(268c) A Personal Perspective on Viscoplasticity

Viscoplasticity refers to the rheological behavior of a subclass of materials with memory in which stresses (or hyper-stresses conjugate to higher velocity gradients) are strictly dissipative. That is, all stress working contributes to entropy generation. Beginning in the mid-1980s the present author (hereinafter denoted by "JG") took up the general question as to the restricted class of continuum mechanical models appropriate to such materials, which include non-Brownian suspensions of rigid particles in viscous fluids and dense granular media dominated by frictional inter-particle contacts.

The purpose of the present note is to trace the subsequent evolution of ideas which finally culminates in an exact and general model based on the much neglected theory of dissipation potentials and non-linear Onsager symmetry laid down by DGB Edelen (Int. J. Eng, Sci.,10:6,,481-490,1972 and many related articles). While the focus here is mechanics, it should be noted that Edelen’s theory also applies to many physicochemical processes, such as chemical kinetics and the various transport processes that are commonly modeled as strictly dissipative.

In his earliest treatment (J. Non-Newtonian Fluid Mech.14, 141-60, 1984 and Acta Mechanica, 63, 3-13, 1986), JG proposed an extension of the generalized Newtonian fluid by means of a fourth-rank viscosity tensor depending on the past history of deformation rate. For isotropic history-independent materials this gives the Cauchy stress as second degree tensor polynomial in deformation rate, as dissipative version of the Reiner-Rivlin fluid, with scalar coefficients that depend on the isotropic invariants of deformation rate.

As a simplified representation of history-dependent anisotropy, the viscosity tensor may assumed to depend on a second rank evolutionary fabric tensor which satisfies a time-scale independent Lagrangian ODE in terms of deformation rate. For suspensions of rigid particles in Newtonian fluids one obtains, by analogy to the Cowin model of linear anisotropic elasticity, a joint tensor polynomial in deformation rate and fabric which is linear in the former (J. Fluid Mech., 568, 1-17, 2006). Truncation at the third order in fabric gives a history dependent anisotropic model that reproduces the pioneering experiments of Gadala Maria and Acrivos on transient stress response on reversal of steady shear in particle suspensions (J. Rheol., 24:6, 799-814, 1980).

The foregoing dissipative model exhibits a fortuitous Onsager symmetric by virtue of the symmetry borrowed from an elastic (Helmholtz) strain energy, a matter resolved by the analogous Edelen dissipation potential, as discussed next.

According to Edelen’s theory, as applied to viscoplasticity by the present author (Acta Mechanica, 225:8, 2239-59,2014), the Cauchy stress tensor in a strictly dissipative simple continuum is given by the gradient with respect to the deformation-rate tensor of a dissipation potential depending on the latter, plus a "powerless" or gyroscopic stress with zero stress power. Whenever the gyroscopic stress vanishes identically we obtain Onsager’s linear symmetry when the dissipation potential has (Rayleigh) quadratic form or, otherwise, Edelen’s non-linear Onsager symmetry. In either case, the symmetry of the viscosity tensor arises from the ("Maxwellian”) symmetry of cross derivatives, which to some extent justifies the above analogy to elasticity. For non-linear Onsager symmetric systems the loss of convexity of the dissipation potential gives rise generally to quasistatic material instability with loss of ellipticity in the field equations. This is the dissipative analog of the loss of loss of convexity of elastic strain energy which results in phase change.

If we allow additional dependence of the dissipation potential on one or more evolutionary parameters, such as the fabric tensor, we obtain a continuum theory that confirms and rigorously extends all the above-cited ad hoc representations except for fabric evolution. As the theory stands it allows one to posit the nonlinear Onsager symmetry of various viscoplastic drag laws discussed by Kamrin & JG (Proc. Roy. Soc. A,470:2172, 20140434, 2014).

As for fabric evolution, we may assume this to be a dissipative process for which the dissipation potential depends also on an objective time rate of change of fabric. Thus, we obtain a conjugate fabric stress given by the gradient of the potential with respect to this rate of fabric change. Such a stress can be envisaged as the additional generalized force necessary to change fabric at constant deformation, a force that could realized by external fields, such as electrostatic or magnetostatic. In the absence of such fields the the fabric stress may be taken as zero, which gives an implicit equation for its rate of fabric change as function of the deformation-rate and fabric tensors. This represents a Lagrangian differential equation that governs fabric evolution.

As a final point, it is worth noting that dissipation and non-linear Onsager symmetry may be viewed, at least approximately, as the result of continual relaxation of elastic energy, as proposed by JG and K Kamrin, (Proc. Roy. Soc. A, 475:2226,20190144},2019). Hence, given an underlying elastic (Helmholtz) free energy F, as function F(E) of an appropriate strain tensor E, the corresponding dissipation potential is given by the scaling F(tau D)/tau, where D is the rate of deformation tensor and tau is a relaxation time. This relation immediately yields Maxwell's celebrated relation between viscosity and elasticity. This picture, with dissipation, being nothing more than relaxing elasticity, implies that all associated constitutive parameters must ultimately be derived from elastic forces and relaxation times.

In closing, it is worth raising an interesting and outstanding question as to whether there is any general origin of Edelen's gyroscopic forces that break dissipative symmetry. It seems unlikely given disparate phenomena such as the effect of magnetic fields on electrical resistance, as portrayed by Onsager's linear theory on the one hand, and, on the other hand, the ostensible breakdown of symmetry for mass-action chemical kinetics pointed out by JG (Combustion Sci. Tech.,195:15, 3627-3637, 2022, which overlooks similar works by Edelen).

Acknowledgement
Words cannot convey my appreciation of the manifold contributions of Professor Andreas Acrivos to my professional career, first as my PhD advisor at UC Berkeley, and subsequently as a colleague and friend who offered invaluable recommendations and advice during my subsequent academic career. My interest in continuum mechanics was to some extent sparked by a discussion with a young Dr. Acrivos during his somewhat surreptitious reading at UCB of one of the celebrated papers of Stone and Ericksen on the non-existence of rectilinear flow of certain fluids in conduits with non-circular section.