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ORAL PRESENTATION
Session: 1 Non-equilibrium thermodynamics and Statistical Mechanics
(scheduled: Sunday, 19:10 )
Non-Affine deformation - The basis of material irreversibility
K. Valanis
Endochronics, 544 NW View Ridge Way Cams, WA 98607, USA
The notion of deformation as an affine map is the foundation of constitutive theory in continuum mechanics. However, while (non-thermal) reversible processes are evolutions of affine maps, irreversibility, other than heat conduction, i.e., 'material irreversibility' for lack of a better term, owes its origin to non-affine deformation of material neighborhoods. In effect, material irreversibility occurs at the local level. These ideas do not translate directly to fluids, since laminar flows are in fact dissipative.The basis of the idea begins with the observation that the reference domain of a material need not be Euclidean, in general, but may have a material metric G, either initially or as a result of subsequent deformation In such a case, the distance squared ds2 between two proximal material points, in the domain, with initial co-ordinates xi and xi + dxi is thus Gijdxidxj. The central idea then, is that the origin of material irreversibility is the in-constancy of material metric G, in the course of deformation.The physical process that underlies the in-constancy of G is the breakdown in the prescribed order that is assigned to the material particles in their reference configuration. Specific processes that cause such breakdown are, slip, dislocation formation and motion, damage and/or other types of loss of connectivity of the material domain. All such processes are associated with loss of free energy and are thus a primary source of dissipation. Because of the evolving loss of connectivity, the deformation is no longer one-to-one and on-to and thus non-affine. The material integrity at the neighborhood level is lost. This idea has not yet been fully recognized or exploited in the development of constitutive equations of solid domains. In this work we show that G plays the role of an internal variable q, thus giving a geometric meaning to this fundamental thermodynamic variable. In their simplest form the ensuing (isothermal) equations are: ψ = ψ(C, G, θo) (equation of state), η = - δψ/δθo (entropy equation), τ = 2 δψ/δC ( stress-deformation relation),- (δψ/δG).dG >= (dissipation inequality),(δψ/δG) + b.δG/δt = 0 (equation of evolution), where θo is the (constant) temperature, η the entropy density, τ is the second Piola-Kirchhoff stress tensor, C is the (Right) Cauchy Green deformation tensor and b a fourth order viscosity tensor. In the ensuing analysis with multiple metrics for multiple material sub-spaces, specific constitutive equations for polymers, appropriate to large deformation, are derived. © IWNET 2006
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and Kleanthi for IWNET 2006