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A Geometric Theory of Nonlinear Morphoelastic Shells

Arash_Yavari's picture

We formulate a geometric theory of nonlinear morphoelastic shells that can model the time evolution of residual stresses induced by bulk growth. We consider a thin body and idealize it by a representative orientable surface. In this geometric theory, bulk growth is modeled using an evolving referential configuration for the shell (material manifold). We consider the evolution of both the first and second fundamental forms in the material manifold by considering them as dynamical variables in the variational problem. Their evolution can be used to model both surface growth and remodeling (the evolution of spontaneous curvatures). We use a Lagrangian field theory to derive the governing equations of motion. In the case where growth can be modeled by a Rayleigh potential, we also find the governing equations for growth in the form of kinetic equations coupling the evolution of the first and the second fundamental forms with the state of stress of the shell. As examples, we first look at the stress-free growth fields of a planar sheet and find non-trivial configurations. We also study a morphoelastic infinitely long circular cylindrical shell subject to time-dependent internal pressure, and a morphoelastic planar circular shell. We  obtained numerically the evolution of growth and compute the induced residual stresses.

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