Nonlinear rate-dependent spectral constitutive equation for viscoelastic solids with residual stresses

Abstract

A spectral constitutive equation for finite strain viscoelastic bodies with residual stresses is developed using spectral invariants, where each spectral invariant has a clear physical meaning. A prototype constitutive equation containing single-variable functions is presented; a function of a single invariant with a clear physical interpretation is easily manageable and is experimentally attractive. The effects of residual stress and viscosity are studied via the results of some boundary value problems, and some of these results are compared with experimental data.

Appendices

Appendix A: Advantages of spectral invariants

One advantage of spectral formulations is that they are more general than classical invariant formulations in the sense that, since all classical invariants [20] depend explicitly on spectral invariants, classical-invariant formulations can be easily and explicitly converted into spectral formulations but not vice versa.

Until now, most viscoelastic models used Spencer [20] classical invariants, or their variants, to describe their constitutive equations [19, 28, 29, 35]. Classical invariants have played an important role in the development of constitutive laws in continuum mechanics since 1940’s . Classical invariants are trace-based invariants, and hence, they are convenient and easy to evaluate. However, in many theoretical models, where such invariants are used, there is no interest about fitting with experimental data, the issue of propagation of error, nor being consistent with physics and the infinitesimal theory. Since most classical invariants do not have an immediate physical meaning, they are not attractive in seeking to design a rational programme of experiments. For example, it is not straightforward to design an experiment [41, 42] (denoted by R-experiment), where in order to rigorously construct a specific functional form of the energy function, it requires to capture the behaviour of a body in terms of a single classical invariant while keeping the remaining (classical) invariants fixed. We note that an R-experiment requires the number of independent invariants in the set of invariants of the corresponding minimal integrity or irreducible basis. It is shown in Refs. [32, 33] that the number of independent invariants is generally less than the number of invariants in the corresponding minimal integrity or irreducible basis and is far less if the number of classical invariants in a minimal integrity or irreducible basis is large. Because of the unclear physical meaning of the classical invariants, it is not clear how to select the relevant independent classical invariants from the set of invariants in the corresponding minimal or irreducible basis. In addition to this, researchers are not sure which invariants are best needed for a given problem, and for simplicity, a reduced number of invariants is commonly considered, which may create problems in order to capture the response of the material [4, 31, 43, 44]. However, it is shown by Shariff [41] that spectral invariants, each one with a clear physical meaning, are easy to analyse and attractive for use in R-experiment. Furthermore, evaluating the number of independent classical invariants in a minimal integrity basis is not straightforward due to the difficulty in constructing relations (syzygies) among classical invariants. However, relations among the spectral variables are easily constructed [32, 33], and hence, the number of independent spectral invariants can be easily obtained.

We further elaborate our claim by considering the strain energy of a transversely isotropic elastic solid with the preferred direction $\varvec{a}$ in the reference configuration. The strain energy function depends on five independent classical invariants:

$$\begin{aligned} I_1=\text{ tr }\,\varvec{C}\, , { \,\,\,\,}I_2={\displaystyle \frac{1}{2}}\left( (\text{ tr }\,\varvec{C})^2-\text{ tr }\,\varvec{C}^2\right) \, , { \,\,\,\,}I_3=\det \varvec{C}\, , { \,\,\,\,}I_4=\varvec{a}\cdot \varvec{C}\varvec{a}\, , { \,\,\,\,}I_5 =\varvec{a}\cdot \varvec{C}^2\varvec{a}\, . \end{aligned}$$

(A1)

Any attempt to construct a good constitutive equation using the invariants in (A1) is hampered due to the restriction that the invariants in (A1) depend explicitly on $\varvec{C}$, and the strain energy function generally depends explicitly on these invariants: This does not allow the modeller to use a more general invariants (that cannot be explicitly expressed in terms of $\varvec{C}$) which could facilitate the construction of a good constitutive equation. In addition to this, except for $I_3$ and $I_4$, the invariants in (A1) do not have a direct physical interpretation; hence, they are not experimentally attractive. For example, there are fairly good functions for the strain energy ${W}_{(I)}$ of an incompressible isotropic elastic solids that used the invariants $I_1$ and $I_2$, however, in 1967, Valanis and Landel [22] suggested that an efficient function of the strain energy had not been found before because of difficulties inherent in its dependence on classical strain invariants; functions ${\displaystyle \partial { {W}_{(I)}}/\partial {I_1}}$ and ${\displaystyle \partial { {W}_{(I)}}/\partial {I_2}}$ might be very complex, and it is not easy to design experiments in which $I_1$ and $I_2$ are not interrelated (see also reference [45]). They hence proposed the well-known Valanis and Landel strain energy function (48) that does not depend explicitly on $\varvec{C}$. Ogden [24] and Shariff [25] have successfully used (48) in constructing specific forms for ${W}_{(I)}$. One of the main reasons that the form (48) is attractive is that the ${W}_{(e)}$ depends explicitly on the spectral invariants (principal stretches) which have an immediate physical interpretation and in view of this an appropriate experiment was constructed [46].

