$$ \pmb {\sigma} = \frac {\partial U}{\partial \pmb {\varepsilon}}, $$ where ¾ and " are any work conjugate stress and strain measures. This form of elasticity model is generally used to model elastomers: materials whose long-term response to large deformations is fully recoverable (elastic). The hyperelasticity modeling provided in ABAQUS is described in \`\`Large-strain elasticity,'' Section 4.6. The hyperelasticity models cannot be used with the plastic deformation models in the program, but can be combined with viscoelastic behavior, as described in \`\`Finite-strain viscoelasticity,'' Section 4.7.2. The plasticity models offered in ABAQUS are discussed in general terms in \`\`Plasticity overview,'' Section 4.2. Both rate-independent and rate-dependent models, with and without yield surfaces, are offered. Models are included in the program that are intended for applications to metals ( \`\`Metal plasticity,'' Section 4.3) as well as some nonmetallic materials such as soils, polymers, and crushable foams (\`\`Plasticity for non-metals,'' Section 4.4). The jointed material model (\`\`Constitutive model for jointed materials,'' Section 4.5.3) and the concrete model (\`\`An inelastic constitutive model for concrete,'' Section 4.5.1) also include plasticity modeling. The constitutive routines in ABAQUS exist in a library that can be accessed by any of the solid or structural elements. This access is made independently at each "constitutive calculation point." These points are the numerical integration points in the elements. Thus, the constitutive routines are concerned only with a single calculation point. The element provides an estimate of the kinematic solution to the problem at the point under consideration. These kinematic data are passed to the constitutive routines as the deformation gradient-- F--or, more typically, as the strain and rotation increments--¢" and ¢R. The constitutive routines obtain the state at the point under consideration at the start of the increment from the "material point data base." The state variables include the stress and any state variables used in the constitutive models--plastic strains, for example. The constitutive routines also look up the constitutive definition. Their function then is to update the state to the end of the increment and, if the procedure uses implicit time integration and if Newton's method is being used to solve the equations, to define the material contribution to the Jacobian matrix, @¾=@". For material models that are defined in rate form and, therefore, require integration (such as incremental plasticity models), this Jacobian contribution depends on the model and also on the integration method used for the model. Its derivation is, therefore, discussed in some detail in the sections that define such models. # 4.2 Plasticity overview # 4.2.1 Plasticity models: general discussion Incremental plasticity theory is based on a few fundamental postulates, which means that all of the elastic-plastic response models provided in ABAQUS (except the deformation theory model in ABAQUS/Standard, which is primarily provided for fracture mechanics applications) have the same general form. The basic equations of the models are defined in their general form in this section. Plasticity models are written as rate-independent models or as rate-dependent models. A rate-independent model is one in which the constitutive response does not depend on the rate of deformation--the response of many metals at low temperatures relative to their melting temperature and at low strain rates is effectively rate independent. In a rate-dependent model the response does depend on the rate at which the material is strained. Examples of such models are the simple metal "creep" models provided in ABAQUS/Standard with the \*CREEP option and the \*RATE DEPENDENT option that is used to describe the behavior of metals at higher strain rates. Because these models have similar forms, their numerical treatment is based on the same technique. A basic assumption of elastic-plastic models is that the deformation can be divided into an elastic part and an inelastic (plastic) part. In