- Email: [email protected]
Comments on anisotropic plastic flow and incompressibility
hf. I Engng Scr Vol. 21. No. 3. PP. ?ll-215. Prmted in Great Bntam
002&7225/83/03021145SO3.0010 Pergamon Press Ltd.
1983
COMMENTS ON ANISOTROPIC PLASTIC FLOW AND INCOMPRESSIBILITY? S. R. BODNER
Faculty of MechanicalEngineering,Technion-IsraelInstitute of Technology,Haifa, Israel
and D. C. STOUFFER
Departmentof AerospaceEngineeringand AppliedMechanics,Universityof Cincinnati,Cincinnati OH 45221,U.S.A. Abstract-The fully anisotropic plasticity theory of Stouffer and Bodner with stress history dependent internal state variablesis modifiedto enforce plastic incompressibilitywhich makesthat theory consistent with stabilityand thermodynamicprinciples.Pressuredependenceof plasticflowcouldbe includedthrough its influenceon the hardeningvariables.An intermediate anisotropic plasticity theory is described which is based on anisotropic work hardening and incremental isotropy of the flow law. A method is suggested for obtaining an effective scalar hardening parameter from the anisotropic components. The incremental isotropic formulation is simpler for numerical calculations and may be adequate for initially isotropic materials.
INTRODUCTION IN GENERALIZINGa
set of elastic-viscoplastic constitutive equations[l, 21 to include induced plastic anisotropy, Stouffer and Bodner [3] proposed an anisotropic hardening law to be used in conjunction with the equation for plastic deformation rate. The reference constitutive equations[l, 21 are characterized by the absence of an explicit yield criterion and unloading conditions. They include history dependent internal state variables to represent the work
hardened state. The form of the deformation rate equation used in [3] was an extension to the anisotropic case of the Prandtl-Reuss flow equation of classical plasticity theory. That is, the equation related the plastic deformation rate to the deviatoric stress by a linear transformation (a fourth order tensor) whose components are functions of stress and stress history. In the general anisotropic form, that equation does not automatically lead to plastic incompressibility, and it previously appeared difficult to impose it as an added condition. For this reason, and to explore the consequences, the formulation of [3] was developed permitting plastic volume changes. The procedure adopted in [3] was to transform the anisotropic flow equation into six-dimensional vector form with the matrix of coefficients becoming a 6 x 6 second order tensor in that space. Upon diagonalization of that tensor, the component equations become uncoupled which enables the coefficients to incorporate the hardening variables in a simple manner. Those variables, in turn, are obtained from a proposed anisotropic work hardening law. The resulting anisotropic plastic flow theory [3] has been used as the basis for the solution of a number of dynamic penetration and impact problems [4]. Although the results obtained for the stress and strain fields using the theory appear to be realistic, the calculated volume changes do not seem to be physically reasonable. In addition, the anisotropic formulation with plastic compressibility can be shown to violate Drucker’s stability postulate as extended by Ponter[S] to include internal state variables. In the present paper, the anisotropic plasticity theory of [3] is revised to satisfy plastic incompressibility by enforcing the condition initially stated by Olszak and Urbanowski[6] and more recently by Hi11[71.This modification is a relatively simple addition to the basic procedure and hardening law given in [3]. In addition, pressurk dependence of plastic flow could be tThe research reported in this paper has been sponsored by the Air Force Wright Aeronautical Laboratory, Wright-Patterson AFB, under contract F33615-78-5199,and by the Air Force Office of Scientific Research through the EOARD under Grant No. AFOSR-8C-0214. 211
212
S. R. BODNER and D. C. STOUFFER
incorporated through the evolution equations for the hardening variables leading to a rate dependent, incompressible plasticity theory with pressure dependent, anisotropic hardening. The resulting theory does satisfy the extended Drucker’s postulate. Some numerical exercises were performed using the revised incompressible anisotropic theory in the examples of [4]. These showed that the stress and strain fields calculated by the theory of [3] were essentially unaffected by the use of the modified theory. Plastic volume changes became zero, of course, in the new calculations. In some particular examples, e.g. uniaxial stress, the incompressible anisotropic theory leads to incremental isotropy of the flow law, i.e. it reduces to the standard Prandtl-Reuss form with a scalar coefficient. This coefficient is a function of the load history dependent anisotropic hardening variables. For general loading states, it is possible to formulate an intermediate anisotropic plasticity theory in which the hardening variables are obtained from the anisotropic hardening law, and a scalar function of those variables is used in the isotropic flow equation. A method for obtaining an effective scalar hardening parameter is given in this paper. The resulting equations are less general than the fully anisotropic theory, but are simpler to use in numerical calculations and may be adequate for initially isotropic materials that undergo anisotropic plastic deformation. ANALYSIS
Consider the anisotropic generalization of the Prandtl-Reuss flow law
where d{ is the plastic deformation rate, ok1is the direct Cauchy stress, and A# is a fourth order linear transformation that is positive semi-definite (at least) and satisfies the following symmetry and reciprocity conditions hijk,
=
hjikl
=
hij[k = hklij.