As for the case of a transversely isotropic solid, since the invariants in (A1), except for $I_3$ and $I_4$, do not have a direct physical interpretation, they are not suitable for a rigorous R-experiment [42]. However, using the spectral invariants $\lambda _i$ and $a_i$, developed for transversely isotropic solids, Shariff [41] has shown that it is possible to construct an R-experiment.

Hence, it is obvious that for an anisotropic residually stressed viscoelastic solid that requires a large amount of classical invariants [20], i.e. 28 classical invariants (we have shown that only 15 of them are independent) to characterize the constitutive equation, it is far more difficult to construct an R-experiment using classical invariants than using spectral invariants.

Appendix B: P-property

The spectral invariants are simply the components of the tensors $\varvec{C}$, $\dot{\varvec{C}}$ and $\varvec{T}_R$ with respect to the basis $\{\varvec{u}_1,\varvec{u}_2,\varvec{u}_3 \}$. The potential functions are described by the spectral invariants that depend on the eigenvalues $\lambda _i$ and eigenvectors $\varvec{u}_i$ of the symmetric tensor $\varvec{U}$. A general anisotropic scalar function $\Phi$, such as that given in (7) and (9), where its arguments are expressed in terms spectral invariants with respect to the basis $\{ \varvec{u}_1,\varvec{u}_2, \varvec{u}_3 \}$ can be written in the form:

$$\begin{aligned} \Phi = \tilde{W}(\lambda _1,\lambda _2,\lambda _3,\varvec{u}_1,\varvec{u}_2,\varvec{u}_3), \end{aligned}$$

(B1)

with the symmetrical property

$$\begin{aligned} \tilde{W}(\lambda _1,\lambda _2,\lambda _3,\varvec{u}_1,\varvec{u}_2,\varvec{u}_3) = \tilde{W}(\lambda _2,\lambda _1,\lambda _3,\varvec{u}_2,\varvec{u}_1,\varvec{u}_3) = \tilde{W}(\lambda _3,\lambda _2,\lambda _1,\varvec{u}_3,\varvec{u}_2,\varvec{u}_1) \, . \end{aligned}$$

(B2)

In view of the non-unique values of $\varvec{u}_i$ and $\varvec{u}_j$ when $\lambda _i=\lambda _j$, a function $\tilde{W}$ should be independent of $\varvec{u}_i$ and $\varvec{u}_j$ when $\lambda _i=\lambda _j$, and $\tilde{W}$ should be independent of $\varvec{u}_1$, $\varvec{u}_2$ and $\varvec{u}_3$ when $\lambda _1=\lambda _2=\lambda _3$. Hence, when two or three of the principal stretches have equal values, the scalar function $\Phi$ must have any of the following forms:

$$\begin{aligned} \Phi = \left\{ \begin{array}{cc} {W}_{(a)}(\lambda ,\lambda _k,\varvec{u}_k) \, , &{} \text{ when } \, \lambda _i=\lambda _j=\lambda \, , i\ne j \ne k \ne i, \\ {W}_{(b)}(\lambda ) \, , &{} \text{ when } \, \, \lambda _1=\lambda _2=\lambda _3=\lambda . \end{array}\right. \end{aligned}$$

(B3)

As an example of (B3), consider ${\displaystyle \Phi =\sum _{i=1}^3 \zeta _ir_2(\lambda _i)}$, taking note that,

$$\begin{aligned} \sum _{i=1}^3 \zeta _i = \text{ tr }(\varvec{T}_R) \, , { \,\,\,\,}\zeta _i = \varvec{u}_i\cdot \varvec{T}_R\varvec{u}_i \, . \end{aligned}$$

(B4)

If $\lambda _1=\lambda _2=\lambda$, we have

$$\begin{aligned} \Phi = {W}_{(a)}(\lambda ,\lambda _3,\varvec{u}_3) =r_2(\lambda )(\text{ tr }\varvec{T}_R - \varvec{u}_3\cdot \varvec{T}_R\varvec{u}_3) + r_2(\lambda _3)\varvec{u}_3\cdot \varvec{T}_R\varvec{u}_3 \, . \end{aligned}$$

(B5)

In the case of $\lambda _1=\lambda _2=\lambda _3=\lambda$,

$$\begin{aligned} \Phi = {W}_{(b)}(\lambda ) = \varvec{T}_R r_2(\lambda ) \, . \end{aligned}$$

(B6)

It is worth emphasizing that, all the classical invariants described in Spencer [20] satisfy the P-property. In Refs. [47] and [10], the P-property described here is extended to non-symmetric tensors such as the two-point deformation tensor $\varvec{F}$.