its most general form this statement is written as Equation 4.2.1-1 $$ \mathbf {F} = \mathbf {F} ^ {e l} \cdot \mathbf {F} ^ {p l}, $$ where F is the total deformation gradient, ${ \bf F } ^ { e l }$ is the fully recoverable part of the deformation at the point under consideration $( [ \mathbf { F } ^ { e l } ] ^ { - 1 }$ is the deformation that would occur if, after the deformation F, inelastic response were somehow prevented but at the same time the stress at the point reduced to zero), and $\mathbf { F } ^ { p l }$ is defined by ${ \bf F } ^ { p l } = [ { \bf F } ^ { e l } ] ^ { - 1 } \cdot { \bf F }$ . The rigid body rotation at the point can be included in the definition of either ${ \bf F } ^ { e l }$ or $\mathbf { F } ^ { p l }$ or can be considered separately before or after either part of the decomposition: this makes no difference except in the convenience of the basis for writing the parts of the deformation. This decomposition can be used directly to formulate the plasticity model. Historically, an additive strain rate decomposition, Equation 4.2.1-2 $$ \dot {\pmb {\varepsilon}} = \dot {\pmb {\varepsilon}} ^ {e l} + \dot {\pmb {\varepsilon}} ^ {p l}, $$ has been used in its place. Here "\_ is the total (mechanical) strain rate, $\dot { \pmb { \varepsilon } } ^ { e l }$ is the elastic strain rate, and $\dot { \varepsilon } ^ { p l }$ is the plastic strain rate. It is shown in \`\`The additive strain rate decomposition, '' Section 1.4.4, that Equation 4.2.1-2 is a consistent approximation to Equation 4.2.1-1 when the elastic strains are infinitesimal (negligible compared to unity) and when the strain rate measure used in Equation 4.2.1-2 is the rate of deformation: $$ \dot {\pmb {\varepsilon}} = \mathrm{sym} \left[ \frac {\partial}{\partial \mathbf {x}} \right]. $$ Equation 4.2.1-2, with the rate of deformation used as the definition of total strain rate, is used in all of the plasticity models that are implemented in ABAQUS. Rice's argument implies that the elastic response must always be small in problems in which these models are used. In practice this is the case: plasticity models are provided for metals, soils, polymers, crushable foams, and concrete; and in each of these materials it is very unlikely that the elastic strain would ever be larger than a few percent (and even this would be quite unusual in a metal). Thus, the use of Equation 4.2.1-2 does not appear to be objectionable for the models in question, at least from a formal point of view. However, the user who needs to develop user subroutine UMAT or VUMAT for a different material model in which the elastic strains and the inelastic strains may both be large should consider using Equation 4.2.1-1 directly. The elastic part of the response is assumed to be derivable from an elastic strain energy density potential, so the stress is defined by Equation 4.2.1-3 $$ \pmb {\sigma} = \frac {\partial U}{\partial \pmb {\mathcal {E}} ^ {e l}}, $$ where U is the strain energy density potential. Since we assume that, in the absence of plastic straining, the variation of elastic strain is the same as the variation in the rate of deformation, conjugacy arguments define the stress measure ¾ as the "true" (Cauchy) stress. All stress output in ABAQUS is given in this form. In some materials the elastic response is essentially incompressible. But this is not usually the case for the materials whose inelastic deformation is considered with the models provided in ABAQUS, so Equation 4.2.1-3 can be taken to define the stress completely. However, the inelastic response is often assumed to be approximately incompressible (in metals, for example, or in soils undergoing large plastic flows), so the user must be careful to ensure that the elements chosen can accommodate incompressible response without exhibiting "locking" problems when the model is three-dimensional, plane strain, or axisymmetric. This requires the use of hybrid or fully or selectively reduced integration elements. For several of the plasticity models provided in ABAQUS the elasticity is linear, so the strain energy density potential has a very simple form. For the