4.37. Phase Field F^eF^p Modular Model

Phase Field F^eF^p Modular is an implementation of a generalized, modular, cohesive variational phase-field fracture model developed for the purpose of modeling brittle and ductile fracture. The model is based upon the Phase Field F^eF^p model, which in turn is based upon the F^eF^p model, a hyperelastic analogue of the J₂ plasticity model, and features a von-Mises yield surface and isotropic hardening. However, the modular version allows for more generalized hardening models, including rate-dependent and kinematic hardening. As with Phase Field F^eF^p, Phase Field F^eF^p Modular is meant to be used in conjunction with a phase-field solver. The use of Sierra/SM’s reaction-diffusion solver is strongly recommended. The model defines several auxiliary classes and departs from the existing LAMÉ hardening capabilities in order to have a unified interface between the elastic, plastic, and damage components of the model.

4.37.1. Theory

As much of the theory and derivation is similar to the Phase Field F^eF^p theory, interested readers are referred to that model’s documentation for more detailed information as well as further references, while deviations will be noted here. As in that model, a multiplicative decomposition of the deformation gradient into elastic and plastic terms (\(F_{ij} = F^e_{ik} F^p_{kj}\)) is used. The plastic deformation tensor is updated with \(\dot{F}^p_{ij} = \dot{Q}_a M_{aik} F^p_{kj}\), where \(\dot{Q}_i\) is a vector of internal variables and \(M_{ijk}\) is a third-order flow tensor (which can be thought of as a vector of flow tensors, one for each internal variable). Additionally, like in Phase Field F^eF^p, there is a phase field variable \(\phi\) which tracks the amount of damage; the convention used here is that \(\phi = 1\) is intact and \(\phi = 0\) is fully damaged.

To further explain the internal variables and flow tensor, this is a generalization of standard plasticity, where there is one internal variable \(\bar{\epsilon}^p\) and a second-order flow tensor \(N^p_{ij}\). The generalization is useful in cases like crystal plasticity, where the existence of multiple slip planes leads to \(N\) internal variables, one per slip plane. For more details, consult Ortiz and Stainier [1].

Following similar steps to Phase Field F^eF^p, the free energy of the system can decomposed into an elastic component \(\psi^e\), a plastic component \(\psi^p\), and a fracture component \(\psi^f\). The total energy \(\psi\) can then be written as in (4.258). Further, degradation functions \(g^e\) and \(g^p\) are introduced to model the effect of damage on the elastic and plastic behavior. Unlike in Phase Field F^eF^p, these degradation functions are allowed to be different, allowing for more complex modeling of real-world behavior. The elastic and plastic potentials are defined in (4.259) and (4.260), where \(\tilde{\psi}^e\) and \(\tilde{\psi}^p\) capture the behavior of the undamaged material. We can also define kinetic (rate-dependent) potentials \(\psi^*_e\) (elastic kinetic), \(\psi^*_p\) (plastic kinetic), and \(\psi^*_f\) (damage kinetic) and, similar to the regular potentials, we define the first two using multiplicative decompositions (with respect to phase) in (4.261), (4.262).

(4.258)\[\psi = \psi^e\left(F^e_{ij},\phi\right) + \psi^p\left(F^p_{ij},Q_i,\phi\right) + \psi^f\left(\phi,\phi_{,i}\right)\]

(4.259)\[\psi^e\left(F^e_{ij},\phi\right) = g^e\left(\phi\right)\tilde{\psi}^e\left(F^e_{ij}\right)\]

(4.260)\[\psi^p\left(F^p_{ij},Q_i,\phi\right) = g^p\left(\phi\right)\tilde{\psi}^p\left(F^p_{ij},Q_i\right)\]

(4.261)\[\psi_e^{*}\left(\dot{F}_{ij};F_{ij},\phi\right) = g^e\left(\phi\right)\tilde{\psi}_e^{*}\left(\dot{F}_{ij};F_{ij}\right)\]

(4.262)\[\psi_p^{*}\left(\dot{Q}_i;Q_i,\phi\right) = g^p\left(\phi\right)\tilde{\psi}_p^{*}\left(\dot{Q}_i;Q_i\right)\]

Additionally, we allow most model parameters to vary with temperature and allow for thermal coupling (see “Usage Guide” for details). This means that temperature is an implicit argument to all potentials and kinetic potentials; this can be used to vary material behavior with respect to temperature. Furthermore, the separation of rate-independent behavior (potentials) from rate-dependent behavior (kinetic potentials) is different from other models and should be noted.

4.37.1.1. Notation Simplifications

The theory above uses the general vector of internal variables \(Q_i\) and third-order flow tensor \(M_{ijk}\). This is with an eye towards future integration of a crystal plasticity-type model with multiple slip planes and multiple internal variables. However, all currently-implemented plasticity models use one internal variable. Accordingly, \(Q_0 = \bar{\epsilon}^p\) and \(M_{0ij} = N^p_{ij}\) and the latter notation will be used from here on out when describing model options and the verification tests.

4.37.1.2. Model Options

Unlike in the case of Phase Field F^eF^p, every component here — the potentials, kinetic potentials, degradation functions, and tension-compression split — is treated as an interface, making it easy to add additional variants of each component without needing to modify the modular framework itself.

The following models are available for each component (the model parameters and state variables are given in the “Usage Guide” section):

• Elastic Potentials

  • Hencky Elastic Potential : \(\tilde{\psi}^e = \mu \epsilon^e_{ij}\epsilon^e_{ij} + \left(\frac{\kappa}{2} - \frac{\mu}{3}\right)\epsilon^e_{kk}\epsilon^e_{\ell\ell}\) where \(\epsilon^e_{ij} = \frac{1}{2}\left[\log[C^e_{mn}]\right]_{ij}\) and \(C^e_{ij} = F^e_{ki}F^e_{kj}\)

• Plastic Potentials : \(\beta_{TQ}\) is the Taylor-Quinney factor which controls how much plastic energy is dissipated (and therefore does not contribute to the driving force for the phase field evolution)

  • No Plastic Potential : \(\tilde{\psi}^p = 0\)

  • Linear Plastic Potential : \(\tilde{\psi}^p = \left(1 - \beta_{TQ}\right)\left(\sigma_y\bar{\epsilon}^p + \frac{h\bar{\epsilon}^{p^2}}{2}\right)\)

  • Power Law Plastic Potential : \(\tilde{\psi}^p = \left(1 - \beta_{TQ}\right)\left(\sigma_y\bar{\epsilon}^p + \frac{a}{b + 1}\max\left(0,\bar{\epsilon}^p - \epsilon_{\ell}\right)^{b + 1}\right)\)

  • Power Law Simple Plastic Potential : \(\tilde{\psi}^p = \frac{n\left(1 - \beta_{TQ}\right)\sigma_y\epsilon_0}{n + 1}\left(\left(1 + \frac{\bar{\epsilon}^p}{\epsilon_0}\right)^{(n + 1) / n} - 1\right)\)