Appendix C

In this appendix, we give details on the derivation of (19).

$$\begin{aligned} \varvec{C}= \sum _{i=1}^3 \lambda _i^2 \varvec{u}_i\otimes \varvec{u}_i \, , \end{aligned}$$

(C1)

which implies

$$\begin{aligned} \dot{\varvec{C}} = \sum _{i=1}^3 2\lambda _i \dot{\lambda }_i \varvec{u}_i\otimes \varvec{u}_i + \lambda _i^2 (\dot{\varvec{u}}_i\otimes \varvec{u}_i + \varvec{u}_i\otimes \dot{\varvec{u}} _i ) \, . \end{aligned}$$

(C2)

Note that

$$\begin{aligned} \varvec{u}_i\cdot \varvec{u}_j = \delta _{ij} \, , \end{aligned}$$

(C3)

which implies that

$$\begin{aligned} \Omega _{ij} = \varvec{u}_i\cdot \dot{\varvec{u}}_j = -\varvec{u}_j\cdot \dot{\varvec{u}}_i= -\Omega _{ji} , { \,\,\,\,}i \ne j , { \,\,\,\,}\varvec{u}_i\cdot \dot{\varvec{u}}_i = 0 \, , { \,\,\,\,}i { \,\,\,\,}\text{ not } \text{ summed } \, . \end{aligned}$$

(C4)

Expressing

$$\begin{aligned} \dot{\varvec{C}} = \sum _{i,j=1}^3 (\varvec{u}_i\cdot \dot{\varvec{C}}\varvec{u}_j) { \,\,\,\,}\varvec{u}_i\otimes \varvec{u}_j \, , \end{aligned}$$

(C5)

and in view of (C2) and (C4), we obtain (19).

Appendix D

In infinitesimal strain

$$\begin{aligned} \Vert \varvec{F}-\varvec{I}\Vert = \left\| {\displaystyle \frac{\partial \varvec{u}}{\partial \varvec{X}}} \right\| = O(e) , \end{aligned}$$

(D1)

where $\varvec{u}$ is the displacement vector, $\Vert \bullet \Vert$ is an appropriate norm, and the magnitude of e is much less than unity. In this case, we have $\lambda _i-1 = e_i$ is O(e), where $e_i$ is the eigenvalues of the infinitesimal strain $\varvec{E}$.

The potential function (36) can be made to represent a class of nonlinear Kelvin-Voigt materials by imposing the conditions:

$$\begin{aligned} r_3(x),r_4(x) > 0 \, , { \,\,\,\,}r_3(1)=r_4(1) = 1 . \end{aligned}$$

(D2)

If we ignore the O(e) terms in (36), we obtain, for infinitesimal strain of non-residually stress viscoelastic solid, the standard Kelvin–Voigt model

$$\begin{aligned} \varvec{T}= 2\mu \varvec{E}+ \eta \varvec{D}\, , \end{aligned}$$

(D3)

where $\eta$ is the viscosity coefficient and

$$\begin{aligned} \eta = \mu (\nu _1 + 4\nu _2) > 0 \, . \end{aligned}$$

(D4)

We could propose ${W}_{(v)}$ to also depend on the residual stress $\varvec{T}_R$ by replacing $r_3$, $r_4$ in (36) by

$$\begin{aligned} r_3(\lambda _i) + \kappa _3\xi _i s_3(\lambda _i) \, , { \,\,\,\,}r_4(\lambda _i) + \kappa _4\xi _i s_4(\lambda _i) \, , { \,\,\,\,}\xi _i \ge 0 \, , \end{aligned}$$

(D5)

respectively, where $s_3$ and $s_4$ have the same properties as $r_3$ and $r_4$ given in Section 5, $\kappa _3,\kappa _4 \ge 0$ and, $\sqrt{{1}/{\kappa _3}}$ and $\sqrt{{1}/{\kappa _4}}$ have a stress dimension. For this paper, we shall not consider the functions in (D2) and (D5) in our formulation of ${W}_{(v)}$.