.49) throughout by i, one obtains

50


τ

z z − a2 2

= τ zy + iτ zx .

(3.2.52)

It is useful to note that the left hand side of Eq. (3.2.52) coincides, when the nominal tensile stress σ is substituted for the nominal shear stress τ , with the stress function Z used by Westergaard to solve the plane problem of an infinite cracked plate under tension (Westergaard, 1939). Starting from Eq. (3.2.52) and following Westergaard’s suggestions, one obtains the following expressions ⎧ ⎛ ϕ + ϕ2 ⎞ ⎫ sin ⎜ ϕ − 1 ⎟ ⎪ ⎧τ zx ⎫ τr ⎪ ⎝ 2 ⎠ ⎪⎪ ⎨ ⎬= ⎨ ⎬ r1r2 ⎪cos⎛ ϕ − ϕ1 + ϕ 2 ⎞⎪ ⎩τ zy ⎭ ⎜ ⎟ 2 ⎠⎭⎪ ⎩⎪ ⎝

(3.2.53)

the geometrical parameters being those defined in Figure (3.2.9). If the crack length a is much greater than the zone of validity of Eq. (3.2.53), then some simplifications are possible r2 ≅ 2a

r≅a

ϕ2 ≅ ϕ ≅ 0 .

(3.2.54)

and Eq. (3.2.53) assume the typical form used in LEFM ⎧τ zx ⎫ ⎨ ⎬= ⎩τ zy ⎭

⎧ ⎧ ⎛ ϕ ⎞⎫ ⎛ ϕ ⎞⎫ − sin ⎜ 1 ⎟⎪ − sin ⎜ 1 ⎟⎪ ⎪ ⎪ τa ⎪ ⎝ 2 ⎠⎪ ⎝ 2 ⎠⎪ = K III ⎪ ⎬ ⎨ ⎬ ⎨ r1 2a ⎪ cos⎛ ϕ1 ⎞ ⎪ 2πr1 ⎪ cos⎛ ϕ1 ⎞ ⎪ ⎜ ⎟ ⎜ ⎟ ⎪⎩ ⎪⎩ ⎝ 2 ⎠ ⎪⎭ ⎝ 2 ⎠ ⎪⎭

(3.2.55)

τzy τzx y

r2 ϕ2

r ϕ x

x = −a z

ϕ1

r1

x=a

Figure 3.2.9. Geometrical parameters for the determination of the location of the generic point on a cracked component.

In polar coordinates, substituting ϕ1 with ϕ and r1 with r, the two stress

51

Linear Elastic Solutions For Notches Under Torsion components become ⎧τ zr ⎫ ⎡ cos ϕ ⎨ ⎬=⎢ ⎩τ zϕ ⎭ ⎣− sin ϕ

⎧ ⎛ ϕ ⎞⎫ sin sin ϕ ⎤ ⎧τ zx ⎫ K III ⎪⎪ ⎜⎝ 2 ⎟⎠ ⎪⎪ ⎬ ⎨ ⎬= ⎨ cos ϕ ⎥⎦ ⎩τ zy ⎭ 2πr ⎪cos⎛ ϕ ⎞⎪ ⎜ ⎟ ⎪⎩ ⎝ 2 ⎠⎪⎭

(3.2.56)

3.2.6 Semicircular circumferential notch in an axisymmetric shaft under torsion

In principle, the elliptic coordinate system used to obtain the previous solutions is no longer valid when a degenerating-to-circle ellipse is considered (a=b). Indeed, the semicircular notch results in a mathematical discontinuity for the adopted coordinate system, because the foci become coincident with the centre of the curve so that the constant c is zero. Strictly speaking, it is not possible to describe the semicircular notch case with the solution already obtained for a semi-elliptical notch. Nevertheless, it is worth noting that the previous solution continues to be efficient even when the a/b ratio is very close to one, allowing us to treat the semicircular notch case as the limiting condition for a/b tending to one. This is confirmed by the results of FE analyses carried out on a quasi-infinite shaft (a/R=0.005) weakened by a semielliptical notch with a/b=1.001, as shown in Figures 3.2.10 and 11. 2.5

τzy / τ

2

Mt

1.5

a=1 R=200 a/b=1.001

1

Mt

0.5

0

a

2R

Eq. (3.2.46) FEM 1

10

100

1000

x [mm]

Figure 3.2.10. Plot of the stress component τ zy along x-direction. The stress is

normalised with respect to the nominal stress.

52


2.5

τzy / τ

τzj / τ

2

Mt

a=1 R=200 a/b=1.001

1.5

τzx / τ

Eq (3.2.46a, b) FEM FEM

1

a

Mt

2R

0.5

0

0

10

20

30

40

50

60

70

80

90

η [degrees]

Figure 3.2.11. Plot of the stress components τ zy and τ zx along the notch edge.

Both stresses normalised with respect to the nominal stress. 3.2.7 Boundary conditions for a finite size shaft

When the finite size effect cannot be neglected, this happen when the a/R ratio is greater than 0.05, the boundary conditions imposed for z → ∞ are no longer valid. So it is necessary to write a new system of boundary conditions. The general stress fields are sinh 2ξ sin 2η + B2 cosh 2ξ − cos 2η cosh 2ξ − cos 2η . sinh 2ξ sin 2η τ zy = − A2 − B2 + B1 cosh 2ξ − cos 2η cosh 2ξ − cos 2η

τ zx = A1 + B1

(3.2.57)

The new boundary conditions to determine the complex parameters A and B in a simple way are

τ zx η = π = A1 + B1 2

sinh 2ξ =0 cosh 2ξ + 1

τ zy ξ =ξπ = − A2 − B2 0

η=

2

τ zy ξ =ξ = − A2 − B2 0

η =0

(3.2.58)

b sinh 2ξ0 = − A2 − B2 = 0 a cosh 2ξ0 + 1

(3.2.59)

sinh 2ξ0 a = − A2 − B2 = τ max cosh 2ξ0 − 1 b

(3.2.60)

53


Since Eq. (3.2.58) should be verified independently of the value of ξ, this immediately results in A1 = B1 = 0 .

(3.2.61)

Moreover, by using Eqs. (3.2.59) and (3.2.60), one obtains ⎧ τ maxb 2 = A ⎪⎪ 2 c2 ⎨ ⎪ B2 = − τ max ab ⎪⎩ c2

(3.2.62)

The stress fields can be linked to the theoretical stress concentration factor by ⎧ ⎞ τ K t , gross ⎛ sinh 2ξ − b ⎟⎟ ⎜⎜ a ⎪τ zy = a − b 1 + a ⎝ cosh 2ξ − cos 2η ⎠ ⎪ ⎪ b ⎨ K sin 2η ⎪ τ zx = − τa t , gross ⎪ a − b 1 + a cosh 2ξ − cos 2η ⎪⎩ b If the shaft can be considered as infinite, K t = 1 +

(3.2.63)

a , and then Eq. (3.2.63) b

matches Eq. (3.2.43). The linear decrease of the shear stress as a function of distance from the axis of symmetry can be taken into account by modifying Eq. (3.2.63) to the following forms ⎧ ⎞⎛ c cosη (cosh ξ − cosh ξ 0 ) ⎞ τ K t , gross ⎛ sinh 2ξ ⎟⎟ − b ⎟⎟⎜⎜1 − ⎜⎜ a ⎪τ zy = a ( ) + − ξ η η ξ − − R a c cos cosh a b cosh 2 cos 2 ⎠⎝ 0 ⎠ ⎪ 1+ ⎝ ⎪ b ⎨ ⎛ c cosη (cosh ξ − cosh ξ 0 ) ⎞ sin 2η ⎪ τ = − τa K t , gross ⎜1 − ⎟ zx ⎪ R + (a − c cosη cosh ξ 0 ) ⎟⎠ a − b 1 + a cosh 2ξ − cos 2η ⎜⎝ ⎪⎩ b (3.2.64) The maximum shear stress, τmax, can be estimated by simply using a global equilibrium condition on the net section of the shaft. In the x direction, where

η=0, we have cosh ξ = So

x c

2

;

⎛ x⎞ sinh ξ = ⎜ ⎟ − 1 . ⎝c⎠

(3.2.65)

54


cosh 2ξ = cosh 2 ξ + sinh 2 ξ =

2x2 −1 c2

(3.2.66)

2

2x ⎛ x ⎞ sinh 2ξ = 2 cosh ξ ⋅ sinh ξ = ⎜ ⎟ −1 . c ⎝c⎠

(3.2.67)

Furthermore: 2

2x ⎛ x ⎞ ⎜ ⎟ − 1 x 22 x 2 − c 2 c ⎝c⎠ sinh 2ξ = = c = 2 2 2 cosh 2ξ − 1 2 ⎛ x⎞ x −c 2⎜ ⎟ − 2 c2 ⎝c⎠

(

)

x x2 − c2

.

(3.2.68)

The stress component τ zy evaluated along the x direction is then τzy =

⎞ ⎛ (x − a ) ⎞ b τ max ⎛ ax ⎜ − b ⎟⎟ ⋅ ⎜1 − ⎟ 2 2 ⎜ 2 2 a −b ⎝ x −c R ⎠ ⎠ ⎝

(3.2.69)

Introducing the auxiliary variable t, being

t = R+a−x,

(3.2.70)

we can re-write Eq. (3.2.69) in the form bτmax ⎛⎜ a 2 − b2 ⎜ ⎝

τ zy =

a (Rg − t )

⎞t − b⎟ , (Rg − t ) 2− c 2 ⎟⎠ R

(3.2.71)

where Rg is the radius of the gross area. Considering the polar coordinate system (t, θ, z) centred on the axis of the shaft and shown in Figure 3.2.12, we can write the equilibrium equation on the net section by equating the contribution given by the τ zy stress and that due to the nominal shear stress τnom ranging from τ* to 0. In particular

τ nom ( t ) = τ *

t . R

(3.2.72)

The equilibrium condition is

∫τ

nom

2π

R

A

∫0 ∫0

tdA = ∫ τ zy tdA A

τ*

t3 2π R dtdθ = ∫ 0 ∫ 0 τ zy t 2 dtdθ R

In a more explicit form

(3.2.73) (3.2.74)

55


(

)

2 ⎛ ⎞ 2 ⎧ a ⎜ R g − R − c − Rg + R ⎟ ⎨ ⋅ ln⎜ ⎟× ⎩8 ⎜ Rg2 − c 2 − Rg ⎟ ⎝ ⎠ 2 2 2 2 2 4 × Rg − c 18 Rg − 3 Rg − c − 15 Rg +

bK R4 τ = τ * 2 t ,net2 4 a −b *

[(

] ( )) ⋅ [13R (R − c ) − 15 R

)(

(

)

(

]

)

2 a 2 2 3 2 2 2 2 3 Rg − R − c 2 g g g + 3R Rg − c − 5 RRg − 2 R Rg − 2 R − 8 a b ⎫ Rg2 − c 2 13Rg Rg2 − c 2 − 15 Rg3 − R 4 ⎬ 8 4 ⎭

[

(

)

]

(3.2.75) By substituting (Rg–R=a) and ( a 2 − c 2 = b 2 ), Eq. (3.2.75) can be re-arranged into the following

τ*

R4 R4 b ⎛ a a⎞ =τ* Kt ,net 2 C⎜ , ⎟ c ⎝R b⎠ 4 4

(3.2.76)

where the parameter C is: ⎞ ⎛ ⎟ ⎜ b a − ⎟ ⎜ ⎛ a a ⎞ ⎧a R R ⎟× C ⎜ , ⎟ = ⎨ ⋅ ln⎜ ⎝ R b ⎠ ⎩ 2 ⎜ ⎛ a ⎞2 ⎛ c ⎞2 ⎛ a ⎞ ⎟ ⎜ ⎜1 + ⎟ − ⎜ ⎟ − ⎜ 1 + ⎟ ⎟ ⎜ ⎝ R⎠ ⎝ R⎠ ⎝ R⎠⎟ ⎠ ⎝ ⎡⎛ ⎛ a ⎞ 2 ⎛ c ⎞ 2 ⎞⎛ ⎛ a ⎞ 2 × ⎢⎜ ⎜1 + ⎟ − ⎜ ⎟ ⎟⎜18⎜1 + ⎟ − ⎢⎣⎜⎝ ⎝ R ⎠ ⎝ R ⎠ ⎟⎠⎜⎝ ⎝ R ⎠ 4 ⎛ ⎛ a ⎞2 ⎛ c ⎞2 ⎞ ⎞ ⎛ a⎞ ⎤ 3⎜ ⎜1 + ⎟ − ⎜ ⎟ ⎟ ⎟ − 15⎜1 + ⎟ ⎥ + ⎜⎝ R ⎠ ⎝ R ⎠ ⎟⎟ ⎝ R ⎠ ⎥⎦ ⎝ ⎠⎠ 2 2 3 ab ⎡ ⎛ a ⎞⎛ ⎛ a ⎞ ⎛ c ⎞ ⎞⎟ ⎛ a⎞ + ⋅ ⎢13⎜1 + ⎟⎜⎜ ⎜1 + ⎟ − ⎜ ⎟ − 15⎜1 + ⎟ + 2 R ⎢⎣ ⎝ R ⎠⎝ ⎝ R ⎠ ⎝ R ⎠ ⎟⎠ ⎝ R⎠

⎛ ⎛ a ⎞2 ⎛ c ⎞2 ⎞ ⎛ a ⎞2 ⎛ a ⎞ ⎤ 3 ⎜ ⎜1 + ⎟ − ⎜ ⎟ ⎟ − 5⎜1 + ⎟ − 2⎜1 + ⎟ − 2⎥ − ⎜⎝ R ⎠ ⎝ R ⎠ ⎟ ⎝ R ⎠ ⎝ R⎠ ⎦ ⎠ ⎝ 2

a ⎛ a⎞ ⎛c⎞ ⎜1 + ⎟ − ⎜ ⎟ 2 ⎝ R⎠ ⎝R⎠

2

3 ⎡ ⎛ a ⎞⎛ ⎛ a ⎞ 2 ⎛ c ⎞ 2 ⎞ ⎛ a ⎞ ⎤ ⎫⎪ ⎢13⎜1 + ⎟⎜ ⎜1 + ⎟ − ⎜ ⎟ ⎟ − 15⎜1 + ⎟ ⎥ − b⎬ ⎝ R ⎠ ⎥⎦ ⎪⎭ ⎢⎣ ⎝ R ⎠⎜⎝ ⎝ R ⎠ ⎝ R ⎠ ⎟⎠

(3.2.77) Simplifying the common terms, the stress concentration factors on the net or the gross section are then:

56


z y θ

x

t

Figure 3.2.12. Reference frame used to set the equilibrium condition on the net section of the shaft. K t , net

c2 = ⋅ b

1 ; ⎛ a a⎞ C⎜ , ⎟ ⎝R b⎠

K t , gross

⎛R = K t , net ⋅ ⎜⎜ g ⎝ R

⎞ ⎟⎟ ⎠

3

(3.2.78a,b)

Figure 3.2.13 shows a comparison between the values of the stress concentration factor calculated by means of Eq. (3.2.78b) and results from FE analyses for different values of the a/R and a/b ratios. The maximum error in the evaluation of the stress concentration factor was found, in the worst case of a/R = 0.1, to be less than 5%.

