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Microstructure-based constitutive law of plastic deformation
Computational Materials Science 25 (2002) 200–206 www.elsevier.com/locate/commatsci
Microstructure-based constitutive law of plastic deformation R. Sedl acek *, W. Blum Institut f€ ur Werkstoffwissenschaften, Lehrstuhl I Universit€at Erlangen-N€urnberg, Martensstrasse 5, 91058 Erlangen, Germany
Abstract A macroscopic constitutive law of plastic deformation which accounts for the heterogeneous microstructure of the deforming material is proposed. It consists of a kinematic framework, kinetic equations, and structure evolution equations. The kinematic framework treats the material as a composite consisting of (plastically) soft cells or subgrains and hard cell walls or subgrain boundaries. The kinetic and structure evolution equations remain unspecified at first, the law being developed in a general way towards a system of ordinary differential equations. The proposed framework connects the heterogeneous microstructure with the macroscopic response independently of deformation mode and under arbitrarily varying deformation conditions. Two application examples are presented. Ó 2002 Elsevier Science B.V. All rights reserved. JEL classification: 61.72; 62.20 Keywords: Plastic deformation; Dislocation microstructure; Composite model
1. Introduction For engineering use, macroscopic work-hardening curves of technical materials are often fitted to empirical equations, the simplest example of them probably being the power law r ¼ km : Strictly speaking, an extrapolation of the computed flow stress r beyond the experimental range of strains , temperatures T, and/or deformation conditions used for fitting the constants k and m is not justified. This holds even for the much more
*
Corresponding author. Current address: Lehrstuhl f€ ur Werkstoffkunde und Werkstoffmechanik, TU-Mu¨nchen, Boltzmannstrasse 15, 85747 Garching, Germany. Tel.: +49-89-28915264; fax: +49-89-289-15248. E-mail address: [email protected] (R. Sedla´cˇek).
elaborate empirical laws used in practice. Therefore, materials science came up with phenomenological models based on the assumption that the flow stress r at strain rate _ and temperature T depends on a parameter which describes the current structure of the material [1,2]. Typically, the mean dislocation density q is used as the structure parameter, r ¼ rðq; _; T Þ:
ð1Þ
Then, the kinetic equation (1) is complemented with an evolution equation for the dislocation density, dq ¼ f ðq; _; T Þ: d
ð2Þ
This approach appears very useful in describing plastic flow in different modes of deformation. For example, the model is capable of predicting the
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 2 6 4 - 1
R. Sedlacek, W. Blum / Computational Materials Science 25 (2002) 200–206
creep (that is deformation at constant stress) behaviour of a material from its work-hardening (to be understood as deformation at constant strain rate) behaviour [1,2]. In practice, the one-parameter model is often unable to describe the plastic behaviour of materials with required accuracy and/or to account for experimentally observed features of deformation curves, especially those following changes in deformation conditions. This is because the dislocation microstructure of a plastically deformed material is in general heterogeneous––typically consisting of dislocation cells or subgrains––and its influence on the material behaviour cannot be adequately described by the average dislocation density alone. However, while modelling the macroscopic plastic response, one is usually not interested in resolving the spatial arrangement of the heterogeneous microstructure in detail (this would lead to description in terms of partial differential equations or even to a discrete dislocation simulation) and prefers to retain the description which utilizes mean quantities and leads to a set of ordinary differential equations. A variety of microstructure-based models have been developed that make use of additional parameters, e.g. mean cell/subgrain size, dislocation densities in cell/subgrain walls and interiors, misorientation between cells/subgrains, etc. (cf. Refs. [3–13]). In technical materials, even more parameters may be required, for example to account for particle strengthening. Models of this kind can either be applied to simulate macroscopic stress–strain curves of simple uniaxial tests where the specimen geometry does not play any substantial role, or they can be implemented as microstructure-based constitutive laws of plastic deformation in FEM codes to simulate the plastic response of components with complex geometry or under complex loading conditions as diverse as rolling of aluminium and creep in rock-salt formations. In general, uniaxial tests may be used to fit the model parameters for FEM applications. There is general agreement neither on the number of parameters to be used nor on the form of equations describing their evolution, nor even on the form of the kinetic equations. This is comprehensible, since the models aim at describing
201
