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### 5.7.2 Kinematic “Hardening” Response

The framework presented in Section 5.7.1↑ has considered an elastic-perfectly plastic response, i.e., all the bonds yield when a single yield threshold \Phi_{m} is met. However, a wealth of experimental results show a more progressive yielding, rather than a sudden onset, and an increase in the stress with increasing plastic deformation, a phenomenon alternately termed strain hardening or work hardening. Real materials typically exhibit the Bauschinger effect, where loading to yield in one direction changes the yield threshold in the reverse direction. The hardening behavior that accounts for this effect is known as kinematic hardening; for a load reversal, it predicts yielding occurs when the change in load achieves twice the monotonic yield strength. The reactive plasticity framework can be extended to allow for kinematic hardening by introducing multiple families of bonds. In the current FEBio implementation each bond family \beta shares the same yield criterion \Phi but distinct associated yield thresholds \Phi_{m\beta} , and it follows the elastic-perfectly plastic behavior for multiple generations outlined in Section 5.7.1↑. The superposition of multiple bond families \beta in parallel naturally develops behavior consistent with kinematic hardening, as different bond families yield at different thresholds.

We consider n_{f} \beta=0,\dots,n_{f}-1 , which may yield under different thresholds, where each bond family may evolve over multiple generations \sigma . This framework requires us to update our notation to include a subscript \beta for suitable variables introduced in the presentation above. In particular, the reference configuration of generation \sigma in bond family \beta is now denoted by \mathbf{X}_{\beta}^{\sigma} and the corresponding deformation gradient is \mathbf{F}_{\beta}^{\sigma} . We assume that the free energy density of each bond family \beta is \Psi_{\beta r}=J_{\beta}^{\sigma s}\Psi_{0}\left(\mathbf{F}_{\beta}^{\sigma}\right) , when the mixture consists entirely of bonds of family \beta in generation \sigma . The master reference configuration of all bond families remains \mathbf{X}^{s} and the associated (total) deformation gradient is still \mathbf{F}^{s} . Therefore, each bond family \beta requires a constitutive relation for the function of state \mathbf{F}_{\beta}^{\sigma s} in the updated form of eq.(2.8.1-2), such as that given in eq.(5.7.1-4), where each term should now include a subscript \beta .

*bond families* The referential mass density of bond family \beta is \rho_{r\beta} , such that the mixture referential mass density is given by \rho_{r}=\sum_{\beta}\rho_{r\beta} . The referential mass density of generation \sigma in bond family \beta is \rho_{r\beta}^{\sigma} , which satisfies \sum_{\sigma}\rho_{r\beta}^{\sigma}=\rho_{r\beta} , as per eq.(2.8.2-3). For convenience, we define \begin{equation} w_{\beta}\equiv\frac{\rho_{r\beta}}{\rho_{r}}\,,\quad\sum_{\beta}w_{\beta}=1\,,\label{eq:multiplebonds-family-mass-balance} \end{equation} which represents the mass fraction of each bond family \beta within the constrained solid mixture, and \begin{equation} w_{\beta}^{\sigma}=\frac{\rho_{r\beta}^{\sigma}}{\rho_{r\beta}}\,,\quad\sum_{\sigma}w_{\beta}^{\sigma}=1\,,\label{eq:multiplebonds-bond-species-mass-fractions} \end{equation} which represents the mass fraction of each generation \sigma within the bond family \beta . From these definitions, it follows that bond family mass fractions w_{\beta} are time-invariant (thus user-selected for a particular material response), whereas generation mass fractions w_{\beta}^{\sigma} evolve with bond-breaking-and-reforming reactions.

