Influence of preliminary mechanical and thermal treatment on the ...

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on the level of plastic prestraining in the course of preliminary mechanical and thermal treatment are described analytically. The results are obtained within the ...
Materials Science, Vol. 40, No. 2, 2004

INFLUENCE OF PRELIMINARY MECHANICAL AND THERMAL TREATMENT ON THE STEADY-STATE CREEP OF METALS A. K. Rusynko

UDC 539.376

The dependences of the rate of steady-state creep for metals with different stacking-fault energies on the level of plastic prestraining in the course of preliminary mechanical and thermal treatment are described analytically. The results are obtained within the framework of the generalized integrated theory of plasticity and creep. The analytic results are in good agreement with the experimental data, which enables one to predict the heat resistance of materials as a function of the level of plastic prestraining.

The procedure of mechanical and thermal treatment (MTT) combines the action of mechanical and thermal factors on the materials and can be used to increase the heat resistance of metals. As a result of this treatment, we observe the formation of ordered dislocation structures in the material inhibiting the motion of crystallographic defects, in particular, linear. The classical theories of plasticity and creep [1] do not take into account the influence of MTT on the rate of steady-state creep, which forces us to use the most advanced theories of irreversible deformation. In the present work, we propose a mathematical model used for the analytic description of the influence of MTT on the rate of steady-state creep of metals. Phenomenon and Its Physical Nature The procedure of MTT includes plastic deformation of a specimen subjected to treatment with subsequent annealing in the prerecrystalization temperature range (Fig. 1) [2]. As follows from Fig. 2 [3, 4], the dependences of the rate of steady-state creep ε˙ on the level of plastic prestraining ε0 for copper and aluminum are different and not monotonic (the temperature T1 and time t1 of annealing and the temperature T and stress σ x of creep are constant). The variations of the rate of steady-state creep after MTT were observed for nickel and its alloys, iron, molybdenum, and stainless steel [2, 5]. In the process of plastic deformation, original relatively perfect crystals split into fragments whose sizes and orientations depend of the level of strains. The boundaries of the fragments form a three-dimensional grid of subboundaries and play the role of sites of dislocation pileups. In the process of heating of a cold-hardened metal, some defects annihilate and the remaining dislocations are redistributed into energy-favorable configurations [6]. The dislocation walls are fixed by point defects. The substructure formed in the course of MTT inhibits the processes running in the matrix of the metal under the conditions of creep and decreases the creep rate [2–5]. As a result of MTT, the material is hardened to a certain level ε0 (≈ 1.5% for aluminum and ≈ 3.2% for copper) and then the creep rate approaches its initial value (Fig. 2). This is explained by the fact that, beginning with a certain value of ε0 , the substructure formed in the course of MTT loses its thermomechanical resistance. This leads to the intensification of the processes of recrystallization, weakening of interactions between dislocations in the subboundaries, and passing of dislocations from the walls of subboundaries into the bulk of the crystal. “L’vivs’ka Politekhnika” National University, Lviv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 2, pp. 59–66, March–April, 2004. Original article submitted October 24, 2003. 1068–820X/04/4002–0223

© 2004

Springer Science+Business Media, Inc.

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Fig. 1. Schematic diagram of successive mechanical (σ – ε coordinates) and thermal ( T– t coordinates) loading in the course of MTT.

Fig. 2. Dependences of the rate of steady-state creep on the level of preliminarily induced strains in tension: (a) aluminum (for the creep stress σ x = 9.6 MPa and creep temperature T = 260°C); annealing in the course of MTT at temperature T1 = 260°C for t1 = 1 h; (b) copper (for the creep stress σ x = 15 MPa and creep temperature T = 500°C); annealing at T1 = 500°C for t1 = 1 h; the curves are plotted according to the numerical results; ( 䊉 ) correspond to the experimental data from [3] (a) and [4] (b).

