Abstract

In this lecture, we will start by a concise classification of multi-scale computational methods. We will concentrate on computational methods that allow for concurrent computing at multiple scales. Difficulties that relate to the efficient and accurate coupling between the various subdomains will be highlighted, with an emphasis on the coupling of domains that are modelled by dissimilar field equations, such as continuum mechanics and molecular dynamics. Two main approaches can be distinguished for resolving interfaces and evolving discontinuities. Within the class of discrete models, cohesive-surface approaches are probably the most versatile, in particular for heterogeneous materials. However, limitations exist, in particular related to stress triaxiality, which cannot be captured well in standard cohesive-surface models. In this lecture, we will present an elegant enhancement of the cohesive-surface model to include stress triaxiality, which preserves the discrete character of cohesive-surface models.

Among the recent developments in continuum approaches we mention the phase-field theories, and we will relate them to gradient damage models. In particular, we will elaborate a phase-field approach for cohesive-surface models, which, although being a continuum approach, results in a well-posed boundary value problem, and is therefore free of mesh dependence.

Whether a discontinuity is modelled via a continuum model, or in a discrete manner, advanced discretisation methods are needed to model the internal free boundary. The most powerful methods are finite element methods that exploit the partition-of-unity property of the shape functions, and isogeometric analysis. Examples will be given, including analyses that include coupling of evolving discontinuities with non-mechanical fields such as moisture and thermal flow.