Anisotropic ductile fracture Part II: theory

Acta Materialia 52 (2004) 4639–4650 www.actamat-journals.com

Anisotropic ductile fracture Part II: theory A.A. Benzerga a

a,*

, J. Besson b, A. Pineau

b

Department of Aerospace Engineering, Texas A&M University, 3141 TAMU, College Station, TX 77843-3141, USA b Ecole des Mines de Paris, Centre des Mat
eriaux, UMR CNRS 7633, BP 87, F91003 Evry Cedex, France Received 18 December 2003; received in revised form 8 June 2004; accepted 11 June 2004 Available online 15 July 2004

Abstract A theory of anisotropic ductile fracture is outlined and applied to predict failure in a low alloy steel. The theory accounts for initial anisotropy and microstructure evolution (plastic anisotropy, porosity, void shape, orientation and spacing) and is supplemented by a recent micromechanical model of void-coalescence. A rate-dependent version of the theory is employed to solve boundary value problems. The application to the studied steel relies on material parameters inferred from quantitative metallography measurements. The quantitative prediction of damage accumulation and crack initiation in notched bars is achieved without any adjustable factor and is discussed under various stress states and loading orientations. Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Alloys; Anisotropic plasticity; Fracture; Void coalescence; Micromechanics

1. Introduction In this article we address the issue of the quantitative prediction of ductile fracture. To that end, an anisotropic theory is presented, which is based on the mechanics of porous plastic materials. The theory is applied to model the anisotropic failure of a low alloy medium strength steel characterized in Part I of this work [1]. The aim of the prediction is precisely to be quantitative, that is matching experiments without fitting parameters. So far, modeling based on micromechanics has essentially involved isotropic models [2,3] and has been successful to some extent [4–9]. While many of the qualitative aspects of ductile fracture have been explained by isotropic approaches, quantitative predictions are still a challenge. The isotropic model, e.g. [3], accounts for pressure-sensitivity through a mechanism of dilational plasticity, which is (i) homogeneous in the *

Corresponding author. Tel.: +1-979-845-1602; fax: +1-979-8456051. E-mail address: [email protected] (A.A. Benzerga).

elementary volume and (ii) neglects microstructure evolution, namely plastic anisotropy and void shape. It is now established that a competing localized dilational mechanism of plasticity, associated with void coalescence, needs to be accounted for; see [10–13]. It is also established that the concept of pressure-sensitivity needs refinement through the incorporation of void shape effects [14–17]. Earlier attempts to incorporate void shape effects in micromechanical models, e.g. [18], were of an empirical character and, as noted by their authors, were restrictive. A theory that incorporates the void shape effect is necessarily anisotropic. But anisotropy in ductile materials enters in many ways. In initially isotropic materials, it enters through the deformation-induced evolution of microstructural features, which can be individual (grain shape, pore shape and size) or collective (texture and spatial distribution of pores). But, of course, any of these features can be initially anisotropic as a result of the metal-working or pre-deformation history so that directionality arises in most mechanical properties of the material. Modeling fracture anisotropy is largely unexplored. Yet both toughness and ductility of many

1359-6454/$30.00 Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.06.019

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A.A. Benzerga et al. / Acta Materialia 52 (2004) 4639–4650

structural alloys depend on the loading orientation (for steel see [1] and references therein; for aluminum and titanium alloys see [19,20]). Anisotropy of fracture properties in ductile materials is usually attributed to the shape of second-phase particles or inclusions such as manganese sulfide in steel. This effect was analyzed in [16] using a model for porous materials incorporating void shape effects [15]. Anisotropy in materials with only equiaxed particles [21] suggests, however, that the particle spatial distribution and/or material’s texture may also be of significance. These two effects were theoretically investigated in [17] and [22,23], respectively. It was confirmed that any anisotropy in void distribution has an influence on void coalescence, not on void growth, with a possible net effect on fracture toughness [17]. Also, in [23] it was found that the effect of plastic anisotropy can explain the difference in ductility between steel and aluminum alloys when other known effects are kept the same. Therefore, a complete theory of ductile fracture should take account of all the material aspects listed above: inclusion and void shape, void distribution anisotropy and plastic anisotropy as a macroscopic effect of material texture or grain elongation. So far each aspect was analyzed separately [16,17,23]. Here the three features are assembled in a general approach based on the two-mechanism plasticity model [11,12] with an explicit illustration of how the approach can be implemented and assessed against experiments.