On passing the minimum, the curves ε˙ = f (ε0) for copper and aluminum exhibit different types of behavior in the course of subsequent cold hardening. This is explained by the fact that, starting from a certain level of prestraining (about 3% for aluminum), the role played by point defects in the process of hardening of the material becomes positive because their density becomes sufficient for the efficient fixing of dislocation walls. However, this type of hardening is observed only for aluminum. This is explained by different mechanisms of recovery in the course of creep in materials with different stacking-fault energies γ ( γAl = 0.2 J / m2, γCu = 0.04 J / m2, i.e., γAl / γCu ≈ 5). Copper, as a metal with low stacking-fault energy, suffers softening mainly as a result of recrystallization. At the same time, aluminum undergoes softening as a result of polygonization [7]. In the process of recrystallization, we observe the formation of a defect-free structure, which removes the hardening effect of point defects and, hence, in copper, the dependence ε˙ = f (ε0) possesses a single minimum. It is worth noting that, in aluminum, the rate ε˙ = f (ε0) again increases for ε0 > 10% due to the presence of microcracks (formed as a result of preliminary cold hardening). Mathematical Model The dependence of the rate of steady-state creep on the level of plastic prestraining is analytically described within the framework of the integrated theory of plasticity and creep [8] combining the Butdorf –Budianski con-

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 cept of slip with the Sanders theory of yield. The load is specified by the vector of stresses S whose compo nents are expressed via the components of the tensor of stress deviators [8]. We restrict ourselves to the case S ∈ S3 , where S3 is a three-dimensional subspace of the five-dimensional space S5 of Il’yushin deviators. The surface of plasticity in S3 has the form of a sphere corresponding to the Huber – von-Mises condition of plasticity. The radius of the sphere is given by the formula 2 R =

2 2 σT , 3

where σT is the plasticity limit of the material under active loading σ S. If the load is constant as a function of time, then σ T = σ P, where σ P is the creep limit of the material. The planes tangential to each point of the sphere correspond to a certain slip system [8].  The position of the plane in S5 is specified by the unit vector N normal to the surface and the distance H from the origin of coordinates. In the subspace S3 , the orientation of planes (traces of plates from S5 ) is speci   fied by a vector n with coordinates ( sin β, cos β cos α, cos β sin α ) and, moreover, N = n cos λ, where λ is   the angle between the vectors n and N [8]. Under a load responsible for the formation of irreversible strains, a certain set of planes moves. This can be described by the following analytic relation [8]:     H = S ⋅ N = S ⋅ n cosλ .

(1)

 The planes introduced by the vector S move parallel to themselves. The motion of each of these planes corresponds to an elementary increment of plastic strains. The vector of plastic strains is perpendicular to the moving plane. The total strains are computed by finding the sum (as a result of taking a triple integral) of increments of plastic strains over all moving planes. In the course of plastic deformation, the loading surface transforms and turns into the envelope of all planes tangential to this surface. In the case of uniaxial tension, we have σx > 0 and the remaining components of the stress tensor are equal to zero ( S1 =

2 / 3 σ x , S2 = S3 = 0 [8]). Hence, in view of relation (1), we obtain H = S1 n1 cos λ = S1 sin β cos λ =

2 σ x sin β cos λ . 3

(2)

Within the framework of the integrated theory, the components of the vector of irreversible strains under the conditions of uniaxial tension are given by the formulas [8] e1 = 2π

∫∫

ϕ sin β cos β cos λ dβ dλ ,

e2 = 0,

e3 = 0.