Mt

10

a/b=3

a

Kt, gross

2R

12

Mt 8

a/b=2

6 Eq. (3.2.78) FEM

4

2

0

0.2

0.4

0.6

0.8

a/R

Figure 3.2.13. Comparison between FE results and Eq. (3.2.78) for different a/b and a/R ratios.

57


3.2.8 Stresses in polar coordinates

Consider a polar coordinate system (r, ϕ, z) with its origin at the centre of the ellipse, as shown in Figure 3.2.4b. Remembering that

τ zr = τ zx cos( x, r ) + τ zy cos( y, r )

(3.2.79)

τ zϕ = τ zx cos( x, ϕ ) + τ zy cos( y, ϕ )

(3.2.80)

we have

τ zr = τ zx cos ϕ + τ zy sin ϕ

(3.2.81)

τ zϕ = τ zy cos ϕ − τ zx sin ϕ

(3.2.82)

and substituting Eqs. (3.2.63) ⎧ τ K t ,gross ⎡ a (sinh 2ξ sin ϕ − sin 2η cos ϕ) − b sin ϕ⎤⎥ ⎪ τ zr = ⎢ a a − b 1 + ⎣ cosh 2ξ − cos 2η ⎦ ⎪ ⎪ b . ⎨ K ⎡ ⎤ a τ t , gross ⎪τ = (sinh 2ξ cos ϕ + sin 2η sin ϕ) − b cos ϕ⎥ a ⎢⎣ cosh 2ξ − cos 2η ⎪ zϕ a − b ⎦ 1+ ⎪⎩ b (3.2.83) The curvilinear coordinates (ξ, η, z) and the physical ones (r, ϕ, z) are linked by the following relationships ⎧r cos ϕ = c cosh ξ cos η ⎨ ⎩r sin ϕ = c sinh ξ sin η

(3.2.84)

3.2.9 Strain and displacement fields

Substitution of Eq. (3.2.64) into Eq. (3.2.2) results in

τ zx

1 τa K t , gross sin 2η × G G a − b 1 + a cosh 2ξ − cos 2η b ⎛ c cosη (cosh ξ − cosh ξ 0 ) ⎞ ⎟ × ⎜⎜1 − R + (a − c cosη cosh ξ 0 ) ⎟⎠ ⎝

γ zx =

γ zy =

τ zy G

=−

=

1 τ K t , gross G a − b 1+ a b

(3.2.85)

⎛ ⎞ sinh 2ξ ⎜⎜ a − b ⎟⎟ × ⎝ cosh 2ξ − cos 2η ⎠

⎛ c cosη (cosh ξ − cosh ξ 0 ) ⎞ ⎟ × ⎜⎜1 − R + (a − c cosη cosh ξ 0 ) ⎟⎠ ⎝

.

(3.2.86)

58


Furthermore Eq. (3.2.5) gives 2 Re{H ( z )} = H ( z ) + H ( z ) = 2Gw . Remembering that H(z) = Ac cosh ζ + Bc sinh ζ and

A1 = 0, B1 = 0, A 2 =

τ max b2 τ ab , B2 = − max2 , 2 c c

(3.2.87)

the new expression for the analytical function is ⎛ e ξ + iη − e − ξ − iη ⎞ ⎛ ξ + iη + e − ξ − iη ⎞ ⎟ + iA 2c⎜ e ⎟ H(z) = iB 2c sinh ζ + iA 2c cosh ζ = iB2c⎜ ⎜ ⎜ ⎟ ⎟ 2 2 ⎝ ⎠ ⎝ ⎠ ⎡ eξ ⎤ e −ξ (cos η − i sin η)⎥ + = icB2 ⎢ (cos η + i sin η) − 2 ⎢⎣ 2 ⎥⎦ ⎡ eξ ⎤ e −ξ (cos η − i sin η)⎥ iA 2c ⎢ (cos η + i sin η) + 2 ⎣⎢ 2 ⎦⎥

= (− cB2 cosh ξ sin η − A 2c sinh ξ sin η) + i(cA 2 cosh ξ cos η + B2c sinh ξ cos η)

(3.2.88) Finally, w=

Re{H ( z )} bτ max = sin η(a cosh ξ − b sinh ξ) G cG

(3.2.89)

1

0.8

τzy / τmax

Mt

a=30 R=100 a/b=2

0.6

a

0.4 Mt

2R

0.2 Eq. (3.2.64) FEM 0

30

40

50

60

70

80

90

100

110

120

130

x [mm]

Figure 3.2.14. Plot of the stress component τ zy along the η=0 direction. Stress

normalised with respect to the maximum shear stress.

59


1

0.8

τzy / τmax

Mt

a=50 R=100 a/b=3

0.6

a

0.4 2R

Mt

0.2

0

Eq. (3.2.64) FEM 50

60

70

80

90

100

110

120

130

140

150

x [mm]

Figure 3.2.15. Plot of the stress component τ zy along the η=0 direction. Stress

normalised with respect to the maximum shear stress. 3.2.10 Numerical results

Figures 3.2.14 and 3.2.15 show a comparison of the stress component τzy along the x direction provided by Eqs. (3.2.64) and (3.2.78), and the results of FE analyses carried out on finite size shafts. The agreement is satisfactory. 3.2.11 Circumferential elliptic notch with an arbitrary orientation angle β For the sake of simplicity only the case of infinite shaft is considered herein. If β is the arbitrary orientation angle, as shown in Figure 3.2.16, we have to find a new system of boundary conditions to replace the previous one. The stress fields in the reference system (x, y, z) are sinh 2ξ sin 2η + B2 cosh 2ξ − cos 2η cosh 2ξ − cos 2η . sinh 2ξ sin 2η τ zy = − A2 − B2 + B1 cosh 2ξ − cos 2η cosh 2ξ − cos 2η

τ zx = A1 + B1

The new boundary conditions are 1. At z → ∞ ; τ zy ' = τ zy , nom = τ and τ zx ' = 0 2. τ zx

η =0 ξ =ξ 0

=0;

(3.2.90)

60

Linear Elastic Solutions For Notches Under Torsion 3. τ zy η =± π = 0 . 2

ξ =ξ0

Taking advantage of the equations ⎧⎪τ zx ' ⎫⎪ ⎡ cos β ⎨ ⎬=⎢ ⎪⎩τ zy ' ⎪⎭ ⎣− sin β

sin β ⎤ ⎧τ zx ⎫ ⎨ ⎬ cos β ⎥⎦ ⎩τ zy ⎭

(3.2.91)

the boundary conditions become

⎧ ( A1 + B1 )cos β − ( A2 + B2 )sin β = 0 ⎪− ( A + B )sin β − ( A + B )cos β = τ 1 1 2 2 ⎪ ⎪ a , ⎨ A1 + B1 = 0 b ⎪ ⎪ A +B b =0 2 2 a ⎩⎪

(3.2.92)

The general solution is then

τa ⎧ ⎪ A1 = − a − b sinβ ⎪ τb ⎪ A2 = cosβ ⎪ a −b . ⎨ τb ⎪ B1 = sinβ a −b ⎪ ⎪B = − τa cosβ ⎪⎩ 2 a −b

(3.2.93)

One can also consider some special feature: a) β= −π/2

With the usual notation of a and b representing the ellipse semi-axes in the xand y-directions, respectively Eq. (3.2.93) becomes ⎧ A2 = 0 ⎪ B =0 ⎪ 2 ⎪ − τa . ⎨ A1 = b−a ⎪ ⎪B = τb ⎪⎩ 1 b − a

(3.2.94)

so that the stress distribution is

τ zx =

⎞ τ ⎛ b sinh 2ξ ⎜⎜ − a ⎟⎟ b − a ⎝ cosh 2ξ − cos 2η ⎠

⎞ τb ⎛ sin 2η ⎟ ⎜⎜ τ zy = b − a ⎝ cosh 2ξ − cos 2η ⎟⎠

.

(3.2.95)

61


At the notch tip, where η =

⎛ ⎝

π 2

, one obtains

b⎞ a⎠

τ zx ξ =ξπ = τ max = τ ⎜1 + ⎟ . 0

η=

2

(3.2.96)

⎛ b⎞ The theoretical stress concentration factor is then equal to ⎜1 + ⎟ . ⎝ a⎠ The solution obtained, which is exact only in the case of an infinite diameter shaft, can be applied to finite size shafts when the ratio a/R is less than 0.05; in this case the error in the determination of τmax is less than 10 %. The global decreasing Coulomb effect can be taken into account by modifying the stress fields in the following way τzx =

⎞ ⎛ c sin η(sinh ξ − sinh ξ0 ) ⎞ τ ⎛ b sinh 2ξ ⎟ − a ⎟⎟ ⋅ ⎜⎜1 − ⎜⎜ R + b − c sinh ξ0 sin η ⎟⎠ b − a ⎝ cosh 2ξ − cos 2η ⎠ ⎝

⎛ c sin η(sinh ξ − sinh ξ0 ) ⎞ τb sin 2η ⎟ ⋅ ⎜⎜1 − τzy = R + b − c sinh ξ0 sin η ⎟⎠ b − a cosh 2ξ − cos 2η ⎝

(3.2.97)

b) β= π/4

If β =

π 4

⎧ ⎪ A1 = − ⎪ ⎪ A = ⎪ 2 ⎨ ⎪ B1 = ⎪ ⎪ ⎪B2 = − ⎩

, then Eq. (3.2.93) becomes

τa

2 (a − b) τb 2 (a − b) . τb 2 (a − b) τa 2 (a − b)

(3.2.98)

Then the stress fields in the (x, y, z) coordinate system are τzx =

⎡ ⎛ ⎞⎤ τ b sinh 2ξ sin 2η − a⎜⎜1 + ⎟⎥ ⎢ 2 (a − b) ⎣ cosh 2ξ − cos 2η ⎝ cosh 2ξ − cos 2η ⎟⎠⎦

⎡ ⎛ ⎞⎤ τ a sinh 2ξ sin 2η τzy = + b⎜⎜ − 1⎟⎟⎥ ⎢ 2 (a − b) ⎣ cosh 2ξ − cos 2η ⎝ cosh 2ξ − cos 2η ⎠⎦

(3.2.99)

62


τzy’ y

τzx’

y’

x

β x’

Figure 3.2.16. Elliptic notch inclined of the angle β with respect to the x’ direction.

The stresses τ zy and τ zx from Eq. (3.2.99) are shown in Figure 3.2.17 as a function of the distance from the notch root, in with the x direction, and are compared with FE results. The agreement is again satisfactory. 2.5 a=1 R=200 a/b=2 β = π/4

τzj / τ

2

1.5

τzy / τ

1

0.5

0

τzx / τ

0

1

2

3

4

5

6

Distance from the notch tip [mm]

Figure 3.2.17. Plots of the stress components τ zy and τ zx along the η=0 direction.

Both stresses normalised with respect to the nominal stress. 3.2.12 A comparison with the solution by Smith

Smith demonstrated that, for distances along the notch bisector line much smaller than the notch root radius, the stress distribution strictly depends on the root radius ρ and the distance d* from the notch tip, and not on the notch shape. In

63


Smith’s equation the shear stress component normalised to its maximum value at the notch tip was according to the expression τ zy τ max

=1−

d* ρ

(3.2.100).

In Figure 3.2.18 are the results for the normalised shear stress component as a function of the normalised distance from the notch tip. The distance over which stress distributions are independent of the notch shape is really very limited, approximately 0.05ρ. This is the region where the results match Smith’s solution. These are quite different to the stress distributions in notched components under mode I loading, where the limiting distance was much greater, about 0.2-0.3ρ (Nui et al., 1994, Atzori et al., 2001). 1 Semielliptic notch, a=2 R = 200 Semicircular

τzy / τmax

0.9

0.8

Smith's solution (2004a)

0.7

(2α=0°) Parabolic notch Hyperbolic notch (2α=60°) Hyperbolic notch (2α=120°) Hyperbolic notch (2α=135°) Hyperbolic notch (2α=150°)

0.6

R = 200 Notch tip radius ρ=0.5 mm Notch opening angle 2α

0.5 0

0.2

0.4 0.6 0.8 Distance from the notch tip / ρ

1

Figure 3.2.18. Plots of the shear stress normalized with respect to the peak stress as a function of the normalised distance from the notch tip. Notches of different geometry, with ρ = 0.5 mm.

