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Material Growth in Thermoelastic Continua: Theory, Algorithmics, and Simulation


Within the medical community, there has been increasing interest in understanding material growth in biomaterials. Material growth is the capability of a biomaterial to gain or lose mass. This research interest is driven by the host of health implications and medical problems related to this unique biomaterial property. Health providers are keen to understand the role of growth in healing and recovery so that surgical techniques, medical procedures, and physical therapy may be designed and implemented to stimulate healing and minimize recovery time. With this motivation, research seeks to identify and model mechanisms of material growth as well as growth-inducing factors in biomaterials.

To this end, a theoretical formulation of stress-induced volumetric material growth in thermoelastic continua is developed. The theory derives, without the classical continuum mechanics assumption of mass conservation, the balance laws governing the mechanics of solids capable of growth. Also, a proposed extension of classical thermodynamic theory provides a foundation for developing general constitutive relations. The theory is consistent in the sense that classical thermoelastic continuum theory is embedded as a special case. Two growth mechanisms, a kinematic and a constitutive contribution, coupled in the most general case of growth, are identified. This identification allows for the commonly employed special cases of density-preserving growth and volume-preserving growth to be easily recovered. In the theory, material growth is regulated by a three-surface activation criterion and corresponding flow rules. A simple model for rate-independent finite growth is proposed based on this formulation. The associated algorithmic implementation, including a method for solving the underlying differential/algebraic equations for growth, is examined in the context of an implicit finite element method. Selected numerical simulations are presented that showcase the predictive capacity of the model for both soft and hard biomaterials.

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