(3)

Ω( λ , β )

Here, ϕ is the intensity of strains, which can be found from the following differential equation [8]: d ψ = r dϕ – K ψ dt,

(4)

where r is a material constant, dt is an increment of time, K is a function of the homologous temperature Θ and stresses,

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 σ   σ  K3 K = f Θ, x = K1 exp ( K2Θ) x ,  σP   σP 

(5)

Ki are material constants, and ψ is the intensity of defects [8]: ψ (H ) = H2 −

2σ T2 . 3

(6)

For the angles β and λ satisfying the inequality H >

2 σ x sin β cos λ , 3

we set ψ = 0. The dimensions of the quantities in relation (4) are as follows: [ ψ ] = MPa2 , [ ϕ ] = 1, [ r ] = MPa2 , and [ K ] = sec– 1. The intensity of defects ψ corresponds to the mean continuous measure of defects (dislocations, vacancies, interstitial atoms, etc.) in a uniform continuous medium. Defects of the slip system in a  microvolume corresponding to the plane with normal n are regarded as defects in this plane. The quantity H characterizes the strain hardening of the material. As follows from relations (2) and (6), the higher the level of stresses, the larger the quantity H and, hence, the higher the intensity of defects ψ. This observation is confirmed by the fact that the density of defects of the crystal lattice of the material increases with the load, which complicates the process of subsequent plastic deformation. The representation of the ϕ – ψ dependence in the form (4) enables us to describe plastic and creep strains by using a single determining relation. Thus, for plastic deformation, we have dt = 0. As a result, in view of relations (2) and (6), relation (4) and integral (3) take the form (the quantities corresponding to plastic deformation are marked with subscript “0” ):

ψ 0 = r 0 ϕ0 ,

e10

4π = 3r0

π / 2 λ10

∫ ∫ {(σ x0 sin β cos λ)

β10

2

− σ 2S } sin β cos λ cos β dλ dβ ,

σx0 > σS.

(7)

0

In the case of steady-state creep, we have S˙1 = 0 and, hence, relation (4) takes the form rϕ˙ = K ψ = const,

(8)

where ϕ˙ is the strain-intensity rate. Thus, within the framework of the integrated theory, the strains of steadystate creep are formed in the system of immobile planes. According to relations (2), (6), and (8), for the rate of steady-state creep strain, we get 4π K e˙ = 3r

λ1 π / 2

∫ ∫

{(σ x sin β cos λ )2 − σ 2P } sin β cos β cos λ dλ dβ ,

σx ( t ) = const > σP .

(9)

0 β1

Thus, the larger the number of defects participating in the formation of creep, the higher the creep rate. The limits of integration in (7) and (9) are determined from the condition ψ = 0 [8]:

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cos λ1 0 ( β ) =

σS , σ x0 sin β

THE

STEADY-STATE CREEP OF METALS

cos λ1 ( β ) =

σP , σ x sin β

227

(10)

and the conditions λ 1 0 = 0 and λ1 = 0: sin β1 0 =

σS , σ x0

sin β1 =

σP . σx

(11)

For – π / 2 ≤ β < β1 0 and λ1 0 < λ, we have H = R and ψ = 0. Integrating relations (7) and (9), we obtain

a0 =

πσ 2S , 9r0

2

a =

πσ P , 9r

e10 = a0 Φ (sin ( β1 0 )),

(12)

e˙1 = K a Φ (sin ( β1 )),

(13)

Φ (ξ) =

1 ξ2

1 + 1 − ξ2   2 2 2 4 2 1 5 1 3 ξ ξ ξ ξ ln − − − +  .   ξ

The relationships between the components of the vector of strains e1 and strain tensor εx have the form [8] εx0 =

2 e1 3 0

and

ε˙ x =

2˙ e1 . 3

If the process of plastic deformation is followed by unloading which terminates the process of growth of irreversible strains (dϕ = 0), then the differential equation (4) takes the form d ψ = – K ψ dt. The solution of this equation is given by the formula ψ = ψ0 exp ( – K t ) =

2 ((σ x0 sin β cos λ)2 − σ 2S ) exp (− Kt ) , 3

(14)

where ψ0 is the intensity of defects accumulated in the process of plastic deformation and t is the time of holding in the unloaded state. Relation (14) describes the relaxation of defects after unloading (thermally activated annihilation of dislocation of the opposite signs, annihilation of vacancies on dislocations and interstitial atoms, etc.). It follows from relations (6) and (14) that, after unloading, the planes move to the origin of coordinates. The motion of each of these planes terminates when the distance H becomes equal to the radius of the sphere R. The velocity of motion is determined by the function K. To take into account the influence of the substructure formed as a result of the preliminary MTT on the rate of steady-state creep, we replace relation (6) with the following formula: 2 ψ ( H ) = H 2 − HM ,