soils and foam models the volumetric elastic strain is proportional to the logarithm of the equivalent pressure stress. For the concrete model damaged elasticity is used to account for crack opening after the concrete has cracked: in that case the elasticity model is more complex. The rate-independent plasticity models in ABAQUS and one of the rate-dependent models all have a region of purely elastic response. The yield function, f , defines the limit to this region of purely elastic response and is written so that Equation 4.2.1-4 $$ f (\pmb {\sigma}, \theta , H _ {\alpha}) < 0 $$ for purely elastic response. Here µ is the temperature, and $H _ { \alpha }$ are a set of hardening parameters. The subscript ® is introduced simply to indicate that there may be several hardening parameters, $H _ { \alpha }$ : the range of ® is not specified until we define a particular plasticity model. The hardening parameters are state variables that are introduced to allow the models to describe some of the complexity of the inelastic response of real materials. In the simplest plasticity model ("perfect plasticity") the yield surface acts as a limit surface and there are no hardening parameters at all: no part of the model evolves during the deformation. Complex plasticity models usually include a large number of hardening parameters. The models provided in ABAQUS are generally not the most complex models and use only a few such parameters (only one is used in the isotropic hardening metal model and in the Cam-clay model; six are used in the simple kinematic hardening model). In two of the plasticity models in ABAQUS (the concrete model and the jointed material model) the yield behavior is modeled with several independent inelastic flow systems. For this case Equation 4.2.1-4 is generalized to $$ f _ {i} < 0 $$ for purely elastic response in system i, where $f _ { i } ( \pmb { \sigma } , H _ { i , \alpha } )$ is one of the yield functions and $H _ { i , \alpha }$ are the hardening parameters for the ith system. For generality in this discussion we assume the model uses such a system of independent yield functions. In the simpler systems with a single yield function i can only take the value 1. Stress states that cause the yield function to have a positive value cannot occur in rate-independent plasticity models, although this is possible in a rate-dependent model. Thus, in the rate-independent models we have the yield constraints $$ f _ {i} = 0 $$ during inelastic flow. When the material is flowing inelastically the inelastic part of the deformation is defined by the flow rule, which we can write as Equation 4.2.1-5 $$ d \pmb {\varepsilon} ^ {p l} = \sum_ {i} d \lambda_ {i} \frac {\partial g _ {i}}{\partial \pmb {\sigma}}, $$ where $g _ { i } ( \pmb { \sigma } , \theta , H _ { i , \alpha } )$ is the flow potential for the ith system and $d \lambda _ { i }$ is the rate of change of time, $d t ,$ for a rate-dependent model or is a scalar measuring the amount of the plastic flow rate on the ith system, whose value is determined by the requirement to satisfy the consistency condition $f _ { i } = 0 ,$ , for plastic flow of a rate-independent model. The summation is over only the actively yielding systems: $d \lambda _ { i } = 0$ for those systems for which $f _ { i } < 0$ . The form in which the flow rule is written above assumes that there is a single flow potential, ${ \mathit { g } } _ { i } ,$ , in the ith system. More general plasticity models might have several active flow potentials at a point. This is, for instance, the case in the concrete and jointed material models built into ABAQUS. For some rate-independent plasticity models the direction of flow is the same as the direction of the outward normal to the yield surface: $$ \frac {\partial g _ {i}}{\partial \pmb {\sigma}} = c _ {i} \frac {\partial f _ {i}}{\partial \pmb {\sigma}}, $$ where $c _ { i }$ is a scalar. Such models are called "associated flow" plasticity models. Associated flow models are useful for materials in which dislocation motion provides the fundamental mechanisms of plastic flow when there are no sudden changes in the direction of