Determination of representative volume elements for small cracks in heterogeneous, linear-elastic domains

Abstract

A method is introduced for determining the size of a representative volume element for microstructurally small cracks (RVEMSC) in heterogeneous, linear-elastic domains. RVEMSC is defined as the minimum volume required around a crack front such that crack-front parameters are converged with respect to volume size. This method uses finite element models of idealized microstructures, where microstructural constituents are represented by varying the elastic modulus (E). RVEMSC is expressed using two parameters: d1,MSC (the required distance from a crack front to the sides of a volume) and d2,MSC (the required distance from a crack to the top and bottom of a volume), both expressed in terms of the number of microstructural constituents (e.g. grains). RVEMSC is found for cases involving different crack sizes, error tolerances, and boundary conditions. For a given type of boundary condition, a conservative estimate of RVEMSC is provided as a closed-form function of crack size, average grain size, and desired error tolerance. The solutions for RVEMSC are validated numerically using models of synthetic, realistic microstructures, whose volumes are determined from the RVEMSC equations. The equations are shown to be applicable for a variety of crack sizes and error tolerances. The convergence rates for macroscale parameters are compared with convergence rates of crack-front parameters, and RVEMSC is found to lead simultaneously to converged macroscale parameters. Furthermore, a correlation analysis is performed between microstructural features and required volumes for converged crack-front parameters, and the results indicate that highly misoriented neighboring grains (implicitly represented by large differences in E) located near a crack front lead to larger required volumes to guarantee crack-front parameter convergence. A combination of constraint and microstructural effects are found to contribute to RVEMSC, where smaller cracks tend to be more sensitive to microstructural effects while larger cracks tend to be more sensitive to constraint effects.