  • Voce Plastic Potential : \(\tilde{\psi}^p = \left(1 - \beta_{TQ}\right)\left(\sigma_y\bar{\epsilon}^p + \sum_{i = 0}^n h_i\bar{\epsilon}^p + \frac{h_i}{\alpha_i}\left(\exp\left(-\alpha_i\bar{\epsilon}^p\right) - 1\right)\right)\)

  • Johnson-Cook Plastic Potential : \(\tilde{\psi}^p = \left(1 - \beta_{TQ}\right)\left(a\bar{\epsilon}^p + \frac{b}{n + 1}\bar{\epsilon}^{p^{n + 1}}\right)\left(1 - \theta^q\right)\) where \(\theta = \frac{T - T_0}{T_m - T_0}\)

• Damage Potentials

  • No Damage Potential : \(\psi^f = 0\)

  • Gradient Damage Potential : \(\psi^f = \frac{3G_c}{8\ell}\left(\left(1 - \phi\right) + \ell^2\phi_{,i}a_{ij}\phi_{,j}\right)\)

  • Classical Gradient Damage Potential : \(\psi^f = \frac{G_c}{4\ell}\left(\left(1 - \phi\right)^2 + 4\ell^2\phi_{,i}a_{ij}\phi_{,j}\right)\)

  • Wu Damage Potential : \(\psi^f = \frac{G_c}{C_0\ell}\left(\xi\left(1 - \phi\right) + \left(1 - \xi\right)\left(1 - \phi\right)^2 + \ell^2\phi_{,i}a_{ij}\phi_{,j}\right)\)

• Plastic Kinetic Potentials : \(\tilde{\sigma}\) and \(\tilde{\tau}\) are the Jacobian and Hessian (respectively) of the selected plastic potential with \(\beta_{TQ} = 1\) and \(dt\) is the instantaneous change in time

  • No Plastic Kinetic Potential : \(\tilde{\psi}^*_p = \frac{\beta_{TQ} dt}{2}\tilde{\tau}\left(\dot{\bar{\epsilon}}^p\right)^2\) (this is non-zero to ensure that the Tayor-Quinney factor is properly taken into account)

  • Power Law Plastic Kinetic Potential : \(\tilde{\psi}^*_p = \frac{\beta_{TQ} dt}{2}\tilde{\tau}\left(\dot{\bar{\epsilon}}^p\right)^2 + \frac{m\sigma_y\dot{\epsilon}_0}{m + 1}\left(\frac{\dot{\bar{\epsilon}}^p}{\dot{\epsilon}_0}\right)^{(m + 1)/m}\)

  • Johnson-Cook Plastic Kinetic Potential : \(\tilde{\psi}^*_p = \frac{\beta_{TQ} dt}{2}\tilde{\tau}\left(\dot{\bar{\epsilon}}^p\right)^2 + c\tilde{\sigma}\left(\dot{\bar{\epsilon}}^p\log\frac{\dot{\bar{\epsilon}}^p}{\dot{\epsilon}_0} - \dot{\bar{\epsilon}}^p + \dot{\epsilon}_0\right)\)

  • Power Law Breakdown Plastic Kinetic Potential : \(\tilde{\psi}^*_p = \frac{\beta_{TQ} dt}{2}\tilde{\tau}\left(\dot{\bar{\epsilon}}^p\right)^2 + \tilde{\sigma}\dot{\bar{\epsilon}}^p\left(\text{asinh}{\left(\gamma^{1/m}\right)} - \frac{\gamma^{1/m}}{m + 1} {}_2F_1\left(g,m\right)\right)\)

• Degradation Functions

  • No Degradation Function : \(g\left(\phi\right) = 0\)

  • Quadratic Degradation Function : \(g\left(\phi\right) = \phi^2\)

  • Talamini Degradation Function : \(g\left(\phi\right) = \left[\frac{\phi}{1 + \left(\frac{3G_0}{16\ell\psi_c} - 1\right)\left(1 - \phi\right)}\right]^2\)

  • Lo Degradation Function : \(g\left(\phi\right) = s\left[1 - \left(\frac{s - 1}{s}\right)^{\phi^2}\right]\)

  • Wu Degradation Function : \(g\left(\phi\right) = \frac{\phi^p}{\phi^p + \sum_ia_i\left(1 - \phi\right)^i}\)

• Tension-Compression Split

  • No Tension-Compression Split : Everything is degraded.

  • Deviatoric-Volumetric Tension-Compression Split : Deviatoric part is always degraded. Volumetric part is degraded if the trace of the strain is positive.

  • Star-Convex Tension-Compression Split : Deviatoric part is always degraded. Volumetric part is fully degraded if the trace of the strain is positive; otherwise, the volumetric part is degraded by a factor \(\gamma\).

4.37.2. Implementation

There are two possible solvers available to solve this minimization problem for \(Q_i\) and \(M_{ijk}\) (in standard plasticity, these are \(\bar{\epsilon}^p\) and \(N^p_{ij}\)): a standard return-mapping algorithm and a sequential quadratic programming (SQP) algorithm. Return-mapping is described in the Phase Field F^eF^p documentation, while SQP is described below.

The overall approach to the SQP solver is as follows. We first write the discretized difference in total energy density between the state at time \(t_{n+1}\) and the state at time \(t_n\) (with \(dt = t_{n+1} - t_n\)) as (4.263), with \(\Delta\psi\) given by (4.264). The objective is to find \(\left(Q_{n+1}\right)_i\) and \(\left(M_{n+1}\right)_{ijk}\) such that this functional is minimized, which only requires minimizing the final term (the rest of (4.263) is not dependent on \(\left(Q_{n+1}\right)_i\) and \(\left(M_{n+1}\right)_{ijk}\)).

(4.263)\[\begin{split}\begin{split} W\left(\left(F_{n+1}\right)_{ij};\left(F_n\right)_{ij},\phi_n,\left(Q_n\right)_i\right) &= \Delta t g^e\left(\phi_n\right)\tilde{\psi}_e^{*}\left(\frac{\left(F_{n+1}\right)_{ij} - \left(F_n\right)_{ij}}{\Delta t}; \left(F_n\right)_{ij}\right) + \Delta t\psi_f^{*}\left(\frac{\phi_{n+1} - \phi_n}{\Delta t}\right) \\ {} + \min_{\left(Q_{n+1}\right)_i,\left(M_{n+1}\right)_{ijk}}&\left\{\Delta\psi\left(F_{ij},\phi,\phi_{,i},F^p_{ij},Q_i\right) + \Delta t g^p\left(\phi_n\right)\tilde{\psi}_p^{*}\left(\frac{\left(Q_{n+1}\right)_i - \left(Q_n\right)_i}{\Delta t};\left(Q_n\right)_i\right)\right\} \end{split}\end{split}\]

