## 4 Anisotropic Damage-Healing Formulations  
Damage and healing of brittle materials are generally simulated by conventional continuum damage-healing mechanics in which scalar damage and healing variables are used to describe the relationship between nominal stress and effective stress; isotropic damage models. In addition, anisotropic damage-healing formulations were also recently proposed and studied using second-order and fourth-order damage and healing tensors. [PERSON] [160] was the first who generalized the multi-axial anisotropic formulation of the description of the material degradation. Although most of the works conducted on CDHM are based on isotropic presentation, an anisotropic CDHM was also investigated by the introduction of a healing tensor. According to the formulation [161], the damage variable tensor is expressed as  
\[\phi_{ij}n_{i}=\frac{\left(dAn_{j}-d\bar{A}n_{j}\right)}{dA};0\leq\left(\phi_ {ij}\phi_{ij}\right)^{1/2}\leq 1, \tag{41}\]  
where \(dAn_{j}\) and \(d\bar{A}n_{j}\) are the damage and effective fictitious area vectors, respectively. When the healing is introduced into the material, the effective area increases. Figure 12 shows the anisotropic damage and healing configurations [161]. According to the presentation in Figure 12, the authors in [161] proposed a second rank anisotropic healing variable tensor as follows:  
\[h_{ij}n_{i}^{d}=\frac{\left(\phi_{jk}dAn_{k}-dA^{h}n_{j}^{h}\right)}{dA^{d}}; 0\leq\left(h_{ij}h\right)^{1/2}\leq 1, \tag{42}\]  
where \(h_{ij}\) describes the relationship between the damaged area vector \(dAn_{i}\) and the effective healed area vector. In the same paper, \(k_{ijkl}\) is denoted the fourth-order anisotropic damage variable tensor and describes the elastic modulus degradation as follows:\[\left\{ \begin{array}{l}k_{ijkl}^{(1)}=\left(\tilde{E}_{ijmn}-E_{ijmn}^{d}\right) \tilde{E}_{mnkl}^{-1},\\ k_{ijkl}^{(2)}=\left(\tilde{E}_{mnkl}-E_{mnkl}^{d}\right)\tilde{E}_{ijmn}^{-1}, \end{array} \right. \tag{43}\]  
where \(\tilde{E}_{ijkl}\) and \(E_{ijkl}^{d}\) are the undamaged and damaged elastic tensors. The subscripts in Equation (43) represent the two different mathematical tensorial expressions of the damage tensor. In addition, a new fourth rank healing tensor \(h_{ijkl}^{{}^{\prime}(1)}\) was also defined to measure the elastic modulus recovery as follows:  
\[\left\{ \begin{array}{l}h_{ijkl}^{{}^{\prime}(1)}=\left(E_{ijmn}^{h}-E_{ijmn} ^{d}\right)E_{mnkl}^{d^{-1}},\\ h_{ijkl}^{{}^{\prime}(2)}=\left(E_{nmkl}^{h}-E_{mnkl}^{d}\right)E_{ijmn}^{d^{ -1}},\end{array} \right. \tag{44}\]  
where \(E_{ijkl}^{h}\) is the healed elastic modulus. It was assumed that the material is undamaged when \(h_{ijkl}^{{}^{\prime}}=0_{ijkl}\) is the fourth rank zero tensor) and is fully healed when \(h_{ijkl}^{{}^{\prime}max}=k_{ijkl}^{max}\). The generalization of the relational between the effective stress and nominal stress of Equation (9) is expressed in the case of anisotropic materials as follows [64; 74]:  
\[\bar{\sigma}_{ij}=M_{ijkl}\sigma_{kl}, \tag{45}\]  
where \(M_{ijkl}\) represents the fourth-rank damage effect tensor. \(\bar{\sigma}_{ij}\) and \(\sigma_{kl}\) are the effective and Cauchy stress tensors, respectively. The relationship between M and \(\varphi\) was investigated in the literature [66; 87; 162] and expressed as  
\[M=\frac{1}{1-\varphi}. \tag{46}\]  
In the case of anisotropic damage-healing mechanics, Equations (35) and (36) become  
\[\bar{\sigma}_{ij}=\left[M_{ijkl}^{-1}+\left(I_{ijmn}-M_{ijmn}^{-1}\right):H_{ mmkl}^{-1}\right]^{-1}\sigma_{kl}, \tag{47}\]  
\[\bar{\sigma}_{ij}=\left[I_{ijmn}-\left(I_{ijmn}-M_{ijmn}^{-1}\right)\left(I_{ mmlp}-\left(n+1\right)H_{mmlp}^{-1}\right)\left(I_{lpll}+\left(n+1\right) \left(I_{nmsf}-M_{mssf}^{-1}\right)H_{sfl}^{-1}\right]^{-1}\sigma_{kl}, \tag{48}\]  
where \(I_{ijmn}\) is the fourth-rank identity tensor and H is the fourth-rank healing tensor. This equation was obtained by assuming that the tensor H corresponds to 1/h [66; 87; 162].