**Bridging the scales - multiscale nature of materials**

The observable properties of solids and structures are only the macroscopic manifestation of very many complex interaction mechanisms involving structural components on numerous scales smaller than those we perceive with the naked eye. A material can be, e.g., crystalline or amorphous, pure or alloyed, single- or multi-phase, single- or polycrystalline; and every such information about the material's structure implies a significant impact on the overall material properties. The material's properties of interest comprise, e.g., its mechanical performance (such as its elastic moduli, damping capacity, hardening or softening behavior, hardness, toughness and ductility, or its fracture resistance), its acoustic behavior (i.e., its ability to propagate or attenuate acoustic waves), its electro-magnetic properties (e.g., the eletrical conductivity or the magnetic permeability), its optical properties, mass density, and many others. All of these macrosopic material characteristics stem from the material's composition across various length scales (and the various interaction mechanisms occur on various time scales). All these jointly determine the effective, macroscopic properties of the material. Therefore, it is of enormous importance to develop techniques that explore the multiscale nature of materials in order to accurately describe, to fundamentally understand, to reliably predict, and - ultimately - to advantageously control the mechanics and physics of solids.

*Figure: schematic illustration of the mechanical stress-strain relation in metallic materials highlighting various physical material properties of interest.*

One of the most important classes of materials for applications in science and engineering are crystalline solids, for which several important scales can be identified. Observable structures commonly reside on the macroscopic level, which can range from a few millimeters to kilometers, depending on the specific problem. This is the scale that we commonly observe without technical equipment. A light microscope reveals characteristics on the first length scale invisible to the naked eye, often called the mesoscale. On the mesoscale, one notices that the seemingly homogeneous body consists of a large number of grains with different crystallographic orientations, whose usual diameters typically range from 0.05 - 0.3 mm (and lower in case of nanocrystalline metals). These grains form the polycrystal. Composites form a special class of materials: several different materials are combined into one material with new distinct properties. The arrangement of the different constituent phases in the material may be regarded as another example of mesoscopic characteristics. Composites also provide an excellent example for the close link between structural design and macroscopic properties, since the overall properties can qualitatively and quantitatively be linked to the constituents' properties and their arrangement. With the aid of transmission electron microscopy (TEM) one can reach the next length scale, often referred to as the microscale (with typical dimensions of 0.1-3 micrometers). Here, within each grain, complex configurations of defects in the regular atomic lattice (most importantly, the so-called dislocations) become visible, forming a complicated network. Together with all other crystal defects, the dislocation network forms the microstructure. Dislocations, as discussed later, are crucial to accommodate the plastic deformation of a crystalline solid, and they are investigated using, e.g., discrete dislocation dynamics. Finally, the lowest scale in our research (others may dig deeper into the structure of matter), the atomic scale, reveals the molecular structure of the body with typical extensions of a few angstroms (0.1 nm). In metals (as in all crystalline solids), the atomic level displays a regular lattice, whose characteristic distances and orientations determine, e.g., the elastic constants but also the electro-magnetic properties. The following graphic gives a schematic overview of the scales involved in metals.

Understanding a material's microstructure is not only important in order to successfully model its effective, macroscopic properties, but it is also a key means towards designing the properties of new materials. Today's challenges in science and engineering call for materials with an ever improving performance: materials with exceptionally high stiffness, hardness, toughness, damping, or with remarkably low density — or combinations of the aforementioned, in particular in modern applications of air and spacecraft design.

Research in our group focuses on the link between macroscopic properties and microstructural mechanisms in various classes of materials with a particular focus on crystalline solids (metals and ceramics), composites, and architected (structural) solids. On the one hand, the investigation of the (nano/micro/meso-) structural components and their interaction mechanisms on various scales (starting from atomistic simulations and reaching up to the continuum level) yields essential insight into the properties of engineering materials. On the other hand, such insight can facilitate the design of novel materials with optimized properties. For example, modifications in the thermomechanical processing history of metals (such as magnesium) can give rise to improved performance and increased lifetime. In particular, a fundamental understanding of deformation and failure mechanisms at the smaller scales is essential for the architecture of Nano- and Micro-Electro-Mechanical Systems (NEMS/MEMS) to be found in all of our electronic equipment. Further, the design of engineered composite materials systems can result in extreme overall properties if the composite is appropriately tuned. Overall, once the link between microstructure and macro-properties is understood, models enable us to make reliable predictions and to use those predictions for the design of novel materials with desirable properties.

We develop and employ a variety of computational techniques to bridge across length and time scales (we primarily use our own in-house codes). Theoretical-computational modeling techniques being used in our group include (among others):

- the finite element method and meshless methods (see, e.g., the modeling of lattice mechanics)
- computational homogenization techniques (where the effective behavior of the lower scale is obtained from a so-called representative volume element),
- Bloch wave analysis (which characterizes the propagation of waves in solids),
- numerical FE
*n*techniques (where all*n*scales are modeled via the finite element method and the effective properties are passed between the respective levels), - relaxation theory (where a fine-scale microstructure is inherent in the thermodynamic framework and computes via minimum principles at each material point),
- phase field models for ferroelectrics and plasticity/twinning,
- continuum dislocation theory (to describe microscale mechanisms in metals),
- the quasicontinuum (QC) method (which aims at atomistic accuracy with finite-element efficiency),
- molecular dynamics and coarse-grained molecular dynamics,