(4.264)\[\begin{split}\begin{split} \Delta\psi\left(F_{ij},\phi,\phi_{,i},F^p_{ij},Q_i\right) &= g^e\left(\phi_{n+1}\right)\tilde{\psi}^e\left(\left(F_{n+1}\right)_{ik}\left(F^{p^{-1}}_{n+1}\right)_{kj}\right) - g^e\left(\phi_n\right)\tilde{\psi}^e\left(\left(F_n\right)_{ik}\left(F^{p^{-1}}_n\right)_{kj}\right) \\ &{} + g^p\left(\phi_{n+1}\right)\tilde{\psi}^p\left(\left(F^p_{n+1}\right)_{ij},\left(Q_{n+1}\right)_i\right) - g^p\left(\phi_n\right)\tilde{\psi}^p\left(\left(F^p_n\right)_{ij},\left(Q_n\right)_i\right) \\ &{} + \psi^f\left(\phi_{n+1},\left(\phi_{n+1}\right)_{,i}\right) - \psi^f\left(\phi_n,\left(\phi_n\right)_{,i}\right) \end{split}\end{split}\]

This leads us to the objective function given in (4.265). All unknowns are grouped into a vector \(x_i\) ((4.267) and (4.268)) and all constraints are grouped into a vector \(c_i\) (along with associated Lagrange multipliers \(\lambda_i\)); in this notation, the assumption is that there are \(n\) internal variables, so \(10n\) total unknowns. The Lagrangian is then as (4.266), leading to the optimization problem (4.269). As an aside, taking the derivative of (4.265) with respect to \(Q_i\) yields a generalized yield equation (4.270), which is effectively what the radial return method solves [2].

[2]

Hardcoding the constraints and assuming associative flow fixes the flow tensor analytically. Doing this reduces the set of unknowns to \(Q_i\), meaning we can take the derivative of the objective function with respect to \(Q_i\) and set it equal to 0 to find the minimizer. Doing this and comparing to the radial return method shows that the equations are identical.

(4.265)\[\begin{split}\begin{split} F &= g^e\left(\phi_{n+1}\right)\tilde{\psi}^e\left(\left(F^e_{n+1}\right)_{ij}\right) + g^p\left(\phi_{n+1}\right)\tilde{\psi}^p\left(\left(F^p_{n+1}\right)_{ij},\left(Q_{n+1}\right)_i\right) \\ &{}+ \Delta t g^p\left(\phi_{n+\alpha}\right)\tilde{\psi}_p^{*}\left(\frac{\left(Q_{n+1}\right)_i - \left(Q_n\right)_i}{\Delta t};\left(Q_{n+1}\right)_i\right) \end{split}\end{split}\]

(4.266)\[\mathcal{L} = F - \lambda_ic_i\]

(4.267)\[x_i = \left(Q_{n+1}\right)_i\quad i\in\left\{0..n-1\right\}\]

(4.268)\[x_{n+9a+3b+c} = \left(M_{n+1}\right)_{abc}\quad a\in\left\{0..n-1\right\},b\in\left\{0..2\right\},c\in\left\{0..2\right\}\]

(4.269)\[\min_{x_j}\mathcal{L} = \min_{x_j}\left(F - \lambda_ic_i\right)\]

(4.270)\[0 = \left[g^p\left(\phi\right)\frac{\partial\tilde{\psi}^p}{\partial F^p_{ij}} + g^e\left(\phi\right)\frac{\partial\tilde{\psi}^e}{\partial F^e_{kl}}\frac{\partial F^e_{kl}}{\partial F^p_{ij}}\right]\frac{\partial F^p_{ij}}{\partial\bar{\epsilon}^p} + g^p\left(\phi\right)\frac{\partial\tilde{\psi}^p}{\partial\bar{\epsilon}^p} + \Delta t g^p\left(\phi\right)\frac{d \tilde{\psi}_p^{*}}{d\bar{\epsilon}^p}\]

A Taylor expansion ((4.271)) is performed around the \(x_i\) that minimizes \(\mathcal{L}\). Then the original minimization problem ((4.269)) can be converted into a minimum over \(p_i\), the deviation from the minimum ((4.272)). In addition, we perform a Taylor expansion around \(x_i\) for the constraint equations as well, leading to the linearized constraint equations given in (4.273) and (4.274). We have introduced the notation here that \(\mathcal{E}\) refers to the indices of the equality constraints and \(\mathcal{I}\) refers to the indices of the inequality constraints — \(\mathcal{E}\cup\mathcal{I} = \left\{0..M\right\}\), where \(M\) is the total number of constraints.

(4.271)\[\mathcal{L}\left(x_i + p_i\right) \approx \mathcal{L}\left(x_i\right) + p_j\frac{\partial\mathcal{L}}{\partial x_j}\left(x_i\right) + \frac{1}{2}p_jp_k\frac{\partial^2\mathcal{L}}{\partial x_j\partial x_k}\left(x_i\right)\]

(4.272)\[\min_{x_i}\mathcal{L} = \min_{p_i}\left[\mathcal{L}\left(x_i\right) + p_j\frac{\partial\mathcal{L}}{\partial x_j}\left(x_i\right) + \frac{1}{2}p_jp_k\frac{\partial^2\mathcal{L}}{\partial x_j\partial x_k}\left(x_i\right)\right]\]

(4.273)\[0 = p_j\frac{\partial c_m}{\partial x_j}\left(x_i\right) + c_m\left(x_i\right)\quad m\in\mathcal{E}\]

(4.274)\[0 \leq p_j\frac{\partial c_m}{\partial x_j}\left(x_i\right) + c_m\left(x_i\right)\quad m\in\mathcal{I}\]

The SQP algorithm then solves (4.275) iteratively until the unknowns \(Q_i\) and \(M_{ijk}\) and the Lagrange multipliers \(\lambda_i\) are converged. Inequality constraints are dealt with through an “active-set” approach. More details can be found in Nocedal and Wright [3].

(4.275)\[\begin{split}\begin{pmatrix}\mathcal{L}_{,\vec{x}\vec{x}} & -\vec{c}_{,\vec{x}} \\ \vec{c}_{,\vec{x}} & \textbf{0}\end{pmatrix}\begin{pmatrix}\vec{p}_k \\ \vec{l}_k\end{pmatrix} = \begin{pmatrix}-\mathcal{L}_{,\vec{x}} \\ -\vec{c}\end{pmatrix}\end{split}\]

4.37.2.1. Model Options

These model options are related to the solver:

• Plastic Solver

  • Radial Return : Standard J2 associative plasticity solve

  • SQP : Generalized nonlinear fully-coupled solver for internal variables and flow tensors

• Constraints (Only applies to SQP)

  • Flow Tensor Constraint : Set the flow tensor explicitly

  • Flow Tensor Norm Constraint : Set the norm of the flow tensor

  • Flow Tensor Trace Constraint : Set the trace of the flow tensor

  • Flow Tensor Symmetric Constraint (XY, YZ, ZX) : Set the symmetry of the flow tensor across specified components

• Kinetic Constraints (Only applies to SQP)

  • Internal Variable Rate Kinetic Constraint : Ensure that the internal variable can only increase

4.37.3. Verification

We pursue both unit tests and regression tests. Regression tests are actual Sierra runs being tested against known solutions and benchmarks whereas unit tests exercise individual code elements against known solutions and correspond to simple single-element deformations (more information below).

