Elasticity, Growth, and Stability

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GorielyDMS-0605029 The general theme of the project is the study of growth,structure, and function in physical and biological systemsthrough the use, and development of, nonlinear elasticity theoryand the associated methods of applied mathematics. The projectis divided into three main lines of interconnected research: (i)the analysis and modeling of elastic growth in soft materials,such as biological tissue; (ii) the biomechanics of microbialsystems including the study of force-driven penetrationmechanisms, such as those arising in fungi that destroy crops;and (iii) the dynamics and stability of various elastic rods,cables, and pipes having important engineering applications. Forthe first theme, the investigators and their students study themechanical consequences of growth and its potential to eithergenerate instabilities through changes in geometry and stresses,or to act as a mechanism to stabilize and regulate physicalproperties. The goal of these studies is to gain insight intothe fundamental coupling between growth and stress in manybiological systems. The second theme of microbial biomechanicsinvolves the mathematical modeling of both bacterial and fungalsystems. The investigators formulate and analyze mechanicalmodels of growing micro-organisms in order to understand theiroverall structure (e.g. the formation of appressoria in therice-blast fungus) and their ability to invade host tissues bypenetration (as found in many fungi), as well as the growth andstructural changes exhibited by filamentary bacteria, such asthose that are a natural source of antibiotics. The descriptionof these organisms combines a general formulation of membranegrowth (based on the first research theme), the elastic modelingof cell walls undergoing large deformations, and the use ofplasticity and fracture theory to describe penetration processes. The third theme of elastic rod dynamics involves the study ofelastic tubes conveying fluids, and the dynamics of rods withconstitutive coupling between twist and tension (hemitropic rods)-- a functionality that is relevant to the design of crane cableand other braided structures. The analysis of filamentinstabilities is carried out through the use and extension of thenonlinear analysis techniques developed by the investigators intheir previous work. The project concerns the dynamics of filamentary structuressuch as pipes and rods and the physical analysis of growthmechanisms and invasion appearing in biological systems. It ishighly interdisciplinary in nature, cutting across the fields ofapplied mathematics, mechanical engineering, microbiology, andbiomechanics, and addresses questions of both practicalimportance and mathematical interest. The themes have a broadrange of applicability in biology (morphogenesis), in biomedicalengineering (analysis of soft tissues, their mechanicalregulation and function), in understanding fundamental processesin biological invasion such as fungal penetration of tissues, andin classical engineering problems (pipes conveying fluids,instabilities in drilling). The project also provides manyattractive training experiences at different levels suitable forgraduate and undergraduate students from diverse backgrounds. These include opportunities to synthesize mathematical modelingwith hands-on experimentation through the use of the AppliedMathematics Program's unique experimental facilities.