the plastic strain rate at a point. They are generally not accurate for materials in which the inelastic deformation is primarily caused by frictional mechanisms. The metal plasticity models in ABAQUS (except cast iron) and the Cam-clay soil model use associated flow. The cast iron, granular/polymer, crushable foam, Mohr-Coulomb, Drucker-Prager/Cap, and jointed material models provide nonassociated flow with respect to volumetric straining and equivalent pressure stress. The concrete model uses associated flow. The rate form of the flow rule is essential to incremental plasticity theory, because it allows the history dependence of the response to be modeled. The final ingredient in plasticity models is the set of evolution equations for the hardening parameters. We write these equations as Equation 4.2.1-6 $$ d H _ {i, \alpha} = d \lambda_ {i} h _ {i, \alpha} (\pmb {\sigma}, \theta , H _ {i, \beta}), $$ where $h _ { i , \alpha }$ is the (rate form) hardening law for $H _ { i , \alpha }$ . In complex plasticity models--for example, models used to describe the cyclic behavior of metals used for high temperature applications--these evolution laws have complicated forms, since such complexity is required to match the experimentally observed behavior. The plasticity models offered in ABAQUS use simple evolution equations: isotropic hardening, Prager-Ziegler kinematic hardening, and the location of the center of the yield surface along the equivalent pressure stress axis in the Cam-clay model. The independence of the yield systems designated by the subscript i is implicit in the assumption in Equation 4.2.1-6 above that the evolution of the $H _ { i , \alpha }$ depends only on other hardening parameters, $H _ { i , \beta }$ , in the same (ith) system. Equation 4.2.1-1 to Equation 4.2.1-6 define the general structure of all of the plasticity models in ABAQUS. Since the flow rule and the hardening evolution rules are written in rate form, they must be integrated. The general technique of integration is discussed in \`\`Integration of plasticity models,'' Section 4.2.2. The sections immediately following that discussion describe the details of the specific plasticity models that are provided in ABAQUS. # 4.2.2 Integration of plasticity models The plasticity models provided in ABAQUS have been described in general terms in \`\`Plasticity models: general discussion,'' Section 4.2.1. The only rate equations are the evolutionary rule for the hardening, the flow rule, and the strain rate decomposition. The simplest operator that provides unconditional stability for integration of rate equations is the backward Euler method: applying this method to the flow rule (Equation 4.2.1-5) gives Equation 4.2.2-1 $$ \Delta \varepsilon^ {p l} = \sum_ {i} \Delta \lambda_ {i} \frac {\partial g _ {i}}{\partial \pmb {\sigma}}, $$ and applying it to the hardening evolution equations, Equation 4.2.1-6, gives $$ \Delta H _ {i, \alpha} = \Delta \lambda_ {i} h _ {i, \alpha}. $$ Equation 4.2.2-2 In these equations and throughout the remainder of this section any quantity not specifically associated # Mechanical Constitutive Theories with a time point is taken at the end of the increment (at time $t + \Delta t )$ . The strain rate decomposition, Equation 4.2.1-2, is integrated over a time increment as $$ \Delta \varepsilon = \Delta \varepsilon^ {e l} + \Delta \varepsilon^ {p l}, $$ where $\Delta \varepsilon$ is defined by the central difference operator: $$ \Delta \varepsilon = \mathrm{sym} \left[ \frac {\partial \Delta \mathbf {x}}{\partial (\mathbf {x} _ {t} + \frac {1}{2} \Delta \mathbf {x})} \right]. $$ We integrate the total values of each strain measure as the sum of the value of that strain at the start of the increment, rotated to account for rigid body motion during the increment, and the strain increment. The rotation to account for rigid body motion during the increment is defined approximately using the algorithm of Hughes and Winget (1980). This integration allows the strain rate decomposition to be integrated into Equation 4.2.2-3 $$ \varepsilon = \varepsilon^ {e l} + \varepsilon^ {p l}. $$ From a computational