(15)

where HM is the distance to the planes after MTT reflecting the resistance of the pretreated material to the deformation of creep. The quantity ψ is a decreasing function of HM and, hence, in view of relations (8) and (9), the same is true for the rate of steady-state creep. To find the quantity HM , we successively use relations (6) and

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(14). However, the function K appearing in these formulas cannot describe the relaxation of defects caused by  cold hardening in the process of annealing of the unloaded material because K ( S = 0 ) = 0. Moreover, to deduce analytic expressions for the influence of preliminary cold hardening and stacking-fault energy on the thermomechanical resistance of the substructure, we use the following function [instead of the function K specified by relation (5)]: KM

 Hmax − S ˜ ˜ = K + K ( Hmax , γ ) , Hmax (16)

K˜ ( H˜ max , γ ) =

 ˜ A  Hmax + exp 1 + γ / 4Γ 

2

   10 γ  ˜ 2    −   Γ  Hmax + C   , 3

where Hmax is the maximum distance from the planes for the entire history of loading at a given temperature, H − σS , H˜ max = max σS A and C are constants, the parameter Γ = 1 J / m2 is used to guarantee the consistency of dimensions [ H˜ max ] = 1, [ A ] = sec– 1, [ C ] = 1 ⇒ [ K ] = [ KM ] = [ K˜ ] = sec– 1. Under the conditions of uniaxial tension, the quantity Hmax can be found from relation (2) with λ = 0 and β = π / 2, namely, Hmax =

2 σx . 3 0

In the course of annealing of the unloaded specimen, we have K M = K˜ ( H˜ max , γ ) . It is clear that e10 can increase only in the case where σx0 and, hence, H˜ max increase. This means that the function K˜ implicitly (via H˜ ) depends on the degree of preliminary strain-hardening. max

If the temperatures of annealing and creep are equal, then relations (6), (14), and (16) give the following result: [(σ sin β cos λ )2 − σ 2S ] exp (− K˜ ( H˜ max , γ ) t1 ) + σ 2P 2  x0 =  3 2 σP

2 HM

for Ω10 (β, λ ) ,

(17)

for Ω1*0 (β, λ ) ,

where Ω10 ( β, λ ) is the range of the angles β and λ, i.e., β1 0 ≤ β ≤ π / 2, 0 ≤ λ ≤ λ 1 0 , Ω1*0 (β, λ) denotes the

region – π / 2 ≤ β < β1 0 , λ1 0 < λ, and the limiting angles can be found from relations (10) and (11). We denote ) . As follows from relation (16) and Fig. 3, the function the expression (σ x sin β cos λ )2 – σ 2 by F( H˜ 0

S

max

K˜ ( H˜ max , γ ) behaves as a power function whose power is equal to 1 / 2 but possesses a perturbation described by an exponential function within the interval [ H , H ] of variation of the quantity H˜ . The parameter γ, first, a

b

max

plays the role of a scaling factor (the larger γ, the smaller the value of K˜ ) and, second, specifies the magnitude of perturbation and its duration (the larger γ, the faster the function K˜ deviates from the power law in the interval [ Ha , Hb ] and the larger this deviation).

