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Article Dans Une Revue International Journal of Engineering Science Année : 2014

A micromechanical approach to volumetric and surface growth in the framework of shape optimization

Résumé

We herewith adopt the physically and micromechanically motivated point of view that growth (resp. resorption) occurs as the expansion (resp. contraction) of initially small tissue elements distributed within a host surrounding matrix, due to the interfacial motion of their boundary. We postulate that the shape of the tissue element evolves in such a way as to minimize a certain functional amongst all possible admissible shapes. The driving force for the motion of the interface describing the surface growth models at the scale of the growing tissue elements is obtained from the identification of the driving force resulting from the shape derivative of the functional to be minimized. The shape derivative expresses as a surface integral involving the surface growth velocity and a conjugated driving force, in line with Hadamard’s structure theorem. Focusing on the total potential energy, the motion of the interface is described within the framework of gradient flows, whereby the interfacial velocity is related to a driving force identified to the jump of the normal component of Eshelby stress at the interface of the growing tissue elements. This formalism is exemplified in the situation of the avascular growth of spheroidal tumors and to bone external remodeling. This brings a dual viewpoint to the macroscopic volumetric growth models which consider production of new mass as a hidden mechanism occurring at a smaller scale, and traduced by a local increase of either density or volume. Volumetric growth at a macroscopic scale appears as the averaging of surface growth mechanisms occurring at the microscopic scale of the growing tissue elements.
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Dates et versions

hal-01095471 , version 1 (15-12-2014)

Identifiants

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Jean-François Ganghoffer, Jan Sokolowski. A micromechanical approach to volumetric and surface growth in the framework of shape optimization. International Journal of Engineering Science, 2014, 74, pp.207-226. ⟨10.1016/j.ijengsci.2013.10.001⟩. ⟨hal-01095471⟩
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