The material models used in simulations are often a major source of uncertainty in the quantification of performance margins. Traditionally, codes were baselined against relevant full-system data and combined with many simplifying assumptions. The resulting empirical models were interpolative in nature and could not reliably be applied outside the range of the calibration data set. Likewise, current engineering material models are empirical in nature and often result in unacceptable levels of uncertainty in calculated performance margins. Worse still, the phenomenological approach does not offer a systematic means of increasing fidelity — and reducing uncertainty — in material modelling. Thus, the introduction of more complex ad hoc response functions invariably introduces additional empirical constants, often without a clear physical meaning, which themselves become a new source of uncertainty. The problem is compounded by the need to describe material behavior under extreme conditions of strain rate, deformation, temperature and pressure, often outside the range of direct laboratory testing.
Within this context, Multiscale Modelling of Materials (MMM) may be viewed as a paradigm for systematically reducing uncertainty in simulations involving complex material behavior. The ultimate goal of MMM is to enable the simulation of full-scale systems without empirical parameters or phenomenological relations, i.e., on the sole basis of fundamental theories such as quantum mechanics. The attainability of this goal notwithstanding, the fidelity of the codes may systematically be enhanced through the gradual addition of improved physics at all relevant length scales. Despite recent progress, the effective bridging of length scales and the linking of microstructure and material behavior in complex materials systems such as described remains one of the central computational and mathematical challenges of our time. Most MMM approaches to date are algorithmic and computational, and there is a critical need for the development of a rigorous mathematical foundation on which to base computational approaches.
The main two strategies in Multiscale Mathematics are: i) Bottom-up: Coarse-graining of first-principles descriptions of material behavior; ii) Top-down: Informing macroscopic models with physics gleaned from the lower scales. The work-horses among bottom-up approaches are constrained minimization at zero temperature and statistical mechanics at finite temperature. The work-horses among top-down approaches are homogeneization, relaxation, and weak convergence. I plan to illustrate the challenges and opportunities that this modeling and analysis framework offers in the particular area of multiscale modeling of mechanical properties of materials, including strength, fracture and fragmentation. In particular, I plan to highlight how modern concepts from the calculus of variations underlie — explicitly or tacitly — most multiscale, enhanced or enriched finite-element methods; and how notions or relaxation and weak convergence provide a powerful and versatile tool — largely unknown to the engineering community — for understanding the properties and limitations of such finite element methods.