64


3.3

Hyperbolic and parabolic notches in round shafts under torsion and uniform antiplane shear loadings

3.3.1 Further comments on the fundamental complex potentials in antiplane elasticity

Consider an axisymmetric body weakened by a circumferential notch of a generic shape, made of an isotropic and homogenous material obeying the theory of linear elastic deformations. Consider also a Cartesian reference system (x, y, z) having the origin at an appropriate distance from the notch tip. Suppose now that the body is loaded by a remote shear stress τ, resulting only in displacements w in the z direction, normal to the plane of the notch characterized by the x and y axes, as shown in Figure 3.3.1. y

τ τyz τxz

τzx τzy x

z τ

Figure 3.3.1. Axi-symmetric body weakened by a circumferential notch and subjected to antiplane shear stresses.

Under these conditions we have proved before that the following relationships for stresses and strains are valid (Lazzarin et al., 2007):

τ zx − iτ zy = H ' ( z )

(3.3.1)

H ' (z) γ zx − iγ zy = G

(3.3.2)

in Cartesian coordinates and: τ zr − iτ zϕ = e iϕ H ' ( z ) .

γ zr − iγ zϕ =

eiϕ H '(z) G

(3.3.3) (3.3.4)

65


in polar coordinates. Moreover the displacement w in the z direction can be determined as (Lazzarin et al., 2007):

w=

Re{H ( z )} G

(3.3.5)

The function H(z) is a holomorphic function of arbitrary form, which will vary from case to case depending on the relevant boundary conditions. Since stresses and displacements cannot change with a change of origin, in the absence of body forces expressions (3.3.1)-(3.3.5) are independent of the choice of it. See, for example, Stevenson (1945) for plane problems. So in the following study we shall translate the origin of the reference system in the x-direction as a function of the notch shape and the transformation used, without any loss of generality. Finally it is worth noting that, according to the past literature, the symbol “z” is used in this paper with two different meanings that must not be confused; indeed it denotes both the complex variable z = x + iy and the antiplane coordinate in the Cartesian reference system (x,y,z).

3.3.2 Hyperbolic coordinate system

To solve this problem, we shall use an orthogonal hyperbolic coordinate system generated by the transformation (Filon, 1900, Timoshenko and Goodier, 1970):

z = c cosh ζ

(3.3.6)

where c is a constant and z = x + iy and ζ = ξ + iη are complex variables in the physical and the transformed planes, respectively. In terms of the components x and y, Eq. (3.3.6) becomes: x = c cosh ξ cos η

y = c sinh ξ sin η

(3.3.7)

and so: cosh ξ =

x y sinh ξ = c cos η c sin η

(3.3.8)

Elimination of ξ requires: x2 y2 − =1 c 2 cos 2 η c 2 sin 2 η

(3.3.9)

66

Linear Elastic Solutions For Notches Under Torsion Different values of η result in a family of hyperbolae, all characterised by the

same foci, see figure 3.3.2a: x = ± c 2 ⋅ (cos 2 η + sin 2 η) = ±c .

(3.3.10)

When varying ξ , for η = η0 and η = - η0 in the first and fourth quadrants of the Cartesian plane, respectively, Eq. (3.3.9) describes a particular hyperbola of the family, see figure 3.3.2b. y

η=cost

y

y=

b x a η=η0

η0 x

x η=-η0

(a)

b y=− x a (b)

Figure 3.3.2. Family of hyperbolae with the same foci (a); hyperbolic profile (1st and 4th quadrants) (b).

Consider a hyperbolic profile characterized by the asymptotes y = ±

b x, a

intersecting the x-axis at the value x=a. The comparison between eq. (3.3.9), setting η = η0 , and the canonical equation of an hyperbola results in: ⎧a = c cos η0 . ⎨ ⎩ b = c sin η0

(3.3.11)

Furthermore the following relations are also valid: ⎧b = c2 − a 2 ⎪ ⎨ η = arctan ⎛ b ⎞ ⎜ ⎟ ⎪⎩ 0 ⎝a⎠

(3.3.12)

Inverting eq. (3.3.6), being x and y both positive, results in (Lazzarin et al., 2007):

67

Linear Elastic Solutions For Notches Under Torsion 2 ⎡z ⎤ ⎛z⎞ ⎛z⎞ ζ = ξ + iη = arccos h⎜ ⎟ = ln ⎢ + ⎜ ⎟ − 1⎥ ⎢c ⎥ ⎝c⎠ ⎝c⎠ ⎣ ⎦

(3.3.13)

and so: 2 ⎧ ⎡z ⎤ ⎫⎪ ⎪ ⎢ ⎛z⎞ η = Im ⎨ln + ⎜ ⎟ − 1⎥ ⎬ . ⎥⎪ ⎝c⎠ ⎪⎩ ⎢⎣ c ⎦⎭

(3.3.14)

Remembering that (Lazzarin et al., 2007): 2 2 ⎞ ⎛z ⎤ ⎡z ⎛z⎞ ⎛z⎞ 2 2 ⎜ ⎥ ⎢ ln + ⎜ ⎟ − 1 = ln z + z − c − ln c + i ph + ⎜ ⎟ − 1 ⎟ ⎟ ⎜c ⎝c⎠ ⎝c⎠ ⎥⎦ ⎢⎣ c ⎠ ⎝

(3.3.15) and that: β β⎞ ⎛ z 2 − c 2 = A⎜ cos + i sin ⎟ , 2 2⎠ ⎝

(3.3.16)

being: A = 4 (x 2 − y 2 − c 2 ) + 4 x 2 y 2 2

(3.3.17)

and ⎧ ⎞ ⎛ 2xy ⎟ ⎪arctan⎜⎜ 2 2 2 ⎟ x − y −c ⎠ ⎪ ⎝ β=⎨ ⎞ 2xy ⎪arctan⎛⎜ ⎟+π 2 2 2 ⎟ ⎜ ⎪⎩ ⎝ x −y −c ⎠

if x 2 − y 2 − c 2 > 0 (3.3.18) if x − y − c < 0 2

2

2

the following expressions immediately result: β⎞ ⎛ ⎜ y + A sin ⎟ 2⎟ η = arctan ⎜ ⎜⎜ x + A cos β ⎟⎟ 2⎠ ⎝

(3.3.19)

⎛ y ⎞ ξ = arcsinh ⎜⎜ ⎟⎟ ⎝ c sin η ⎠ Note that, if x and y are both positive, 0 ≤ ξ < ∞ and 0 < η ≤

(3.3.20) π . 2

68


3.3.3 Hyperbolic-parabolic coordinate system We shall also use an orthogonal curvilinear coordinate system generated by the transformation (Neuber, 1958, Lazzarin and Tovo, 1996): z = wq

(3.3.21)

where z = x + iy and w = u + iv are complex variables in the physical and the transformed planes, respectively, and q is a real number related to the opening angle of the curve 2α: q=

2π − 2α 2 γ = . π π

(3.3.22)

Eq. (3.3.21) can be re-written as: w q = (u + iv ) = re iϕ = r (cos ϕ + i sin ϕ ) q

(3.3.23)

and so: 1 ⎧ ϕ q ⎪u = r cos ⎪ q ⎨ 1 ⎪ v = r q sin ϕ ⎪⎩ q

(3.3.24)

The angle between the radial vector r and the normal vector nu to the curve u=cost is then equal to −

ϕ (see figure 3.3.3b). q

Moreover eq. (3.3.24) results in: r = (u 2 + v 2 ) 2 q

(3.3.25)

The curvilinear coordinate system introduced here allows one to completely describe the hyperbolic (1>ρ and x ' τ 0 , the constancy of the Strain Energy Density It is evident that, when τ max

at the notch tip is no longer assured. When the notch root radius is equal or tends to zero, the local energies should be compared over a control volume characterised by a radius R (Lazzarin and Zambardi, 2002). Due the high stress and strain concentration effects caused in the plastic zone by the crack, the contribution to the strain energy density by the elastic term is small when compared to that of the plastic term, and the local

148

NonLinear Solutions For Mode III Loaded Notches

behaviour of the material is very close to that of a pure power law. Then, by using e Eq. (4.2.65) and setting K 3eρ = K III , one obtains:

γ

1

⎛ γ ⎞n n τ n +1 WP = τ 0 ∫ ⎜⎜ ⎟⎟ dγ = γ n + 1 Gτ 0 n −1 0⎝ 0 ⎠ 1 1 ⎫ ⎧ n +1 ~ n +1 e 2 ⎤ ⎡ ⎡ F⎤ ⎪ n 1 ⎪ n (K III ) n −1 τ0 ⎥ ⎢ ⎥ ⎬ = n −1 ⎨ ⎢ n + 1 Gτ 0 ⎪ ⎣⎢ π(n + 1) ⎦⎥ ⎣ r ⎦ ⎪ ⎩ ⎭

n +1

~ n F (KeIII )2 r −1 = 2 πG (n + 1)

(4.2.92)

2

The total energy evaluated over the control volume turns out to be: ~ n2 F

Rπ

E p = ∫ Wp dA = 2 ∫ ∫ A

=

00

2n 2 πG (n + 1)

2

πG (n + 1)

2

(K ) R ⋅ I e III

(K ) r

e 2 −1 rdrdϕ = III

(4.2.93)

2

3p

being: π

~ I 3 p = ∫ F dϕ

(4.2.94)

0

The strain energy averaged over the control volume becomes:

(K eIII ) I . 2n 2 Wp = = 3p πR 2 π 2G (n + 1)2 R 2

Ep

(4.2.95)

Setting n=1 into Eq. (4.2.95), I 3p = π , and then: 1 (K eIII ) 1 + ν (K eIII ) = We = 2G πR πE R 2

2

(4.2.96)

in agreement with Lazzarin et al. (2004). Eq. (4.2.95) normalized with respect to Eq. (4.2.96) gives: Wp 4n 2 = I 3p . We π(n + 1)2

(4.2.97)

This equation gives the increment of the plastic Strain Energy Density over the control volume with respect to the elastic one. Table 4.2.1 gives I 3p and Wp /We for some values of n.

Then to assure the constancy of the strain energy density, it is necessary to change the control volume:


(KeIII ) I = 1 (KeIII ) 2n 2 3p 2 2G πR e π 2G (n + 1) R p 2

Rp =

149

2

(4.2.98)

4n 2 I R 2 3p e π(n + 1)

(4.2.99)

and the plastic control radius, Rp, results some times greater than the elastic one, Re.

Table 4.2.1. Values of I3p and Wp /We obtained by a numerical integration.

n

I3p

Wp We

1

3.14159

1

2.5

2.09436

1.36052

4

1.7726

1.44445

8.33

1.44376

1.46531

10

1.38556

1.45797

12

1.33475

1.44806

Increasing applied load

Figure 4.2.9. Shape of the plastic zone for a rounded notch under small and large scale yielding.

4.2.6 Further remarks on the plastic zone shape

In the close neighbourhood of the notch apex the behaviour of a narrow hyperbolic notch matches that of a parabolic one (Zappalorto et al., 2008). While analysing the problem of narrow hyperbolic notches under torsion, Davis and Tuba (1963) found numerically a circular elastic-plastic boundary for small load

150


magnitudes, thus confirming the circular shape obtained in this work. Further on Davis and Tuba (1963, ibid. Fig. 4.2.2) showed that there is a transition from the circular plastic zone at small applied loads to an elongated plastic zone for higher loads (Fig. 4.2.9), the phenomenon being dependent on the plastic modulus G'' used to describe the material behaviour (that is on the exponent n of the powerhardening material). The elongated shape of the plastic zone was obtained also in the crack case (Koskinen 1963, Rice 1966, 1967a, Gdoutos 1990), confirming so the strong analogy characterizing rounded and sharp notches when subjected to antiplane loading. This analogy exists also in linear elastic conditions (Zappalorto et al., 2008).

Davis and Tuba (1963) recommended not to use Neuber’s relation under torsion, due to the absence of a uniformly stressed region which is necessary for determining the stress concentration factor. Coherently with these findings, the present authors wrote that the results obtained under uniform antiplane shear conditions in linear elasticity should not be applied to the torsion case without introducing a correction factor able to take into account the different nominal stress distribution on the transverse sectional area (Lazzarin et al., 2007, Zappalorto et al., 2008). This means that Eqs. (4.2.65, 4.2.72, 4.2.74) should be considered valid only as a first approximation when applied to a torsion case. Further on, in the solution proposed here the local stresses have been obtained as a function of a single leading term, while higher order terms must be added for a complete solution. Yang et al. (1996) showed that an excellent agreement between analytical and FEM results could be obtained for the crack case by using the first three terms of an asymptotic expansion of plastic stress fields. A singleterm-based solution was found to result in errors even at small distances from the crack tip, especially outside the notch bisector line. Nevertheless, Figures 4.2.10-4.2.12 show that the agreement between the analytical results obtained according to the present analytical frame and the FE results is good enough, with errors for stresses, along the notch bisector line, which are less than 3% in the cases tested herein. In particular Figure 4.2.11a-c shows that the accuracy of the solution, based on the leading order term of stress distribution, decreases as r increases; this is more evident outside the notch


151

bisector line, for increasing values of the radial angle ϕ, confirming for the parabolic notch the trend obtained by Yang et al. (1996) dealing with the crack case. Then, the greater the notch root radius, the narrower the range of validity of the present solution becomes (being always, for a parabolic notch, r>ρ/2). Finally, it has to be noted that in the previous sections we have found that for a parabolic notch the product τ ziγ zi is always proportional to the term 1 / r , independently of the material constitutive law. In this sense the parabolic notch is a special feature. Any other feature may, in principle, create different stress distributions, as shown in (Lazzarin et al., 2007) dealing with linear elastic stress distributions due to parabolic, semi-elliptic, semi-circular and hyperbolic notches. The stress distributions had universal features only within a distance from the notch tip less than ρ/20,. as measured along the notch bisector. Besides, for the crack case, some papers have highlighted the existence of higher order singularities (namely -3/2, -5/2…) in the elastic zone surrounding the plastic core (Hui and Ruina, 1995, Chen and Hasebe, 1997), where r is always nonzero. These stronger singularities are, on the contrary, commonly omitted in a complete linear elastic analysis, since ‘almost all researchers recognised the arguments that the strain energy density as well as the displacements in the near tip region should be bounded’ (Chen and Hasebe, 1997). Unlike those contributions, the present work is mainly focused on the stress components inside the plastic core, ahead of the notch tip, rather than in the outer elastic region, and for this reason we have used only the leading order term of the stress distribution. Not only the calculations related to the maximum plastic stress, but also those for the strain energy density over a control volume should be confined within the plastic zone. Higher order singularities exist at crack tips also when a strain gradient plasticity model is adopted in micro-scale analyses (see Zhang et al., 1998, Radi,

2007), instead of the flow plasticity theory used here dealing with macro-scale notch problem.