4.37.3.1. Unit testing

In order to ensure thorough verification of all possible combinations of different models in a variety of different regimes, a comprehensive unit testing framework was developed for this model (which can be found with the rest of the LAME unit tests). Every possible combination of different model choices is tested in uniaxial tension, uniaxial compression, and biaxial tension-compression.

The overall test workflow is as follows. For any given test, the inputs are \(\bar{\epsilon}^p\), the flow tensor \(N^p_{ij}\) (known from the mechanical configuration), and the desired stress state \(\sigma_{ij}\) (also known from the mechanical configuration). Coupled with the model choices, this allows the calculation of the input elasic deformation tensor \(F^e_{ij}\). Given that the plastic deformation tensor \(F^p_{ij}\) is already known from the inputs, this allows the calculation of the total deformation tensor \(F_{ij}\). This deformation tensor is passed as the input to the material model and the calculated stress and \(\bar{\epsilon}^p\) are compared with the reference quantities.

The mechanical configurations tested are:

• Uniaxial tension, elastic regime

• Uniaxial tension, plastic regime

• Uniaxial compression, elastic regime

• Uniaxial compression, plastic regime

• Biaxial tension-compression, elastic regime

• Biaxial tension-compression, plastic regime

4.37.3.1.1. Overview

Given the mechanical configuration, the final stress state is known. Coupled with the TCS, this leads to the derivation of effective elastic moduli (such that the desired stress state is achieved even with the degradation function’s impact). In the elastic case, \(\bar{\epsilon}^p \equiv 0\) and \(F^p_{ij} \equiv \delta_{ij}\). In the plastic case, \(\bar{\epsilon}^p = 0.05\) and \(F^p_{ij} = \left[\exp\left(0.05[N^p_{mn}]\right)\right]_{ij}\). All of this is used to solve for the elastic strain such that the generalized yield equation (4.270) is satisfied. A quantity \(C_p\) is defined in (4.276) that groups together all of the plastic quantities.

(4.276)\[C_p = \frac{\partial\tilde{\psi}^p}{\partial F^p_{ij}}\frac{\partial F^p_{ij}}{\partial\bar{\epsilon}^p} + \frac{\partial\tilde{\psi}^p}{\partial\bar{\epsilon}^p} + \Delta t \frac{d\tilde{\psi}_p^{*}}{d\bar{\epsilon}^p}\]

Additionally, there are several code branches which are all tested:

• Radial Return

• SQP with flow tensor norm, trace, and symmetry constraints

• SQP with flow tensor constraint (setting the flow tensor explicitly)

4.37.3.1.2. Uniaxial Tension/Compression

In the case of Uniaxial Tension or Compression, the log strain is given in (4.277) and the stress state is given in (4.278), where \(\Upsilon > 0\) for uniaxial tension and \(\Upsilon < 0\) for uniaxial compression. \(E^{*}\) and \(\nu^{*}\) are a modified Young’s modulus and Poisson’s ratio (respectively) derived based on the specific tension-compression split.

(4.277)\[\begin{split}[E^e_{ij}] = \begin{pmatrix}\Upsilon & 0 & 0 \\ 0 & -\nu^{*}\Upsilon & 0 \\ 0 & 0 & -\nu^{*}\Upsilon\end{pmatrix}\end{split}\]

(4.278)\[\begin{split}\frac{\partial\tilde{\psi}^e}{\partial[E^e_{ij}]} = E^{*}\Upsilon\begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}\end{split}\]

The flow tensor \(N^p_{ij}\) is given by (4.279) for tension and (4.280) for compression.

(4.279)\[\begin{split}[N^p_{ij}] = \begin{pmatrix}1 & 0 & 0 \\ 0 & -0.5 & 0 \\ 0 & 0 & -0.5\end{pmatrix}\end{split}\]

(4.280)\[\begin{split}[N^p_{ij}] = \begin{pmatrix}-1 & 0 & 0 \\ 0 & 0.5 & 0 \\ 0 & 0 & 0.5\end{pmatrix}\end{split}\]

Solving the yield equation yields the value of \(\Upsilon\) ((4.281)). As mentioned above, \(\Upsilon\) is positive in uniaxial tension and negative in uniaxial compression.

(4.281)\[\Upsilon = \pm\frac{g^pC_p}{E^{*}}\]

4.37.3.1.3. Biaxial Tension-Compression

In the biaxial tension-compression case, the log strain is given by (4.282) and the stress state is given by (4.283), where \(\mu^{*}\) is a modified shear modulus derived based on the specific tension-compression split. Likewise, the flow tensor is given by (4.284).

(4.282)\[\begin{split}[E^e_{ij}] = \begin{pmatrix}\Upsilon & 0 & 0 \\ 0 & -\Upsilon & 0 \\ 0 & 0 & 0\end{pmatrix}\end{split}\]

(4.283)\[\begin{split}\frac{\partial\tilde{\psi}^e}{\partial[E^e_{ij}]} = 2\mu^{*}\Upsilon\begin{pmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0\end{pmatrix}\end{split}\]

(4.284)\[\begin{split}[N^p_{ij}] = \frac{\sqrt{3}}{2}\begin{pmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0\end{pmatrix}\end{split}\]

Putting it all together and solving the yield equation yields the value of \(\Upsilon\) ((4.285)).

(4.285)\[\Upsilon = \frac{g^pC_p}{2 \sqrt{3}\mu^{*}}\]

4.37.3.2. Regression testing

There are a variety of regression tests for this material model that test different kinds of behavior. The 1d_phaseBC_fefp and control_1d_extension tests are identical to the tests of the same name implemented for Phase Field F^eF^p, so refer to the relevant sections under that documentation for more information.