(2)
For general constitutive equations that do not require a yield criterion, the components Aiikl would be functions of stress, temperature, and current values of stress history dependent internal state variables, e.g. [3]. Plastic incompressibility therefore requires that Aiikr= 0 for arbitrary values of ok{ and general loading paths. It then follows from the symmetry conditions that the requirement of plastic incompressibility for the flow law (1) is Aiikl= Aijkk = 0.
(3)
Equation (3) was derived initially by Olszak and Urbanowski[6] and more recently by Hill[7] for the case of a flow law associated with an anisotropic yield criterion (plastic potential). In that case the Aijklare constants, but, as indicated above, stress history dependent coefficients also require eqn (3) to be satisfied for incompressibility. As shown by Hill[7], eqn (3) leads to consistency with Drucker’s Postulate for the anisotropic yield surface theory. It can also be demonstrated that the incompressible version of the anisotropic, state variable plasticity theory will also satisfy the extended Drucker’s Postulate of Ponter[5]. In that case, the inequality condition given in [5] for stability will be satisfied since no work will be done by the hydrostatic component of the stress. Rewriting eqn (1) in terms of the deviatoric stress sij = oij - (o,J3)Sij, gives (4) where the second term becomes zero on the basis of (3). Plastic incompressibility therefore infers that the flow law (1) should contain only deviatoric stresses and be independent of a separate pressure term. However, pressure dependence of plastic flow could be represented in a state variable, non-yield-surface theory by its effect on the hardening variables which are included in the A,,, components. For example, the hardening variable Z introduced in [l] would then be a function of the stress invariant I, as well as plastic work W, and would have to
Comments on anisotropic plastic flow and incompressibility
213
satisfy the stability condition given in [5], namely dZ dZ 10 for constant stress states. Inclusion of pressure dependence in this manner will lead to a theory in general conformity with recent experimental results by Richmond and his colleagues[S, 91 that plastic flow of strong metals exhibits definite pressure dependence but relatively small plastic volume changes. As stated previously, the incompressibility condition (3) was not enforced in the anisotropic plastic hardening theory of [3]. Developing a hardening law for a state variable theory that is both physically realistic and consistent with (3) appears to be a difficult procedure. An alternative method would be to initially formulate the hardening law, as in [3], and then enforce incompressibility. The flow law will initially contain coefficients Xirkrobtained from the proposed anisotropic hardening law which may not satisfy eqn (3) as in [3]. A term pij could then be added to the flow law where the pij are determined as functions of A;,kland ski on the basis of setting dz = 0. These relations will not be unique and a possible procedure would be to set pij = PSij SO that the flow law becomes
Setting ds = 0 and solving for p leads to
which obviously satisfies (3). Equation (6) is equivalent to simply calculating the volume change due to the coefficients and then subtracting one third the value from each direct plastic deformation rate. In certain special cases, the enforcement of incompressibility leads to incremental isotropy of the flow law, i.e. the matrix reduces to a scalar function of the components. One such case is uniaxial longitudinal stress u in a transversely isotropic rod with coefficients A, in the longitudinal direction and A2= A3in the transverse directions. The anisotropic flow law would then be hijkl
Aijkf
dP, = (2/3)A,a,
d!, = d:, = - (1/3)Aza
(7)
