Elasticity dissipation structure

The evolution of the elastic-plastic structural state in time is provided a weak formulation using Hamilton s principle. It is shown that a certain class of structures called reciprocal structures has a mixed weak formulation in time involving Lagrangian and dissipation functions. The new form of the Lagrangian developed in this work involves... 

This apparent time dependent cell disruption is caused because of the statistically random distribution of the orientation of the cells within a flow field and the random changes in that distribution as a function of time, the latter is caused as the cells spin in the flow field in response to the forces that act on them. In the present discussion this is referred to as apparent time dependency in order to distinguish it from true time-dependent disruption arising from anelastic behaviour of the cell walls. Anelastic behaviour, or time-dependent elasticity, is thought to arise from a restructuring of the fabric of the cell wall material at a molecular level. Anelasticity is stress induced and requires energy which is dissipated as heat, and if it is excessive it can weaken the structure and cause its breakage. 

In this book, elastic strain and plastic deformation will be differentiated by both words and symbols. Elastic strain is given the usual symbols e and y for extensional and shear elastic strains, respectively. For plastic shear deformation. 8 will be used, e and 8 are physically different entities, e and y are conservative quantities which store internal energy. 8 is not conservative. The work done to create it is dissipated as heat and structural defects. 

As an example we consider the flow of a fluid/adsorbate mixture through the big pores of a skeleton, thought like an elastic solid with an ellipsoidal microstructure, and propose suitable constitutive equations to study the coupling of adsorption and diffusion under isothermal conditions in particular, we insert the concentration of adsorbate and its gradient in the usual variables, other than microstructural ones. Finally, the expression of the dissipation shows clearly its dependence on the adsorption and the diffusion, other than on the micro-structural interactions. The model was already applied by G. and Palumbo [7] to describe the transport of pollutants with rainwater in soil. 

The expression for foam viscosity Eq. (8.1 la) contains two terms x 0/y which is the elasticity component, related to the demolition of foam structure, and r, which is a dissipation term, related to the liquid flow through films and borders during the deformation process. The models of Khan and Armstrong [14] and Kraynik and Hansen [43] imply that the continuous phase is in the films, no liquid exchange occurs and the film surfaces are mobile, thus predicting a very small contribution of the viscous dissipation in the films, rj = 13[Pg.584]

The second model introduced by Hunter and cowor)ters (20,21) is the elastic floe model. In this case, the structural units (which persist at high shear rates) are assumed to be small floes of particles (called floccules) which are characterised by the ability of the particle structure to trap some of the dispersion medium. In this energy dissipation is considered to arise from two processes, namely the viscous flow of the suspension medium around the floes (which are the basic flow units) and the energy involved in stretching the floes to brealc the floe doublets apart so that the amount of structure in the system is preserved inspite of the floc-floc collision. This model gives the following expression for the yield value. 

Many materials, particularly polymers, exhibit both the capacity to store energy (typical of an elastic material) and the capacity to dissipate energy (typical of a viscous material). When a sudden stress is applied, the response of these materials is an instantaneous elastic deformation followed by a delayed deformation. The delayed deformation is due to various molecular relaxation processes (particularly structural relaxation), which take a finite time to come to equilibrium. Very general stress-strain relations for viscoelastic response were proposed by Boltzmann, who assumed that at low strain amplitudes the effects of prior strains can be superposed linearly. Therefore, the stress at time t at a given point in the material depends both on the strain at time t, and on the previous strain history at that point. The stress-strain relations proposed by Boltzmann are [4,39] ... 

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