A study of the homogenized behavior of defected composites coupling instabilities and contact

The interaction between instabilities and contact at the microscopic level is an important problem in the homogenization theory of defected composite, since macroscopic failure often results from instability phenomena induced by the presence of microcracks at the constituent scale. The determination of the exact microstructural instability mechanisms including the effects of contact between crack faces in a heterogeneous periodic solid requires a full-scale direct stability analysis, taking into account a precise description of the solid micro-structure, performed on a apriori unknown unit cell assembly. However this approach is computationally expensive, due to nonlinearities arising from finite changes in geometric and constitutive properties and contact. Consequently, it is preferable to take advantage from the stability information that can be extracted by examining the homogenized properties of the solid determined by means calculations performed on a unit cell. However the validity of the stability analysis based on a homogenized model of the solid, must be appropriately verified by examining the interrelations between instabilities on the macro- and micro-scales. A class of macroscopic constitutive stability measures has been introduced in and their relations with microscopic stability have been investigated with reference to reinforced composites and cellular solids, showing that a conservative estimation of the microscopic critical load can be obtained. The objective of this paper is to analyze some aspect of the stability behavior of the homogenized response of elastic periodic composite solids containing microscopic cracks in unilateral frictionless contact condition. This condition is not taken into account in usual homogenization process of finitely deformed composite solids. The stability problem of 2D hyperelastic models of composites with discontinuous reinforcements containing interface debonding and defected cellular solids is analyzed numerically by adopting a coupled finite element approach, developed for a micro-structure driven along prescribed monotonic macro-strain paths. A total Lagrangian finite element formulation is implemented to determine the nonlinear solution path and to solve the coupled eigenvalue stability problem for the examined composite microstructures subjected to periodic boundary conditions. The stability analysis at the microstructural level is developed including contact starting from the results of Nguyen, leading to a Hill’s type stability criterion. A linear comparison problem is also proposed in order to provide bounds to microscopic instability. Macroscopic conditions of constitutive stability are introduced and the sequence of critical points relative to micro- and macroinstabilities is determined.