so that de = (2/3)~r(A,- AJ. An anisotropic hardening law would usually require that hardening be strongest in the stress direction for this case, which would give A,> A, and dE $0. The procedure described above can be used to enforce incompressibility. Writing eqn (5) for uniaxial loading of a transversely isotropic rod similar to eqn (7) and imposing incompressibility gives p = (- 2/9)(h, - A+. Then it is easy to show that the flow law, eqn (5), becomes isotropic with a single scalar coefficient defined by A = (1/3)(2A,+ A?). Motivated by results for particular cases of loading and geometry, an “intermediate” anisotropic plasticity theory could be formulated based on anisotropic hardening according to 131and the use of a scalar function of the anisotropic hardening coefficients in the isotropic (incompressible) flow law. Such an approximation for general loading conditions appears to be a reasonable one for initially isotropic materials where ankotropy is induced by plastic deformation. DERIVATION
OF AN ALTERNATIVE
FLOW LAW
In order to calculate an effective scalar coefficient A, it is useful to review the formulation of [3]. The deviatoric stress and deformation rate components are first expressed in vector form in a 6-D space, TA and DL, which are related to the standard tensor components by a simple transformation, e.g. (3.5) of [31. The anisotropic flow law (1) is then 0: = f: A&T;, !3=I
(8)
where A&, is a 6 x 6 transformation matrix (a second order tensor in the 6-D space whose components are real and symmetric). The tensor is then diagonalized so that the transformed deformation rate D, and stress deviator 7’, form vectors in the 6-D space whose bases & are
214
5, R. B~~~~~
and D, c, ~~~~~~~~
the eigenvectors, and the efements of the new mrttrixWG:the eigeavalues A,, (cy=I1, I . qt 6; no sum on a). The base vectors $ depend, in general, on the anisotropic state of the material which is a consequence of the initial state and the loading Iristary. For initially isotropic incompressible materials, and also for incompressible transversely isotropic materials, the base vectors are constant for arbitrary loading paths, In these two cases the A.&tensor maintains its diagonal form during loading, so that A& -+AcIL1 and the base vectors can be represented by the unity matrix, In diagonal form, the component flow equations are completely uncaupled which consider+ ably simplifies the computation of the six coefficients A,,. Those are taken to have the same functional dependence as in the isotropic case, ~~rn~~y
where J2 is the second invariant of the stress deviators, Z,, is the associated h~~~~i~g variable, and & and n are constants, The terms ZWaare the di~~~~~Ie~erne~tsof a second order tensor in the eigen~basis& SOthat Z,, are Jso the e.i~~~~~~~~ of Z& in the basis &. It is noted that the complete b~de~i~g variabIe~~~~res~~t~~by Z& has twelve terms corresponding to both positive and ~e~a~ve values of each direct stress ~orn~o~e~t.Direct stress is taken to be the reference for anisotropi~h~~~e~i~gia this dispassion. Those hardening terms co~res~o~d~ngto the signs of the foment stress ~orn~o~entsare used in (9) to evaluate A,, at each stage of ioadi~g.The evol~~~o~~q~at~~usfor Z&, as described in 131and f47,are
be ~x~~~~s~~ irktern% of This has the property that X uf = Xand that the quantity X2$ COIL stress in~ariants fl and &,
Comments on anisotropic plastic flow and incompressibility
215
The hardening components Z:a can be obtained by integrating (10) over the loading path, and it is noted that &, is generally dependent on the initial state and load history. At any stage, an effective scalar hardening term for initially isotropic incompressible materials can be obtained from the expression
zeff(t)= zo+ 9
I’ i(T) dr + (1 - q) i 0
Cl=1
u,(i) [ i(T)u,(T) d7
(15)
where z,,is the initial hardness value. The effective hardening parameter zeffis then used for z in (11) and also to obtain the scalar coefficient A in the isotropic flow law, namely