152


τzy / τmax

1 0.9

ρ/2 = 1

0.8 20

99

0.7

Markers: FEA Solid lines: theoretical τzy

0.6

Material

n

0.5

Elastic A B C

1 2.5 4 8.33

0.4 0.01

K [MPa] σ0 [MPa] E [MPa] 2000 600 950

166 125 450

206000 206000 206000 206000

0.1 1 Distance from the notch tip [mm]

10

Figure. 4.2.10. Plot of the stress component τ zy along the notch bisector line for different materials with hardening exponent n, yield stress σ0, and strength coefficient K . The stress component is normalized with respect to the maximum shear stress. All sizes in mm. 1.2

τzy / τmax

1 Material B: n=4 K=600 MPa σ0=125 MPa

0.8 0.6

Parabolic notch r = 1.01 ϕ

Markers: FEA Solid line: Theoretical τzy

0.4

O' ρ/2 = 1 10

0.2 0

0

1

2

O' 300

3

4

5

6

ϕ [degrees]

Figure 4.2.11. Plot of the stress component τ zy along circular paths of radii 1.01 mm centred on the notch focus. The stress component is normalized with respect to the maximum shear stress. Material B with hardening exponent n, yield stress σ0, and strength coefficient K. All sizes in mm.

153

NonLinear Solutions For Mode III Loaded Notches 1.2

τzy / τmax

1 0.8 0.6 0.4

r = 1.1 ϕ O'


0.2 0

Parabolic notch

Material B: n=4 K=600 MPa σ0=125 MPa

0

5

O' ρ/2 = 1 300

10

10

15

20 ϕ [degrees]

25

30

(a)

τzy / τmax

1.2 1 0.8

Parabolic notch

Material B: n=4 K=600 MPa σ0=125 MPa

0.6

r = 1.3 ϕ

0.4 0.2 0

O'


0

10

O' ρ/2 = 1

10

20

300

30 40 ϕ [degrees]

50

(b) Figure 4.2.12. Plot of the stress component τ zy along circular paths of radii 1.1 mm (a), 1.3 mm (b), centred on the notch focus. The stress component is normalized with respect to the maximum shear stress. Material B with hardening exponent n, yield stress σ0, and strength coefficient K. All sizes in mm.

60

154


4.2.7 Final observations

The main conclusions based on the developed analytical frame can be drawn as follows: 1. the elastic-plastic boundary has a circular shape, as it happens in the crack case; 2. the entire stress field in the plastic zone surrounding the notch apex can be expressed as a function either of the plastic Notch Stress Intensity Factor or the maximum plastic stress. In the limiting cases of n=1 (ideally linear elastic material) and n=∞ (elastic-perfectly plastic material), the proposed solution matches the Creager-Paris elastic solution and the Sokolovskii’s slip-lines model, respectively; 3. a simple relationship links elastic and plastic Notch Stress Intensity Factors (NSIFs); 4. by using elastic and plastic NSIFs, Neuber’solution for blunt notches under antiplane shear can be found again, but only under some particular conditions, which have been discussed in detail; 5. finally, starting from Glinka and Molski’s Equivalent Strain Energy Density criterion, the NSIFs have been used to give the increment of the strain energy density at the notch tip under antiplane shear loads, both for parabolic notches and sharp, zero radius, cracks.

4.3

Plastic stress fields for pointed V-notches under torsion

4.3.1 Basic equations for materials with a nonlinear powerlaw behaviour

In a body weakened by a pointed V-notch and subjected to anti-plane shear loading, see Fig. 4.3.1, only the out-of-plane displacement in the z direction is different from zero. As a consequence only the shear strain γzx and γzy are different from zero and satisfy the compatibility equation: ∂γ zx ∂γ zy − =0 ∂y ∂x

(4.3.1)

In the context of the usual assumptions, isotropic material and small deformations, all stresses vanish, except the shear stress τzx and τzy . Such stress components satisfy the equilibrium equation:

155

NonLinear Solutions For Mode III Loaded Notches ∂τ zx ∂τ zy + =0 ∂x ∂y

(4.3.2)

T y

y

z r

ϕ

τzϕ r

x

2α

a)

τzy τzr

τzx

ϕ x

b)

Figure 4.3.1. Stresses and coordinates at pointed V-notch under antiplane shear load T; 3D view (a) and cross sectional view (b)

Consider the following nonlinear elastic material law: γ ⎛τ⎞ =⎜ ⎟ γ 0 ⎜⎝ τ0 ⎟⎠

n

(4.3.3)

where n is positive and greater than unity ( 1 ≤ n ≤ ∞ ), γ0 is the reference shear strain, τ0 is the reference shear stress, whereas τ and γ are the modulus of the stress and strain vector, respectively, in agreement with the expressions: τ = τ 2zx + τ 2zy

;

γ = γ 2zx + γ 2zy .

(4.3.4)

The components of the stress and strain tensor are related by the following relationship: γ zi ⎛ τ ⎞ =⎜ ⎟ γ 0 ⎜⎝ τ0 ⎟⎠

n −1

τ zi τ0

(4.3.5)

The basic nonlinear field equations resulting from the adoption of a nonlinear material law may be converted to linear equations by regarding ‘physical’ coordinates as functions of stresses or of strains (Hult and McClintock, 1956, Rice, 1967). In this way, the equilibrium and compatibility equations can be rewritten as:

156

NonLinear Solutions For Mode III Loaded Notches ∂x ∂y + =0 ∂τ zx ∂τ zy

(4.3.6)

∂x ∂y − =0 ∂γ zy ∂γ zx

(4.3.7)

provided that the Jacobian operator of the transformation is nonzero. Following Hult and McClintock (1956), the transformation used in this work is the following: x=−

∂ψ ∂τ zy

y=

∂ψ ∂τ zx

(4.3.8)

which is similar to the hodograph transformation of the fluid mechanics (Rice, 1967). Then, by introducing a polar reference system in the shear stress plane (see Fig. 4.3.2) (Rice 1967, Gdoutos, 1990): τ zx = -τ sin ϕ

τ zy = τ cos ϕ

(4.3.9)

Eq. (4.3.8) can be rewritten in the form: x=−

∂ψ ∂τ zy

=−

∂ψ ∂ψ sin ϕ cos ϕ + ∂τ ∂ϕ τ

(4.3.10)

∂ψ ∂ψ ∂ψ cos ϕ =− y= sin ϕ − ∂τ zx ∂τ ∂ϕ τ −τ zx

C

∞

A

Iso-line with constant τ ~ ∂ψ =0 ∂ϕ A

α y

τ

B 2α D

C

r

ϕ

B≡D

ϕ

~=0 ψ

E

x

τ zy

α ~=0 ψ

E

C

∞

(a) (b) Figure 4.3.2. Coordinates in the physical plane (a) and mapping in the transformed plane (b), (after Hult and MacClintock, 1956 and Rice, 1967).


157

By inserting x and y from Eq. (4.3.10) into Eq. (4.3.7) and by expressing stresses and strains according to Eq. (4.3.3), the following differential equation is found (Euler’s equation): 1 ∂ 2 ψ 1 ∂ψ 1 ∂ 2 ψ + + = 0. n ∂τ 2 τ ∂τ τ 2 ∂ϕ 2

(4.3.11)

In the context of the present work, which is based on linear and nonlinear NSIFs, the solutions to Eq. (4.3.11) can be sought in the following form: ~ (ϕ) ψ( τ, ϕ ) = τ m ψ (4.3.12) and not in a series-expansion form as done by Rice (1967) and, more recently, by Wang and Kuang (1999); consequently, only the leading order term in stress distributions will be considered herein. In Eq. (4.3.12) m is a rational constant and ψ~ (ϕ ) is an arbitrary function. Substitution of Eq. (4.3.12) into Eq. (4.3.11) gives: 2~ ~ ( ϕ ) ⎡ 1 m(m − 1) + m⎤ + ∂ ψ( ϕ ) = 0 ψ ⎢⎣ n ⎥⎦ ∂ϕ 2

(4.3.13)

Eq. (4.3.13) is an ordinary differential equation which admits solutions in the form: ~ ( ϕ ) = (C sin ω ϕ + C cos ω ϕ) ψ 1 2

(4.3.14)

where ω=

1 m(m − 1) + m n

(4.3.15)

The boundary conditions τzϕ=−τzx sinϕ + τzy cosϕ=0 when ϕ=π-α and τzx=0 on the notch bisector (ϕ=0), can be transferred to the τzx-τzy plane providing the following system: ~ ( ϕ ) = (C sin ωϕ + C cos ωϕ ) ψ 1 2 ~ ⎛⎜ ∓ ⎛ π − α ⎞ ⎞⎟ = 0 ψ ⎜ ⎟ ⎠⎠ ⎝ ⎝2 ~ ∂ψ(0) =0 ∂ϕ

(4.3.16)

Then: C1 = 0

⎛π ⎞ C 2 cos ω ⎜ − α ⎟ = 0 ⎝2 ⎠

ω=

π π − 2α

(4.3.17)

158


Finally, we get from Eq. (4.3.15): m=

1 − n − ( n − 1) 2 + 4ω2 n 2

(4.3.18)

It has to be noted that this equation matches the first order solution proposed by Wang and Kuang (1999) when the damage index in their solution is set equal to zero (ibid. Eq. (44), pag. 310). By introducing the condition C1=0 into Eq. (4.3.14) one obtains: ψ( τ, ϕ ) = τ m C 2 cos ωϕ

(4.3.19)

By introducing this expression into Eq. (4.3.10), the following expressions can be derived: x =−

∂ψ ∂ψ sin ϕ cos ϕ + ∂τ ∂ϕ τ

= −C 2 τ m −1 (m cos ωϕ cos ϕ + ωsin ωϕ sin ϕ )

(4.3.20)

∂ψ ∂ψ cos ϕ y =− sin ϕ − ∂τ ∂ϕ τ = −C 2 τ m −1 (m cos ωϕ sin ϕ − ωsin ωϕ cos ϕ ) Then, the radial distance r can be expressed as: r = C 2 τ (m −1) m 2 cos 2 ωϕ + ω2 sin 2 ωϕ

(4.3.21)

and viceversa the shear stress τ : ⎡C τ=⎢ 2 ⎣ r

1

⎤ 1− m m cos ωϕ + ω sin ωϕ ⎥ ⎦ 2

2

2

2

(4.3.22)

Finally, by inserting τ from Eq. (4.3.22) into Eq. (4.3.9) the shear stress components turn out to be: 1 ⎤ 1−m

⎡C τ zx = -τsin ϕ = − ⎢ 2 m 2 cos 2 ω ϕ + ω 2 sin 2 ω ϕ ⎥ ⎣ r ⎦ ⎡ − mC2 = −⎢ ⎣ r

1

~ ⎤ 1−m F⎥ sin ϕ ⎦ 1 ⎤ 1−m

⎡C τ zy = τcos ϕ = ⎢ 2 m 2 cos 2 ω ϕ + ω 2 sin 2 ω ϕ ⎥ ⎣ r ⎦ ⎡ − mC2 =⎢ ⎣ r

sin ϕ

1

~ ⎤ 1−m F⎥ cos ϕ ⎦

(4.3.23) cos ϕ


159

where m is negative and ω2 ~ 2 F = cos ωϕ + 2 sin 2 ωϕ m

(4.3.24)

In parallel the strain components can be obtained by inserting Eq. (4.3.23) into Eq. (4.3.5): γ z,y

γ ⎡ − mC2 = n0 ⎢ τ0 ⎣ r

γ z,x = −

γ0 τ 0n

n

~ ⎤ 1− m F⎥ cos ϕ ⎦

⎡ − mC2 ⎢⎣ r

~⎤ F⎥ ⎦

n 1− m

(4.3.25)

sin ϕ

Furthermore, thanks to Eq. (4.3.20), it is also possible to obtain an analytical link between the polar angle in the physical (x,y) plane, ϕ, and the polar angle in the stress (τzy, τzx) plane, ϕ : ⎛ω ⎞ ϕ = ϕ − arctan ⎜ tan ωϕ ⎟ ⎝m ⎠

(4.3.26)

The determination of constant C2 needs an additional condition. It can be linked to a nonlinear stress field parameter K3 defined here according to the following expression: K 3 = 2 π lim r →0 r

1 1− m

τ zy (r , ϕ = 0)

(4.3.27)

1

where the term r 1− m is introduced to cancel the singularity of stress fields, as made by Gross and Mendelson (1972) dealing with mode I and mode II notch stress intensity factors. Subsequently it is possible to express C2 as a function of K3 1⎛ K ⎞ C 2 = − ⎜⎜ 3 ⎟⎟ m ⎝ 2π ⎠

1− m

(4.3.28)

and then to write the stress and strain components in the form: 1

~ K 3 ⎛ F ⎞ 1− m ⎜ ⎟ cos ϕ τ zy = 2π ⎜⎝ r ⎟⎠

1

τ zx

~ K 3 ⎛ F ⎞ 1− m ⎜ ⎟ sin ϕ =− 2π ⎜⎝ r ⎟⎠

(4.3.29)

160


n

n

γ zy

⎛ K ⎞ γ ⎡ 1 ~ ⎤ 1− m = ⎜⎜ 3 ⎟⎟ n0 ⎢ F⎥ cos ϕ ⎝ 2π ⎠ τ 0 ⎣ r ⎦ n

⎛ K ⎞ γ ⎡1 ~⎤ γ zx = −⎜⎜ 3 ⎟⎟ n0 ⎢ F⎥ ⎝ 2π ⎠ τ 0 ⎣ r ⎦

n 1− m

(4.3.30)

sin ϕ

y

1.5

2α=0

1+ n

⎛ K3 ⎞ ⎜⎜ ⎟⎟ ⎝ τ 2π ⎠

1

n=1 n=4

0.5

Crack edge

n=100 -1

-1.5

-0.5

0.5

1

1.5

-0.5

x 1+ n

-1

⎛ K3 ⎞ ⎜⎜ τ 2π ⎟⎟ ⎠ ⎝

-1.5

(a) 1.5

y

2α=120°

1− m

⎛ K3 ⎞ ⎜⎜ ⎟⎟ ⎝ τ 2π ⎠

1

Notch edge

0.5

n=1 n=4 n=100

-1.5

1

-0.5

0.5

1

1.5

-0.5

x 1− m

-1

⎛ K3 ⎞ ⎜⎜ ⎟⎟ ⎝ τ 2π ⎠

-1.5

(b) Figure 4.3.3. Plotting of the normalised shear stress iso-lines, according to Eq. (4.3.31), for the crack case (a) and a pointed notch with 2α=120° (b).