different materials, deformation modes, temperature ranges, etc. and, moreover, the dislocation mechanisms themselves are not yet fully understood. However, a general framework is needed, which connects the heterogeneous microstructure to the macroscopic response. In most of the above mentioned approaches, the composite model [14] is implemented for this purpose. However this is usually done in a reduced or incomplete form. Sometimes, the full composite model is used, but in a way which allows to consider only one specific deformation mode––typically deformation either at constant strain rate or at constant stress. Thus, the above mentioned flexibility of the one-parameter model [1,2], Eqs. (1) and (2) is lost and the possibility to use these models under continuously varying deformation conditions in FEM applications is restricted. For the numerical integration of the differential equations resulting from the multi-parameter models, various ad hoc methods with unclear mathematical properties (convergence, accuracy, stability) are often used. This is because in these models, attention is usually paid to the details of the dislocation mechanisms and to the fitting to experimental curves, but not to the kinematic framework and mathematical structure. Typically, the simplest integration method, i.e. the ‘‘Euler’s method, which is, however, not recommended for any practical use’’ [15], is implemented. The stepsize control in these implementations is typically empirical, though physically motivated. For example, if the increase of plastic strain or change in dislocation density in one step is greater than a certain limit, the stepsize is reduced. The limit is determined so that the integration of a typical deformation curve succeeds, no attention being paid to the accuracy of the computation. The purpose of this paper is to present a general and flexible kinematic framework for multiparameter microstructure-based models of timedependent plastic deformation based on the full composite model [14]. It connects the heterogeneous microstructure with the macroscopic response independently of deformation mode and under arbitrarily varying deformation conditions. Various kinetic and structure evolution equations may be used and tested within it. It has a simple
202
R. Sedlacek, W. Blum / Computational Materials Science 25 (2002) 200–206
mathematical structure leading to a system of ordinary differential equations which can be solved by standard methods of numerical mathematics. In these methods, the stepsize is governed by the required accuracy of the solution. Also, special methods for stiff systems can be readily utilized if necessary [15].
s þ
rs ¼ e; E
ð4Þ
h þ
rh ¼ e: E
ð5Þ
The average stress equilibrium (Albenga’s law) takes the form of the rule of mixtures, r ¼ fs rs þ fh rh ;
2. Microstructure-based constitutive law The composite model [14] is reviewed in Section 2.1. It serves as kinematic framework for the proposed macroscopic constitutive law of plastic deformation. It is completed by microscopic constitutive relations for the rate of plastic deformation and for the evolution of microstructure parameters in Section 2.2. These relations are considered in a general form, thus allowing for later implementation of diverse specific laws into the proposed framework. In Section 2.3, a general macroscopic constitutive law results in the form of a set of ordinary differential equations. It can be solved for prescribed time evolutions of applied stress or strain. 2.1. Kinematic framework The original composite model [14] is timeindependent. It treats the dislocation cell structure as a composite consisting of plastically hard walls with high dislocation density and soft cell interiors with comparatively low density of dislocations. To arrive at a mathematically tractable formulation in terms of averaged quantities, the continuummechanics equations describing the non-homogeneous plastic deformation at the microscale are considered in a simplified form. Material strain e is additively decomposed into its plastic part and the elastic part, the latter being expressed in terms of Hooke’s law, r e¼þ ; ð3Þ E where r is the applied axial stress, E is Young’s modulus. Instead of strain compatibility, iso-strain at the microscale is considered, i.e. material strains in the soft (s) and hard (h) regions equal the overall material strain e,
ð6Þ
with the volume fractions of the hard and soft regions fulfilling the condition fs þ fh ¼ 1. The overall mean plastic strain then follows from Eqs. (3)–(6), ¼ fs s þ fh h :
ð7Þ
We note in passing that the above formulation leads in a natural way to the build-up of internal stresses (kinematic hardening) [14]. To arrive at a macroscopic time-dependent constitutive law, the kinematic framework will be completed by (microscopic) constitutive equations. 2.2. Constitutive equations At the microscale, kinetic and evolution equations will be considered, similar to Eqs. (1) and (2) of the one-parameter model. Kinetic equations can be set up for the local plastic strain rates in the soft and hard regions as functions of the local stresses, the current microstructure, and temperature T, d s ¼ _s ðt; rs ; X ; T Þ; dt
ð8Þ
d h ¼ _h ðt; rh ; X ; T Þ: dt
ð9Þ
Evolution equations must be defined for the set X of microstructural parameters. It is commonly assumed that the microstructure evolves with plastic strain, d X ¼ X ð; X ; T Þ: d
ð10Þ
The symbol ( ) denotes plastic-strain derivative. Although there may be also some explicitly timedependent evolution of the microstructure, especially in high-temperature deformation (e.g. static recovery of dislocation density or Ostwald ripening of particles), we retain Eq. (10) without loss of
R. Sedlacek, W. Blum / Computational Materials Science 25 (2002) 200–206
generality for the purpose of setting up the kinematic framework. Evolution of the microstructure with time, d X ¼ X _; dt
ð11Þ
is formulated using the plastic strain rate _ which results from the time derivative of Eq. (7) in the form _ ¼
fs _s þ fh _h
:
ð12Þ
1 fh ðh s Þ We note that the volume fractions generally are functions of the plastic-strain-dependent microstructure parameters, fh ¼ fh ðX ðÞÞ.