(16) The calculation A = h(z,e) from (16) is generally not identical to that obtained from eqn (6) for those cases in which (6) does reduce to the incremental isotropic form. For the example of uniaxial stress in a transversely isotropic rod, (6) would give A = (1/3)(2A,+ AJ, so that A depends on both the longitudinal and transverse hardening terms (corresponding to the current sign of the stress). For this example, z,,, eqn (16) reduces simply to Z,, (with the appropriate value for either tension or compression), and the transverse hardening variable is not a factor. Either formulation, eqn (6) or eqn (16) is applicable in such cases with some of the material constants having different values for each procedure. In the preceding discussion, as in [3], both isotropic and anisotropic hardening are based on the same scalar function of plastic work and are therefore coupled with a single constant q dividing the relative contributions. This assumption may be over simplistic from the physical viewpoint in some applications, and it may be more realistic to consider isotropic and anisotropic hardening as uncoupled with different reference functions governing their respective evolution laws. Nevertheless, the basic formulation developed in [3] and modified in this paper should be applicable for the treatment of anisotropic hardening. CONCLUSIONS
A simple modification of the rate dependent anisotropic hardening theory of 131enforces plastic incompressibility which makes that theory consistent with stability and thermodynamic principles. Stress and strain fields calculated by the revised theory have been found to be essentially the same as those obtained from the original theory141 with the exception that the plastic volume changes become zero. The resulting state variable, incompressible theory could be further developed to include pressure dependence of plastic flow through the influence of pressure on the hardening variables. An “intermediate” theory of the same general class can be formulated which considers anisotropic work hardening combined with the isotropic (incompressible) form of the flow law. A method is suggested for obtaining an effective scalar hardening parameter to be used in the isotropic flow law on an incremental basis. The latter theory would be simpler to use in finite element programs and may be sufficiently accurate for initially isotropic materials that experience anisotropic work hardening. Acknowledgements-The authors are grateful to Prof. Jacob Aboudi of Tel Aviv University and to Dr. Jose J. A. Rodal of Kaman AviDyne, Burlington, Mass. for their helpful discussions during the development of this paper. REFERENCES [ll S. R. B,ODNERand Y. PARTOM, ASME J. Appl. Mech. 42, 385 (1975). [21 S. R. BODNER, I. PARTOM and Y. PARTOM, ASME J. Appl. Mech. 46, 805 (1979). [31 D. C. STOUFFER and S. R. BODNER, Int. J. Engng Sci. 17,737 (1979). [41J. ABOUDI and S. R. BODNER, Inr. J. Engng Sci. 18, 801 (1980). [51 A. R. S. PONTER, Int. J. Solids Structures 16, 793 (1980). 161W. OLSZAK and W. URBANOWSKI, Arch. Mech. Stos. 8, 671 (1956). 171R. HILL, Math. Pm. Camb. Phil. Sot. 85, 179(1979). 181W. A. SPITZIG, R. J. SOBER and 0. RICHMOND, Met. Trans. A. 7A, 1703(1976). [91 0. RICHMOND and W. A. SPITZIG, Pm. Int. Conj. Theor. Appl. Mech. (IUTAM), Toronto, Canada (1980). (Received 24 February 1982)
002&7225/83/03021145SO3.0010 Pergamon Press Ltd.
1983
COMMENTS ON ANISOTROPIC PLASTIC FLOW AND INCOMPRESSIBILITY? S. R. BODNER
Faculty of MechanicalEngineering,Technion-IsraelInstitute of Technology,Haifa, Israel
and D. C. STOUFFER
Departmentof AerospaceEngineeringand AppliedMechanics,Universityof Cincinnati,Cincinnati OH 45221,U.S.A. Abstract-The fully anisotropic plasticity theory of Stouffer and Bodner with stress history dependent internal state variablesis modifiedto enforce plastic incompressibilitywhich makesthat theory consistent with stabilityand thermodynamicprinciples.Pressuredependenceof plasticflowcouldbe includedthrough its influenceon the hardeningvariables.An intermediate anisotropic plasticity theory is described which is based on anisotropic work hardening and incremental isotropy of the flow law. A method is suggested for obtaining an effective scalar hardening parameter from the anisotropic components. The incremental isotropic formulation is simpler for numerical calculations and may be adequate for initially isotropic materials.
