Mathematical modeling of volumetric material growth in thermoelasticity

Growth (resp. atrophy) describes the physical processes by which a material of solid body increases (resp. decreases) its size by addition (resp. removal) of mass. In the present contribution, we propose a sound mathematical analysis of growth, relying on the decomposition of the geometric deformation tensor into the product of a growth tensor describing the local addition of material and an elastic tensor which is characterizing the reorganization of the body. The Blatz-Co hyperelastic constitutive model is adopted for an isotropic body, satisfying convexity conditions (resp. concavity conditions) with respect to the transformation gradient (resp. temperature). The evolution law for the transplant is obtained from the natural assumption that the evolution of the material is independent of the reference frame. It involves a modified Eshelby tensor based on the specific free energy density. The heat flux is dependent upon the transplant. The model consists of the constitutive equation, the energy balance, and the evolution law for the transplant. It is completed by suitable boundary conditions for the displacement, temperature and transplant tensor. The existence of locally unique solutions is obtained, for sufficiently smooth data close to the stable equilibrium. The quesJ.F. Ganghoffer LEMTA, Nancy Université. 2,Avenue de la Forêt de Haye. BP 160. TSA 60604. 54518 Vandoeuvre Cedex. France Tel.: +33-3-83595724 Fax: +33-3-83595551 E-mail: jean-francois.univ-lorraine.fr J. Soko lowski Institut Élie Cartan Nancy, UMR7502 (Université Lorraine, CNRS, INRIA), Laboratoire de Mathématiques, Université de Lorraine, B.P.239, 54506 Vandoeuvre-lès-Nancy Cedex, France Tel.: +33-3-83 684580 Fax: +33-3-83 684534 E-mail: [email protected] P.I. Plotnikov Lavrentyev Institute of Hydrodynamics, Lavrentyev pr. 15,630090 Novosibirsk, Russia Tel.: +33-3-83595724 Fax: +33-3-83595551 E-mail: [email protected] 2 Jean-Franois Ganghoffer et al. tion of the global existence is examined in the simplified situation of quasistatic isothermal equations of linear elasticity under the assumption of isotropic growth.