rh ¼ r Efs ðh s Þ:
203
ð17Þ
The system (13)–(15) is solved for the local plastic strains s ðtÞ, h ðtÞ and the set of microstructure parameters X ðtÞ. The plastic response ðtÞ is found with the help of Eq. (7). The material strain eðtÞ follows from Eq. (3). Strain control: In a strain controlled test, the material strain eðtÞ as a function of time is prescribed. The local stresses rs , rh entering Eqs. (8) and (9) follow from Eqs. (4) and (5) as functions of the local plastic strains s , h , rs ¼ Eðe s Þ;
ð18Þ
rh ¼ Eðe h Þ:
ð19Þ
2.3. Macroscopic constitutive law We aim at the macroscopic time-dependent relation between stress rðtÞ and plastic strain ðtÞ, which can be computed by solving the kinetic Eqs. (8) and (9) and the evolution equations (11). The kinematic framework, Eqs. (3)–(7) enables us to re-formulate these equations by specifying the dependence of rs , rh and on r, e, s , h and X so that a closed system of differential equations results in the form _s ¼ _s ðt; s ; h ; X ; T Þ;
ð13Þ
_h ¼ _h ðt; s ; h ; X ; T Þ;
ð14Þ
X_ ¼ X ðs ; h ; X ; T Þ_ðt; s ; h ; X ; T Þ;
ð15Þ
where _ is given by Eq. (12). The dependence of in Eq. (11) on s , h and X is given by Eq. (7). In specifying the dependence of rs , rh in Eqs. (8) and (9) on r, e, s , h and X, we have to distinguish between the following deformation modes. 2.3.1. Deformation modes Stress control: In a stress controlled test, the applied stress rðtÞ as a function of time is prescribed. The local stresses rs , rh entering Eqs. (8) and (9) are derived from Eqs. (3)–(6) as functions of the local plastic strains s , h , and the microstructure-dependent volume fractions fs , fh , rs ¼ r Efh ðs h Þ;
ð16Þ
The system (13)–(15) is solved for the local plastic strains s ðtÞ, h ðtÞ and the set of microstructure parameters X ðtÞ. The plastic strain ðtÞ follows from Eq. (7) and the applied stress rðtÞ from Eq. (3).
3. Application As an example, a simplified version of the model introduced in [7,8], which was developed to simulate deformation of aluminium at constant applied stress and elevated temperature (creep), has been implemented into the proposed framework. The model is briefly summarized in the Appendix A. The resulting system of ordinary differential equations turns out to be stiff while approaching steady state (saturation of the microstructure parameters). To avoid instabilities during computation, semiimplicit numerical methods [15] were chosen to integrate it in such a case. To demonstrate the flexibility of the proposed framework, tests at varying deformation conditions were simulated. Fig. 1 shows a creep test with a strain controlled loading part. The starting point is steady state deformation of aluminium at applied stress r ¼ 5 MPa and temperature T ¼ 573 K. In the strain controlled deformation mode, the strain rate is rapidly increased and then held constant. The stress controlled creep test branches off
204
R. Sedlacek, W. Blum / Computational Materials Science 25 (2002) 200–206
Fig. 1. Creep test with a strain controlled loading part. Applied stress r, plastic strain rate _, internal stresses rs , rh ; spacing of free dislocations sf and volume fraction of subgrain boundaries fh normalized with the corresponding initial values plotted as functions of plastic strain .