161


By using in combination Eqs. (4.3.20) and (4.3.28), it is easy to draw the isolines characterised by the same value of the

K 3 / τ ratio, the relevant

equations being: x 1− m

⎛ K3 ⎞ ⎜⎜ ⎟⎟ ⎝ τ 2π ⎠ y

1− m

⎛ K3 ⎞ ⎜⎜ ⎟⎟ ⎝ τ 2π ⎠

ω ⎞ ⎛ = ⎜ cos ωϕ cos ϕ + sin ωϕ sin ϕ ⎟ m ⎠ ⎝ ω ⎞ ⎛ = ⎜ cos ωϕ sin ϕ − sin ωϕ cos ϕ ⎟ m ⎠ ⎝

(4.3.31)

The isolines are plotted in Fig. 4.3.3 for the cases 2α=0 and 2α=120° considering different values of n. It is evident that the shape of these curves strictly depends on the notch opening angle and the hardening exponent n. In particular: •

In the crack case, the isolines are circles centred at the crack tip when n=1; as

n increases the isolines continue to be circles but no longer centred at the crack tip. For n=∞ the left-hand side of the circle pass through the crack tip (according to Hult and McClintock, 1956); •

For 2α ≠ 0 the isolines are circle centred on the notch tip for n=1; as n increases the isolines are no longer represented by circles but assume a groinlike shape. Finally, for n=∞ the cusp of the groin falls on the notch tip.

4.3.2 Linear-elastic stress fields When n=1 the governing equation of the problem becomes a Laplace equation, whose solution allows us to determine the linear elastic stress fields. In the close neighbourhood of the notch tip, the stresses can be written in terms of the elastic notch stress intensity factor K 3, e

⎧τ zx (r , ϕ )⎫ K 3,e r λ3 −1 ⎧− sin (1 − λ 3 )ϕ ⎫ ⎬ ⎨ ⎬= ⎨ ( ) r , τ ϕ 2π ⎩ cos (1 − λ 3 )ϕ ⎭ zy ⎩ ⎭

(4.3.32)

where K 3,e = 2π lim r →0 r 1−λ3 τ zy (r , ϕ = 0)

(4.3.33)

λ3 being the first eigenvalue of the linear-elastic V-notch problem. It can be easily

162


determined from the shear-stress-free boundary condition on the notch flanks: sin[2λ 3 ( π − α)] = 0

λ3 =

π 2 π − 2α

(4.3.34)

It has be noted that, thanks to Eqs. (4.3.17, 4.3.34), a simple relationship can be found between the first linear eigenvalue, λ3, and the first nonlinear eigenvalue

ω. It states that ω=

λ3 1 − λ3

(4.3.35)

4.3.3 Elastic-plastic fields

The hardening behaviour of materials can be treated in a simplified way by considering an elastic response up to the yield limit τ0 and thereafter a power law for stresses and strains in the plastic region (Rice 1967, Rice and Rosengren 1968, Wang and Kuang, 1999, Unger 2001, ibid. page 53), Fig. 4.3.4. 2 n=1 n=2

τ/τ0

1.5

n=8.33 n=100

1

⎧γ = G ⋅ τ n ⎪ ⎨ γ = ⎛⎜ τ ⎞⎟ ⎜ ⎟ ⎪γ ⎩ 0 ⎝ τ0 ⎠

0.5

0

0

0.5

1

1.5

γ/γ0

if γ ≤ γ 0 if γ ≥ γ 0

2

G=

τ0 γ0

2.5

Figure 4.3.4. Simplified stress-strain law for power-hardening material; with

hardening exponent n. As the matter of fact the solution for stresses and strains in the plastic zone takes the same form of the general nonlinear solution obtained in the previous section of the present paper.

3

163


Then, in the plastic core region, it is natural to define the plastic NSIF in analogy to the linear elastic case: K 3,p = 2π lim r →0 r

1 1− m

τ zy (r, ϕ = 0 )

(4.3.36)

By simply replacing K3 in Eq. (4.3.28) with K3,p, one obtains: 1 ⎛ K 3,p ⎞⎟ C2 = − ⎜ m ⎜⎝ 2π ⎟⎠

1− m

(4.3.37)

so that all stress and strain components can be given as a function of K 3,p : 1

1

K 3,p ⎛ ~ F ⎞1− m ⎜⎜ ⎟⎟ cos ϕ τzy = 2π ⎝ r ⎠

K 3,p ⎛ ~ F ⎞1− m ⎜⎜ ⎟⎟ sin ϕ τzx = − 2π ⎝ r ⎠

(4.3.38)

n

n

⎛ K ⎞ γ ⎡ 1 ~ ⎤ 1− m γ zy = ⎜⎜ 3,p ⎟⎟ n0 ⎢ F⎥ cos ϕ ⎝ 2π ⎠ τ 0 ⎣ r ⎦ n

⎛ K ⎞ γ ⎡1 ~⎤ γ zx = −⎜⎜ 3,p ⎟⎟ n0 ⎢ F⎥ ⎝ 2π ⎠ τ 0 ⎣ r ⎦

n 1− m

(4.3.39)

sin ϕ

4.3.4 Link between plastic and elastic NSIFs under small scale yielding conditions

According to the previous literature (from Irwin, 1960, to Unger, 2001), the radius rp of the plastic zone can be found, as a first approximation, by substituting the linear-elastic solution for stresses into the yield condition (with the shear yield limit τ0): τ 2zx + τ 2zy = τ 0

2

(4.3.40)

Combining Eqs. (4.3.32) and Eq. (4.3.41) and solving by r leads to: ⎛ K 3,e ⎞ ⎟ r p = ⎜⎜ ⎟ ⎝ 2π τ0 ⎠

1 1− λ 3

.

(4.3.41)

It is worth noting that this holds true only within the small scale yielding regime, which assures a plastic zone small as compared to the notch depth. We will use the following Legendre transformation (Zwillinger, 1989), whose appropriate differentiation leads to Eqs. (4.3.8):

164

NonLinear Solutions For Mode III Loaded Notches ψ + φ = − xτ zy + yτ zx

(4.3.42)

Here φ is a real potential function such as: ∂φ ; ∂y

τzx =

τzy = −

∂φ ∂x

(4.3.43)

Based on Eq. (4.3.32), it is easy to demonstrate that for a pointed V-notch under linear-elastic conditions, the potential φ can be written as: φ=−

K 3,e 2 πλ 3

r λ 3 cos λ 3ϕ

(4.3.44)

Under small scale yielding, we can evaluate Eq. (4.3.42) on the elastic-plastic boundary at the V-notch bisector (r=rp, ϕ = ϕ = 0 ), where τzy=τ=τ0 and τzx=0. By using in combination Eqs. (4.3.19, 4.3.20, 4.3.41, 4.3.42, 4.3.44) the parameter C2 turns out to be: C 2 τ0 m −

C2 =

K 3,e 2πλ 3

rp λ3 = mC 2 τ0 m 1 ⎞1−λ 3

⎛K rp 1 ⎜ 3,e ⎟ = m 1 − ⎜ 2π ⎟ 1 − m λ 3τ 0 ⎝ ⎠

(4.3.45) 1 λ 3τ0m +ω

(1 − m )

Finally, by comparing Eq. (4.3.37) and Eq. (4.3.45), one can find the relationship that links plastic and elastic NSIF: 1

K 3,p

1 ⎡ ⎤ 1− m 1− λ 3 ⎛ ⎞ K m 1 ⎢ ⎥ ⎜ 3,e ⎟ = 2π ⎢− . m+ω ⎥ ⎜ ⎟ λ 3 (1 − m ) ⎝ 2π ⎠ τ0 ⎢⎣ ⎥⎦

(4.3.46)

It has to be noted that the effect of the stress redistribution in the plastic zone is fully included in Eq. (4.3.46). The stress redistribution is unavoidable when the material law shown in Figure 4.3.4 is adopted, because in correspondence of the point (γ0, τ0) the behaviour of the material abruptly changes from the elastic to the power hardening law. The influence on K3p of stress-strain laws different from that shown in Figure 4.3.4 will not be discussed here but in a next contribution.

4.3.5 Scale effect

Consider two similar pointed V-notch specimens, geometrically scaled by a factor

165


f so that the absolute dimension of the specimen B is f times greater than those of the specimen A, both of them being subjected to antiplane shear loading. The shear stress components τz,i (i=x,y) along the notch bisector line are related as follows: A B τ z,i (r) = τ z,i (r/f)

(4.3.47)

Two structural components A and B scaled in geometrical proportion have the NSIFs obeying of the following equation: 1

K 3B,p = K 3A,p f 1− m

(4.3.48)

Plastic NSIF ratio, K3,p,B / K3,p,A

1 n=20 0.9

8.33 4

0.8 2 0.7 1 0.6 0.5

Shear-loaded V-notch, 2α=135°

1

2

3

4

5 6 7 8 9 10 Geometrical scale factor, f Figure 4.3.5. Scale effect for different values of the strain hardening exponent n; shear-loaded V-notch with 2α=135°

Eq. (4.3.48) states that the scale effect is fully included in the notch stress distribution characterised by the NSIF. For linear-elastic conditions (n=1), Eq. (4.3.48) becomes: K 3,e,B = K 3,e,A f 1−λ 3

(4.3.49)

The ratio of the plastic NSIFs, Eq. (4.3.48), is plotted in Fig. 4.3.5 versus the scale factor f for different hardening exponents n. It is evident that the scale effect reduces as n increases, so that no scale effect occurs with an elastic perfectlyplastic material.

166


4.3.6 Plastic zone size under small scale yielding

The actual shape and intensity of the stress isolines within the plastic zone can be evaluated as a function of the plastic NSIF by using in combination Eqs. (4.3.20, 4.3.37). 1− m

⎛ K ⎞ x = ⎜⎜ 3,p ⎟⎟ ⎝ τ 2π ⎠

1− m

⎛ K ⎞ y = ⎜⎜ 3,p ⎟⎟ ⎝ τ 2π ⎠

ω ⎛ ⎞ ⎜ cos ωϕ cos ϕ + sin ωϕ sin ϕ ⎟ m ⎝ ⎠ ω ⎛ ⎞ ⎜ cos ωϕ sin ϕ − sin ωϕ cos ϕ ⎟ m ⎝ ⎠ 1− m

⎛ K ⎞ R (ϕ ) = ⎜⎜ 3,p ⎟⎟ ⎝ τ 2π ⎠

cos2 ωϕ +

(4.3.50)

ω2 sin 2 ωϕ 2 m

Eq. (4.3.50) is plotted in Fig 4.3.6 for 2α=0° and 2α=120°. In the crack case, 2α=0°, the lines are circles with different centres. According to Eq. (4.3.50) the radius of each circle corresponds to the maximum value of y, which occurs at

ϕ = π / 4 . Then the radius of the circles and the distance between their centre and the crack tip are: 1− m

r (τ ) = y max

n + 1 ⎛ K 3,p ⎞ ⎜ ⎟ = 2n ⎜⎝ τ 2 π ⎟⎠

(4.3.51)

1− m

n − 1 ⎛ K 3,p ⎞ ⎜ ⎟ X (τ ) = x max − r (τ ) = 2 n ⎜⎝ τ 2 π ⎟⎠ y

2α=0°

y

2α=120°

r (τ )

X (τ )

x

x

1− m

⎛ K ⎞ x (τ ) = ⎜⎜ 3,p ⎟⎟ ⎝ τ 2π ⎠

(a) (b) Figure 4.3.6. Qualitative plotting of the shear stress iso-lines for the crack case (a) and a pointed notch with 2α=120° (b).


