A Theory of Volumetric Growth for Compressible Elastic Biological Materials

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Abstract
A general theory of volumetric growth for compressible elastic materials is presented. The authors derive a complete set of governing equations in the present configuration for an elastic material undergoing a continuous growth process. In particular, they obtain two constitutive restrictions from a work-energy principle. First, the authors show that a growing elastic material behaves as a Green-elastic material. Second, they obtain an expression that relates the stress power due to growth to the rate of energy change due to growth. Then, the governing equations for a small increment of growth are derived from the more general theory. The equations for the incremental growth boundary-value problem provide an intuitive description of the quantities that describe growth and are used to implement the theory. The main features of the theory are illustrated with specific examples employing two strain energy functions that have been used to model biological materials.