at the stress level r ¼ 10 MPa. Fig. 2 shows a stress relaxation test with a stress controlled loading part. Starting from the same point as above, the stress is slowly increased in the stress controlled deformation mode. The stress relaxation test at constant material strain e branches off at the point of maximum plastic strain rate. We note in passing that the advanced integration
method allows to take large time steps while keeping the desired accuracy. In both cases, we note the build-up of internal stresses during loading which is typical of the composite model [14]. Similar to the one-parameter model [1,2], the increase of dislocation density (i.e. the decrease of dislocation spacing sf ) contributes to the work-hardening, along with the
R. Sedlacek, W. Blum / Computational Materials Science 25 (2002) 200–206
205
Fig. 2. Stress relaxation test with a stress controlled loading part. Applied stress r, plastic strain rate _, internal stresses rs , rh ; spacing of free dislocations sf and volume fraction of subgrain boundaries fh normalized with the corresponding initial values plotted as functions of time tðsÞ.
decreasing subgrain size which is reflected in the variation of the volume fraction fh . Refer to [5–8] for the physical background of the model and for further details, as well as for discussion on relation between computed and experimentally measured curves.
4. Conclusions The proposed constitutive law of plastic deformation accounts for the heterogeneous microstructure of the deforming material. It consists of the kinematic framework (‘composite model’),
206
R. Sedlacek, W. Blum / Computational Materials Science 25 (2002) 200–206
kinetic equations, and structure evolution equations. The latter two could be specified independently of the general structure. The law takes the form of a set of ordinary differential equations and yields macroscopic time-dependent stress–strain relations. Standard methods of numerical mathematics can be used for its integration. It works independently of the deformation mode and under arbitrarily varying deformation conditions. Acknowledgements RS gratefully acknowledges helpful discussions with Dr. A. Hampel and Dr. F. Roters as well as financial support by the Deutsche Forschungsgemeinschaft. Appendix A. Specific constitutive equations The kinetic and evolution equations corresponding to a simplified version of the composite model introduced in [7,8] are briefly summarized and values of the parameters used are given in the following. Kinetic Eqs. (8) and (9) are considered in the form b Qi bDai ri _i ¼ v0i exp sinh Ms2f kT MkT i ¼ s; h
ðA:1Þ
which is based on the Orowan equation and assumes thermally activated glide of dislocations in subgrains (soft phase) and subgrain boundaries (hard phase). Here b ¼ 0:286 nm is the Burgers vector modulus, M ¼ 3 is the Taylor factor, k is the Boltzmann constant. Activation energies Qs ¼ 1:6 eV, Qh ¼ 1:5 eV and pre-factors v0s ¼ 107 m/s, v0h ¼ 105 m/s were obtained by fitting the computed results to experimentally measured creep curves [7,8]. Structure evolution Eq. (10) are considered in the form
Xj ¼ Xj lnð10Þ
logðXj Þ logðXj1 Þ kj
ðA:2Þ
which describes evolution of the parameters Xj towards experimentally determined stress-depen-
dent steady state values Xj1 . The four microstructure parameters used are: spacing of free dislocations sf , subgrain size w, width of the subgrain boundaries a, spacing of dislocations in subgrain boundaries sb . The corresponding evolution constants were used: kf ¼ 0:0075, kw ¼ 0:059, ka ¼ 0:00006, kb ¼ 0:0075. The steady state values are: sf1 ¼ 3:7bðG=rÞ0:82 , w1 ¼ 23bG=r, 0:25 sb ¼ 11bðG=rÞ ; G ¼ 25:4ð1 0:5ðT 300Þ=933Þ GPa is shear modulus, Young’s modulus is E ¼ 80:637ð1 0:5ðT 300Þ=933Þ GPa. As the volume fraction fh ¼ 2a=w, the assumed steady state fh1 ¼ 0:05 determines a1 . The free dislocation spacing enters Eq. (A.1), determines the activation area Das ¼ bsf and effective stress rs ¼ rs aGb=sf (a ¼ 0:19) in the soft phase. In the hard phase, the boundary dislocation spacing determines the activation area Dah ¼ bsh , the effective stress is rh ¼ rh .
