A computational homogenization framework for non-ordinary state-based peridynamics

Abstract

Peridynamic theory has been shown to possess the capabilities of describing phenomena that theories based on partial differential equations are not capable of describing. These phenomena include nonlocal interactions and presence of singularities in system responses. To exploit the capabilities offered by peridynamics in the homogenization of heterogenous media, a nonlocal computational homogenization theory based on peridynamic correspondence model (non-ordinary state-based peridynamics) is proposed. To set the development of the theory on a rigorous mathematical framework and to ensure consistency with the nonlocal nature of the peridynamic theory, a nonlocal vector calculus was used in the analysis of the nonlocal homogenization theory. The proposed theory is a two-scale micro–macro-homogenization strategy in which the constitutive relation at the macroscale is derived from explicit solution of a nonlocal volume constraint problem at the microscale. To justify the coupling between the two scales, nonlocal analogues of the stress and strain average theorems as well as the Hill–Mandel macrohomogeneity condition were derived. Validation of the proposed theory is achieved via numerical solution of Representative Volume Elements (RVE) from composite materials and comparing the results with those obtained by means of established methodologies.

Peridynamic computational homogenization theory for materials with evolving microstructure and damage

“…Let be in static equilibrium when a constant stress tensor is applied on Ω RVE Γ . The nonlocal average stress theorem [65] allows the following preposition to hold true: if the heterogeneous body Ω RVE is subjected to a traction boundary condition generated by the constant stress tensor , then irrespective of the complexity of stress field ( (x)) within Ω RVE , the stress averaged over Ω RVE is the same as . If is chosen to be * , then the statement of the average stress theorem can be written mathematically as…”

“…The next very important task in developing this homogenization framework is to determine the admissible fields * and * that will satisfy the energy equivalence criteria (9). This task is achieved by determining the condition under which * and * will satisfy the so-called nonlocal Hill's lemma which was proved in [65] to be where S s x k is a nonlocal symmetric tensor operator (see [65] and [76] for explanation on nonlocal integral operators). It is obvious that for the Hill's lemma to satisfy the macrohomogeneity condition (9) will require the integral in (14) to vanish.…”

“…A family of PD homogenization methods based on strain energy equivalence have been proposed and utilized in [64][65][66][67][68]. The key idea in these methods is that two materials with different microstructure are taken to be constitutively equivalent if the strain energy stored in both materials under affine deformation is the same.…”

“…In [65], a mathematically rigorous homogenization framework that is compatible with the nonlocal nature of PD was proposed. This method is called peridynamic computational homogenization theory (PDCHT).…”

“…Motivated by the development of the PDCHT in [65], the contribution of this work is to extend the application of the PDCHT to a class of materials with evolving microstructure. In other words, this study is interested in examining the influence of evolving microstructure on the effective response of materials at the macroscale.…”