INTRODUCTION IN GENERALIZINGa
set of elastic-viscoplastic constitutive equations[l, 21 to include induced plastic anisotropy, Stouffer and Bodner [3] proposed an anisotropic hardening law to be used in conjunction with the equation for plastic deformation rate. The reference constitutive equations[l, 21 are characterized by the absence of an explicit yield criterion and unloading conditions. They include history dependent internal state variables to represent the work
hardened state. The form of the deformation rate equation used in [3] was an extension to the anisotropic case of the Prandtl-Reuss flow equation of classical plasticity theory. That is, the equation related the plastic deformation rate to the deviatoric stress by a linear transformation (a fourth order tensor) whose components are functions of stress and stress history. In the general anisotropic form, that equation does not automatically lead to plastic incompressibility, and it previously appeared difficult to impose it as an added condition. For this reason, and to explore the consequences, the formulation of [3] was developed permitting plastic volume changes. The procedure adopted in [3] was to transform the anisotropic flow equation into six-dimensional vector form with the matrix of coefficients becoming a 6 x 6 second order tensor in that space. Upon diagonalization of that tensor, the component equations become uncoupled which enables the coefficients to incorporate the hardening variables in a simple manner. Those variables, in turn, are obtained from a proposed anisotropic work hardening law. The resulting anisotropic plastic flow theory [3] has been used as the basis for the solution of a number of dynamic penetration and impact problems [4]. Although the results obtained for the stress and strain fields using the theory appear to be realistic, the calculated volume changes do not seem to be physically reasonable. In addition, the anisotropic formulation with plastic compressibility can be shown to violate Drucker’s stability postulate as extended by Ponter[S] to include internal state variables. In the present paper, the anisotropic plasticity theory of [3] is revised to satisfy plastic incompressibility by enforcing the condition initially stated by Olszak and Urbanowski[6] and more recently by Hi11[71.This modification is a relatively simple addition to the basic procedure and hardening law given in [3]. In addition, pressurk dependence of plastic flow could be tThe research reported in this paper has been sponsored by the Air Force Wright Aeronautical Laboratory, Wright-Patterson AFB, under contract F33615-78-5199,and by the Air Force Office of Scientific Research through the EOARD under Grant No. AFOSR-8C-0214. 211
212
S. R. BODNER and D. C. STOUFFER
incorporated through the evolution equations for the hardening variables leading to a rate dependent, incompressible plasticity theory with pressure dependent, anisotropic hardening. The resulting theory does satisfy the extended Drucker’s postulate. Some numerical exercises were performed using the revised incompressible anisotropic theory in the examples of [4]. These showed that the stress and strain fields calculated by the theory of [3] were essentially unaffected by the use of the modified theory. Plastic volume changes became zero, of course, in the new calculations. In some particular examples, e.g. uniaxial stress, the incompressible anisotropic theory leads to incremental isotropy of the flow law, i.e. it reduces to the standard Prandtl-Reuss form with a scalar coefficient. This coefficient is a function of the load history dependent anisotropic hardening variables. For general loading states, it is possible to formulate an intermediate anisotropic plasticity theory in which the hardening variables are obtained from the anisotropic hardening law, and a scalar function of those variables is used in the isotropic flow equation. A method for obtaining an effective scalar hardening parameter is given in this paper. The resulting equations are less general than the fully anisotropic theory, but are simpler to use in numerical calculations and may be adequate for initially isotropic materials that undergo anisotropic plastic deformation. ANALYSIS
Consider the anisotropic generalization of the Prandtl-Reuss flow law
where d{ is the plastic deformation rate, ok1is the direct Cauchy stress, and A# is a fourth order linear transformation that is positive semi-definite (at least) and satisfies the following symmetry and reciprocity conditions hijk,
=
hjikl
=
hij[k = hklij.
(2)
For general constitutive equations that do not require a yield criterion, the components Aiikl would be functions of stress, temperature, and current values of stress history dependent internal state variables, e.g. [3]. Plastic incompressibility therefore requires that Aiikr= 0 for arbitrary values of ok{ and general loading paths. It then follows from the symmetry conditions that the requirement of plastic incompressibility for the flow law (1) is Aiikl= Aijkk = 0.
