Viscoelasticity

Abstract

Viscoelastic processes are history dependent processes in which some energy is stored and some energy is dissipated. A simple viscoelastic relaxation process is one in which the state variables respond over time to constant control variables. For example, in creep, the stress is held constant, and in stress relaxation, the strain is held constant. Each viscoelastic material has a long-term behavior towards which a relaxation process tends. The structure of a viscoelastic material is perturbed only slightly during time-dependent responses. This is in contrast to a viscoplastic material (see Chapter 5) which involves mechanisms, such as dislocation multiplication, that permanently change the structure of the material. The creep of a non-aging polymer is a qualitatively different phenomenon than that of a metal creep, which may involve mechanisms such as dislocation climb. Other than relaxation processes such as creep or stress relaxation, viscoelastic process may be forced, such as in rate-dependent loading or in sinusoidal loading. The goal is to describe the application of the maximum dissipation evolution construction to produce a nonlinear viscoelastic model that can represent both relaxation and forced loadings. The class of models developed here assumes that dissipation occurs in a viscoelastic material without affecting the equilibrium, or long-term state.

Appendix: Evolution Equation When the Strain Energy Is a Function of a Tensor

Let E be a second order tensor of state variables, and let H be the second order tensor of conjugate control variables. Assume that the energy function has the form

$$\psi(\textbf{E}; \textbf{H}) = \phi(\textbf{E}) + \textbf{E}:\textbf{H}.$$

The affinities are the second order tensor, $\textbf{X}= \partial \psi/\partial \textbf{E}$. The affinities are a function of E, so that $\textbf{X}= h(\textbf{E})$. Assume that h is invertible so that $\psi(\textbf{E}; \textbf{H})$ can be written in terms of the affinities for fixed control variables, $\bar{\psi}(\textbf{X}; \textbf{H})$.

The gradient dissipation condition is that the evolution of the affinities obeys $\dot{\textbf{X}}= -k\partial \bar{\psi}/\partial \textbf{X}$ for fixed controls, where $\dot{\textbf{X}}$ is a second order tensor. By the chain rule,

$$\dot{\textbf{X}} = \frac{\partial \textbf{X}}{\partial \textbf{E}} \dot{\textbf{E}}.$$

((64))

Here $\partial \textbf{X}/\partial \textbf{E}$ is a fourth order tensor. Again by the chain rule,

$$\frac{\partial \bar{\psi}}{\partial \textbf{X}}= \frac{\partial \psi}{\partial \textbf{E}} \frac{\partial \textbf{E}}{\partial \textbf{X}}.$$

((65))

Substitution of (64) and (65) into $\dot{\textbf{X}}= -k\partial \bar{\psi}/\partial \textbf{X}$ yields the evolution equation in terms of the state variables,

$$\dot{\textbf{E}} = -k\left( \frac{\partial \textbf{X}}{\partial \textbf{E}} \right)^{-2} \frac{\partial \psi}{\partial \textbf{E}} =-k\left( \frac{\partial \textbf{X}}{\partial \textbf{E}} \right)^{-2} \left( \frac{\partial \phi}{\partial \textbf{E}} + \textbf{H} \right).$$

((66))

This form reduces to that of Eq. (19) in the case that E has only diagonal non-zero entries. Then H also has only diagonal non-zero entries. The affinity is a second order tensor defined by

$$(\textbf{X})_{ij}= \left(\frac{\partial \psi}{\partial \textbf{E}} \right)_{ij} = \frac{\partial \psi}{\partial \textbf{E}}_{ij}.$$

((67))

If ψ only depends on $E_{11}, E_{22}$, and E ₃₃, then the only non-zero terms of X are the three terms X _ii .

The fourth order tensor $\partial \textbf{X}/\partial \textbf{E}$ is defined by

$$\left(\frac{\partial \textbf{X}}{\partial \textbf{E}} \right)_{ijkm} = \frac{\partial \textbf{X}_{ij}}{\partial \textbf{E}_{km}}.$$

((68))

The only non-zero term of this tensor, if only the diagonal entries of E are non-zero, have the form

$$\left(\frac{\partial \textbf{X}}{\partial \textbf{E}} \right)_{iikk} = \frac{\partial \textbf{X}_{ii}}{\partial \textbf{E}_{kk}} = \frac{\partial^2 \psi}{\partial \textbf{E}_{ii}\partial \textbf{E}_{kk}}.$$

((69))

The evolution equation therefore has the form of Eq. (19).