167

Finally, the equation of the isolines for the crack case is:

[x − X( τ)] 2 + y2 = r 2 (τ ) . By using in combination Eqs. (4.3.46) and (4.3.50) and setting τ = τ 0 , it is possible to draw the elastic-plastic boundary as a function of the elastic NSIF: ⎛ K 3,e m ⎜ xp = − λ 3 (1 − m ) ⎜⎝ τ0 2π yp = −

⎛ K 3,e m ⎜ λ 3 (1 − m ) ⎜⎝ τ0 2π

1

⎞1−λ 3 ⎛ ω ⎟ ⎜ cos ωϕ cos ϕ + sin ωϕ sin ϕ ⎞⎟ ⎟ ⎝ m ⎠ ⎠ ⎞ ⎟ ⎟ ⎠

1 1−λ 3

⎛ K 3,e m ⎜ R p (ϕ ) = − λ 3 (1 − m ) ⎜⎝ τ0 2π

ω ⎛ ⎞ ⎜ cos ωϕ sin ϕ − sin ωϕ cos ϕ ⎟ m ⎝ ⎠

(4.3.52)

1

⎞1−λ3 ω2 2 ⎟ cos ωϕ + 2 sin 2 ωϕ ⎟ m ⎠

The extension of the plastic zone along the notch bisector line (where ϕ = 0 ) is then: ⎛ K 3,e m ⎜ xp = − λ 3 (1 − m ) ⎜⎝ τ0 2π

1

⎞1−λ3 m ⎟ =− rp ⎟ ( ) λ 1 − m 3 ⎠

(4.3.53)

For a crack we have 2α=0, and then Eq. (4.3.53) gives x p = rp ⋅ 2n/(n + 1 ) according to Rice (1967). Equation (4.3.53) shows that the actual extension of the plastic zone is larger than that suggested by the elastic parameter rp, see Eq. (4.3.41). The difference is obviously due to the stress redistribution provoked by the plastic deformation in the elastic field and can be estimated as (see also section 4.2): ⎛ −m ⎞ Δ p = x p − rp = ⎜⎜ − 1⎟⎟ rp ⎝ λ 3 (1 − m ) ⎠

(4.3.54)

Koskinen (1963) and Rice (1967) showed that, for a crack, the plastic zone is circular only for low applied loads (small scale yielding), whereas higher loads cause an elongated shape. This is simply due to the fact that as the elastic-plastic boundary gets far from the notch tip due to the high load level, some higher order terms of the stress distribution becomes much more meaningful, especially out of the notch bisector line. On the contrary Eq. (4.3.50) is valid only in the vicinity of the notch tip, at least when the notch depth is finite. However, the size of the zone of validity for Eq. (4.3.50) strongly depends on the V-notch angle: it increases as

168


the V-notch angle increases. For the mathematical abstraction of infinite-deep notch, but in the presence of a finite transverse sectional area, a single term-based stress distribution is sufficient to exactly describe the entire stress field, from the notch tip to the axis of symmetry. In the case of an elastic perfectly-plastic material, n=∞, Eqs. (4.3.18) and (4.3.26) give m = −∞ and ϕ = ϕ , respectively. Then, according to Eq. (4.3.52): xp =

rp

,

λ3

R (ϕ ) =

rp λ3

cos ωϕ

(4.3.55a-b)

For the crack-case, 2α=0, ω=1 and λ3=0.5, so that: x p = 2rp , R (ϕ ) = 2rp cos ϕ

(4.3.56)

in agreement with Hult and McClintock’s solution (1956). The extension of the plastic zone can be determined also on the basis of a local balance condition, applied to the shear stress, by extending to V-notches the approach suggested by Irwin for the crack. The local stress redistribution results in: x p = C P rp

(4.3.57)

where Cp can be easily determined according to the following relationship: rp

Cp =

∫ τ zy 0

ϕ= 0

rp τ zy

dr

r = rp

1 K 3ρ λ 3 rp λ 3 2π 1 = = K 3ρ λ 3 λ3 rp 2π

(4.3.58)

It is evident that combining Eqs. (4.3.57) and (4.3.58) results in Eq.(4.3.55a).

4.3.7 Platic J-integral under mode III Considering the case of two-dimensional deformation field (plane stress, generalised plane strain and antiplane strain), Rice (1968) defined the parameter Jintegral in the form: ∂u ⎞ ⎛ J = ∫ ⎜ Wdy − T ds ⎟ , ∂x ⎠ Γ⎝

(4.3.59)

In agreement with the schematic representation of notch free edges shown in Fig. 4.3.7a, we can write:

169


R

R

J 23 = J 31 = ∫ Wp [ϕ = ±( π − α)]dy = ∫ Wp [ϕ = ±( π − α)]sin( π − α) dr . 0

(4.3.60)

0

This holds true because the traction vector T is zero along the free-of-load notch edges. Then R

J 3,p = 2 ∫ Wp [ϕ = ±( π − α)]sin( π − α) dr

(4.3.61)

0

Under antiplane shear loading, the strain energy density is given by: γ

W = ∫ τdγ .

(4.3.62)

0

R

1

R

3 π−α

2 A = (π − α ) ⋅ R 2

(a) (b) Figure 4.3.7. Geometrical parameters of circular sections for and J-integral (a) averaged SED (b). By taking advantage of Eq. (4.3.3) and due to the fact that in a finite size volume of material surrounding the notch tip the stresses are expected to be much more greater than the yield limit, it is possible to estimate the local strain energy density in a simplified form: ⎛ γ Wp = ∫ τ 0 ⎜⎜ 0 ⎝ γ0 γ

1

⎞n n τ n +1 ⎟⎟ dγ = n + 1 Gτ 0 n −1 ⎠

(4.3.63)

By using Eq. (4.3.38), this equation can be rewritten as a function of the plastic NSIF: Wp =

n +1

n τ n + 1 Gτ 0 n −1

⎧ n 1 ⎪ K 3,p = ⎨ n + 1 Gτ 0 n −1 ⎪ 2 π ⎩

1 ~ 1− m ⎫ ⎡ F⎤ ⎪ ⎢r⎥ ⎬ ⎣ ⎦ ⎪ ⎭

n +1

(4.3.64)

170


By introducing Eq. (4.3.64) into Eq. (4.3.61) one obtains: 1 ⎫ ⎧ ~ K 1 − R 2n 1 ⎪ 3,p ⎡ F( ϕ = π / 2 − α) ⎤ m ⎪ J 3,p = ∫ ⎨ ⎬ ⎢ ⎥ 0 n + 1 Gτ n −1 r ⎦ ⎪ 2π ⎣ ⎪ 0 ⎩ ⎭ 2n (m − 1) sin(π − α) × sin(π − α) dr = × (n + 1)(n + m ) Gτ0 n −1 1 ⎞ ⎛K 3,p ~ F( ϕ = π / 2 − α) 1−m ⎟ ×⎜ ⎜ 2π ⎟ ⎝ ⎠

[

]

n +1

n +1

×

(4.3.65)

n+m R m −1

In a more synthetic form:

(K ) = B ( 2α , n )

n +1

J 3,p

3, p

J

Gτ 0

n+m m −1

R

n −1

(4.3.66)

where

[

]

1 ⎛ ~ 2n (m − 1) ⎜ F( ϕ = π / 2 − α) 1− m BJ ( 2α, n ) = (n + 1)(n + m ) ⎜⎜ 2π ⎝

⎞ ⎟ ⎟ ⎟ ⎠

n +1

sin( π − α)

(4.3.67)

Table 4.3.1 gives the parameter BJ ( n, 2α ) for some values of the notch opening angles 2α and of the exponent n. When 2α=0, Eq. (4.3.67) simplifies 1 ~ both because π − α = π, F( ϕ = π / 2) = , m = − n and because the limit of n

(m − 1) ) sin(π − α ) n+m

(which is a zero-to-zero ratio) has a finite value:

(n + 1) m −1 lim sin( π − α) = π α→ 0 n + m 4n

2

Then: n +1⎛ 1 ⎞ BJ ( 2α, n ) = ⎜ ⎟ 2n ⎝ 2π ⎠

n +1

π

(4.3.68)

and J 3,p

π n + 1 ⎛ K 3,p ⎞ ⎜ ⎟ = n −1 2n ⎜⎝ 2π ⎟⎠ Gτ 0

n +1

(4.3.69)

171


Table 4.3.1. Parameter BJ (n, 2α ) as a function of notch opening angle 2α and hardening exponent n. BJ ( n, 2α) [10-4] n 1 2 4 8.33 10

30°

2α=0°

60°

90°

120°

135°

150°

5000.00 4531.16 3978.87 3376.19 2756.64 2450.67 2152.25 1496.03 1483.05 1428.36 1332.72 1199.39 1120.37 1034.40 198.418 202.69 203.23 199.78 192.03 186.403 179.528 3.32543 3.34 3.32 3.29 3.27 3.27 3.29 0.70392 0.702 0.693 0.682 0.6789 0.683 0.694

Finally, it is trivial to note that the substitution n=1 immediately gives: 2

K J III = 3,e 2G

(4.3.70)

according to Rice (1968). Only in the case 2α=0 the J-integral is pathn+m

independent. On the other hand, it has to be noted that the ratio J 3,p / R m −1 does not depend on the integration radius, but only on the notch opening angle and the material properties. A convenient parameter can be defined, as made for Mode I by Lazzarin et al. (2002), such as: J

L 3,p

=

J 3,p R

n+m m −1

(K ) = B ( 2α, n )

n +1

3,p

J

Gτ 0

(4.3.71)

n −1

Note that this relationship is coherent with the classic J-integral expression for a crack. For an arbitrary notch opening angle, linear-elastic conditions (n=1) result in: λ 1 sin( π − α) ~ F = 1, m = − 3 , BJ ( 2α) = 1 − λ3 2λ 3 − 1 2π so that sin( π − α) (K e3 ) 2 λ 3 −1 = R , 2π(2λ 3 − 1) G

J 3,e

sin( π − α) (K e3 ) = 2π(2λ 3 − 1) G

2

2

J

L 3,e

(4.3.72a-b)

It is now possible to normalise J 3,p , Eq. (4.3.66), with respect to the linearelastic value J 3,e , Eq. (4.3.72a), for any notch opening angle. It has to be noted

172


that in small-scale yielding the plastic NSIF can be given in terms of the elastic NSIF, see Eq. (4.3.46). However, there is no possibility to eliminate the dependence of the elastic NSIF in the J 3,p / J 3,e ratio, except for the crack case; then, for an arbitrary notch opening angle, this ratio depends not only on the material properties and on the integration path, but also on the load intensity. Since the path radius R works as the microstructural support length in Neuber’s approach (Neuber, 1968), a convenient choice of the circular or semicircular path should be made on the basis of the material’s properties (Livieri, 2008). Finally, it has to be noted that several plots for Jp have been recently reported in the literature with reference to U-shaped notches under mode I loading (Berto et al., 2007). Rice (1968) demonstrated that under small scale yielding the elastic-plastic Jintegral for the antiplane shear loading matches the value determined under linear elastic conditions, Eq. (4.3.70). Fully accepting this idea, it is possible to give an analytical link between the plastic NSIF (small scale yielding) and the elastic NSIF, by simply equating Eqs. (4.3.69) and (4.3.70). Doing so: 1

K 3,p

⎡ n 2 ⎤ n +1 n −1 = 2π ⎢ τ 0 K 3, e ⎥ ⎦ ⎣ π(n + 1)

(4.3.73)

This equation matches our Eq. (4.3.46) when 2α=0 (ω=1, λ3=0.5, m = −n).

4.3.8 Averaged strain energy density By taking advantage of Eq. (4.3.64) the total energy evaluated over a circular sector (see Fig. 4.3.7b) turns out to be: R π−α

E p = ∫ WpdA = 2 ∫ A

0

∫ 0

n +1

⎧ n 1 ⎪ K 3,p ⎨ n + 1 Gτ0 n −1 ⎪ 2 π ⎩

n +1

1 ~ 1− m ⎫ ⎡ F⎤ ⎪ ⎢ r ⎥ ⎬ rdrdϕ = ⎣ ⎦ ⎪ ⎭

(4.3.74)

1− n − 2 m π − α 2n ⎛ K 3,p ⎞ 1 1− m ~ 1n−+m1 1− m ⎟ ⎜ R ∫0 F dϕ n + 1 ⎜⎝ 2 π ⎟⎠ Gτ0 n −1 1 − n − 2 m ~ Since F is a function of ϕ , it is convenient to switch the integral in the

=

()

variable ϕ (ranging from 0 to π−α) in an integral in the variable ϕ (ranging from

173


0 to π/2−α, see Fig. 4.3.2). Taking advantage of Eq. (4.3.26) it is possible to write:

ω2 sec 2 (ωϕ ) ∂ϕ = 1 − ∂ϕ ⎛ ω2 tan 2 (ωϕ ) ⎞ ⎟⎟ m⎜⎜1 + m2 ⎝ ⎠

(4.3.75)

and determine Ep and its mean value Wp related to a circular sector of radius R. The final result is

(K ) ( n , 2α )

n +1

Wp = B W

3, p

Gτ 0

n −1

n +1

R m −1

(4.3.76)

where B W (n , 2α) =

×

2n (1 − m ) × γ (n + 1)(1 − n − 2m )

1

( 2π )

n +1

π −α 2

⎡ ⎤ ⎢ ⎥ n +1 ω2 sec 2 (ω ϕ ) ~ ⎢ ⎥ F 1−m ⎢1 − dϕ 2 2 ⎛ ω tan (ω ϕ ) ⎞ ⎥ ⎢ m⎜1 + ⎟⎥ 2 ⎜ ⎟⎥ ⎢⎣ m ⎝ ⎠⎦

∫ () 0

(4.3.77)

Table 4.3.2 gives BW (n, 2α ) for different values of the notch opening angle 2α and the exponent n. When n=1, we have: K 3,p = K 3,e

m=−

(

λ3 1 − λ3

~ F =1

)

BW =

1 4πλ 3

2

K 3,e 1 We = . 4π λ 3 G R 2(1−λ 3 )

(4.3.78)

where We is in agreement with an expression already reported by Lazzarin et al. (2004). Under small scale yielding, it is possible to link Wp to the elastic NSIF by using Eq. (4.3.46), n +1

( )

Wp = B W (n , 2α) 2π

n +1

R

n +1 m −1

Gτ 0

n −1

1 ⎡ ⎤ 1− m 1− λ 3 ⎛ ⎞ K 1 m ⎢ ⎥ 3,e ⎜ ⎟ (4.3.79) m +ω ⎥ ⎢− λ (1 − m ) ⎜ 2π ⎟ τ0 3 ⎝ ⎠ ⎢⎣ ⎥⎦

174


Table 4.3.2. Values of coefficients BW (n, 2α ) for different opening angles 2α and hardening exponents n. BW ( n, 2α) [10-4] n 1 2 4 8.33 10