The mixture strain energy density \Psi_{r} is now given by \begin{equation} \Psi_{r}=\sum_{\beta}w_{\beta}\sum_{\sigma}w_{\beta}^{\sigma}J_{\beta}^{\sigma s}\Psi_{0}\left(\mathbf{F}_{\beta}^{\sigma}\right)\,,\label{eq:multiplebonds-mixture-energy} \end{equation} whereas the mixture stress is \begin{equation} \boldsymbol{\sigma}=\sum_{\beta}w_{\beta}\underbrace{\sum_{\sigma}w_{\beta}^{\sigma}\boldsymbol{\sigma}_{0}\left(\mathbf{F}_{\beta}^{\sigma}\right)}_{\boldsymbol{\sigma}_{\beta}}\,,\quad\boldsymbol{\sigma}_{0}\left(\mathbf{F}_{\beta}^{\sigma}\right)=\frac{1}{J_{\beta}^{\sigma}}\frac{\partial\Psi_{0}}{\partial\mathbf{F}_{\beta}^{\sigma}}\cdot\left(\mathbf{F}_{\beta}^{\sigma}\right)^{T}\,.\label{eq:multiplebonds-mixture-stress} \end{equation} To simplify the remainder of this presentation, we introduce the concept of y , to represent bonds of the current extant generation in a plasticity formulation. The yielded bond fraction for each family \beta is given by \begin{equation} w_{\beta}^{y}=\sum_{\sigma\neq s}w_{\beta}^{\sigma}=1-w_{\beta}^{s}\label{eq:hardening-yielded-mass-fraction} \end{equation} where the summation runs over all possible yielded generations \sigma\neq s . In particular, at time t=u , eq.(5.7.2-5) reduces to the statement w_{\beta}^{y}=w_{\beta}^{u} . We then define the relative deformation gradient of yielded bonds as \mathbf{F}_{\beta}^{y} , which equals \mathbf{F}_{\beta}^{\sigma} for the extant generation \sigma in family \beta . With these notational changes, we may write the yielding reactions in the form \begin{equation} \mathcal{E}^{s}\to\underbrace{\mathcal{E}^{u}\rightarrow\mathcal{E}^{v}\rightarrow\dots}_{\mathcal{E}^{y}}\,.\label{eq:hardening-yielded-rxn} \end{equation} Then the mixture stress in eq.(5.7.2-4) may be rewritten as \boldsymbol{\sigma}=\sum_{\beta}w_{\beta}\boldsymbol{\sigma}_{\beta} where \boldsymbol{\sigma}_{\beta}=w_{\beta}^{s}\boldsymbol{\sigma}_{0}\left(\mathbf{F}^{s}\right)+\left(1-w_{\beta}^{s}\right)\boldsymbol{\sigma}_{0}\left(\mathbf{F}_{\beta}^{y}\right) . We may also define the total fraction w^{s} of intact bonds in the mixture as w^{s}=\sum_{\beta}w_{\beta}w_{\beta}^{s} , and the total fraction of yielded bonds as w^{y}=\sum_{\beta}w_{\beta}w_{\beta}^{y}=1-w^{s} , such that w^{s}+w^{y}=1 . Then \boldsymbol{\sigma}=w^{s}\boldsymbol{\sigma}_{0}\left(\mathbf{F}^{s}\right)+\sum_{\beta}w_{\beta}\left(1-w_{\beta}^{s}\right)\boldsymbol{\sigma}_{0}\left(\mathbf{F}_{\beta}^{y}\right) . The summation in this last expression does not simplify further since \mathbf{F}_{\beta}^{y} is different for each bond family \beta .

*yielded bonds*, denoted by Let each bond family \beta exhibit an elastic-perfectly plastic response, following the model of Section 5.7.1↑. Once the yield threshold \Phi_{m\beta} is reached, all the intact bonds of that family yield at once, such that w_{\beta}^{s}=0 and w_{\beta}^{y}=1 as shown for the mixture stress response in Figure 5.1↓a-c. Now consider that there are three bond families, \beta=0,1,2 which are weighted evenly, w_{\beta}=1/3\,\forall\beta . The stress response for this illustrative example is shown in Figure 5.1↓d-f. Though each bond family is elastic-perfectly plastic, their superposition develops “hardening”-like behavior. At the onset of yielding, when family \beta=0 yields, its bond mass fractions are w_{0}^{s}=0 and w_{0}^{y}=1 , implying that this entire family has yielded. However, since the family has a mass fraction w_{0}=1/3 in the solid mixture, two-thirds of the bonds in the mixture remain intact at this juncture, 1-w_{0}^{s}=2/3 . As subsequent families \beta yield, their bonds transition from intact to yielded generations in the same manner. Though the resulting stress response in Figure 5.1↓d is classically described as a “hardening” behavior, the reactive plasticity mixture framework proposes a different interpretation, namely that there are multiple elastic-perfectly plastic bond families in the material, each with a different threshold of yielding.

For each bond family \beta , the family mass fraction w_{\beta} and the associated yield threshold \Phi_{m\beta} must be provided by constitutive assumption, along with a single constitutive model for \Psi_{0} which applies to all generations of all bond families. The total number n_{f} of bond families must also be provided. Parameters n_{f} and \left\{ w_{\beta},\Phi_{m\beta}\right\} ,\,\beta\in\left[0,n_{f}-1\right] suffice to define an elastoplastic material which exhibits classical kinematic hardening behavior, for a given elastic response \Psi_{0} and yield criterion \Phi .