### Mode-I and Mode-II Crack Tip Fields in Implicit Gradient Elasticity Based on Laplacians of Stress and Strain—Part III: Numerical Simulations [chapter]

Carsten Broese, Jan Frischmann, Charalampos Tsakmakis
2020 Nanomechanics [Working Title]
Models of implicit gradient elasticity based on Laplacians of stress and strain can be established in analogy to the models of linear viscoelastic solids. The most simple implicit gradient elasticity model including both, the Laplacian of stress and the Laplacian of strain, is the counterpart of the three-parameter viscoelastic solid. The main investigations in Parts I, II, and III concern the "three-parameter gradient elasticity model" and focus on the near-tip fields of Mode-I and Mode-II
more » ... e-I and Mode-II crack problems. It is proved that, for the boundary and symmetry conditions assumed in the present work, the model does not avoid the well-known singularities of classical elasticity. Nevertheless, there are significant differences in the form of the asymptotic solutions in comparison to the classical elasticity. These differences are discussed in detail on the basis of closed-form analytical solutions. Part I provides the governing equations and the required boundary and symmetry conditions for the considered crack problems. Keywords: implicit gradient elasticity, Laplacians of stress, Laplacians of strain, micromorphic and micro-strain elasticity, plane strain state a Cartesian coordinate system. It seems that the constitutive law (1) has been introduced for the first time by Altan and Aifantis . These authors (cf. also Georgiadis ) showed that the constitutive Eq. (1) leads to regular strain solutions at the crack tip of Mode-III crack problems. However, the stress field remains 1 singular at the crack tip as in the case of classical elasticity. Moreover, Altan and Aifantis , as well as Georgiadis  , presented an appropriate isotropic energy function for the mechanical model in Eq. (1) in the context of Mindlins gradient elasticity theory (see Mindlin  as well as Mindlin and Eshel ). An alternative approach to this model has been proposed in Broese et al. , where an analogy between gradient elasticity models and linear viscoelastic solids is established. According to this analogy, Eq. (1) is regarded as the gradient elasticity counterpart of the Kelvin viscoelastic solid. The short hand notation "KG-Model" in Eq. (1) stands for "Kelvin-Gradient-Elasticity-Model." Now, the question arises, if a gradient elasticity model including both, the Laplacian of stress and the Laplacian of strain, could remove both, the singularities of stress and the singularities of strain at the crack tip (cf. Gutkin and Aifantis ). The most simple generalization of Eq. (1), including the Laplacians of stress and strain, reads as follows: