Keywords: multiscale analysis, multi-physics, discontinuities, finite element method, fracture, phase transformation, level sets.

The past two decades have seen a tremendous development in computational science. On the one hand, computational possibilities have vastly increased. At the same time, new concepts have been proposed and algorithms have been elaborated in computational mechanics and computational mathematics that sometimes reduce computation times by an order of magnitude, and for other cases enable computations that were not possible, or even conceivable, hitherto. An example is the tracking of evolving discontinuities such as solid-solid phase boundaries, that move through the body during the deformation process. Widely used discretisation concepts such as finite element methods, finite difference methods or finite volume methods have been designed to solve continuum problems and cannot readily handle such evolving discontinuities inside bodies. On the other hand, major improvements in experimental techniques have made it possible to visualise physical processes and measure relevant parameters at fine scales that were not thought possible only a few decades ago. Progress has also been made at identifying parameters in constitutive models at different scales from measurements that stem from these advanced experiments.

On the technology side, demands have come for more accurate, more reliable and faster predictions and new applications areas have arisen, for micro-electrical mechanical systems (MEMS), biomedical applications, new joining techniques such as friction-stir welding, and an increasing attention on durability and sustainability of materials and systems. In virtually all cases, diffusion problems have to be considered in addition to, and often even simultaneous with, the solution of conventional stress problems. Furthermore, the demand for new or improved materials, e.g. thermal-barrier coating systems which enable higher operating temperatures in engines with an ensuing higher efficiency, has contributed to the drive towards the development of multiscale techniques. The latter class of methods aims at understanding the material behaviour at a lower level of observation, and with this understanding these methods can be a major tool for developing and engineering new materials systems. When considering materials at a lower length scale, the classical concept of a continuum more and more fades away. Where, at the macroscopic level we already have to take into account cracks, shear bands, Lüders bands and Portevin-Le Chatelier bands, to mention some of the most commonly observed discontinuities, at a lower level we encounter for example grain boundaries in crystalline materials, solid-solid phase boundaries as in austenite-martensite transformations, and discrete dislocations. As mentioned before, classical discretisation methods are not greatly amenable to capturing discontinuities. Accordingly, next to multi-physics phenomena and multiscale analysis, the proper capturing of discontinuities is a third major challenge in contemporary computational mechanics of materials.

This paper strives to give an overview of some developments in the three first-mentioned challenges - multi-physics, multiscale analysis, and the capturing of evolving discontinuities, with particular reference to failure of materials. We will start with a rather generic overview of multiscale techniques, where a distinction will be made between upscaling and concurrent computing at multiple scales. Next, the capturing of discontinuities will be addressed. Basically, two methods exist to handle them: by distributing them over a finite distance, or by treating them as true discontinuities. The first method has been a subject of much research in the past two decades and will be discussed only briefly here. Recently, however, methods that treat discontinuities in a truly discrete manner have emerged. They will be discussed and the most promising of these techniques, which exploits the partition-of-unity property of finite element shape functions, will be examined in more detail. Examples are given of several types of discontinuities, including fast crack propagation in heterogeneous materials and solid-solid phase transformations. The last section turns towards multiphysics. Here, all three challenges come together, since we discuss how to analyse flow in cracks as well as in the surrounding porous medium in a truly two-scale approach.

A discussion like this cannot avoid being biased and incomplete. For instance, the discussion is limited to continuum mechanics concepts, and important developments in molecular dynamics, coupling atomistics to continua, or coupling atomistics to discrete dislocations have not been included.

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