4.37.4. User Guide

BEGIN PARAMETERS FOR MODEL PHASE_FIELD_FEFP_MODULAR
  # Overall options
  ELASTIC POTENTIAL = HENCKY_ELASTIC_POTENTIAL
                    (HENCKY_ELASTIC_POTENTIAL)
  PLASTIC POTENTIAL = NO_PLASTIC_POTENTIAL
                    | LINEAR_PLASTIC_POTENTIAL
                    | POWER_LAW_PLASTIC_POTENTIAL
                    | POWER_LAW_SIMPLE_PLASTIC_POTENTIAL
                    | VOCE_PLASTIC_POTENTIAL
                    | JOHNSON_COOK_PLASTIC_POTENTIAL
  DAMAGE POTENTIAL = NO_DAMAGE_POTENTIAL
                   | GRADIENT_DAMAGE_POTENTIAL
                   | CLASSICAL_GRADIENT_DAMAGE_POTENTIAL
                   | WU_DAMAGE_POTENTIAL
                   (GRADIENT_DAMAGE_POTENTIAL)
  ELASTIC KINETIC POTENTIAL = NO_ELASTIC_KINETIC_POTENTIAL
                            (NO_ELASTIC_KINETIC_POTENTIAL)
  PLASTIC KINETIC POTENTIAL = NO_PLASTIC_KINETIC_POTENTIAL
                            | POWER_LAW_PLASTIC_KINETIC_POTENTIAL
                            | POWER_LAW_BREAKDOWN_PLASTIC_KINETIC_POTENTIAL
                            | JOHNSON_COOK_PLASTIC_KINETIC_POTENTIAL
                            (NO_PLASTIC_KINETIC_POTENTIAL)
  DAMAGE KINETIC POTENTIAL = NO_DAMAGE_KINETIC_POTENTIAL
                           (NO_DAMAGE_KINETIC_POTENTIAL)
  ELASTIC DEGRADATION FUNCTION = NO_DEGRADATION_FUNCTION
                               | QUADRATIC_DEGRADATION_FUNCTION
                               | TALAMINI_DEGRADATION_FUNCTION
                               | LO_DEGRADATION_FUNCTION
                               | WU_DEGRADATION_FUNCTION
                               (TALAMINI_DEGRADATION_FUNCTION)
  PLASTIC DEGRADATION FUNCTION = NO_DEGRADATION_FUNCTION
                               | QUADRATIC_DEGRADATION_FUNCTION
                               | TALAMINI_DEGRADATION_FUNCTION
                               | LO_DEGRADATION_FUNCTION
                               | WU_DEGRADATION_FUNCTION
                               (TALAMINI_DEGRADATION_FUNCTION)
  TENSION COMPRESSION SPLIT = NO_TCS
                            | DEVIATORIC_VOLUMETRIC_TCS
                            | STAR_CONVEX_TCS
                            (NO_TCS)
  THERMAL SOFTENING MODEL = ADIABATIC | COUPLED
  PLASTIC WORK DRIVING ENERGY PORTION = <real> (1.0)
  CONDITIONING COEFFICIENT = <real> (0.0)
  NONLOCAL SOLVER = NONE | ARIA | RXNDIFF (RXNDIFF)
  PLASTIC SOLVER = RADIAL_RETURN | SQP (RADIAL_RETURN)
  CONSTRAINTS = FLOW_TENSOR_CONSTRAINT
              | FLOW_TENSOR_TRACE_CONSTRAINT
              | FLOW_TENSOR_NORM_CONSTRAINT
              | FLOW_TENSOR_SYMMETRIC_CONSTRAINT_XY
              | FLOW_TENSOR_SYMMETRIC_CONSTRAINT_YZ
              | FLOW_TENSOR_SYMMETRIC_CONSTRAINT_ZX
  FLOW TENSOR = <reals>
  KINETIC CONSTRAINTS = INTERNAL_VARIABLE_RATE_KINETIC_CONSTRAINT

  # Hencky Elastic Potential properties
  YOUNGS MODULUS          = <real>
  POISSONS RATIO          = <real>
  BULK MODULUS            = <real>
  SHEAR MODULUS           = <real>
  YOUNGS MODULUS FUNCTION = <string>
  POISSONS RATIO FUNCTION = <string>
  BULK MODULUS FUNCTION   = <string>
  SHEAR MODULUS FUNCTION  = <string>

  # Linear Plastic Potential properties
  YIELD STRESS               = <real>
  HARDENING MODULUS          = <real>
  TAYLOR QUINNEY             = <real>
  YIELD STRESS FUNCTION      = <string>
  HARDENING MODULUS FUNCTION = <string>
  TAYLOR QUINNEY FUNCTION    = <string>

  # Power Law Plastic Potential properties
  YIELD STRESS                = <real>
  HARDENING CONSTANT          = <real>
  HARDENING EXPONENT          = <real>
  LUDERS STRAIN               = <real>
  TAYLOR QUINNEY              = <real>
  YIELD STRESS FUNCTION       = <string>
  HARDENING CONSTANT FUNCTION = <string>
  HARDENING EXPONENT FUNCTION = <string>
  LUDERS STRAIN FUNCTION      = <string>
  TAYLOR QUINNEY FUNCTION     = <string>

  # Power Law Simple Plastic Potential properties
  REFERENCE STRAIN            = <real>
  YIELD STRESS                = <real>
  HARDENING EXPONENT          = <real>
  TAYLOR QUINNEY              = <real>
  REFERENCE STRAIN FUNCTION   = <string>
  YIELD STRESS FUNCTION       = <string>
  HARDENING EXPONENT FUNCTION = <string>
  TAYLOR QUINNEY FUNCTION     = <string>

  # Voce Plastic Potential properties
  YIELD STRESS                     = <real>
  HARDENING MODULUS                = <reals>
  EXPONENTIAL COEFFICIENT          = <reals>
  TAYLOR QUINNEY                   = <real>
  YIELD STRESS FUNCTION            = <string>
  HARDENING MODULUS FUNCTION       = <string>
  EXPONENTIAL COEFFICIENT FUNCTION = <string>
  TAYLOR QUINNEY FUNCTION          = <string>

  # Johnson-Cook Plastic Potential properties
  JOHNSON COOK A          = <real>
  JOHNSON COOK B          = <real>
  JOHNSON COOK N          = <real>
  JOHNSON COOK Q          = <real>
  JOHNSON COOK T0         = <real>
  JOHNSON COOK TM         = <real>
  TAYLOR QUINNEY          = <real>
  JOHNSON COOK A FUNCTION = <string>
  JOHNSON COOK B FUNCTION = <string>
  JOHNSON COOK N FUNCTION = <string>
  JOHNSON COOK Q FUNCTION = <string>
  TAYLOR QUINNEY FUNCTION = <string>

  # Gradient Damage Potential properties
  GC                             = <real>
  FRACTURE LENGTH SCALE          = <real>
  ANISOTROPY TENSOR 11           = <real> (1.0)
  ANISOTROPY TENSOR 12           = <real> (0.0)
  ANISOTROPY TENSOR 13           = <real> (0.0)
  ANISOTROPY TENSOR 21           = <real> (0.0)
  ANISOTROPY TENSOR 22           = <real> (1.0)
  ANISOTROPY TENSOR 23           = <real> (0.0)
  ANISOTROPY TENSOR 31           = <real> (0.0)
  ANISOTROPY TENSOR 32           = <real> (0.0)
  ANISOTROPY TENSOR 33           = <real> (1.0)
  GC FUNCTION                    = <string>
  FRACTURE LENGTH SCALE FUNCTION = <string>