viewpoint the problem is now algebraic: we must solve the integrated equations of the constitutive model for the state at the end of the increment. The set of equations that define the algebraic problem are the strain decomposition, Equation 4.2.2-3; the elasticity, Equation 4.2.1-3; the integrated flow rule, Equation 4.2.2-1; the integrated hardening laws, Equation 4.2.2-2; and for rate independent models, the yield constraints Equation 4.2.2-4 $$ f _ {i} = 0, $$ for active systems (systems in which $f _ { i } < 0$ have $\Delta \lambda _ { i } = 0 )$ . We assume that the flow surface is sufficiently smooth so that its (second) derivatives with respect to stress and the hardening parameters are well-defined. This is generally true for the models in ABAQUS: the exceptions occur at corners or vertices of the surfaces. These special cases are handled individually when they arise. For some plasticity models the algebraic problem can be solved in closed form. For other models it is possible to reduce the problem to a one variable or a two variable problem that can then be solved to give the entire solution. For example, the Mises yield surface--which is generally used for isotropic metals, together with linear, isotropic elasticity--is a case for which the integrated problem can be solved exactly or in one variable (see \`\`Isotropic elasto-plasticity,'' Section 4.3.2). For other rate-independent models with a single yield system the algebraic problem is considered to be a problem in the components of $\Delta \varepsilon ^ { p l }$ . Once these have been found--the elasticity--together with the integrated strain rate decomposition--define the stress. The flow rule then defines $\Delta \lambda$ and the hardening laws define the increments in the hardening variables. We now derive the equations for the Newton solution of the integrated problem for the case of rate-independent plasticity with a single yield system. The rate-dependent problem with a single yield system is solved in a similar way. For the particular cases of multiple, independent, yield systems (concrete and jointed material) particular techniques are used for this algebraic solution, taking advantage of the simplifications available in those particular models. The concrete model and its integration are described in \`\`An inelastic constitutive model for concrete, '' Section 4.5.1, and the jointed material model is described in \`\`Constitutive model for jointed materials, '' Section 4.5.3. During the solution, the elasticity relationship and the integrated strain rate decomposition are satisfied exactly, so that Equation 4.2.2-5 $$ \mathbf {c} _ {\sigma} = - \mathbf {D} ^ {e l}: \mathbf {c} _ {\varepsilon}, $$ where $\mathbf { c } _ { \sigma }$ is the correction to the stress, $\mathbf { c } _ { \varepsilon }$ is the correction to the plastic strain increments, and $$ \mathbf {D} ^ {e l} = \frac {\partial^ {2} U}{\partial \pmb {\varepsilon} ^ {e l} \partial \pmb {\varepsilon} ^ {e l}} $$ is the tangent elasticity matrix. The hardening laws are also satisfied exactly (because the increments of the hardening parameters are defined from these laws) so that $$ c _ {\alpha} = h _ {\alpha} c _ {\lambda} + \Delta \lambda \left(\frac {\partial h _ {\alpha}}{\partial \pmb {\sigma}}: \mathbf {c} _ {\sigma} + \frac {\partial h _ {\alpha}}{\partial H _ {\beta}} c _ {\beta}\right), $$ where $c _ { \alpha }$ is the correction to $\Delta H _ { \alpha }$ and $c _ { \lambda }$ is the correction to $\Delta \lambda$ . This set of equations can be rewritten Equation 4.2.2-6 $$ c _ {\alpha} = \hat {H} _ {\alpha} c _ {\lambda} + \hat {\mathbf {w}} _ {\alpha}: \mathbf {c} _ {\sigma}, $$ where $$ \hat {H} _ {\alpha} = \left[ \delta_ {\alpha \beta} - \Delta \lambda \frac {\partial h _ {\beta}}{\partial H _ {\alpha}} \right] ^ {- 1} h _ {\beta} $$ and $$ \hat {\mathbf {w}} _ {\alpha} = \Delta \lambda \left[ \delta_ {\alpha \beta} - \Delta \lambda \frac {\partial h _ {\beta}}{\partial H _ {\alpha}} \right] ^ {- 1} \frac {\partial h _ {\beta}}{\partial \pmb {\sigma}}. $$ The flow rule is not satisfied exactly until the solution has been found, so it gives