Heterogeneity, spatial correlations, size effects and dissipated energy in brittle materials

The present study was inspired by recent experiments performed by the second author and co-workers where heterogeneity in a brittle material was investigated by ultrasonic scanning. The experiments showed an irregular pattern of material heterogeneity even before any external load was applied on the specimens. Under increasing uniaxial compressive load, the initial heterogeneity pattern evolved, and finally a macro crack network formed. In a previous study the finite element method was used in conjunction with a theory for distributed damage to study the effects of material heterogeneity numerically. Both experiments and finite element analysis showed that initially "strong" regions dissipated energy at a much higher rate than "weak" ones. However, the FEM is more suitable and efficient when material response can be homogenized and deformation gradients are not prominent. If this is not the case, or if one is interested in understanding and/or modeling the effects of heterogeneity and crack network formation, a lattice approach may be more suitable. In this study, the predictions of the lattice-based numerical approach are compared to experimental data on crack formation. Recently, a branch of statistical physics has focused on statistical modeling of materials. Here, it is attempted to "connect" this approach to continuum solid mechanics theories. Important connections are believed to be spatial correlations and localization phenomena in materials, as discussed herein. Using random fields to represent initial heterogeneity, the effects of spatial correlations on size effects, dissipated energy and on crack formation are studied. Results from two random field generation algorithms are reported, and, surprisingly, dissipated energy and crack network characteristics were dependent on the algorithm. Both provide a good description of the well-known size effect observed in brittle materials. However, notable differences between the two algorithms with respect to localization of deformation were identified.