(3)
Equation (3) was derived initially by Olszak and Urbanowski[6] and more recently by Hill[7] for the case of a flow law associated with an anisotropic yield criterion (plastic potential). In that case the Aijklare constants, but, as indicated above, stress history dependent coefficients also require eqn (3) to be satisfied for incompressibility. As shown by Hill[7], eqn (3) leads to consistency with Drucker’s Postulate for the anisotropic yield surface theory. It can also be demonstrated that the incompressible version of the anisotropic, state variable plasticity theory will also satisfy the extended Drucker’s Postulate of Ponter[5]. In that case, the inequality condition given in [5] for stability will be satisfied since no work will be done by the hydrostatic component of the stress. Rewriting eqn (1) in terms of the deviatoric stress sij = oij - (o,J3)Sij, gives (4) where the second term becomes zero on the basis of (3). Plastic incompressibility therefore infers that the flow law (1) should contain only deviatoric stresses and be independent of a separate pressure term. However, pressure dependence of plastic flow could be represented in a state variable, non-yield-surface theory by its effect on the hardening variables which are included in the A,,, components. For example, the hardening variable Z introduced in [l] would then be a function of the stress invariant I, as well as plastic work W, and would have to
Comments on anisotropic plastic flow and incompressibility
213
satisfy the stability condition given in [5], namely dZ dZ 10 for constant stress states. Inclusion of pressure dependence in this manner will lead to a theory in general conformity with recent experimental results by Richmond and his colleagues[S, 91 that plastic flow of strong metals exhibits definite pressure dependence but relatively small plastic volume changes. As stated previously, the incompressibility condition (3) was not enforced in the anisotropic plastic hardening theory of [3]. Developing a hardening law for a state variable theory that is both physically realistic and consistent with (3) appears to be a difficult procedure. An alternative method would be to initially formulate the hardening law, as in [3], and then enforce incompressibility. The flow law will initially contain coefficients Xirkrobtained from the proposed anisotropic hardening law which may not satisfy eqn (3) as in [3]. A term pij could then be added to the flow law where the pij are determined as functions of A;,kland ski on the basis of setting dz = 0. These relations will not be unique and a possible procedure would be to set pij = PSij SO that the flow law becomes
Setting ds = 0 and solving for p leads to
which obviously satisfies (3). Equation (6) is equivalent to simply calculating the volume change due to the coefficients and then subtracting one third the value from each direct plastic deformation rate. In certain special cases, the enforcement of incompressibility leads to incremental isotropy of the flow law, i.e. the matrix reduces to a scalar function of the components. One such case is uniaxial longitudinal stress u in a transversely isotropic rod with coefficients A, in the longitudinal direction and A2= A3in the transverse directions. The anisotropic flow law would then be hijkl
Aijkf
dP, = (2/3)A,a,
d!, = d:, = - (1/3)Aza
(7)
so that de = (2/3)~r(A,- AJ. An anisotropic hardening law would usually require that hardening be strongest in the stress direction for this case, which would give A,> A, and dE $0. The procedure described above can be used to enforce incompressibility. Writing eqn (5) for uniaxial loading of a transversely isotropic rod similar to eqn (7) and imposing incompressibility gives p = (- 2/9)(h, - A+. Then it is easy to show that the flow law, eqn (5), becomes isotropic with a single scalar coefficient defined by A = (1/3)(2A,+ A?). Motivated by results for particular cases of loading and geometry, an “intermediate” anisotropic plasticity theory could be formulated based on anisotropic hardening according to 131and the use of a scalar function of the anisotropic hardening coefficients in the isotropic (incompressible) flow law. Such an approximation for general loading conditions appears to be a reasonable one for initially isotropic materials where ankotropy is induced by plastic deformation. DERIVATION
OF AN ALTERNATIVE
FLOW LAW
In order to calculate an effective scalar coefficient A, it is useful to review the formulation of [3]. The deviatoric stress and deformation rate components are first expressed in vector form in a 6-D space, TA and DL, which are related to the standard tensor components by a simple transformation, e.g. (3.5) of [31. The anisotropic flow law (1) is then 0: = f: A&T;, !3=I
(8)
where A&, is a 6 x 6 transformation matrix (a second order tensor in the 6-D space whose components are real and symmetric). The tensor is then diagonalized so that the transformed deformation rate D, and stress deviator 7’, form vectors in the 6-D space whose bases & are
214
5, R. B~~~~~
and D, c, ~~~~~~~~
