## 4 Anisotropic Damage-Healing Formulations
Based on CDHM, it can be observed from Equation (47) that the parameter \(\varphi(1-h)\) is generalized to become \((I_{ijmn}-M_{ijmn}^{-1})(I_{mnkl}-H_{mnkl}^{-1})\). Figure 12: (**a**) fictitious damaged state; (**b**) fictitious healed state; (**c**) fictitious effective healed material state; and (**d**) fictitious damaged state [161]. (Copyright, 2012, The Royal Society). The proposed healing tensor was later decomposed to healing tensor for cracks and healing tensor for voids in [163] along the lines of the decomposition theory applied on scalar based healing definition in [73; 163]. For more details on the definition of the anisotropic damage variable tensors that were defined based on cross-section area reduction and elastic stiffness degradation, the reader can refer to [75; 143]. The same authors proposed an anisotropic presentation of new damage variables that are called Fabric Tensors [74; 87; 162]. Later on, the same authors of [161] extended their work and proposed a coupled viscoplastic-viscodamage-viscohealing model to study the irregular behavior of glassy polymers [129]. Power function was added to the Frederick-Armstrong-Philips-Chaboche (FAPC) model in the expression of the dynamic recovery of the hardening function. This latter results in increase of back stress evolution that cannot describe the irregular responses associated with the inelastic responses of glassy polymers. A thermodynamic viscoplastic-viscodamage-viscohealing framework was presented where the healing was assumed to be coupled or uncoupled. The same coupled and uncoupled healing systems were investigated previously using a thermodynamic framework of elasto-plastic-damage-healing problems [164]. Asphalt concrete is a multiphased material and exhibits complicated mechanical behavior and multiple modes of degradation. In [165], the authors developed a viscoelastic-viscoplastic model coupled to anisotropic damage in which a second-order tensor damage tensor was introduced in order to relate the nominal and effective stresses. The damage tensor was divided into permanent and non-permanent parts. The first part represents the classical damage process, while the second part represents the self-healing during unloading and rest period. A creep recovery test was simulated using the healing model. The rest concerns the application of a pressure of 1 MPa during 800 s. Afterwards, unloading period of 50 s and rest period of 3000 s were imposed. A reduction of the degradation was observed during the unloading and rest period (Figure 13). Some investigations were also carried out in which anisotropic damage is coupled with a scalar healing variable. In this regard, the authors in [99; 156] developed a thermodynamic damage healing model applied to salt rock with alternative fabric descriptors. Later on, the anisotropy induced by the healing was also modeled in [103]. The effect of crack opening, closure and healing on the stiffness evolution was described by means of a multiscale model. Fabric tensors are used to relate the microcrack evolution with the macroscopic deformation rate. In [25; 145], the anisotropic Cosserat continuum model was used to simulate the damage, healing and plastic of granular materials. Combination of damage and healing was defined in terms of undamaged and damaged elastic moduli tensors. Other investigations on the anisotropic definition of damage variable based on elastic modulus tensor degradation were undertaken. This concerns the decomposition of the stiffness [73], definition of anisotropic damage tensors based on Poisson's ratio, shear modulus and bulk modulus degradation [87], and description of damage in series and damage in parallel [166]. Figure 13: Effect of creep recovery load on evolution of degradation (**left**); evolution of axial strain due to healing parameter effect resulting from the creep-recovery test (**right**) [165]. (Copyright, 2016, Springer Nature).