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Fig. 3. Qualitative representation of a function K˜ ( H˜ max , γ ) for two values of γ : γ1 > γ2 . For large values of γ within the ranges H˜ max ∈ [0, Ha(1) ] and H˜ max > Hb(1) the function HM increases together with the function F because the influence of exp (− K˜ ( H˜ , γ ) t ) is insignificant. The increase in H max

1

M

in these intervals illustrate the MTT-induced hardening of the material corresponding to the descending branches in the dependence ε˙ = f (ε0) (Fig. 2a). For H˜ max ∈ [ Ha(1) , Hb(1) ] , the function K˜ rapidly increases. Therefore, exp (− K˜ ( H˜ , γ ) t ) → 0 and the value of H decreases to 2 σ (the peak of the perturbation caused by max

1

M

3

P

the exponent is determined by the material constant C ). It is easy to see that, for HM = 23 σ P , relation (15) coincides with (6), which corresponds to the creep rate in the absence of preliminary MTT. The decrease in H M reflects the loss of the thermomechanical stability of the substructure formed as a result of MTT. Hence, the rate of steady-state creep approaches its initial value as a function of the degree of preliminary cold hardening. For low values of the parameter γ, the function K˜ increases more intensely and the perturbation of the function exp (− K˜ ( H˜ max , γ ) t1 ) in the interval H˜ max ∈ [ Ha(2) , Hb(2) ] is insignificant. This is why the product F( H˜ ) exp (− K˜ ( H˜ , γ ) t ) first increases but then, for large values of H˜ , decreases and tends to zero. max

max

1

max

This type of behavior of the quantity HM reflects the MTT-induced hardening of the material (descending branch in Fig. 2b). However, this effect disappears when the degree of cold hardening is high and the substructure loses its thermomechanical stability (ascending branch in Fig. 2b). We now compute the rate of steady-state creep in the metal subjected to MTT. For this purpose, we substitute the values of the distance to the planes HM given by relation (17) in relation (15). This gives

ψ =

2 HM

 (σ sin β cos λ )2 − [(σ x sin β cos λ )2 − σ 2S ] exp (− K˜ ( H˜ max ) t1 ) − σ 2P 0 2 x =  3 2 2  (σ x sin β cos λ ) − σ P

for Ω10 ,

(18)

for Ω1 ,

where σx are time-independent stresses and Ω1 ( β, λ) is the domain specified by the inequalities – λ1 0 ≤ λ ≤ λ1 and β1 ≤ β ≤ β1 0 . The angles β1 and λ1 can be found from relations (10) and (11). Clearly, under force load ing, when the planes are located at the end of the vector of loading, we can write H max = S . Therefore, it follows from relation (16) that KM = K . Substituting the parameter ψ given by relation (18) in relations (8) and (9), we find the rate of steady-state creep e˙1M in the material subjected to MTT as follows:

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e˙1M

2π K = r

4πK = 3r

λ1 π / 2

∫ ∫

ψ sin β cos β cos λ dλ dβ

0 β1

λ1 π / 2

∫ ∫ {(σ x sin β cos λ)

2

− σ 2P} sin β cos β cos λ dλ dβ

0 β1

4π – K exp (− K˜ ( H˜ max , γ ) t1 ) 3r

λ 10 π / 2

∫ ∫

0 β10

{(σ x0 sin β cos λ )2 − σ 2S } sin β cos β cos λ dλ dβ .

The first integral on the right-hand side of this equality coincides with relation (9) and specifies the rate e˙1 under normal conditions (without MTT). The second integral coincides with relation (7), i.e., determines the degree of plastic prestraining e10 : e˙1M (e10 ) = e˙1 − K

K = K1 exp( K2Θ)

r0 exp (− K˜ ( H˜ max ) t1 ) e10 , r

 2 σ x  K3 = const,  3 σP 

K˜ ( H˜ max , γ ) = K˜ (e10 , γ ) = var.