the eigenvectors, and the efements of the new mrttrixWG:the eigeavalues A,, (cy=I1, I . qt 6; no sum on a). The base vectors $ depend, in general, on the anisotropic state of the material which is a consequence of the initial state and the loading Iristary. For initially isotropic incompressible materials, and also for incompressible transversely isotropic materials, the base vectors are constant for arbitrary loading paths, In these two cases the A.&tensor maintains its diagonal form during loading, so that A& -+AcIL1 and the base vectors can be represented by the unity matrix, In diagonal form, the component flow equations are completely uncaupled which consider+ ably simplifies the computation of the six coefficients A,,. Those are taken to have the same functional dependence as in the isotropic case, ~~rn~~y
where J2 is the second invariant of the stress deviators, Z,, is the associated h~~~~i~g variable, and & and n are constants, The terms ZWaare the di~~~~~Ie~erne~tsof a second order tensor in the eigen~basis& SOthat Z,, are Jso the e.i~~~~~~~~ of Z& in the basis &. It is noted that the complete b~de~i~g variabIe~~~~res~~t~~by Z& has twelve terms corresponding to both positive and ~e~a~ve values of each direct stress ~orn~o~e~t.Direct stress is taken to be the reference for anisotropi~h~~~e~i~gia this dispassion. Those hardening terms co~res~o~d~ngto the signs of the foment stress ~orn~o~entsare used in (9) to evaluate A,, at each stage of ioadi~g.The evol~~~o~~q~at~~usfor Z&, as described in 131and f47,are
be ~x~~~~s~~ irktern% of This has the property that X uf = Xand that the quantity X2$ COIL stress in~ariants fl and &,
Comments on anisotropic plastic flow and incompressibility
215
The hardening components Z:a can be obtained by integrating (10) over the loading path, and it is noted that &, is generally dependent on the initial state and load history. At any stage, an effective scalar hardening term for initially isotropic incompressible materials can be obtained from the expression
zeff(t)= zo+ 9
I’ i(T) dr + (1 - q) i 0
Cl=1
u,(i) [ i(T)u,(T) d7
(15)
where z,,is the initial hardness value. The effective hardening parameter zeffis then used for z in (11) and also to obtain the scalar coefficient A in the isotropic flow law, namely
(16) The calculation A = h(z,e) from (16) is generally not identical to that obtained from eqn (6) for those cases in which (6) does reduce to the incremental isotropic form. For the example of uniaxial stress in a transversely isotropic rod, (6) would give A = (1/3)(2A,+ AJ, so that A depends on both the longitudinal and transverse hardening terms (corresponding to the current sign of the stress). For this example, z,,, eqn (16) reduces simply to Z,, (with the appropriate value for either tension or compression), and the transverse hardening variable is not a factor. Either formulation, eqn (6) or eqn (16) is applicable in such cases with some of the material constants having different values for each procedure. In the preceding discussion, as in [3], both isotropic and anisotropic hardening are based on the same scalar function of plastic work and are therefore coupled with a single constant q dividing the relative contributions. This assumption may be over simplistic from the physical viewpoint in some applications, and it may be more realistic to consider isotropic and anisotropic hardening as uncoupled with different reference functions governing their respective evolution laws. Nevertheless, the basic formulation developed in [3] and modified in this paper should be applicable for the treatment of anisotropic hardening. CONCLUSIONS
A simple modification of the rate dependent anisotropic hardening theory of 131enforces plastic incompressibility which makes that theory consistent with stability and thermodynamic principles. Stress and strain fields calculated by the revised theory have been found to be essentially the same as those obtained from the original theory141 with the exception that the plastic volume changes become zero. The resulting state variable, incompressible theory could be further developed to include pressure dependence of plastic flow through the influence of pressure on the hardening variables. An “intermediate” theory of the same general class can be formulated which considers anisotropic work hardening combined with the isotropic (incompressible) form of the flow law. A method is suggested for obtaining an effective scalar hardening parameter to be used in the isotropic flow law on an incremental basis. The latter theory would be simpler to use in finite element programs and may be sufficiently accurate for initially isotropic materials that experience anisotropic work hardening. Acknowledgements-The authors are grateful to Prof. Jacob Aboudi of Tel Aviv University and to Dr. Jose J. A. Rodal of Kaman AviDyne, Burlington, Mass. for their helpful discussions during the development of this paper. REFERENCES [ll S. R. B,ODNERand Y. PARTOM, ASME J. Appl. Mech. 42, 385 (1975). [21 S. R. BODNER, I. PARTOM and Y. PARTOM, ASME J. Appl. Mech. 46, 805 (1979). [31 D. C. STOUFFER and S. R. BODNER, Int. J. Engng Sci. 17,737 (1979). [41J. ABOUDI and S. R. BODNER, Inr. J. Engng Sci. 18, 801 (1980). [51 A. R. S. PONTER, Int. J. Solids Structures 16, 793 (1980). 161W. OLSZAK and W. URBANOWSKI, Arch. Mech. Stos. 8, 671 (1956). 171R. HILL, Math. Pm. Camb. Phil. Sot. 85, 179(1979). 181W. A. SPITZIG, R. J. SOBER and 0. RICHMOND, Met. Trans. A. 7A, 1703(1976). [91 0. RICHMOND and W. A. SPITZIG, Pm. Int. Conj. Theor. Appl. Mech. (IUTAM), Toronto, Canada (1980). (Received 24 February 1982)