References [1] H. Mecking, U.F. Kocks, Acta Metallurgica 29 (1981) 1865. [2] Y. Estrin, H. Mecking, Acta Metallurgica 32 (1984) 57. [3] F.B. Prinz, A.S. Argon, Acta Metallurgica 32 (1984) 1021. [4] W.D. Nix, J.C. Gibeling, D.A. Hughes, Metallurgical Transactions A 16A (1985) 2215. [5] W. Blum, A. Rosen, A. Cegielska, J.L. Martin, Acta Metallurgica 37 (1989) 2439. [6] W. Blum, S. Vogler, M. Biberger, A.K. Mukherjee, Materials Science and Engineering A 112 (1989) 93. [7] M. Meier, W. Blum, Materials Science and Engineering A 164 (1993) 290. [8] U. Hofmann, W. Blum, in: Y. Hosoi, H. Yoshinaga, H. Oikawa, K. Maruyama (Eds.), Proceedings of the 7th JIM International Symposium on Aspects of high temperature deformation and fracture in crystalline materials, The Japan Institute of Metals, Sendai, 1993, pp. 625–632. [9] M. Zehetbauer, Acta Metallurgica and Materialia 41 (1993) 589. [10] A.S. Argon, P. Haasen, Acta Metallurgica 41 (1993) 3289. [11] E. Nes, Progress in Materials Science 41 (1998) 129. [12] U. Hunsche, A. Hampel, Engineering geology 52 (1999) 271. [13] F. Roters, D. Raabe, G. Gottstein, Acta Materialia 48 (2000) 4181. [14] H. Mughrabi, Acta Metallurgica 31 (1983) 1367. [15] H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1989.
Microstructure-based constitutive law of plastic deformation R. Sedl acek *, W. Blum Institut f€ ur Werkstoffwissenschaften, Lehrstuhl I Universit€at Erlangen-N€urnberg, Martensstrasse 5, 91058 Erlangen, Germany
Abstract A macroscopic constitutive law of plastic deformation which accounts for the heterogeneous microstructure of the deforming material is proposed. It consists of a kinematic framework, kinetic equations, and structure evolution equations. The kinematic framework treats the material as a composite consisting of (plastically) soft cells or subgrains and hard cell walls or subgrain boundaries. The kinetic and structure evolution equations remain unspecified at first, the law being developed in a general way towards a system of ordinary differential equations. The proposed framework connects the heterogeneous microstructure with the macroscopic response independently of deformation mode and under arbitrarily varying deformation conditions. Two application examples are presented. Ó 2002 Elsevier Science B.V. All rights reserved. JEL classification: 61.72; 62.20 Keywords: Plastic deformation; Dislocation microstructure; Composite model
1. Introduction For engineering use, macroscopic work-hardening curves of technical materials are often fitted to empirical equations, the simplest example of them probably being the power law r ¼ km : Strictly speaking, an extrapolation of the computed flow stress r beyond the experimental range of strains , temperatures T, and/or deformation conditions used for fitting the constants k and m is not justified. This holds even for the much more
*
Corresponding author. Current address: Lehrstuhl f€ ur Werkstoffkunde und Werkstoffmechanik, TU-Mu¨nchen, Boltzmannstrasse 15, 85747 Garching, Germany. Tel.: +49-89-28915264; fax: +49-89-289-15248. E-mail address: [email protected] (R. Sedla´cˇek).
elaborate empirical laws used in practice. Therefore, materials science came up with phenomenological models based on the assumption that the flow stress r at strain rate _ and temperature T depends on a parameter which describes the current structure of the material [1,2]. Typically, the mean dislocation density q is used as the structure parameter, r ¼ rðq; _; T Þ:
ð1Þ
Then, the kinetic equation (1) is complemented with an evolution equation for the dislocation density, dq ¼ f ðq; _; T Þ: d
ð2Þ
This approach appears very useful in describing plastic flow in different modes of deformation. For example, the model is capable of predicting the
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 2 6 4 - 1
R. Sedlacek, W. Blum / Computational Materials Science 25 (2002) 200–206
creep (that is deformation at constant stress) behaviour of a material from its work-hardening (to be understood as deformation at constant strain rate) behaviour [1,2]. In practice, the one-parameter model is often unable to describe the plastic behaviour of materials with required accuracy and/or to account for experimentally observed features of deformation curves, especially those following changes in deformation conditions. This is because the dislocation microstructure of a plastically deformed material is in general heterogeneous––typically consisting of dislocation cells or subgrains––and its influence on the material behaviour cannot be adequately described by the average dislocation density alone. However, while modelling the macroscopic plastic response, one is usually not interested in resolving the spatial arrangement of the heterogeneous