  # Classical Gradient Damage Potential properties
  GC                             = <real>
  FRACTURE LENGTH SCALE          = <real>
  ANISOTROPY TENSOR 11           = <real> (1.0)
  ANISOTROPY TENSOR 12           = <real> (0.0)
  ANISOTROPY TENSOR 13           = <real> (0.0)
  ANISOTROPY TENSOR 21           = <real> (0.0)
  ANISOTROPY TENSOR 22           = <real> (1.0)
  ANISOTROPY TENSOR 23           = <real> (0.0)
  ANISOTROPY TENSOR 31           = <real> (0.0)
  ANISOTROPY TENSOR 32           = <real> (0.0)
  ANISOTROPY TENSOR 33           = <real> (1.0)
  GC FUNCTION                    = <string>
  FRACTURE LENGTH SCALE FUNCTION = <string>

  # Wu Damage Potential properties
  GC                             = <real>
  FRACTURE LENGTH SCALE          = <real>
  WU XI                          = <real>
  ANISOTROPY TENSOR 11           = <real> (1.0)
  ANISOTROPY TENSOR 12           = <real> (0.0)
  ANISOTROPY TENSOR 13           = <real> (0.0)
  ANISOTROPY TENSOR 21           = <real> (0.0)
  ANISOTROPY TENSOR 22           = <real> (1.0)
  ANISOTROPY TENSOR 23           = <real> (0.0)
  ANISOTROPY TENSOR 31           = <real> (0.0)
  ANISOTROPY TENSOR 32           = <real> (0.0)
  ANISOTROPY TENSOR 33           = <real> (1.0)
  GC FUNCTION                    = <string>
  FRACTURE LENGTH SCALE FUNCTION = <string>
  WU XI FUNCTION                 = <string>

  # Power Law Plastic Kinetic Potential properties
  REFERENCE STRAIN RATE          = <real>
  YIELD STRESS                   = <real>
  RATE EXPONENT                  = <real>
  TAYLOR QUINNEY                 = <real>
  REFERENCE STRAIN RATE FUNCTION = <string>
  YIELD STRESS FUNCTION          = <string>
  RATE EXPONENT FUNCTION         = <string>
  TAYLOR QUINNEY FUNCTION        = <string>

  # Johnson-Cook Plastic Kinetic Potential properties
  REFERENCE STRAIN RATE          = <real>
  JOHNSON COOK C                 = <real>
  TAYLOR QUINNEY                 = <real>
  REFERENCE STRAIN RATE FUNCTION = <string>
  JOHNSON COOK C FUNCTION        = <string>
  TAYLOR QUINNEY FUNCTION        = <string>

  # Power Law Breakdown Plastic Kinetic Potential properties
  RATE COEFFICIENT          = <real>
  RATE EXPONENT             = <real>
  TAYLOR QUINNEY            = <real>
  RATE COEFFICIENT FUNCTION = <string>
  RATE EXPONENT FUNCTION    = <string>
  TAYLOR QUINNEY FUNCTION   = <string>

  # Talamini Degradation Function properties
  GC                               = <real>
  FRACTURE LENGTH SCALE            = <real>
  CRITICAL ENERGY DENSITY          = <real>
  GC FUNCTION                      = <string>
  FRACTURE LENGTH SCALE FUNCTION   = <string>
  CRITICAL ENERGY DENSITY FUNCTION = <string>

  # Lo Degradation Function properties
  LO S          = <real>
  LO S FUNCTION = <string>

  # Wu Degradation Function properties
  WU P            = <real>
  WU COEFFICIENTS = <reals>
  WU P FUNCTION   = <string>

  # Star-Convex TCS properties
  STAR CONVEX GAMMA = <real>

END [PARAMETERS FOR MODEL PHASE_FIELD_FEFP_MODULAR]

4.37.4.1. Model Parameters

4.37.4.1.1. Elastic Potentials

Table 4.66 Property Inputs for HENCKY_ELASTIC_POTENTIAL

Name	Description
YOUNGS_MODULUS	Young’s modulus \(E\)
POISSONS_RATIO	Poisson’s ratio \(\nu\)
BULK_MODULUS	Bulk modulus \(\kappa\)
SHEAR_MODULUS	Shear modulus \(\mu\)
YOUNGS_MODULUS_FUNCTION	\(E\)’s temperature-dependent function
POISSONS_RATIO_FUNCTION	\(\nu\)’s temperature-dependent function
BULK_MODULUS_FUNCTION	\(\kappa\)’s temperature-dependent function
SHEAR_MODULUS_FUNCTION	\(\mu\)’s temperature-dependent function

Notes:

• Any two of the above four properties can be specified.

• If temperature-dependent functions are specified, two of the four should be specified; the other properties’ temperature dependence will be automatically calculated.

4.37.4.1.2. Plastic Potentials

Table 4.67 Property Inputs for LINEAR_PLASTIC_POTENTIAL

Name	Description
YIELD_STRESS	Yield stress \(\sigma_y\)
HARDENING_MODULUS	Hardening modulus \(h\)
TAYLOR_QUINNEY	Taylor-Quinney factor \(\beta_Q\)
YIELD_STRESS_FUNCTION	\(\sigma_y\)’s temperature-dependent function
HARDENING_MODULUS_FUNCTION	\(h\)’s temperature-dependent function
TAYLOR_QUINNEY_FUNCTION	\(\beta_Q\)’s temperature-dependent function

Table 4.68 Property Inputs for POWER_LAW_PLASTIC_POTENTIAL

Name	Description
YIELD_STRESS	Yield stress \(\sigma_y\)
HARDENING_CONSTANT	Hardening constant \(a\)
HARDENING_EXPONENT	Hardening exponent \(b\)
LUDERS_STRAIN	Lüder’s strain \(\epsilon_{\ell}\)
TAYLOR_QUINNEY	Taylor-Quinney factor \(\beta_Q\)
YIELD_STRESS_FUNCTION	\(\sigma_y\)’s temperature-dependent function
HARDENING_CONSTANT_FUNCTION	\(a\)’s temperature-dependent function
HARDENING_EXPONENT_FUNCTION	\(b\)’s temperature-dependent function
LUDERS_STRAIN_FUNCTION	\(\epsilon_{\ell}\)’s temperature-dependent function
TAYLOR_QUINNEY_FUNCTION	\(\beta_Q\)’s temperature-dependent function

Table 4.69 Property Inputs for POWER_LAW_SIMPLE_PLASTIC_POTENTIAL

Name	Description
REFERENCE_STRAIN	Reference strain \(\epsilon_0\)
YIELD_STRESS	Yield stress \(\sigma_y\)
HARDENING_EXPONENT	Hardening exponent \(n\)
TAYLOR_QUINNEY	Taylor-Quinney factor \(\beta_Q\)
REFERENCE_STRAIN_FUNCTION	\(\epsilon_0\)’s temperature-dependent function
YIELD_STRESS_FUNCTION	\(\sigma_y\)’s temperature-dependent function
HARDENING_EXPONENT_FUNCTION	\(n\)’s temperature-dependent function
TAYLOR_QUINNEY_FUNCTION	\(\beta_Q\)’s temperature-dependent function

