Coupled continuum and discrete analysis of random heterogeneous materials: Elasticity and fracture

• The heterogeneous distribution of mechanical properties in natural materials induces a toughening mechanism.
• We model an elastic solid with a Young′s modulus distribution described by a Gaussian process, representing the heterogeneity in the system.
• We establish a link between the microscale and the macroscale in the presence of disorder.
• We analyze a flawed discrete particle system and investigate the influence of heterogeneity on the fracture mechanical properties of the solid.
• Solids with disordered elastic fields toughen by a ‘distribution-of-weakness’ mechanism inducing crack arrest and stress delocalization.

Recent work has suggested that the heterogeneous distribution of mechanical properties in natural and synthetic materials induces a toughening mechanism that leads to a more robust structural response in the presence of cracks, defects or other types of flaws. Motivated by this, we model an elastic solid with a Young′s modulus distribution described by a Gaussian process. We study the pristine system using both a continuum and a discrete model to establish a link between the microscale and the macroscale in the presence of disorder. Furthermore, we analyze a flawed discrete particle system and investigate the influence of heterogeneity on the fracture mechanical properties of the solid. We vary the variability and correlation length of the Gaussian process, thereby gaining fundamental insights into the effect of heterogeneity and the essential length scales of heterogeneity critical to enhanced fracture properties. As previously shown for composites with complex hierarchical architectures, we find that materials with disordered elastic fields toughen by a ‘distribution-of-weakness’ mechanism inducing crack arrest and stress delocalization. In our systems, the toughness modulus can increase by up to 30% due to an increase in variability in the elastic field. Our work presents a foundation for stochastic modeling in a particle-based micromechanical environment that can find broad applications within natural and synthetic materials.