2. Formulation of the theory The weak form of the principal of virtual work is written as Z Z Z S : dE dV ¼ T  du dS þ f  du dV ; ð1Þ V





S

V

with  1
T F FI ; ð2Þ




2


is the symmetric second Piola–Kirchoff stress where S
tensor, E is the Green–Lagrange strain, F is the defor

mation gradient, R is Cauchy stress, J ¼ detðF Þ, T and f

are, respectively, the surface tractions and body forces if any, u is the displacement vector and V and S are the volume and surface of the body in the reference configuration. An updated Lagrangian formulation is used [24] which employs objective space frames with the reference configuration being either chosen at the beginning of the increment or at the end of the increment. Unless otherwise stated, the latter option has been adopted in most solutions here so that the stress measure S reduces to the Cauchy stress.
S ¼ J F1  R  FT ;

E¼

The constitutive framework is that of a progressively cavitating anisotropic viscoplastic solid initiated in previous studies [16,17]. The formulation was detailed in [25] but for completeness it is outlined here. In the objective (polar or co-rotational) frame, the strain rate e , and a tensor is written as the sum of an elastic part, D
p viscoplastic part, D . Assuming small elastic strains and
isotropic elasticity, a hypo-elastic law is expressed using the rotated stress P
De ¼ C1 : P_ ;


ð3Þ






P ¼ J XT  R  X;








ð4Þ




where C is the rotated tensor of elastic moduli. If the co rotational frame is used then the rotation tensor X is
identified with the spin Q (skew-symmetric part of the
velocity gradient) so that Jaumann rate of R is used. If
the polar frame is used then X is identified with the
rotation R resulting from the polar decomposition of the
deformation gradient F and the Green–Naghdi rate of R

ðpÞ is used. The viscoplastic part of the strain rate, D , is
obtained by normality from the gauge function: ðpÞ; / ¼ rH  r

ð5Þ

 is the matrix yield stress, p the effective plastic where r strain and rH is an effective matrix stress which is implicitly defined through an equation of the type FðR ; f ; S; ez ; H; rH Þ ¼ 0 with f the porosity, S the shape
 parameter (logarithm of the void aspect ratio W ), ez the void axis and H Hill’s fourth-rank tensor. The potential  F admits two different expressions, FðcÞ and FðcþÞ , prior to and after coalescence, respectively. The flow potential prior to coalescence is given by [15,25] ! 3R0 : H : R0 :R jA



ðcÞ F ðR; f ; S; H; rH Þ ¼ þ 2qw f cosh
 h rH 2r2H  1  q2w f 2 ¼ 0;

ð6Þ

0

where ðÞ refers to the deviator, h is a factor calculated using Hill coefficients, expressed in the basis ½e ¼ ðeL ; eT ; eS Þ pointing onto the principal directions of orthotropy [22,23], as   1=2 2 hL þ hT þ hS 1 1 1 1 h¼ þ ; þ þ 5 hL hT þ hT hS þ hS hL 5 hTS hLS hLT ð7Þ is the void anisotropy tensor expressed in the and A
basis ½e0  ¼ ðex ; ey ; ez Þ associated with the void A ¼ a2 ðex  ex þ ey  ey Þ þ ð1  2a2 Þez  ez ;


ð8Þ

with a2 in (8) and j in (6) being scalar functions of both f and S (Table 1). The coefficient qw is void-shape dependent and was determined by Gologanu [15] to fit unit-cell results: qw ¼ 1 þ ðq  1Þ= cosh S;

