Homogenization of intergranular fracture towards a transient gradient damage model

A nonlocal computational homogenization of softening quasi‐brittle materials

“…It must be noted that by choosing a small value of d I , for example, a value less than 1e − 7, the derivatives of the tangent modulus can be obtained independent of the chosen increment. Equation (36) can be rewritten for the state (2) by employing Equation (41) as…”

“…32 In the present study, the conventional nonlocal gradient enhancement method is utilized to regularize the softening behavior in both the microscale and macroscale levels, although there are several alternative approaches proposed in the literature to obtain the localized damage in the microscale and macroscale levels. [33][34][35][36][37][38][39] It should be mentioned that the term of the RVE proposed throughout this research is referred to as a unit cell, or a microsample, as it is aimed at extending the capability of the continuous homogenization method for softening materials; hence, the unit cell is chosen as an RVE to capture the global response of the system, such as the force-displacement diagram. It is shown here that if the regular microstructure is utilized, the unit cell is, in fact, a real RVE that can be employed to capture the local response of the system, such as the localization band width, local stress, strain, etc.…”

Localizing gradient damage model with decreasing interactions

“…At each macroscopic point, an equivalent strain is introduced as the morphic variable that characterizes the microscopic deformation underlying the embedded micro‐continuum. In , the morphic variable is defined as the continuum description of the intergranular cracks smeared over a unit cell. Apart from the standard stress power contribution, the internal power is appended with additional terms to account for the work performed arising from interactions between micro‐processes, as well as their propagation onto the macro‐scale, through and its gradient.…”

“…Recognizing that the process zone bandwidth localizes towards a macroscopic crack in quasi‐brittle fracture, we adopt the view that interactions between micro‐processes decrease with deformation – a departure from most modified nonlocal models in literature where the length scale parameter increases with deformation. The proposed phenomenological enhancement is also motivated by the homogenization theory for intergranular fracture in , where it is shown that a gradient damage model with decreasing nonlocal interactions is able to fully regularize the material softening response.…”

Modelling of localized ductile fracture with volumetric locking‐free tetrahedral elements

“…Beyond a limiting point, void coalescence occurs to result in more localized process zone(s), leading to the development of macroscopic crack(s). Since the nonlocal interaction at the coarse scale arises from the interactions between active microprocesses, 18 the localizing gradient enhancement in Poh and Sun, 19 based on the micromorphic framework proposed by Forest, 20 is adopted for finite deformation ductile fracture in Xu and Poh. 21 It was shown therewith that the localizing gradient enhancement, where nonlocal interactions decrease with damage, can regularize the softening response, with the development of a localized crack at failure.…”