Table 4.70 Property Inputs for VOCE_PLASTIC_POTENTIAL

Name	Description
YIELD_STRESS	Yield stress \(\sigma_y\)
HARDENING_MODULUS	Hardening moduli \(h_i\)
EXPONENTIAL_COEFFICIENT	Exponential coefficients \(\alpha_i\)
TAYLOR_QUINNEY	Taylor-Quinney factor \(\beta_Q\)
YIELD_STRESS_FUNCTION	\(\sigma_y\)’s temperature-dependent function
HARDENING_MODULUS_FUNCTION	\(h_i\)’s temperature-dependent function
EXPONENTIAL_COEFFICIENT_FUNCTION	\(\alpha_i\)’s temperature-dependent function
TAYLOR_QUINNEY_FUNCTION	\(\beta_Q\)’s temperature-dependent function

Table 4.71 Property Inputs for JOHNSON_COOK_PLASTIC_POTENTIAL

Name	Description
JOHNSON_COOK_A	Johnson-Cook yield stress \(a\)
JOHNSON_COOK_B	Johnson-Cook hardening modulus \(b\)
JOHNSON_COOK_N	Johnson-Cook hardening exponent \(n\)
JOHNSON_COOK_Q	Johnson-Cook thermal exponent \(q\)
JOHNSON_COOK_T0	Johnson-Cook reference temperature \(T_0\)
JOHNSON_COOK_TM	Johnson-Cook melting temperature \(T_m\)
TAYLOR_QUINNEY	Taylor-Quinney factor \(\beta_Q\)
JOHNSON_COOK_A_FUNCTION	\(a\)’s temperature-dependent function
JOHNSON_COOK_B_FUNCTION	\(b\)’s temperature-dependent function
JOHNSON_COOK_N_FUNCTION	\(n\)’s temperature-dependent function
JOHNSON_COOK_Q_FUNCTION	\(q\)’s temperature-dependent function
TAYLOR_QUINNEY_FUNCTION	\(\beta_Q\)’s temperature-dependent function

Notes:

• If the Taylor-Quinney factor is not explicitly set, it defaults to 0.

• If the Lüder’s strain is not explicitly set, it defaults to 0.

• In the Voce case, HARDENING_MODULUS and EXPONENTIAL_COEFFICIENT can be lists of parameters to add multiple Voce hardening terms, but they should both have the same length (if HARDENING_MODULUS has 3 terms, you must also have 3 terms for EXPONENTIAL_COEFFICIENT). The HARDENING_MODULUS_FUNCTION is one function (as is EXPONENTIAL_COEFFICIENT_FUNCTION) and is applied to all Voce hardening terms.

4.37.4.1.3. Damage Potentials

Table 4.72 Property Inputs for GRADIENT_DAMAGE_POTENTIAL

Name	Description
GC	Griffith fracture energy \(G_c\)
FRACTURE_LENGTH_SCALE	Fracture length scale \(\ell\)
ANISOTROPY_TENSOR_{1..3}{1..3}	Components of anisotropy tensor \(a_{ij}\)
GC_FUNCTION	\(G_c\)’s temperature-dependent function
FRACTURE_LENGTH_SCALE_FUNCTION	\(\ell\)’s temperature-dependent function

Table 4.73 Property Inputs for CLASSICAL_GRADIENT_DAMAGE_POTENTIAL

Name	Description
GC	Griffith fracture energy \(G_c\)
FRACTURE_LENGTH_SCALE	Fracture length scale \(\ell\)
ANISOTROPY_TENSOR_{1..3}{1..3}	Components of anisotropy tensor \(a_{ij}\)
GC_FUNCTION	\(G_c\)’s temperature-dependent function
FRACTURE_LENGTH_SCALE_FUNCTION	\(\ell\)’s temperature-dependent function

Table 4.74 Property Inputs for WU_DAMAGE_POTENTIAL

Name	Description
GC	Griffith fracture energy \(G_c\)
FRACTURE_LENGTH_SCALE	Fracture length scale \(\ell\)
WU_XI	\(\xi\) parameter
ANISOTROPY_TENSOR_{1..3}{1..3}	Components of anisotropy tensor \(a_{ij}\)
GC_FUNCTION	\(G_c\)’s temperature-dependent function
FRACTURE_LENGTH_SCALE_FUNCTION	\(\ell\)’s temperature-dependent function
WU_XI_FUNCTION	\(\xi\)’s temperature-dependent function

4.37.4.1.4. Plastic Kinetic Potentials

Table 4.75 Property Inputs for POWER_LAW_PLASTIC_KINETIC_POTENTIAL

Name	Description
REFERENCE_STRAIN_RATE	Reference strain rate \(\dot{\epsilon}_0\)
YIELD_STRESS	Yield stress \(\sigma_y\)
RATE_EXPONENT	Rate exponent \(m\)
TAYLOR_QUINNEY	Taylor-Quinney factor \(\beta_Q\)
REFERENCE_STRAIN_RATE_FUNCTION	\(\dot{\epsilon}_0\)’s temperature-dependent function
YIELD_STRESS_FUNCTION	\(\sigma_y\)’s temperature-dependent function
RATE_EXPONENT_FUNCTION	\(m\)’s temperature-dependent function
TAYLOR_QUINNEY_FUNCTION	\(\beta_Q\)’s temperature-dependent function

Table 4.76 Property Inputs for JOHNSON_COOK_PLASTIC_KINETIC_POTENTIAL

Name	Description
REFERENCE_STRAIN_RATE	Reference strain rate \(\dot{\epsilon}_0\)
JOHNSON_COOK_C	Johnson-Cook rate dependence strength \(c\)
TAYLOR_QUINNEY	Taylor-Quinney factor \(\beta_Q\)
REFERENCE_STRAIN_RATE_FUNCTION	\(\dot{\epsilon}_0\)’s temperature-dependent function
JOHNSON_COOK_C_FUNCTION	\(c\)’s temperature-dependent function
TAYLOR_QUINNEY_FUNCTION	\(\beta_Q\)’s temperature-dependent function

Table 4.77 State Variables for POWER_LAW_BREAKDOWN_PLASTIC_KINETIC_POTENTIAL

Name	Description
RATE_COEFFICIENT	Rate coefficient \(\gamma\)
RATE_EXPONENT	Rate exponent \(m\)
TAYLOR_QUINNEY	Taylor-Quinney factor \(\beta_Q\)
RATE_COEFFICIENT_FUNCTION	\(\gamma\)’s temperature-dependent function
RATE_EXPONENT_FUNCTION	\(m\)’s temperature-dependent function
TAYLOR_QUINNEY_FUNCTION	\(\beta_Q\)’s temperature-dependent function

4.37.4.1.5. Degradation Functions

Table 4.78 Property Inputs for TALAMINI_DEGRADATION_FUNCTION

Name	Description
GC	Griffith fracture energy \(G_c\)
FRACTURE_LENGTH_SCALE	Fracture length scale \(\ell\)
CRITICAL_ENERGY_DENSITY	Critical energy density \(\psi_c\)
GC_FUNCTION	\(G_c\)’s temperature-dependent function
FRACTURE_LENGTH_SCALE_FUNCTION	\(\ell\)’s temperature-dependent function
CRITICAL_ENERGY_DENSITY_FUNCTION	\(\psi_c\)’s temperature-dependent function

Table 4.79 Property Inputs for LO_DEGRADATION_FUNCTION

Name	Description
LO_S	Lo \(s\) parameter
LO_S_FUNCTION	\(s\)’s temperature-dependent function

Table 4.80 Property Inputs for WU_DEGRADATION_FUNCTION

Name	Description
WU_P	Wu \(p\) parameter
WU_COEFFICIENTS	Wu degradation function coefficients \(a_i\)
WU_P_FUNCTION	\(p\)’s temperature-dependent function

Notes:

• If CRITICAL_ENERGY_DENSITY is not defined, it is defaulted to \(\frac{3G_c}{16\ell}\).