microstructure in detail (this would lead to description in terms of partial differential equations or even to a discrete dislocation simulation) and prefers to retain the description which utilizes mean quantities and leads to a set of ordinary differential equations. A variety of microstructure-based models have been developed that make use of additional parameters, e.g. mean cell/subgrain size, dislocation densities in cell/subgrain walls and interiors, misorientation between cells/subgrains, etc. (cf. Refs. [3–13]). In technical materials, even more parameters may be required, for example to account for particle strengthening. Models of this kind can either be applied to simulate macroscopic stress–strain curves of simple uniaxial tests where the specimen geometry does not play any substantial role, or they can be implemented as microstructure-based constitutive laws of plastic deformation in FEM codes to simulate the plastic response of components with complex geometry or under complex loading conditions as diverse as rolling of aluminium and creep in rock-salt formations. In general, uniaxial tests may be used to fit the model parameters for FEM applications. There is general agreement neither on the number of parameters to be used nor on the form of equations describing their evolution, nor even on the form of the kinetic equations. This is comprehensible, since the models aim at describing
201
different materials, deformation modes, temperature ranges, etc. and, moreover, the dislocation mechanisms themselves are not yet fully understood. However, a general framework is needed, which connects the heterogeneous microstructure to the macroscopic response. In most of the above mentioned approaches, the composite model [14] is implemented for this purpose. However this is usually done in a reduced or incomplete form. Sometimes, the full composite model is used, but in a way which allows to consider only one specific deformation mode––typically deformation either at constant strain rate or at constant stress. Thus, the above mentioned flexibility of the one-parameter model [1,2], Eqs. (1) and (2) is lost and the possibility to use these models under continuously varying deformation conditions in FEM applications is restricted. For the numerical integration of the differential equations resulting from the multi-parameter models, various ad hoc methods with unclear mathematical properties (convergence, accuracy, stability) are often used. This is because in these models, attention is usually paid to the details of the dislocation mechanisms and to the fitting to experimental curves, but not to the kinematic framework and mathematical structure. Typically, the simplest integration method, i.e. the ‘‘Euler’s method, which is, however, not recommended for any practical use’’ [15], is implemented. The stepsize control in these implementations is typically empirical, though physically motivated. For example, if the increase of plastic strain or change in dislocation density in one step is greater than a certain limit, the stepsize is reduced. The limit is determined so that the integration of a typical deformation curve succeeds, no attention being paid to the accuracy of the computation. The purpose of this paper is to present a general and flexible kinematic framework for multiparameter microstructure-based models of timedependent plastic deformation based on the full composite model [14]. It connects the heterogeneous microstructure with the macroscopic response independently of deformation mode and under arbitrarily varying deformation conditions. Various kinetic and structure evolution equations may be used and tested within it. It has a simple
202
R. Sedlacek, W. Blum / Computational Materials Science 25 (2002) 200–206
mathematical structure leading to a system of ordinary differential equations which can be solved by standard methods of numerical mathematics. In these methods, the stepsize is governed by the required accuracy of the solution. Also, special methods for stiff systems can be readily utilized if necessary [15].
s þ
rs ¼ e; E
ð4Þ
h þ
rh ¼ e: E
ð5Þ
The average stress equilibrium (Albenga’s law) takes the form of the rule of mixtures, r ¼ fs rs þ fh rh ;
