Scale transitions in space and time for materials

Introduction

A common challenge in several disciplines of physics, engineering and material sciences is the formulation of scale transitions that establish a link between the structure and dynamics of a material at a small (micro) scale and the emerging response at a higher scale. The key challenge in scale transitions is to retain the relevant physical parameters and coarse-grain (or homogenize) the others out.

Different methodological approaches and terminologies are used in various fields, and the respective communities are often not aware of the work done in neighboring areas since they hardly meet.

Empirical or phenomenological constitutive continuum theories, where the small-scale information is generally lost on the macro-level, are well known. While such macroscopic phenomenological models are often used in engineering approaches, they do not provide the required links with the small scale phenomena that are needed e.g. in a micro-based materials design. Upscaling, homogenization or coarse graining works well for idealized cases, like linear elasticity or conventional thermal material properties. However, most methods fail for the majority of (complex) materials, where moving and interacting defects, irreversible processes and memory-effects are dominating for the material behaviour across the scales. Unresolved issues that require scale transitions in space, in time or both involve:

1)     Nonlinear behaviour. Straightforward upscaling has been successful for ideal, mostly linear systems, but not for intrinsically nonlinear systems. The latter, however, are much more relevant for most practical applications (e.g. plasticity and damage in materials, diffusiondriven microstructure evolution, stress-driven martensitic transformations, etc.).

2)     Defects. Elegant existing theories for perfect crystals (e.g. employing the Cauchy–Born rule) cannot be applied for real materials since they contain defects which, e.g., cause scattering of waves.

3)     Discreteness. Many phenomena are carried at the atomic scale or larger scales by subsequent discrete events. Under large deformations, plasticity with localized and intermittent dynamics occurs and cannot be understood without scale transitions.

4)     Interfaces and surfaces. Interfacial properties are governed by processes that are active in very small volumes, but strongly affect the overall behavior of many materials, in particular those with layered microstructures, or in granular materials, due to the peculiar interaction of dissipation in the bonded layers and (de-)adhesion at the interfaces.

5)     Non-affinity. In many systems, like granular materials or biophysical networks, the local

6)     motion of particles or domains is non-affine and there exists no theory that takes this into account on the macro-scale.

7)     Time-dependent behavior. The prediction of the collective behavior of different phases, each with its own time-dependent behavior, requires proper scale transitions (e.g. upscaling viscoelastic effects in a multi-phase microstructure towards macroscopic time-dependent models).

Even though length scales are typically tightly coupled to the time scales of the physical processes taking place at that scale, current coarse-graining or homogenization procedures generally do not involve coarse-graining of time scales. Time scale transitions form the biggest challenge in the field by themselves: Discrete, fast events have to end in a coarse grained time-like approach that handles much larger time variations. A few statistical physics-based approaches have been proposed very recently in order to upscale quantities in space as well as in time, but these are limited to special cases.

Finally, very often, a physical phenomenon is governed by mechanisms at nested length or time scales (e.g. fracture). In that case, subsequent scale transitions are needed to go from the smallest relevant scale to the macroscopic scale via intermediate ones. The identification of the different scales of interest in this transition process is a challenge on its own.

Objective

In this workshop, we aim to identify (with daily focus) different approaches for

1. spatial scale transitions,

2. temporal scale transitions,

3. scale transitions with space-time interactions, and

4. transitions from discrete to continuous theories.

By inviting people from different disciplines like mathematics, physics, engineering and material science, we intend to identify and unravel generic concepts and methodologies that have been developed for certain classes of materials, processes and properties, offering a potential for other disciplines and applications as well.

One main goal of this workshop is thus to bring together researchers of all fields with a research focus on scale transitions, since the work on various materials is often done independently in different disciplines, and researchers do often not know each other across the fields. Many of the invited colleagues would typically know less than half of the other participants (cf. section Participation). On the other hand there is a clear consensus among all participants who agreed to attend that getting to know these colleagues and their work will be beneficial.