An Iterated Homogenization Method to Study Cavitation in Soft Solids

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Lopez-PamiesDMS-1009503 Experimental evidence has shown that loading conditions withsufficiently large triaxialities can induce the sudden appearanceof internal cavities within elastomeric (and other soft) solids. The occurrence of such instabilities, commonly referred to ascavitation, can be attributed to the growth of pre-existingdefects into finite sizes. Because of its close connection withmaterial failure initiation, the phenomenon of cavitation hasreceived much attention from the materials and mechanicscommunities. Cavitation has also been a subject of interest inthe mathematical community because its modeling has prompted thedevelopment of techniques to deal with a broad class ofnon-convex variational problems. While in recent yearsconsiderable progress has been made via energy minimizationmethods to establish existence results, fundamental problemsregarding the quantitative prediction of the occurrence ofcavitation in real material systems remain largely unresolved. In this project, the principal investigator develops a novelframework to study cavitation that: (i) is applicable to largeclasses of nonlinear elastic solids of practical interest, (ii)allows for 3D general loading conditions with arbitrarytriaxiality, (iii) incorporates direct information on the initialshape, spatial distribution, and mechanical properties of theunderlying defects at which cavitation can initiate, and (iv) is,at the same time, computationally tractable. This isaccomplished by means of an innovative iterated homogenizationmethod that allows for the construction of exact solutions forthe mechanical response of nonlinear elastic materials containingrandom distributions of initially infinitesimal cavities (ordefects). These include solutions for the change in size of theunderlying cavities as a function of the applied loadingconditions, from which the onset of cavitation can be determined. In spite of its generality, the analysis of the proposedformulation reduces to the study of tractable Hamilton-Jacobiequations in which the initial size of the cavities plays therole of time and the applied load plays the role of space. This project makes available a fresh methodology radicallydifferent from existing approaches to investigate the influenceof defects in solids. This is a core topic in mechanics, ofgreat importance for understanding and predicting the failure ofreal-world materials. More generally, the project aims todevelop analytical techniques that link the macroscopicproperties of soft heterogeneous materials directly to theirmicroscopic properties and underlying microstructures, a centralissue in many fields of modern science. Beyond putting forwardfundamental understanding of how microscopic behavior influencesmacroscopic behavior, these techniques provide engineers andscientists with mathematical tools to characterize and predictthe mechanical response and failure of a broad spectrum of softcomposite materials, including elastomeric composites (e.g.,filled elastomers) and biological tissues (e.g., arterial walls).