the Newton equations $$ \mathbf {c} _ {\varepsilon} - c _ {\lambda} \frac {\partial g}{\partial \pmb {\sigma}} - \Delta \lambda \left(\frac {\partial^ {2} g}{\partial \pmb {\sigma} \partial \pmb {\sigma}}: \mathbf {c} _ {\sigma} + \frac {\partial^ {2} g}{\partial \pmb {\sigma} \partial H _ {\alpha}} c _ {\alpha}\right) = \Delta \lambda \frac {\partial g}{\partial \pmb {\sigma}} - \Delta \pmb {\varepsilon} ^ {p l}. $$ Using Equation 4.2.2-5 and Equation 4.2.2-6 allows these equations to be rewritten as Equation 4.2.2-7 $$ \left[ \mathbf {I} + \Delta \lambda \hat {\mathbf {N}}: \mathbf {D} ^ {e l} \right]: \mathbf {c} _ {\varepsilon} - \hat {\mathbf {n}} c _ {\lambda} = \Delta \lambda \frac {\partial g}{\partial \pmb {\sigma}} - \Delta \pmb {\varepsilon} ^ {p l}, $$ where $$ \hat {\mathbf {N}} = \frac {\partial^ {2} g}{\partial \pmb {\sigma} \partial \pmb {\sigma}} + \frac {\partial^ {2} g}{\partial \pmb {\sigma} \partial H _ {\alpha}} \hat {\mathbf {w}} _ {\alpha}, $$ and $$ \hat {\mathbf {n}} = \frac {\partial g}{\partial \pmb {\sigma}} + \Delta \lambda \frac {\partial^ {2} g}{\partial \pmb {\sigma} \partial H _ {\alpha}} \hat {H} _ {\alpha}. $$ Likewise, the yield condition is not satisfied exactly during the Newton iteration, so $$ \frac {\partial f}{\partial \pmb {\sigma}}: \mathbf {c} _ {\sigma} + \frac {\partial f}{\partial H _ {\alpha}} c _ {\alpha} = - f. $$ Using Equation 4.2.2-5 and Equation 4.2.2-6 in this equation gives Equation 4.2.2-8 $$ \hat {\mathbf {m}}: \mathbf {D} ^ {e l}: \mathbf {c} _ {\varepsilon} - \frac {\partial f}{\partial H _ {\alpha}} \hat {H} _ {\alpha} c _ {\lambda} = f, $$ where $$ \hat {\mathbf {m}} = \frac {\partial f}{\partial \pmb {\sigma}} + \frac {\partial f}{\partial H _ {\alpha}} \hat {\mathbf {w}} _ {\alpha}. $$ We now eliminate $c _ { \lambda }$ between Equation 4.2.2-7 and Equation 4.2.2-8. Taking Equation 4.2.2-7 along m^ : $\mathbf { D } ^ { e l }$ and using Equation 4.2.2-8 gives $$ c _ {\lambda} = \frac {1}{d} \Delta \lambda \hat {\mathbf {m}}: \mathbf {D} ^ {e l}: \hat {\mathbf {N}}: \mathbf {D} ^ {e l}: \mathbf {c} _ {\varepsilon} - \frac {1}{d} \hat {\mathbf {m}}: \mathbf {D} ^ {e l}: \left(\Delta \lambda \frac {\partial g}{\partial \pmb {\sigma}} - \Delta \pmb {\varepsilon} ^ {p l}\right) + \frac {1}{d} f, $$ where $$ d = \hat {\mathbf {m}}: \mathbf {D} ^ {e l}: \hat {\mathbf {n}} - \frac {\partial f}{\partial H _ {\alpha}} \hat {H} _ {\alpha}. $$ Using this equation in Equation 4.2.2-7 then gives $$ \left[ \mathbf {I} + \Delta \lambda \mathbf {Z}: \hat {\mathbf {N}}: \mathbf {D} ^ {e l} \right]: \mathbf {c} _ {\varepsilon} = \mathbf {Z}: \left(\Delta \lambda \frac {\partial g}{\partial \pmb {\sigma}} - \Delta \pmb {\varepsilon} ^ {p l}\right) + \frac {1}{d} f \hat {\mathbf {n}}, $$ where $$ \mathbf {Z} = \mathbf {I} - \frac {1}{d} \hat {\mathbf {n}} \hat {\mathbf {m}}: \mathbf {D} ^ {e l}, $$ which is a set of linear equations solved for the $\mathbf { c } _ { \varepsilon }$ . The solution is then updated and the Newton loop continued until the flow equation and yield constraint are satisfied. The solution for rate-dependent plasticity models with a single yield function is developed in the same way, the only differences being the lack of a yield constraint and the identification of $\Delta \lambda$ with time. # Tangent matrix The tangent matrix for the material, @¾=@", is required when ABAQUS/Standard is being used for implicit time integration and Newton's method is being used to solve the equilibrium equations. The matrix is obtained directly by taking variations of the integrated equations with respect to all solution parameters, and then solving for the relationship between ¾ and ". The procedure closely follows the derivation used above for the Newton solution: the result is the tangent matrix $$ \boldsymbol {\sigma} = \mathbf {D}: \varepsilon , $$ where $$ \mathbf {D} = \left[ \mathbf {I} + \Delta \lambda \mathbf {D} ^ {e l}: \mathbf {Z}: \hat {\mathbf {N}} \right] ^ {- 1}: \mathbf {D} ^ {e l}: \mathbf {Z}. $$ # 4.3 Metal plasticity # 4.3.1 Metal plasticity models ABAQUS offers several models for metal plasticity analysis. The main