# A Large-Deformation Gradient Damage Model for Single Crystals Based on Microdamage Theory
## 1 Introduction  
Highly anisotropic behavior of single crystals necessitates the consideration of microstructurally motivated life time estimation models. Considering the general use of single crystals in complex geometries under harsh environments such as turbine blades, modeling of crack growth due to the coupling of plasticity and damage becomes essential [1, 2]. The chosen constitutive model should satisfactorily demonstrate the extremely anisotropic and nonlinear behavior of single crystals under severe thermomechanical loading conditions. There are many modeling attempts in the literature which associate damage localization to crystallographic planes and inelastic deformations [3, 4, 5].  
Conventional rate-independent strain based approaches do not effectively model the localization of deformation due to damage within a finite thickness, since they lack an intrinsic length scale. A theory which includes a material length scale is necessary for modeling such an inherently size-dependent phenomenon. Earliest attempts of modeling introducing a length scale are in the plasticity theory [6, 7]. The proposed approach is further extended and numerically implemented in several works [8, 9, 10] and established as so-called gradient plasticity theories with the remarkable works by [PERSON], [PERSON], [PERSON] and [PERSON] [11, 12, 13, 14, 15]. Based on principle of virtual power, a thermodynamically consistent formulation considering nonlocal terms energetic, i.e., not dissipative in character, is presented by [PERSON] and [PERSON] [16] and a micromorphic version of the theory is also demonstrated in [PERSON] et al. [17]. In the same spirit, there are many regularization attempts in the literature for strain softening behavior due to damage [18]. Such a micromorphic continuum based regularization technique has also been proposed for single crystals in order to eliminate the mesh-dependency problem of the results of such simulations of finite elements by [PERSON] et al. [19]. However, those attempts exclude the energetic nature of the nonlocal terms.  
Our main effort in this paper is to demonstrate a detailed development of the gradient damage model developed by [PERSON] and [PERSON] for single crystals under large deformations [18], in parallel with the thermodynamical framework presented in [17]. For that purpose, first of all, the micromorphic damage model which introduces several crystallographic damage mechanisms is demonstrated in Section 2 and the emphasis is placed on the presentation of thermodynamical consistency of the model and identification of the dissipative or energetic character of the gradient variables which is presented in Section 3. The constitutive equations are presented in detail in Section 4 and a specialized modeling case is scrutinized in Section 5.  
The second goal of this paper is to use our theory to report on several numerical simulation outcomes generated from model problems. The proposed theory is implemented for the commercial finite element software Abaqus/Standard by implementing a user element subroutine (UEL). In addition to the standart displacement degrees of freedom, a microdamage variable \({}^{x}d\) is introduced as an additional degree of freedom. The implementation is based on a standard four-node isoparametric element on the interpolated fields with \(C_{0}\) continuity. Representative numerical examples that illustrate the capacity of the theory and its implementation to damage softening response which results in intense localization are provided together with discussions in Section 6.