4.37.4.1.6. Tension-Compression Splits

Table 4.81 Property Inputs for STAR_CONVEX_TCS

Name	Description
STAR_CONVEX_GAMMA	Star-Convex \(\gamma\) parameter

4.37.4.2. State Variables

4.37.4.2.1. Elastic Potentials

Table 4.82 State Variables for HENCKY_ELASTIC_POTENTIAL

Name	Description
YOUNGS_MODULUS	\(E\)
POISSONS_RATIO	\(\nu\)
BULK_MODULUS	\(\kappa\)
SHEAR_MODULUS	\(\mu\)

4.37.4.2.2. Plastic Potentials

Table 4.83 State Variables for LINEAR_PLASTIC_POTENTIAL

Name	Description
YIELD_STRESS	\(\sigma_y\)
HARDENING_MODULUS	\(h\)
TAYLOR_QUINNEY	\(\beta_Q\)
EQPS	\(\bar{\epsilon}^p\)
EQPSDOT	\(\dot{\bar{\epsilon}}^p\)
FP	\(F^p_{ij}\)

Table 4.84 State Variables for POWER_LAW_PLASTIC_POTENTIAL

Name	Description
YIELD_STRESS	\(\sigma_y\)
HARDENING_CONSTANT	\(a\)
HARDENING_EXPONENT	\(b\)
LUDERS_STRAIN	\(\epsilon_{\ell}\)
TAYLOR_QUINNEY	\(\beta_Q\)
EQPS	\(\bar{\epsilon}^p\)
EQPSDOT	\(\dot{\bar{\epsilon}}^p\)
FP	\(F^p_{ij}\)

Table 4.85 State Variables for POWER_LAW_SIMPLE_PLASTIC_POTENTIAL

Name	Description
REFERENCE_STRAIN	\(\epsilon_0\)
YIELD_STRESS	\(\sigma_y\)
HARDENING_EXPONENT	\(n\)
TAYLOR_QUINNEY	\(\beta_Q\)
EQPS	\(\bar{\epsilon}^p\)
EQPSDOT	\(\dot{\bar{\epsilon}}^p\)
FP	\(F^p_{ij}\)

Table 4.86 State Variables for VOCE_PLASTIC_POTENTIAL

Name	Description
YIELD_STRESS	\(\sigma_y\)
HARDENING_MODULUS_*	\(h_i\)
EXPONENTIAL_COEFFICIENT_*	\(\alpha_i\)
TAYLOR_QUINNEY	\(\beta_Q\)
EQPS	\(\bar{\epsilon}^p\)
EQPSDOT	\(\dot{\bar{\epsilon}}^p\)
FP	\(F^p_{ij}\)

Table 4.87 State Variables for JOHNSON_COOK_PLASTIC_POTENTIAL

Name	Description
JOHNSON_COOK_A	\(a\)
JOHNSON_COOK_B	\(b\)
JOHNSON_COOK_N	\(n\)
JOHNSON_COOK_Q	\(q\)
JOHNSON_COOK_THETA	\(\theta\)
JOHNSON_COOK_DTHETADT	\(\td{\theta}{T}\)
TAYLOR_QUINNEY	\(\beta_Q\)
EQPS	\(\bar{\epsilon}^p\)
EQPSDOT	\(\dot{\bar{\epsilon}}^p\)
FP	\(F^p_{ij}\)

4.37.4.2.3. Damage Potentials

Table 4.88 State Variables for GRADIENT_DAMAGE_POTENTIAL

Name	Description
GC	\(G_c\)
FRACTURE_LENGTH_SCALE	\(\ell\)

Table 4.89 State Variables for CLASSICAL_GRADIENT_DAMAGE_POTENTIAL

Name	Description
GC	\(G_c\)
FRACTURE_LENGTH_SCALE	\(\ell\)

Table 4.90 State Variables for WU_DAMAGE_POTENTIAL

Name	Description
GC	\(G_c\)
FRACTURE_LENGTH_SCALE	\(\ell\)
XI	\(\xi\)

4.37.4.2.4. Plastic Kinetic Potentials

Table 4.91 State Variables for POWER_LAW_PLASTIC_KINETIC_POTENTIAL

Name	Description
REFERENCE_STRAIN_RATE	\(\dot{\epsilon}_0\)
YIELD_STRESS	\(\sigma_y\)
RATE_EXPONENT	\(m\)
TAYLOR_QUINNEY	\(\beta_Q\)
EQPS	\(\bar{\epsilon}^p\)
EQPSDOT	\(\dot{\bar{\epsilon}}^p\)
FP	\(F^p_{ij}\)

Table 4.92 State Variables for JOHNSON_COOK_PLASTIC_KINETIC_POTENTIAL

Name	Description
REFERENCE_STRAIN_RATE	\(\dot{\epsilon}_0\)
JOHNSON_COOK_C	\(c\)
TAYLOR_QUINNEY	\(\beta_Q\)
EQPS	\(\bar{\epsilon}^p\)
EQPSDOT	\(\dot{\bar{\epsilon}}^p\)
FP	\(F^p_{ij}\)

Table 4.93 State Variables for POWER_LAW_BREAKDOWN_PLASTIC_KINETIC_POTENTIAL

Name	Description
RATE_COEFFICIENT	\(\gamma\)
RATE_EXPONENT	\(m\)
TAYLOR_QUINNEY	\(\beta_Q\)
EQPS	\(\bar{\epsilon}^p\)
EQPSDOT	\(\dot{\bar{\epsilon}}^p\)
FP	\(F^p_{ij}\)

4.37.4.2.5. Degradation Functions

Table 4.94 State Variables for TALAMINI_DEGRADATION_FUNCTION

Name	Description
GC	\(G_c\)
FRACTURE_LENGTH_SCALE	\(\ell\)
CRITICAL_ENERGY_DENSITY	\(\psi_c\)

Table 4.95 State Variables for LO_DEGRADATION_FUNCTION

Name	Description
LO_S	\(s\)

Table 4.96 State Variables for WU_DEGRADATION_FUNCTION

Name	Description
WU_P	\(p\)