2. Microstructure-based constitutive law The composite model [14] is reviewed in Section 2.1. It serves as kinematic framework for the proposed macroscopic constitutive law of plastic deformation. It is completed by microscopic constitutive relations for the rate of plastic deformation and for the evolution of microstructure parameters in Section 2.2. These relations are considered in a general form, thus allowing for later implementation of diverse specific laws into the proposed framework. In Section 2.3, a general macroscopic constitutive law results in the form of a set of ordinary differential equations. It can be solved for prescribed time evolutions of applied stress or strain. 2.1. Kinematic framework The original composite model [14] is timeindependent. It treats the dislocation cell structure as a composite consisting of plastically hard walls with high dislocation density and soft cell interiors with comparatively low density of dislocations. To arrive at a mathematically tractable formulation in terms of averaged quantities, the continuummechanics equations describing the non-homogeneous plastic deformation at the microscale are considered in a simplified form. Material strain e is additively decomposed into its plastic part and the elastic part, the latter being expressed in terms of Hooke’s law, r e¼þ ; ð3Þ E where r is the applied axial stress, E is Young’s modulus. Instead of strain compatibility, iso-strain at the microscale is considered, i.e. material strains in the soft (s) and hard (h) regions equal the overall material strain e,
ð6Þ
with the volume fractions of the hard and soft regions fulfilling the condition fs þ fh ¼ 1. The overall mean plastic strain then follows from Eqs. (3)–(6), ¼ fs s þ fh h :
ð7Þ
We note in passing that the above formulation leads in a natural way to the build-up of internal stresses (kinematic hardening) [14]. To arrive at a macroscopic time-dependent constitutive law, the kinematic framework will be completed by (microscopic) constitutive equations. 2.2. Constitutive equations At the microscale, kinetic and evolution equations will be considered, similar to Eqs. (1) and (2) of the one-parameter model. Kinetic equations can be set up for the local plastic strain rates in the soft and hard regions as functions of the local stresses, the current microstructure, and temperature T, d s ¼ _s ðt; rs ; X ; T Þ; dt
ð8Þ
d h ¼ _h ðt; rh ; X ; T Þ: dt
ð9Þ
Evolution equations must be defined for the set X of microstructural parameters. It is commonly assumed that the microstructure evolves with plastic strain, d X ¼ X ð; X ; T Þ: d
ð10Þ
The symbol ( ) denotes plastic-strain derivative. Although there may be also some explicitly timedependent evolution of the microstructure, especially in high-temperature deformation (e.g. static recovery of dislocation density or Ostwald ripening of particles), we retain Eq. (10) without loss of
R. Sedlacek, W. Blum / Computational Materials Science 25 (2002) 200–206
generality for the purpose of setting up the kinematic framework. Evolution of the microstructure with time, d X ¼ X _; dt
ð11Þ
is formulated using the plastic strain rate _ which results from the time derivative of Eq. (7) in the form _ ¼
fs _s þ fh _h
:
ð12Þ
1 fh ðh s Þ We note that the volume fractions generally are functions of the plastic-strain-dependent microstructure parameters, fh ¼ fh ðX ðÞÞ.
rh ¼ r Efs ðh s Þ:
203
ð17Þ
The system (13)–(15) is solved for the local plastic strains s ðtÞ, h ðtÞ and the set of microstructure parameters X ðtÞ. The plastic response ðtÞ is found with the help of Eq. (7). The material strain eðtÞ follows from Eq. (3). Strain control: In a strain controlled test, the material strain eðtÞ as a function of time is prescribed. The local stresses rs , rh entering Eqs. (8) and (9) follow from Eqs. (4) and (5) as functions of the local plastic strains s , h , rs ¼ Eðe s Þ;
ð18Þ
rh ¼ Eðe h Þ:
ð19Þ
2.3. Macroscopic constitutive law We aim at the macroscopic time-dependent relation between stress rðtÞ and plastic strain ðtÞ, which can be computed by solving the kinetic Eqs. (8) and (9) and the evolution equations (11). The kinematic framework, Eqs. (3)–(7) enables us to re-formulate these equations by specifying the dependence of rs , rh and on r, e, s , h and X so that a closed system of differential equations results in the form _s ¼ _s ðt; s ; h ; X ; T Þ;
ð13Þ
_h ¼ _h ðt; s ; h ; X ; T Þ;
ð14Þ
X_ ¼ X ðs ; h ; X ; T Þ_ðt; s ; h ; X ; T Þ;
ð15Þ
where _ is given by Eq. (12). The dependence of in Eq. (11) on s , h and X is given by Eq. (7). In specifying the dependence of rs , rh in Eqs. (8) and (9) on r, e, s , h and X, we have to distinguish between the following deformation modes. 2.3.1. Deformation modes Stress control: In a stress controlled test, the applied stress rðtÞ as a function of time is prescribed. The local stresses rs , rh entering Eqs. (8) and (9) are derived from Eqs. (3)–(6) as functions of the local plastic strains s , h , and the microstructure-dependent volume fractions fs , fh , rs ¼ r Efh ðs h Þ;
ð16Þ
The system (13)–(15) is solved for the local plastic strains s ðtÞ, h ðtÞ and the set of microstructure parameters X ðtÞ. The plastic strain ðtÞ follows from Eq. (7) and the applied stress rðtÞ from Eq. (3).
