The mathematics and mechanics of elastic growth; with biological and biomedical applications

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GorielyDMS-0907773 The theme of the project is the study of growth, structure,and function in physiological and biological systems through theuse and development of exact elasticity theory and the associatedmethods of applied mathematics. The project divides intospecific but interconnected lines: the mathematical developmentof a theory of elastic growth in soft materials; the long-timedynamics of growth; the linear and nonlinear stability analysisof growing elastic systems with application to morphogenesis andpattern formation; the problem of cavitation and void-opening ingrowth; and the study of reduced theories. The PrincipalInvestigators study elastic growth within the framework of exactelasticity theory, where growth is modeled through themultiplicative decomposition of the deformation gradient. Themechanical consequences of growth, its long-time dynamics, andits potential to either generate instabilities through changes ingeometry and stresses, or to act as a regulatory mechanism, areexamined. Furthermore, a study of growing tissues is carried outthrough the development and application of the techniques ofweakly nonlinear analysis. The development of reduced theoriesfor rods, membranes, plates, and shells for growing structures isalso undertaken. Various applications in biology and physiologyare studied; in particular the ideas of growth are applied tobacterial and fungal systems, stems, leaves, arteries, and bloodvessels. The various theoretical analyses and specific models ofgrowth are carried out through the use and extension of nonlinearanalysis techniques developed by the Principal Investigators. Growth is a fundamental scientific problem of greatcomplexity coupling biological, chemical, and physical phenomena. The goal of the project is to develop mathematical tools tounderstand and model a variety of problems related to growth,including plant and microbial morphogenesis, the stability ofarteries, the development of aneurysms, and the regular andabnormal functions of organs such as the heart, the esophagus,and the trachea. Despite the great biological diversity of thesesystems, they share common features that can be modeled throughthe use of similar mathematical tools. These models help thescientific community to understand the intimate coupling betweenmechanics and biology in growth processes and to predict betterthe response of many biological and physiological systems.