2α=0°

30°

60°

90°

120°

135°

150°

1591.55 1458.92 1326.29 1193.66 1061.03 994.718 928.404 616.914 594.11 568.234 538.802 505.262 486.758 466.989 91.2289 89.9431 88.601 87.1611 85.531 84.5932 83.5308 1.55106 1.52412 1.50416 1.49778 1.51554 1.53807 1.57269 0.32668 0.319791 0.314452 0.312469 0.317068 0.323423 0.333664 By comparing Eq. (4.3.78) and Eq. (4.3.79), it is evident that the ratio

between the plastic strain energy density and the elastic strain energy density depends not only on the material properties and the notch opening angle, but also on the control volume and the mode III NSIF. When 2α=0, the link can be expressed in the form:

Wp We

=

4n 2 π ~ 4n 2 F d ϕ = I 2 ∫ 2 p π(n + 1) 0 π(n + 1)

(4.3.80)

and the ratio has the property to become independent of R and K 3, e . The values provided by Eq. (4.3.80) are listed in Table 4.3.3 as a function of the hardening exponent n. Once the NSIF of the geometry is known, Eq. (4.3.38) states that the entire stress and strain fields in the close neighbourhood of the notch tip are completely defined. It is also well known that the NSIF evaluation generally needs very fine meshes in the vicinity of the point of singularity and this commonly results in long computational times, which sometimes are incompatible with engineering practical applications. Recently Lazzarin and Berto (2008) demonstrated that the mean value of the local SED is substantially independent of the mesh size, being this true both for a linear-elastic material and a strain hardening material. So, once the mean value W has been determined, the mode III NSIF can be calculated a

posteriori by simply using the following relationship: 1

K 3,p

⎛ Wp G τ0 n −1 ⎞ n +1 1 ⎟ R 1− m =⎜ ⎜ BW ( n , 2 α ) ⎟ ⎠ ⎝

(4.3.81)


175

Table 4.3.3. Values of Ip and Wp /We for some values of n, as determined by integrating numerically Eq. (4.3.80)

n

Ip

Wp We

1

3.1415

1

2.5

2.0944

1.3605

4

1.7726

1.4444

8.33

1.4438

1.4653

10

1.3856

1.4580

12

1.3347

1.44816

Under linear-elastic condition (n=1) this expression simplifies and becomes:

K 3,e = 4π Gλ 3 We R 1−λ3

(4.3.82)

The main advantage in the use of Eq. (4.3.81) is due to the fact that the strain energy density is only slightly influenced by the mesh pattern and this allows us to strongly reduce the running time of the nonlinear FEM analysis by using coarse meshes in the FE analyses. This happens also when super-convergent methods, like those based on J-integral, are used (Szanto and Read, 1992).

4.3.9 Further remarks

Generally commercial FEM softwares manage the strain hardening behaviour of materials by using an equivalent uniaxial stress formulation in combination with the von Mises criterion; this means that the stress strain curve in the plastic zone takes the form: n

⎛σ ⎞ ε eq ≅ ⎜⎜ eq ⎟⎟ , 1 ≤ n < ∞ . ⎝ K ⎠

(4.3.83)

This leads to a strain energy density equal to: n +1

Wp ≅

n σeq n + 1 Kn

(4.3.84)

Under pure antiplane shear deformation, von Mises criterion gives σ eqn = 3τ , so that Eq. (4.3.84) becomes:

176

NonLinear Solutions For Mode III Loaded Notches n +1

3 2 n τ n +1 Wp ≅ n + 1 Kn

(4.3.85)

Equating Eq. (4.3.85) to the expression Wp = ⎛ K = ⎜⎜ Gτ ⎝

n +1 n −1 2 0

3

1 n

⎞ ⎟⎟ , ⎠

n τ n +1 , we obtain: n + 1 Gτ 0 n −1

1

⎛ K n ⎞ n −1 τ0 = ⎜ n +1 ⎟ ⎜ ⎟ ⎝3 2 G⎠

(4.3.86)

Then Eqs. (4.3.66, 4.3.76, 4.3.81) can be rewritten as follows: Wp = 3

K 3,p

n +1 2

(K ) ( n, 2α)

n +1

BW

3,p n

R

K

⎞ ⎛ Wp K n ⎟ ⎜ = ⎜ n +1 ⎟ ⎜ 3 2 B ( n , 2 α) ⎟ W ⎠ ⎝

1 n +1

R

n +1 m −1

1 1− m

(4.3.87)

(4.3.88)

and J 3,p = 3

n +1 2

(K ) B ( 2α, n )

n +1

J

3,p n

K

n+m

R m −1

(4.3.89)

Finally, it is also worth noting that, thanks to Eqs. (4.3.87) and (4.3.89), a well defined analytical link exists between Wp and J 3,p : J 3,p Wp

=

BJ R BW

(4.3.90)

For a crack, under a linear-elastic hypothesis, Eq. (4.3.90) simply gives: J 3,e = πR We

(4.3.91)

where πR is half the path over which J is estimated. This mean that if R is thought of as a material- dependent parameter, as proposed by Lazzarin and Zambardi (2001), a precise analytical link exists between the mode III SED and the mode III J-integral, which is linear in R. However, focusing our attention on the crack case, the slope of the linear trend strongly depends on the hardening exponent n, ranging from 3.14 (when n=1, linear elastic case) to 2.15 (n=10). Then, under mode III, the SED and the J-integral manage plasticity in a different way (the


177

maximum difference in the crack case being of about 30 percent). Further analyses are intended on this topic. It is finally worth noting that in this work the symbol τ0 is the flow (or yield) stress according to the theory (for which a simplified material law has been used), and can be determined as a function of the more conventionally used strength coefficient K by means of Eq. (4.3.86). Most of commercial FE codes use by default the von Mises criterion to manage yielding of materials, so that the shear yield stress results in τ Y = σ Y ,0.2 / 3 . This is not a contradiction with theory but simple a practical consequence. When using the analytical frame presented here one should pay attention to not confuse the two values, which are in principle different. Then in the analytical formulation providing the plastic NSIF as a function of the elastic NSIF (Eq. (4.3.46)) one should use τ0 as obtained from Eq. (4.3.86). This problem is avoided in the SED and J-integral formulation, for which Eqs. (4.3.87-4.3.89) can be used.

4.3.10 Comparison to FE analyses and discussion In principle, the equations reported in the previous sections are valid only for the infinite body with a pointed V-notch under uniform antiplane shear loading; under such circumstances a single term in the nonlinear stress formulation is sufficient to get a complete and exact solution. As the global dimension decreases a single term is no longer sufficient to get a complete solution. However the developed analytical frame can also be used with V-notches in structural components of finite size, provided that it is applied in the vicinity of the notch tip. A further topic to be discussed is the amount of approximation introduced in the analytical derivations by the choice of a simplified material law; it is indeed well know that the actual behaviour of common engineering metals is well described by a Ramberg-Osgood type law, where, in principle, strains are determined by an elastic and a plastic contribution. However, it is the author’s opinion that due the high stress and strain concentration effects caused, in the plastic zone, by the pointed V-notch, the contribution to the local stress and strain fields by the elastic term is small when compared to that of the plastic term, as shown by Hutchinson (1968) and the local

178


behaviour of the material is very close to that of a pure power law. Furthermore in the previous literature (Sharma and Aravas, 1991) it has been shown that when a Ramberg-Osgood law is used to describe the material behaviour, the elastic term becomes influent at least in the second, or higher, order terms of stress and strain distributions, and can then be neglected in a leading order term based analysis. A number of FE analyses have been carried out with the aim to compare theoretical and numerical results and to clarify the degree of accuracy and limits of applicability of the analytical frame. Two materials have been considered with the following properties: •

Steel AISI 1045, E=206 GPa, K=950 MPa, n=8.33, σY=450 MPa, ν=0.3, so 1

that: τ Y = σ Y ,0.2 / 3 ≅ 259 MPa; •

⎡ ⎛ n +1 ⎞⎤ n −1 τ0 = ⎢ K n ⎜⎜ 3 2 G ⎟⎟⎥ ≅ 258 MPa ⎢⎣ ⎝ ⎠⎥⎦

Steel AISI 1008, E=206 GPa, K=600 MPa, n=4, σY=125 MPa, ν=0.3, so that: 1

τ Y = σ Y ,0.2 / 3 ≅ 72 MPa;

⎡ ⎛ n +1 ⎞⎤ n −1 τ 0 = ⎢ K n ⎜⎜ 3 2 G ⎟⎟⎥ ≅ 47 MPa ⎢⎣ ⎝ ⎠⎥⎦

It is worth noting that only AISI 1045 has τY very close to τ0. All the finite element analyses have been carried out by using ANSYS® code, version 9. Three-dimensional models have been used, with 20 node solid finite elements (named Solid95 in the ANSYS® code, Ansys Tutorials, 2005). Very fine mesh patterns have been used in order to achieve accurate values for plastic and elastic NSIFs, the size of the elements close to the notch tip being about 10-4 mm. Results from FE analyses are shown in Figs. 4.3.8-4.3.23 considering rounded bars under torsion loading, the V-notch opening angle being 2α=0 and 2α=120°. Figures 4.3.8 and 4.3.9 show that there exists a well-defined zone along the notch bisector where the plastic notch stress intensity factor can be defined according to Eq. (4.3.36). As soon as the plastic NSIF is determined, it can be used to accurately describe the stress fields in the vicinity of the notch tip. Figs. 4.3.10 and 4.3.11 compare FEM results and theoretical predictions carried out according to Eq. (4.3.38), the shear stresses being plotted along two circular paths of different radius embracing the V-notch and the crack tip. The agreement is

179


found to be very satisfactory also outside the notch bisector and excellent along the notch bisector line (Figs. 4.3.12 and 13). By using in combination Eq. (4.3.38) and the plastic NSIF given by Eq. (4.3.36), it is possible to accurately evaluate the shear stresses in a wide zone ahead of the notch. For high level of stresses (and then for large plastic zones), the accuracy at a certain distance from the notch tip reduces due to the decrease of the nominal stress on the net sectional area; however the predictions remain satisfactory. It is clear that the dimension of the zone where FE results and theoretical predictions agree depends on the geometry, the nominal stress and material

Plastic NSIF, K3,p [MPa mm1/(1+n)]

properties and hence may vary from case to case. 1000 τnn=450 MPa 800 Shear-loaded crack, 2α=0 100 MPa

600

Mt

Steel AISI 1045, n= 8.33 Steel AISI 1008, n=4

400 τnn=100 MPa 200

50 MPa 20 MPa

0 0.001

0.01

0.1 Distance from the notch tip [mm]

1

10

Figure 4.3.8. Plastic NSIF according to Eq. (4.3.36) of a crack for different loading stresses and materials. Figures 4.3.14 and 4.3.15 compare plastic NSIF as estimated according to Eq. (4.3.46) or directly obtained from FE analyses (2α=0 and 2α=120°). It is interesting to note that, when the plastic zone size is small with respect to the notch depth, the accuracy of Eq. (4.3.46) is good even if based on the simplified material law shown in Figure 4.3.4. In particular, for the crack case, the maximum error is less than 10 % for AISI 1008 up to τnn=0.8 τY and less then 4 % for AISI 1045 up to τnn= τY. As far as the V- notch with 2α=120° is concerned, the

180


comparison is found to be even more encouraging; the maximum error here is less than 10 % for AISI 1008 up to about τnn=0.9 τY and less than 4 % for steel AISI

Plastic NSIF, K3,p [MPa mm1/(1-m)]

1045 up to τnn= 1.3 τY. AISI 1045, m= -13.06, 1/(1-m)=0.071 AISI 1008, m= -7.68, 1/(1-m)=0.115 τnn=450 MPa

1000

120°

Mt

150 MPa 500 τnn=140 MPa 90 MPa 50 MPa 0 0.01


10

Figure 4.3.9. Plastic NSIF according to Eq. (4.3.36) of a V-notch (2α=120°) for different loading stresses and materials Shear stresses [MPa]

180 Solid line: theory (Eq. (4.3.38))

Points: FEA

160 140

τzy

2α

τzy

y r

ϕ

Shear-loaded crack, 2α=0

τzx 40 60

x

120

Mt

150

100 80 r=0.5 mm r=0.2 mm

60 40 20 0

0

20

τzx τnn=100 MPa K3,p=245 MPa mm0.2

Steel AISI 1008 σY =125 MPa n=4 K=600 MPa 40

60

80

100

120

140

160

180

ϕ [degrees]

Figure 4.3.10. Shear stresses along two circular path of different radius embracing the crack tip; comparison between FEM results and theoretical analysis.

181


Shear stresses [MPa]

500

y τzy

120°

300

x

τnn=450 MPa K3,p=950 MPa mm0.071

40

τzx

150

r=0.2 mm r=0.5 mm Solid line: theory (Eq. (4.3.38)) Points: FEA

100

0

τzx

Mt

60

200

r ϕ

2α

400

τzy

0

20

40

60 ϕ [degrees]

Steel AISI 1045 σY =450 MPa n=8.33 K=950 MPa

80

100

120

Figure 4.3.11. Shear stresses along two circular path of different radius embracing the V-notch tip; comparison between FEM results and theoretical analysis. 1000

Shear stress τzy [MPa]

Slope: −0.107

Solid line: theory, Eq. (4.3.38) Points: FEA

τnn=400 MPa 100

τY=260 MPa Slope: −0.2 100

τnn=100 MPa τY=72 MPa

50

20

Shear-loaded crack, 2α=0 Mt 40 60

n τY [MPa] AISI 1045 8.33 260 AISI 1008 4 72

150

10 0.001

0.01

0.1

1

10

Distance from the notch tip [mm]

Figure 4.3.12. Shear stresses along the crack bisector for different nominal shear stresses in the net section τnn and different materials; comparison between FEM results and theoretical analysis.

