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Vandœuvre-lès-Nancy, France

Ganghoffer J.-F.,LEMTA
International Journal of Engineering Science | Year: 2012

The surface growth of biological tissues is presently analyzed at the continuum scale of tissue elements, adopting the framework of the thermodynamics of surfaces. Growth is assumed to occur in a moving referential configuration (called the natural configuration), considered as an open evolving domain exchanging mass, work, and nutrients with its environment. The growing surface is endowed with a superficial excess concentration of moles, which is ruled by an appropriate kinetic equation. From a thermodynamic framework of surface growth, the equilibrium equations are derived in material format from a suitable thermodynamic potential, highlighting the material surface forces for growth based on a surface Eshelby stress. Those forces depend upon a surface Eshelby stress, the curvature tensor of the growing surface, the gradient of the chemical potential of nutrients, and a surface force field. Application of the developed formalism to bone external remodeling highlights the interplay between transport phenomena and generation of surface mechanical forces. The model is able to describe both bone growth and resorption, according to the respective magnitude of the chemical and mechanical contributions to the material surface driving force for growth. © 2011 Elsevier Ltd. All rights reserved. Source

Ganghoffer J.-F.,LEMTA
Journal of the Mechanics and Physics of Solids | Year: 2010

The growth of biological tissues is here described at the continuum scale of tissue elements. 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. Evolution laws for growth are written from the expression of the local dissipation in terms of a relation linking the growth velocity gradient to a growth-like Eshelby stress, in the spirit of configurational mechanics. Surface growth is next envisaged in terms of phenomena taking place in a varying reference configuration, relying on the setting up of a surface potential depending upon the surface transformation gradient and to the normal to the growing surface. The balance laws resulting from the stationnarity of the potential energy are expressed, involving surface Eshelby tensors associated to growth. Simulations of surface growth in both cases of fixed and moving generating surfaces evidence the interplay between diffusion of nutrients and the mechanical driving forces for growth. © 2010 Elsevier Ltd. All rights reserved. Source

Ganghoffer J.-F.,LEMTA
Mechanics Research Communications | Year: 2011

Surface growth is presently described as the motion of a moving interface of vanishing thickness, physically representing the generating cells, separating a zone not yet affected by growth from a domain in which growth has occurred. The jump conditions of density, velocity, momentum, energy, and entropy over the moving front are expressed from the general balance laws of open systems in both physical and material format. The writing of the jump of the internal entropy production in material format allows the identification of a driving force for surface growth, thermodynamically conjugated to the material velocity of the moving front. © 2011 Elsevier Ltd. All rights reserved. Source

Ganghoffer J.-F.,LEMTA
International Journal of Engineering Science | Year: 2010

The connections between the notion of Eshelby tensor and the variation of Hamiltonian like action integrals are investigated, in connection with the thermodynamics of continuous open bodies exchanging mass, heat and work with their surrounding. Considering first a homogeneous representative volume element (RVE), it is shown that a possible choice of the Lagrangian density is the material derivative of a suitable thermodynamic potential. The Euler equations of the so built action integral are the state laws written in rate form. As the consequence of the optimality conditions of the resulting Jacobi action, the vanishing of the surface contribution resulting from the general variation of this Hamiltonian action leads to the well-known Gibbs-Duhem condition. A general three-field variational principle describing the equilibrium of heterogeneous systems is next written, based on the zero potential, the stationarity of which delivers a balance law for a generalized Eshelby tensor in a thermodynamic context. Adopting the rate of the grand potential as the lagrangian density, a generalized Gibbs-Duhem condition is obtained as the transversality condition of the thermodynamic action integral, considering a solid body with a movable boundary. The stationnarity condition of the surface part of the thermodynamic action traduces a relationship between the virtual work of the field variables and the virtual work of the material forces at the moving boundary. This framework is applied to the volumetric growth of spherical tissue elements due to the diffusion of nutrients, whereby a growth model relating the growth velocity gradient to a growth like Eshelby stress built from the grand potential is set up. © 2010 Elsevier Ltd. All rights reserved. Source

Bluman G.,Mathematics Building | Ganghoffer J.F.,LEMTA
Archives of Mechanics | Year: 2011

NONLOCALLY RELATED SYSTEMS for the Euler and Lagrange systems of two-dimensional dynamical nonlinear elasticity are constructed. Using the continuity equation, i.e., conservation of mass of the Euler system to represent the density and Eulerian velocity components as the curl of a potential vector, one obtains the Euler potential system that is nonlocally related to the Euler system. It is shown that the Euler potential system also serves as a potential system of the Lagrange system. As a consequence, a direct connection is established between the Euler and Lagrange systems within a tree of nonlocally related systems. This extends the known situation for one-dimensional dynamical nonlinear elasticity to two spatial dimensions. © 2011 by IPPT PAN. Source

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