options are a choice between rate-independent and rate-dependent plasticity; a choice between the Mises yield surface for isotropic materials and Hill's yield surface for anisotropic materials; and, for rate-independent modeling, a choice between isotropic and kinematic hardening. Special plasticity theories are the cast iron model (\`\`Cast iron plasticity,'' Section 4.3.7), the ORNL model for types 304 and 316 stainless steel in nuclear applications (\`\`ORNL constitutive theory,'' Section 4.3.8), and deformation plasticity for fracture mechanics applications (\`\`Deformation plasticity,'' Section 4.3.9). Rate-independent plasticity is used mostly in modeling the response of metals and some other materials at low temperature (typically below half the melting temperature on an absolute scale) and low strain rates. The rate-independent metal plasticity model uses associated flow. Two types of rate-dependent models are offered. In the first type a rate-dependent yield strength is introduced in the material model. This is intended for relatively high strain rate applications, such as dynamic events or metal forming process simulations. This type of rate dependence can be introduced in different ways. One way is to use an overstress power law,overstress $$ \dot {\bar {\varepsilon}} ^ {p l} = D \left(\frac {\bar {\sigma}}{\sigma^ {0}} - 1\right) ^ {p} \quad \mathrm{for} \quad \bar {\sigma} \geq \sigma^ {0}, $$ where $\dot { \bar { \varepsilon } } ^ { p l }$ is the equivalent plastic strain rate; $\bar { \sigma }$ is the yield stress at nonzero plastic strain rate; $\sigma ^ { 0 } ( \varepsilon ^ { p l } , \theta , f _ { i } )$ is the static yield stress (which may depend on the plastic strain $\ -- \varepsilon ^ { p l } \mathrm { \Pi } _ { -- }$ via isotropic hardening, on the temperature--µ--and on other field variables, $f _ { i } ) ;$ ; and $D ( \theta , f _ { i } ) , p ( \theta , f _ { i } )$ are material parameters that can be functions of temperature and, possibly, of other predefined state variables. Another way is to define a yield stress ratio, $\bar { \sigma } / \sigma ^ { 0 }$ , as a function of the equivalent plastic strain rate, $\dot { \bar { \varepsilon } } ^ { p l }$ . Both of these options assume that the shapes of the hardening curves at different strain rates are identical and are activated by using the \*RATE DEPENDENT option in conjunction with the \*PLASTIC option. If the shapes of the hardening curves at different strain rates are different, the static and rate-dependent stress-strain relations can be specified directly on the \*PLASTIC, RATE=option. The yield stress at a given strain rate is interpolated directly from these relations. Finally, the user can describe general rate-dependent isotropic hardening with user subroutine UHARD. See Symonds (1967), Lindholm and Besseny (1969), and Eleiche (1972) for collections of material response measurements or bibliographies of such measurements at high strain dependents. For high temperature "creep" problems, ABAQUS/Standard offers some simple built-in creep laws. But for many practical problems the user must write the uniaxial creep behavior into user subroutine CREEP, because of the complexity of the experimentally measured material response. Creep response under cyclic loading shows significant Bauschinger effects, which cannot be modeled except by introducing sophisticated hardening models. The only capability in ABAQUS for such cases is the "ORNL" option. This option uses simple rules to model the Bauschinger effect and is intended primarily as a design evaluation model for the high temperature response of stainless steel. It does not model the material's response in detail. User subroutine UMAT must be used if that hardening model is not adequate. Isotropic hardening means that the yield function is written $$ f (\pmb {\sigma}) = \sigma^ {0} (\varepsilon^ {p l}, \theta), $$ where $\sigma ^ { 0 }$ is the equivalent (uniaxial) stress, $\varepsilon ^ { p l }$ is the work equivalent plastic strain, defined by $$ \sigma^ {0} \dot {\varepsilon} ^ {p l} = \pmb {\sigma}: \dot {\pmb {\varepsilon}} ^ {p l}, $$ and µ is temperature.