ð9Þ

A.A. Benzerga et al. / Acta Materialia 52 (2004) 4639–4650 Table 1 Coefficients used in Eqs. (6), (8) and (18), analytical expressions and particular values Coefficient

eðSÞ Eðf ; SÞ jðf ; SÞ a2 ðf ; SÞ a1 ðSÞ aG 1 ðSÞ

Prolate cavity (S P 0)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  expð2SÞ E3 e3 ¼f 1  E2 1  e2  1 1 1
pffiffiffi e pffiffiffi þ ð 3  2Þ ln E 3 ln f 1 þ E2 3 þ E4 1 1  e2  tanh1 e 2 2e 2e3 1 3  e2

Limit cases Sphere (S  0)

Cylinder (S ! 1)

0

1

0

1

3 2

pffiffiffi 3

1 3 1 3 1 3

1 2 1 2 1 2

For an arbitrary void shape (spheroid, cone, . . .), v is exactly related to the void spacing ratio, k, through a shape factor c as (see Fig. 1) (

1=3 EZ ¼ ez ; 3c Wf k ð12Þ v¼ f 1=3 k EZ ? ez : W 3c W k In [11] k ¼ 1=3 was used but refinements based on unitcell calculations suggested that k ¼ 1 is more accurate. For an arbitrary loading orientation v admits a more complex expression but k is always defined as the ratio of the Z-spacing to the average radial spacing. The function cðvÞ was introduced in [12] to represent the actual non-spheroidal void shapes observed during coalescence (see e.g. Fig. 9(b) in [1]), with  c v cS ), as in the f ¼ 10 bar (low triaxiality ratio T), then S must be an increasing function of strain. Conversely, if void growth is mostly dilational (cL < cS ) then S must be a decreasing function. Comparison between Fig. 13 in [1] (maximum values) and the present Fig. 9(b) shows that the model not only accounts for that qualitative trend but also picks up the transition from extensional to dilational void growth for elongated cavities. Under T-loading, assuming a spheroidal shape for the voids at the initiation of macroscopic failure and noting that W =W0 is identified with the measured pffiffiffiffiffiffiffiffiffi cL = cT cS , it is also easy to check that the void aspect ratio (as defined here in the local frame) must decrease, irrespective of the stress state. This is precisely what is seen in Fig. 5(c) above, consistent with the experimental trends. Next we proceed to the comparison of the predicted porosities at the onset of coalescence, fc , with the local void area fractions measured in Part I. As noted there, measured porosities comprised between the local average, f , and the local maximum, fmax , should be representative of near-coalescence states. This type of detailed comparison is only possible for an L oriented bar with f ¼ 4 and a T oriented bar with f ¼ 10, for which local measurements were carried out. Fig. 12 shows the predicted porosity, fc (open symbols), the local porosities, f and fmax when available, along with the average void area fraction, fa , defined in [1] as the total void area over the total area analyzed. In Fig. 12 error bars shown for fa include the accuracy of measuring void dimensions and total analyzed area [1] while those shown for f and fmax indicate the standard error with 95% confidence intervals. In particular, the error in estimating fmax is computed using all cells having porosity larger than 0.02. Under L-loading, the predicted fc matches quite well the maximum local porosity, fmax , whereas, under Tloading, the predicted fc falls between f and fmax . Both predictions are remarkable not only because they fall within the relevant experimental range but also because each prediction approximately corresponds to the local maximum in the frequency distribution of the measured

4649

Fig. 12. Measured and predicted porosities at incipient coalescence in notched bars. (a) L orientation. (b) T orientation. fmax : largest local porosity; f : local average; fa : global average.

void area fractions (see Fig. 15 in Part I). Also, under L-loading fc is found to increase with enhancing stress triaxiality and so does the measured average porosity, but to a lesser extent. This trend of an increasing fc with increasing stress triaxiality, within the range considered here, is consistent with previous theoretical predictions [12,17]. It is worth recalling that the measured values correspond to elongated voids. Thus, for consistency, the fc values in Fig. 12 were determined for the same set of initial microstructural variables. In particular, the value corresponding to f ¼ 10 does not correspond to an actual onset of coalescence; it is the value that porosity would have taken at failure initiation in experiments if the elongated cavities were involved in that initiation. For the T orientation case, the porosity at coalescence is not found to vary much within the range of stress triaxiality explored here. However, because of the heuristic character of our extension of the coalescence criterion to non-axisymmetric configurations, that model in particular is open to improvements.

