The history of innovation is linked to the discovery of novel materials with superior properties. From wood to rock, and from rock, to concrete and steel. From glass and metals to plastics. Nowadays, the production of a superior material usually results from extended and combined research endeavors. Regarding mechanical properties –that is, a material’s response to applied stresses– materials scientists and physicists have looked into ever lower length scales to understand their microscopic origins. Mechanical properties are the consequence of the so-called microstructure: the material’s structure at a scale small enough such that its relevant constituents (be it atoms, molecules, grains, fibers, etc.) can be perceived. At the other end of the spectrum, at the macroscale, we perceive a material’s response to stress as elastic, plastic, viscoelastic etc. These macroscale concepts have been described, combined, and exploited since the late 19th century. Nowadays, a primary goal is to establish links between microstructural properties and macroscale behavior. Such links can guide the design of new materials that meet the demands of specific applications and industries. On the other hand, they can enhance our understanding of the small-scale physical mechanisms ultimately responsible for the macroscale, observable behavior.

Historically, plastic deformation has been described as a smooth and deterministic process akin to the laminar flow of a fluid. However, modern technology has enabled researchers to systematically investigate materials’ behavior below the micron scale. A widely different picture becomes apparent at such small scales: plastic deformation is highly non-linear, intermittent, exhibits stochastic features, and gives rise to spatial patterns. Qualitatively similar behaviors have been found in a wide diversity of materials such as, e.g., crystals, glasses, rocks, wood, paper, solid foams, or some complex fluids, among others. The fact that such a diversity of materials share similarities suggests that their plastic deformation results from laws operating at a scale at which microscopic differences between them are not essential. In other words, at a range of scales between the micro and the macroscale, a material’s plastic deformation can be described with great independence of microscopic details. If the scale is not too large, such that we can still perceive spatial variability, that scale is known as the mesocale. Models operating at the mesoscale are an excellent tool for establishing the mentioned links between micro and macroscale descriptions of plastic deformation. Moreover, from a computational performance point of view, they can reach larger spatial and temporal scales than microscale methods. At the same time, they can reproduce fluctuating phenomena missing in macro-scale approaches. Such fluctuations, which can be misinterpreted as noise, are of the utmost importance since their statistics encode most of the information regarding the nature of the plastic deformation process.

The guiding principle of mesoscale modeling is to replace the quasi-infinite details present at the microscale for a coarser, more manageable description. The coarser description must retain only the fundamental features necessary for describing the process of interest, in this case, plastic deformation. To reduce the complexity of the description, mesoscale models divide a material sample into mesoscale subdomains, known as elements. The elements have only a few internal variables, defined in terms of continuum mechanics. This is to be compared with, e.g., models based on the position and velocities of the tens or hundreds of atoms that would inhabit a single mesoscale element. The behavior of the elements is defined according to a set of simple local rules, representing the most relevant characteristics of the material’s microstructure and plastic deformation mechanisms. By doing so, the mesoscale description suffers a loss of information with respect to the microscale one. Such loss means that we have imperfect knowledge about the elements’ state. To capture such statistical uncertainty, the rules governing the behavior of the elements correspond to stochastic processes. Lastly, plastic deformation is represented as a series of individual unit deformation events, or quanta, localized both in space and time. The effects of an event are modeled by the plastic deformation of a single element. Since the elements compose a solid material, they must obey elementary rules of solid mechanics, such as stress equilibrium (i.e., the continuum version of Newton’s 3rd law). Stress equilibrium implies that when an element deforms, it influences all other elements due to the material’s elastic behavior. This elastic influence leads to a highly correlated system that exhibits complex behavior. Eventually, the self-organization of the elements gives rise to the emergence of, e.g., spatial patterns and well-known macroscale laws of plastic deformation.   (a) A disordered material at a scale where atomistic heterogeneity can be appreciated. The sample is discretized into a mesh of mesoscale elements of a length above individual atoms. (b) A shear-like atomistic rearrangement (in red) inducing a distortion in the material matrix. In the mesocale description, the leading effects of the distortion are captured by modifying the mesh nodes accordingly and solving the stress equilibrium equation over the elements. (c) The mesoscale model neglects microscopic details. Thus, the state S of each element is defined in terms of the local elastic fields, local coarse-grained structural variables, and local plastic deformation. These variables evolve according to stochastic rules that may represent different physical scenarios. The rules are stated in terms of local quantities, but collective behavior may emerge due to self-organization induced by elastic interactions.