Comparative study of multiscale computational strategies for materials with discrete microstructures

The evolution of local defects such as dislocations and cracks often determines the performance of engineering materials. For a proper description and understanding of these phenomena, one typically needs to descend to a very small scale, at which the discreteness of the material emerges. Fully-resolved discrete numerical models, although highly accurate, often suffer from excessive computing expenses when used for application-scale considerations. More efficient multiscale simulation procedures are thus called for, capable of capturing the most significant microscopic phenomena while being computationally tractable for macroscopic problems. Two broad classes of methods are available in the literature, which conceptually differ significantly. The first class considers the fully-resolved discrete system, which is subsequently reduced through suitable mathematical tools such as projection and reduced integration. The second class of methods first homogenizes the discrete system into an equivalent continuum formulation, into which the main phenomena are added through specific enrichments. This paper provides a thorough comparison of the two different modeling philosophies in terms of their theory, accuracy, and performance. To this goal, two typical representatives are adopted: the Quasicontinuum method for the first class, and an effective continuum with an embedded cohesive zone model for the second class. Two examples are employed to demonstrate capabilities and limitations of both approaches. In particular, dislocation propagation and pile-up against a coherent phase boundary is considered at the nanoscale level, whereas a three-point bending test of a concrete specimen with crack propagation is considered at the macroscale level. In both cases, the accuracy of the two methods is compared against the fully-resolved discrete reference model. It is shown that whereas continuum models with embedded cohesive zones offer good performance to accuracy ratios, they might fail to capture unexpected more complex mechanical behavior such as dislocation reflection or crack branching. The Quasicontinuum method, on the other hand, offers more flexibility and higher accuracy at a slightly higher computational cost.