Mechanical modeling of growth considering domain variation. Part I: constitutive framework

Mechanical modeling of growth considering domain variation—Part II: Volumetric and surface growth involving Eshelby tensors

“…Our objective in the treatment of volumetric growth exposed in this contribution is not to bring a refinement to the existing models, but rather to focus on the role of material forces and Eshelby stresses in the formulation of growth models. More specifically, the main issue and originality advocated in the present contribution is to provide a model and a mathematical treatment unifying surface and volumetric growth, accounting for the impact of the domain variation of the tissue element, as a follow up of the work of Ganghoffer and Haussy (2005). Specifically, the balance equations for growth shall herewith be rephrased in terms of appropriate Eshelby tensors, considering successively volumetric and surface growth, in line with the spirit and methods of configurational mechanics.…”

“…As mentioned in Ganghoffer and Haussy (2005), consideration of a strain energy density is a convenient way to identify the virtual powers of both internal and external forces in the present situation of a varying domain, using the potential energy as an intermediate object. This however does not preclude dissipation, as reflected especially in the domain variation resulting from growth.…”

“…Relying on a previous work in Ganghoffer and Haussy (2005), the rephrasing of the balance laws for tissue elements under growth in terms of suitable Eshelby tensors is done in the present contribution, considering successively volumetric and surface growth. Balance laws for volumetric growth are written in both compatible and incompatible configurations, highlighting the material forces for growth associated to Eshelby tensors.…”

Thermodynamics of Surface Growth with Application to Bone Remodeling

“…Evolution laws for a growth tensor (the kinematic multiplicative decomposition of the transformation gradient into a growth tensor and an accommodation tensor is adopted) in the context of volumetric growth are formulated, considering the interactions between the transport of nutrients and the mechanical forces responsible for growth. As growth deals with a modification of the internal structure of the body in a changing referential configuration, the language and technique of Eshelbian mechanics (Eshelby, 1951) are adopted and the driving forces for growth are identified in terms of suitable Eshelby stresses (Ganghoffer and Haussy, 2005;Ganghoffer, 2010a). Considering next surface growth, the thermodynamics of surfaces is first exposed as a basis for a consistent treatment of phenomena occurring at a growing surface (section 3), corresponding to the set of generating cells in a physiological context.…”

A nonlinear finite element model of cartilage growth

“…More recent theories have instead represented volumetric growth by a tensor quantity (Skalak et al 1996(Skalak et al , 1997 and were first used to study the growth of vascular tissues (Rodriguez et al 1994;Taber and Eggers 1996;Taber 1998). In addition to our work on the development of cartilage growth mixture models, in recent years there has been much interest in the development of continuum growth models for single constituents (Chen and Hoger 2000;Epstein and Maugin 2000;DiCarlo and Quiligotti 2002;Lubarda and Hoger 2002;Kuhl and Steinmann 2003;Huang 2004;Volokh 2004;Lappa 2005;Menzel 2005), mixtures (Quiligotti 2002;Ganghoffer and Haussy 2005) and mixtures that employ a stress balance hypothesis (Humphrey and Rajagopal 2002;Preziosi and Farina 2002;Breward et al 2003;Garikipati et al 2004).…”