(19)

(20)

We now analyze these formulas. In the absence of preliminary MTT, we have e10 = 0 and e˙1M = e˙1 . As the degree of preliminary cold hardening e10 increases, the product exp (− K˜ ( H˜ max , γ ) t1 ) e10 from relation (19) , γ ) t ) . Thus, the indicated formula qualitatively describes the ) exp (− K˜ ( H˜ behaves as the function F( H˜ max

max

1

plots of the function ε˙ = f ( ε0 ) (Fig. 2) for different values of the stacking-fault energy γ. In order to construct the dependence of the rate of steady-state creep on the degree of plastic prestraining of the material by using relations (19) and (20), it is necessary to specify the material constants σS, σ P , r0 , r, K, and C and the constant of the model A. For this purpose, we must analyze the available experimental data. The constant r0 is chosen in constructing the diagram of plasticity of the material by using relation (12) and the constants r and K are selected in constructing the diagram of ordinary creep of the material under different thermal and force conditions by using relation (13). Note that, within the framework of the integrated theory, the values of plastic and creep strains with and without preliminary MTT are found from the same determining relation (4). Therefore, it suffices to determine the required constants according to the experimental diagrams of plasticity and creep without MTT. The constants C and A remain free for variation and can be selected in constructing the calculated curves by using relations (19) and (20). The calculated dependences of the rate of steady-state creep on the degree of plastic prestraining were constructed for aluminum (Fig. 2a) (with σS = 35 MPa, σP = 6 MPa [9], r 0 = 8 ⋅ 103 MPa2 , r = 1 ⋅ 103 MPa2 , K = 3.7 ⋅ 10– 5 sec– 1, and C = – 15) and copper (Fig. 2b) (with σS = 70 MPa, σ P = 9 MPa [9], r0 = 1.5 ⋅ 105 MPa2 , r = 2.53 ⋅ 104 MPa2 , K = 2.1 ⋅ 10– 5 sec– 1, and C = – 2.09). The constant of the model A = 1.39 ⋅ 10– 3 sec– 1. The numerical calculations were carried out by using the MathCad7-Professional software package. The calculated curves are in good agreement with the experimental data, which means that the results given by relation (19) are reliable both from the qualitative and quantitative points of view.

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CONCLUSIONS We propose a mathematical model of irreversible deformation. By using this model, we construct analytic dependences of the rate of steady-state creep on the degree of the preliminary cold hardening of metals in the process of their mechanical and thermal treatment. The model is constructed within the framework of the integrated theory of plasticity and creep. The formulas deduced in the present work are in good agreement with the experimental data, which enables us to predict the characteristics of heat resistance of the material depending on the degree of plastic prestraining. REFERENCES 1. Yu. N. Rabotnov, Creep of Structural Elements [in Russian], Nauka, Moscow (1966). 2. I. I. Ivanova, Mechanical and Thermal Treatment as a Method for the Enhancement of the Heat Resistance of Metals and Alloys [in Russian], Nauch.-Tekh. Obshch. Mashinostr. Prom, Moscow (1964). 3. G. Ya. Bazelyuk, G. Ya. Kozyrskii, G. A. Petrunin, and I. G. Polotskii, “Influence of preliminary ultrasonic irradiation and mechanical and thermal treatment on the creep resistance of aluminum,” Fiz. Metal. Metalloved., 32, No. 1, 145–151 (1971). 4. G. Ya. Bazelyuk, G. Ya. Kozyrskii, G. A. Petrunin, and I. G. Polotskii, “Influence of preliminary ultrasonic irradiation on the high-temperature creep and microhardness of copper,” Fiz. Metal. Metalloved., 29, No. 3, 508–511 (1970). 5. L. K. Gordienko, Substructural Hardening of Metals and Alloys [in Russian], Nauka, Moscow (1973). 6. I. I. Novikov, Theory of Thermal Treatment of Metals [in Russian], Metallurgiya, Moscow (1973). 7. D. McLean, Mechanical Properties of Metals [Russian translation], Metallurgiya, Moscow (1965). 8. Ya. F. Andrusik and K. N. Rusinko, “Plastic deformation of hardening materials under loading in a three-dimensional subspace of a five-dimensional space of deviators,” Mekh. Tverd. Tela, No. 2, 78–83 (1993). 9. G. V. Samsonov (editor), Physicochemical Properties of Elements. A Handbook [in Russian], Naukova Dumka, Kiev (1965).