## 2 Linear Elasticity with Microracking  
A linear elastic body with isotropic microcracking is perhaps the simplest example of a continuum damage model based on the theory of internal state variables. While we view the damage field as describing the effects of microcracking, the numerical methods and adaptive meshing algorithms we discuss are largely independent of this assumption. That is, we make no claim that these techniques are intrinsically restricted to problems in linear elasticity, or to bodies which fail through microcracking and fracture. Let \(\mathcal{B}\subset\mathbb{R}^{3}\) be a continuous body, and let \(\mathcal{B}_{\kappa}\) denote a chosen reference configuration, which, for convenience, we identify with the configuration at time \(t=0\). By \(\mathcal{B}_{t}\) we denote the deformed configuration at time \(t\), where \(t\in(0,T]\). The position of a material point relative to a fixed origin in an inertial reference frame is denoted by \(X\) and \(x\) for the reference and deformed configurations, respectively. In the reference configuration the body is assumed to have a mass density \(\rho_{\kappa}\). We denote by \(\mathbf{u}\) the displacement field, i.e.,  
\[\mathbf{u}(X,t)=x(X,t)-X. \tag{1}\]  
The deformation and displacement gradients are, respectively,  
\[\mathbf{F}=\mathbf{I}+\mathbf{H}\quad\text{and}\quad\mathbf{H}=\
abla\mathbf{ u}. \tag{2}\]  
Here we follow a Lagrangian formulation, and \(\
abla(\cdot)\) denotes the gradient operator with respect to position in the reference configuration. The finite strain and symmetric small strain are defined, respectively, as  
\[\mathbf{E}=\tfrac{1}{2}\left(\mathbf{H}+\mathbf{H}^{T}+\mathbf{H}^{T}\mathbf{ H}\right)\quad\text{and}\quad\boldsymbol{\varepsilon}=\tfrac{1}{2}\left( \mathbf{H}+\mathbf{H}^{T}\right). \tag{3}\]  
Microcracking is the only failure mechanism considered in this model. This assumption allows for the resulting IBVP to have a simple form, and is well accepted as being successfully modeled by continuum methods (see [5]). Microcracking within the body is assumed to be isotropic, and we use a single scalar field to capture its effect. Let \(\phi(X,t)\) be such a field and, as is often the case in CDM, let \(\phi\) be restricted to the interval  
\[0\leq\phi\leq 1. \tag{4}\]  
Here \(\phi\!=\!0\) and \(\phi\!=\!1\) correspond respectively to the absence of microcracking (a pristine material point), and a totally failed material point. We assume that the growth of microcracks, and hence the damage parameter, is irreversible,  
\[\dot{\phi}\geq 0. \tag{5}\]  
We consider an evolution law for microcracking based on the [PERSON] criterion, as described by [6]. Let the damage energy release rate be defined as  
\[G=-\frac{\partial\psi}{\partial\phi}, \tag{6}\]  
where \(\psi(\mathbf{E},\phi)\) is the Helmholtz free energy per unit volume. For physically admissible damage evolutions, the energy release rate must be a positive quantity, since during such evolutions damage growth is expected to entail the expenditure of stored energy. As with the Griffith criterion, we posit that there is a critical energy release rate \(G_{cr}>0\) that must be met before microcracking can evolve. This concept, coupled with the non-decreasing nature of the damage variable, suggests an evolution law of the form  
\[\dot{\tilde{\phi}}=\eta_{c}\left\langle G-G_{cr}\right\rangle, \tag{7}\]  
where \(\eta_{c}>0\) is a crack propagation parameter, and the operator \(\left\langle\cdot\right\rangle\) returns the value of the argument if the argument is positive, and zero otherwise. In this manner, on a point by point basis, the damage variable can only be an increasing function of time, whose time rate of growth is related to the excess energy release rate required for microcracking to evolve. Notice that the presence of \(G_{cr}\) in the damage evolution law serves the purpose of a built-in nucleation criterion. In addition, notice that the parameters \(\eta_{c}\) and \(G_{cr}\) can be functions of damage.