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In all loading configurations considered here, the values of fc are an outcome of the calculations and not an input. Quantitative comparisons based on isotropic models usually require that critical porosities be adjusted so that the macroscopic failure strains can be fairly predicted. It should be emphasized that commonly used values for the critical porosity, compared with our fc , fall in the range 0.001–0.003 for an initial porosity, f0 , of 0.00075 and can be even smaller for lower values of f0 . Such ‘‘critical porosities’’ are one order of magnitude lower than the values measured in [1] and, for the L orientation case, they are of the order of the global average porosity, fa . But what is critical to the onset of failure in notched bars is the local porosity, not fa .

4. Conclusions In this paper, a general theory of ductile fracture has been proposed and its application to engineering alloys illustrated. The theory first accounts for matrix anisotropy which may result from texture development during the metal working process. Next, the theory accounts for morphological anisotropy which can be initial, due to non-spherical second-phase particles, or induced, due to the evolution of void shape. The theory also accounts for the whole void-coalescence process through a micromechanical model. Accounting for that process leads to an effective yield surface which displays regions of extreme curvature. One outcome of using the developed constitutive relations is a loading response that includes the transition from a pre-coalescence stage to a postcoalescence stage without using adjustable factors. The predictive approach pursued in this study relies on: (i) a careful characterization of the microstructure in the undeformed state using standard quantitative metallography techniques; (ii) recording the mechanical response of the material under various stress states and orientations with a special attention to plastic anisotropy; (iii) a finite-element implementation of the constitutive relations relevant to anisotropy and to different modes of dilational deformation; (iv) a complementary collection of data related to the microstructure in the deformed state with the aim of assessing the approach in an unbiased way. The calculations here quantitatively reproduce the behavior seen in notched bars in remarkable detail. Moreover, the accuracy of the predictions has been achieved under the following conditions: (i) no adjustable parameter has been used (the counterpart for that is the collection of a large amount of data on the initial microstructure); (ii) the quality of the prediction is obtained using a very low value for the initial porosity, which is inferred from measurements; (iii) predictions of global failure strains and local

porosities at incipient simultaneously.

coalescence

are

achieved

Acknowledgements Financial support from Gaz de France (Direction de la Recherche; Program managers R. Batisse and M. Zarea) is greatly acknowledged. References [1] Benzerga AA, Besson J, Pineau A. Anisotropic ductile fracture. Part I: experiments. Acta Mater 2004. doi:10.1016/j.actamat. 2004.06.020, this issue. [2] Rice JR, Tracey DM. J Mech Phys Solids 1969;17:201–17. [3] Gurson AL. J Eng Mater Technol 1977;99:2–15. [4] Beremin FM. In: Nemat-Nasser S, editor. Three-dimensional constitutive relations of damage and fracture. New York: Pergamon Press; 1981. [5] Needleman A, Tvergaard V. J Mech Phys Solids 1984;32:461–90. [6] Tvergaard V. Adv Appl Mech 1990;27:83–151. [7] Pineau A. In: Argon AS, editor. Topics in fracture and fatigue. Berlin: Spinger–Verlag; 1992. [8] Zhang ZL, Niemi E. Fat Frac Eng Mater Struct 1994;17:695–707. [9] Besson J, Steglich D, Brocks W. Int J Solids Struct 2001;38:8259– 84. [10] Gologanu M. Etude de quelques problemes de rupture ductile des m
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