3. Application As an example, a simplified version of the model introduced in [7,8], which was developed to simulate deformation of aluminium at constant applied stress and elevated temperature (creep), has been implemented into the proposed framework. The model is briefly summarized in the Appendix A. The resulting system of ordinary differential equations turns out to be stiff while approaching steady state (saturation of the microstructure parameters). To avoid instabilities during computation, semiimplicit numerical methods [15] were chosen to integrate it in such a case. To demonstrate the flexibility of the proposed framework, tests at varying deformation conditions were simulated. Fig. 1 shows a creep test with a strain controlled loading part. The starting point is steady state deformation of aluminium at applied stress r ¼ 5 MPa and temperature T ¼ 573 K. In the strain controlled deformation mode, the strain rate is rapidly increased and then held constant. The stress controlled creep test branches off
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Fig. 1. Creep test with a strain controlled loading part. Applied stress r, plastic strain rate _, internal stresses rs , rh ; spacing of free dislocations sf and volume fraction of subgrain boundaries fh normalized with the corresponding initial values plotted as functions of plastic strain .
at the stress level r ¼ 10 MPa. Fig. 2 shows a stress relaxation test with a stress controlled loading part. Starting from the same point as above, the stress is slowly increased in the stress controlled deformation mode. The stress relaxation test at constant material strain e branches off at the point of maximum plastic strain rate. We note in passing that the advanced integration
method allows to take large time steps while keeping the desired accuracy. In both cases, we note the build-up of internal stresses during loading which is typical of the composite model [14]. Similar to the one-parameter model [1,2], the increase of dislocation density (i.e. the decrease of dislocation spacing sf ) contributes to the work-hardening, along with the
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205
Fig. 2. Stress relaxation test with a stress controlled loading part. Applied stress r, plastic strain rate _, internal stresses rs , rh ; spacing of free dislocations sf and volume fraction of subgrain boundaries fh normalized with the corresponding initial values plotted as functions of time tðsÞ.
decreasing subgrain size which is reflected in the variation of the volume fraction fh . Refer to [5–8] for the physical background of the model and for further details, as well as for discussion on relation between computed and experimentally measured curves.
4. Conclusions The proposed constitutive law of plastic deformation accounts for the heterogeneous microstructure of the deforming material. It consists of the kinematic framework (‘composite model’),
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kinetic equations, and structure evolution equations. The latter two could be specified independently of the general structure. The law takes the form of a set of ordinary differential equations and yields macroscopic time-dependent stress–strain relations. Standard methods of numerical mathematics can be used for its integration. It works independently of the deformation mode and under arbitrarily varying deformation conditions. Acknowledgements RS gratefully acknowledges helpful discussions with Dr. A. Hampel and Dr. F. Roters as well as financial support by the Deutsche Forschungsgemeinschaft. Appendix A. Specific constitutive equations The kinetic and evolution equations corresponding to a simplified version of the composite model introduced in [7,8] are briefly summarized and values of the parameters used are given in the following. Kinetic Eqs. (8) and (9) are considered in the form b Qi bDai ri _i ¼ v0i exp sinh Ms2f kT MkT i ¼ s; h
ðA:1Þ
which is based on the Orowan equation and assumes thermally activated glide of dislocations in subgrains (soft phase) and subgrain boundaries (hard phase). Here b ¼ 0:286 nm is the Burgers vector modulus, M ¼ 3 is the Taylor factor, k is the Boltzmann constant. Activation energies Qs ¼ 1:6 eV, Qh ¼ 1:5 eV and pre-factors v0s ¼ 107 m/s, v0h ¼ 105 m/s were obtained by fitting the computed results to experimentally measured creep curves [7,8]. Structure evolution Eq. (10) are considered in the form
Xj ¼ Xj lnð10Þ
logðXj Þ logðXj1 Þ kj
ðA:2Þ
which describes evolution of the parameters Xj towards experimentally determined stress-depen-
dent steady state values Xj1 . The four microstructure parameters used are: spacing of free dislocations sf , subgrain size w, width of the subgrain boundaries a, spacing of dislocations in subgrain boundaries sb . The corresponding evolution constants were used: kf ¼ 0:0075, kw ¼ 0:059, ka ¼ 0:00006, kb ¼ 0:0075. The steady state values are: sf1 ¼ 3:7bðG=rÞ0:82 , w1 ¼ 23bG=r, 0:25 sb ¼ 11bðG=rÞ ; G ¼ 25:4ð1 0:5ðT 300Þ=933Þ GPa is shear modulus, Young’s modulus is E ¼ 80:637ð1 0:5ðT 300Þ=933Þ GPa. As the volume fraction fh ¼ 2a=w, the assumed steady state fh1 ¼ 0:05 determines a1 . The free dislocation spacing enters Eq. (A.1), determines the activation area Das ¼ bsf and effective stress rs ¼ rs aGb=sf (a ¼ 0:19) in the soft phase. In the hard phase, the boundary dislocation spacing determines the activation area Dah ¼ bsh , the effective stress is rh ¼ rh .
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