Abstract
Though implicit constitutive relations have been in place for a long time, wherein the stress, the strain (or the symmetric part of the velocity gradient), and their time derivatives have been used to describe the response of viscoelastic and inelastic bodies, it is only recently purely algebraic relationships between the stress and the displacement gradient (or the velocity gradient) have been introduced to describe the response of non-linear fluids and solids. Such models can describe phenomena that the classical theory, wherein the stress is expressed explicitly in terms of kinematical variables, is incapable of describing, and they also present a sensible way to approach important practical problems, such as the flows of colloids and suspensions and the turbulent flows of fluids, and that of the fracture of solids. In this paper we review this new class of algebraic implicit constitutive relations that can be used to describe the response of fluids.
Highlights
Most commonly used fluid models such as the Euler fluid model, the Navier-Stokes fluid model, and the power-law fluid model, as well as models of the differential and integral type assume explicit expressions for the Cauchy stress in terms of the density and appropriate kinematical variables
Rate type fluids on the other hand are usually implicit relationships between the stress, its several objective time derivatives, and the symmetric part of the velocity gradient and its various time derivatives
It is possible that one could have an implicit relationship between the stress, density and the symmetric part of the velocity gradient that can describe several non-Newtonian response characteristics that cannot be adequately described by the Navier-Stokes fluid model
Summary
Most commonly used fluid models such as the Euler fluid model, the Navier-Stokes fluid model, and the power-law fluid model, as well as models of the differential and integral type assume explicit expressions for the Cauchy stress in terms of the density and appropriate kinematical variables. As we shall see, this class of models can be used to characterize a very rich class of material response such as shear thinning/shear thickening, nonlinear creep, pressure dependence of viscosity, “yield”, non-monotonicity of relationship between stress and shear rate, thixotropy, etc. The algebraic class of implicit models being considered includes models that can explain phenomena that cannot be described by the classical linearly viscous or power-law models It includes as a special sub-class fluids whose material moduli depend on the mean value of the stress, a situation that has been amply demonstrated to be the case in numerous experiments (experimental literature prior to 1930 that discuss the dependence of the material properties on the mean value of the stress can be found in Bridgman [16] and a documentation of more recent experimental literature can be found in Bulicek et al [17]). The paper by Perlocova and Prusa [20]
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