182


1000

Shear stress τzy [MPa]

Slope: −0.0711 τnn=450 MPa

Solid line: theory, Eq. (4.3.38) Points: FEA τY=260 MPa

150

Slope: −0.115 τnn=140 MPa 100

90

τY=72 MPa 120°

50

Mt 40

60

n τY AISI 1045 8.33 260 AISI 1008 4 72

150

10 0.01


10

Figure 4.3.13. Shear stresses along the V-notch bisector for different nominal shear stresses in the net section τnn and different materials; comparison between FEM results and theoretical analysis.


1000 Plastic NSIF from elastic NSIF, Eq. (4.3.46) Plastic NSIF from plastic SED, Eq. (4.3.88) 800

Steel AISI 1045, n= 8.33, FEA Steel AISI 1008, n=4, FEA Shear-loaded crack, 2α=0

600

Mt

400

200

0

0.1

0.3

0.5

0.7

0.9

τnn / τY

1.1

1.3

1.5

Figure 4.3.14. Plastic NSIF according to theory in comparison to FE analysis; with nominal shear stress τnn in net section and shear yield limit τY. Shear-loaded crack

183



1000 Steel AISI 1045, m= -13.06, FEA, 1/(1-m)=0.071 Steel AISI 1008, m= -7.68, FEA, 1/(1-m)=0.115 800 120°

Mt

600

400

Plastic NSIF from elastic NSIF, Eq. (4.3.46) Plastic NSIF from plastic SED, Eq. (4.3.88)

200

0

0.5

0.7

0.9

1.1

1.3

τnn / τY

1.5

1.7

1.9

Figure 4.3.15. Plastic NSIF according to theory in comparison to FE analysis; with nominal shear stress τnn in net section and shear yield limit τY. Pointed Vnotch (2α=120°) For higher load levels, Eq. (4.3.46) underestimates the plastic NSIFs and should not be used. In such cases, a good idea is that to determine the plastic NSIFs by directly using the SED value, which is available in most FE codes as an easy-to-call function, in combination with Eq. (4.3.88). Doing so, the results are like those shown in Figs. 4.3.14 and 4.3.15. Figures 4.3.16-4.3.19 show a comparison between Eq. (4.3.87) and results from FE analyses; even in these cases the analytical derivations are fully confirmed by the numerical results. It has to be noted that the plastic strain energy density over a small control volume surrounding the notch tip is greater than the corresponding elastic component; this holds true also under small scale yielding. In the crack case, the ratio Wp / We has been analytically determine in section 6 for the simplified material law (see Eq. (4.3.87) and Table 4.3.3); the theoretically predictions give a Wp / We ratio very close to 1.45 for both the materials and are confirmed by the numerical results. This trend already appeared in some numerical results by Lazzarin and Berto (2008) and represents a major difference with respect to the tension case under

184


plane strain conditions the constancy of the strain energy density was assured under small scale yielding (Lazzarin and Zambardi, 2002). The different trend in the two loading modes should be mainly due to the different shape of energy isolines ahead of the notch. 12 Strain energy density [N mm / mm3]

Shear-loaded crack, 2α=0 10

Mt

R=0.1 mm R

8 6 Plastic SED, Eq. (4.3.87), with K3,p from FEA

0.2

SED from FEA

4

Elastic SED

0.3

Steel AISI 1008

2

R=0.1 0

0.1

0.3

0.5

0.7

τnn / τY

0.9

1.1

1.3

Figure 4.3.16. Strain energy density according to theory in comparison to FE analysis; with nominal shear stress τnn in net section and shear yield limit τY. Crack case; steel AISI 1008. 120 Strain energy density [N mm / mm3]

Shear-loaded crack, 2α=0 100

R=0.1 mm

Mt R

80 60

0.2

Plastic SED, Eq. (4.3.87), with K3,p from FEA SED from FEA

40

0.3

Elastic SED Steel AISI 1045

20 0

0.3

0.5

R=0.1

0.7

0.9 τnn / τY

1.1

1.3

1.5

Figure 4.3.17. Strain energy density according to theory in comparison to FE analysis; with nominal shear stress τnn in net section and shear yield limit τY. Crack case; steel AISI 1045

185


Strain energy density [N mm / mm3]

140 120°

120

R=0.1 mm

Mt

100

R

Steel AISI 1045

80

Plastic SED, Eq. (4.3.87), with K3,p from FEA

60

0.3

SED from FEA 40

0.5

Elastic SED

20 R=0.1 0

0.5

0.7

0.9

1.1

1.3

τnn / τY

1.5

1.7

1.9

Figure 4.3.18. Strain energy density according to theory in comparison to FE analysis; with nominal shear stress τnn in net section and shear yield limit τY. Pointed V-notch (2α=120°); steel AISI 1045

Strain energy density [N mm / mm3]

16 R=0.1 mm

120°

14

Mt

12

R

10

Steel AISI 1008

8

0.3 Plastic SED, Eq. (4.3.87), with K3,p from FEA

6

0.5

SED from FEA Elastic SED

4 2 0

R=0.1 0.5

0.7

0.9

1.1

1.3

τnn / τY

1.5

1.7

1.9

Figure 4.3.19. Strain energy density according to theory in comparison to FE analysis; with nominal shear stress τnn in net section and shear yield limit τY. Pointed V-notch (2α=120°); steel AISI 1008. As the load increases, the ratio also increases, up to about 4.8 and for AISI 1008 and up to about 3.8 for AISI 1045 (with respect to the maximum value of the applied load).

186


2.5 Shear-loaded crack, 2α=0

Mt

JIII [MPa mm]

2

R

1.5 Plastic J-integral, Eq. (4.3.89), with K3,p from FEA

1

J-integral from FEA Elastic J-integral

0.5

0

Steel AISI 1008

0

0.2

0.4

0.6

0.8

1

1.2

1.4

τnn / τY

Figure 4.3.20. J-integral according to theory in comparison to FE analysis; with nominal shear stress τnn in net section and shear yield limit τY. Crack case; steel AISI 1008 60 120°

J3 [MPa mm]

50

Mt

40

R

30 Plastic J-integral, Eq. (4.3.89), with K3,p from FEA J-integral from FEA

20

Elastic J-integral Steel AISI 1045

10 0

0.5

0.7

0.9

1.1

τnn / τY

1.3

1.5

1.7

Figure 4.3.21. J-integral according to theory in comparison to FE analysis; with nominal shear stress τnn in net section and shear yield limit τY. Pointed V-notch (2α=120°); steel AISI 1045

1.9

187



1000 Steel AISI 1045, n= 8.33, FEA Steel AISI 1008, n=4, FEA 800

Linear law Shear-loaded crack, 2α=0

600

Mt

K3,p = 243 (τnn/τY) + 488 400

200

0

0.1

K3,p = 133 (τnn/τY) + 59

0.3

0.5

0.7

0.9

τnn / τY

1.1

1.3

1.5


Figure 4.3.22. Plastic NSIF according to the linear rule in comparison to FE analysis; with nominal shear stress τnn in net section and shear yield limit τY. Shear-loaded crack 1000 Steel AISI 1045, 1/(1-m) = 0.071, FEA Steel AISI 1008, 1/(1-m) = 0.115, FEA 800

Linear law 120°

600

K3,p = 319.5 (τnn/τY) + 382

Mt

400 K3,p = 145 (τnn/τY) + 38.5 200

0

0.5

0.7

0.9

1.1

1.3

τnn / τY

1.5

1.7

1.9

Figure 4.3.23. Plastic NSIF according to the linear rule in comparison to FE analysis; with nominal shear stress τnn in net section and shear yield limit τY. Pointed V-notch (2α=120°) For the V- notch with 2α=120°, the maximum detected values of the Wp / We ratio are about 17.9 and 14.5 for AISI 1008 and AISI 1045, respectively,

188


when R=0.1 mm. Finally, Figures 4.3.20 and 4.3.21 plot the J-integral for the cases 2α=0 and 2α=120°. For the sake of brevity, AISI 1008 is considered in the former case, AISI 1045 in the latter case, but the conclusions aim to be material-independent. Theoretical predictions and numerical results are found to be in good agreement also under large scale yielding conditions where no explicit rule exists to correlate the plastic values of the J-integral to the elastic ones. Plots of the plastic NSIF values as determined from the FE analyses make evident that a simple linear law is valid as a function of the τnn/τY (see Figs. 4.3.22 and 4.3.23). The following convenient form can be chosen: K 3,p = A

τ nn +B τY

(4.3.92)

Two elastic-plastic analyses are then sufficient to determine the parameters A and B for each geometry and material. For doing so, it is only required that the amount of plasticity ahead of the notch tip is sufficient to unambiguously determine the plastic SIF (a tenth of millimeter could be sufficient, as already shown in Figs. 4.3.12 and 4.3.13).

4.3.11 Final observations The main conclusions based on the developed analytical frame can be drawn as follows: −

the elastic-plastic boundary under small scale yielding has a circular shape only in the crack case. When varying the opening angle the boundary assumes a groin-like shape; for n=∞ the cusp of the groin falls on the notch tip;

−

the entire stress field in the plastic zone surrounding the notch apex can be expressed as a function of the plastic Notch Stress Intensity Factor;

−

a simple relationship ties elastic, as obtained by a pure linear elastic analysis, and plastic Notch Stress Intensity Factors (NSIFs) under small scale yielding; in the crack case this relationship matches those already proposed by Rice;

−

the plastic Strain Energy Density averaged over a well-defined circular sector centred at the notch tip can be given in closed form as a function of the plastic NSIF; the plastic SED is amplified with respect to the corresponding linear

189


elastic value even under small scale yielding and the amplification factor can be evaluated analytically by using elastic and plastic NSIFs. In the crack case this amplification depends only on the hardening exponent n and does not depend on the dimension of the volume and the far applied load magnitude; −

the plastic J-integral can be given in closed form as a function of plastic NSIF;

−

A simple linear rule allows one to determined plastic NSIFs both under small and large scale yielding.

−

All the analytical results have shown to be satisfactorily in agreement with the results of a large body of finite element analyses carried out by using Ansys 9.0®.

4.4

A reformulated version of the Neuber rule accounting for the influence of the notch opening angle

4.4.1 Basic equations In order to solve the nonlinear problem of an out-of-plane loaded notched body (see Figure 4.1.1), it is convenient to regard the ‘physical’ coordinates as functions of the strains (Rice, 1967b): x=

∂ψ ∂γ zx

y=

∂ψ ∂γ zy

(4.4.1)

T −γzx

y z r

ϕ

γ x

ϕ γzy (a)

(b)

Figure 4.4.1. Coordinates at a V-notch under antiplane shear load T; physical plane (a) and strain plane(b)

190


The function ψ can be thought of as a nonlinear potential, which plays the same role of the Airy stress function in linear elastic analyses. The equilibrium and compatibility equations can be rewritten in the form (Rice, 1967b): ∂x ∂y + =0 ∂τ zx ∂τ zy

(4.4.2)

∂x ∂y − =0 ∂γ zy ∂γ zx

(4.4.3)

provided that the Jacobian operator of the transformation is nonzero. The solution will then be sought such to satisfy in combination Eq. (4.4.2,4.4.3) and the material law, which can be written in general as: τ = τ(γ) ,

τ zj = γ zj

τ(γ) γ

(4.4.4)

It should be noted that when the shear stress component τzj vanishes on a portion of the mode III-loaded body, so does also γzj. It is convenient to introduce a polar reference system in the shear strain plane (Rice, 1967b; see Figure 4.1.1b): γ zx = -γγsi ϕ

γ zy = γcosϕ

(4.4.5a-b)

where γ is the modulus of the strain vector, γ = γ 2zx + γ 2zy

(4.4.6)

and ϕ is the angle defined according to the expression: ⎛ γ ⎞ ϕ = arctan ⎜ − zx ⎟ ⎜ γ ⎟ ⎝ zy ⎠

(4.4.7)

It is easy to verify by substitution, with the aid also of the Chauchy-Schwarz conditions, that Eq. (4.4.1) inherently satisfies Eq. (4.4.3). Equation (4.4.1) can thus be rewritten in the following form: x=−

∂ψ ∂ψ cos ϕ ∂ψ ∂ψ sin ϕ sin ϕ − , y= cos ϕ − ∂ϕ ∂γ ∂ϕ γ ∂γ γ

(4.4.8a,b)

By inserting Eqs. (4.4.8) into Eq. (4.4.2) and using also Eq. (4.4.4), we obtain the following equation (Rice, 1967b): τ(γ) ∂ 2 ψ 1 ∂ψ 1 ∂ 2 ψ + + =0 γτ'(γ) ∂γ 2 γ ∂γ γ 2 ∂ ϕ 2

(4.4.9)

where τ'(γ) = ∂τ(γ)/∂γ . Eq. (4.4.9) allows the solution of the problem as soon as

191

NonLinear Solutions For Mode III Loaded Notches appropriate boundary conditions are imposed on ψ .

4.4.2 Description of the notch profile Consider now the following transformation (Neuber, 1958): z = wq

(4.4.10)

where z = x + iy and w = u + iv are complex variables in the physical and the transformed planes, respectively (see Fig. 4.4.2), whereas q is a real number

related to the notch opening angle 2α by means of the expression: q=

2π − 2α . π u=u0

(4.4.11)

y

r -ϕ/q

u=0

τzϕ nu

y

τzr

ϕ

2α

x

(a)

r0

x

(b)

Figure 4.4.2. Auxiliary system of curvilinear coordinates (u, v) (a); reference system in the physical plane( b).

By using Euler’s formula, eiϕ=cosϕ + isinϕ, Eq. (4.4.10) can be rewritten in terms of the u and v components: 1 q

ϕ u = r cos , q

1 q

v = r sin

ϕ q

(4.4.12)

Equation (4.4.12) shows that the angle between the vector r and the vector nu normal to the curve u=cost is equal to −

ϕ q

(see Fig. 4.4.2b).

The curvilinear coordinate system allows us to describe hyperbolic notches (1