Experimental evidence has shown that loading conditions with sufficiently large triaxialities can induce the sudden appearance of internal cavities within elastomeric (and other soft) solids. The occurrence of such instabilities, commonly referred to as cavitation, can be attributed to the growth of pre-existing defects into finite sizes. Because of its close connection with material failure initiation, the phenomenon of cavitation has received much attention from the materials and mechanics communities. Cavitation has also been a subject of interest in the mathematical community because its modeling has prompted the development of techniques to deal with a broad class of non-convex variational problems. While in recent years considerable progress has been made via energy minimization methods to establish existence results, fundamental problems regarding the quantitative prediction of the occurrence of cavitation in real material systems remain largely unresolved. In this project, the principal investigator develops a novel framework to study cavitation that: (i) is applicable to large classes of nonlinear elastic solids of practical interest, (ii) allows for 3D general loading conditions with arbitrary triaxiality, (iii) incorporates direct information on the initial shape, spatial distribution, and mechanical properties of the underlying defects at which cavitation can initiate, and (iv) is, at the same time, computationally tractable. This is accomplished by means of an innovative iterated homogenization method that allows for the construction of exact solutions for the mechanical response of nonlinear elastic materials containing random distributions of initially infinitesimal cavities (or defects). These include solutions for the change in size of the underlying cavities as a function of the applied loading conditions, from which the onset of cavitation can be determined. In spite of its generality, the analysis of the proposed formulation reduces to the study of tractable Hamilton-Jacobi equations in which the initial size of the cavities plays the role of time and the applied load plays the role of space.
This project makes available a fresh methodology radically different from existing approaches to investigate the influence of defects in solids. This is a core topic in mechanics, of great importance for understanding and predicting the failure of real-world materials. More generally, the project aims to develop analytical techniques that link the macroscopic properties of soft heterogeneous materials directly to their microscopic properties and underlying microstructures, a central issue in many fields of modern science. Beyond putting forward fundamental understanding of how microscopic behavior influences macroscopic behavior, these techniques provide engineers and scientists with mathematical tools to characterize and predict the mechanical response and failure of a broad spectrum of soft composite materials, including elastomeric composites (e.g., filled elastomers) and biological tissues (e.g., arterial walls).
