A nonlinear electro-elastic model with residual stresses and a preferred direction

Abstract

In this communication, a spectral model is developed for a residually stressed electro-elastic body with a preferred direction. The model uses a total energy function that depends on the right stretch tensor, the residual stress tensor, a preferred direction structural tensor and one of the electric variables. The proposed spectral invariants have a clear physical meaning; using these invariants, we prove that only $13$ of the $37$ classical invariants in the corresponding minimal integrity basis are independent. A method for exclusion or partial exclusion of compressed fibres is proposed. Some boundary value problems with cylindrical symmetry are studied. Results for the inflation of a hollow sphere, where the residual stress is assumed to depend only on the radial position, are also given. The spectral constitutive formulation is useful in a rigorous construction of a specific form of the total energy function via appropriate experiments.

Keywords

Nonlinear elasticity residual stress electro-elastic preferred direction constitutive equation

1. Introduction

The existence of residual stresses in solid materials has been the subject of many publications [1 –3]. The consequences of residual stresses, from a mechanics point of view, are yet to be fully appreciated. This has created considerable interest during the last years and many valuable publications have resulted from attempts to understand the influence of residual stresses on the mechanical behaviour of solid materials [1, 2]. However, the mechanical response of a residually stressed electrosensitive material with a preferred direction (RSEPD) was rarely, possibly never, investigated in the past. An RSEPD deforms under the application of an electric field; it has recently attracted growing interest because of its potential for use in actuators, artificial muscles in robotics and biomedical applications in prostheses [4]. The mathematical modelling of the properties of such materials, however, is at an early stage of development, partly because of a shortage of sufficient experimental data that can be used for material characterization. Hence, a constitutive equation using variables (invariants) with a clear physical meaning is particularly useful for experimental design [5]. With this in mind, we develop a constitutive model for RSEPDs using spectral invariants with a clear physical interpretation. If an RSEPD constitutive model is developed using the traditional classical invariant method, it generally requires a large number of classical invariants [6] to describe its mechanical behaviour, and it is not clear in the literature how many of the numerous classical invariants are independent. Knowing the number of independent invariants facilitates a rigorous construction of a constitutive equation via experiments (see, for example, Shariff [5]) and, hence, an additional aim of this paper is to evaluate the number of independent invariants from the given set of invariants in the minimal integrity basis [6].

The paper is divided into the following sections. In Section 2, we present the basic equations of kinematics of deforming bodies, some properties of the residual stresses and some basic equations and definitions from electrostatics. In Section 3, we present a model using a total energy function [7, 8], where the mechanical behaviour of the material depends on deformation, the preferred direction, the residual stresses and an electric field; the total stress and the electric displacement can be obtained from the proposed total energy function. Special attention is given to obtain a list of independent invariants based on the spectral formulation developed by Shariff and co-workers [5, 9 –12]. In Section 4, a simple form of the total energy function is proposed,¹ which is used in Section 5 to study some cylindrical and spherical boundary value problems. Finally, in Section 6 we give some final remarks.

2. Governing equations

2.1. Preliminaries

In this paper, the summation convention is not used and all subscripts $i$ , $j$ and $k$ take the values $1, 2, 3$ , unless stated otherwise. Only quasistatic deformations and time-independent fields of incompressible solids are considered. The mechanical body forces are assumed to be negligible. More details about the kinematics of deforming bodies and the equation of motion can be found, for example, in Truesdell and Toupin [13].

The deformation gradient and the right Cauchy–Green tensor are denoted by $F = \partial x / \partial X$ and $C = F^{T} F = U^{2}$ , respectively, where $U$ is the right stretch tensor and $x$ and $X$ denote the position vectors in the current and reference configurations, respectively, of a particle in the solid body. The incompressibility constraint implies that det $(F) = 1$ , where det denotes the determinant of a tensor.

2.2. Residual stress

In the absence of an electric field, we assume that there exists a reference configuration, in which there exists an equilibrium stress field with zero traction on the surface. This stress is commonly called the residual stress and we use the notation $T_{R}$ for such a stress. Hence,

Div T_{R} = 0 in B_{r}

(1)

and the boundary condition

T_{R} N = 0 on \partial B_{r},

(2)

where $Div$ is the divergence operator with respect to $X$ , $\partial B_{r}$ is the boundary of $B_{r}$ and $N$ is unit outward normal to $\partial B_{r}$ . In view of equations (1) and (2), the residual stress is inhomogeneous with the mean value

\int_{B_{r}} T_{R} d V = 0 .

(3)

2.3. Electrostatics

If there is no interaction with magnetic fields and there is no distribution of free charges, the simplified forms of the Maxwell equations are

curl (e) = 0, div (d) = 0,

(4)

Where $e$ is the electric field, $d$ is the electric displacement in the current configuration and $div$ and $curl$ are the divergence and curl operators with respect to $x$ , respectively. In vacuum, $e$ and $d$ are related through

d = ε_{0} e,

(5)

where $ε_{0} = 8.85 \times 10^{- 12}$ F/m is the electric permittivity in vacuum.

In condensed matter, equation (5) does not hold in general and must be replaced by a constitutive law, which describes the electric behaviour of the material in question. When an electric field is applied to condensed matter, a certain kind of charge is generated. This is conveniently described by a vector, known as the electric polarization, denoted by $p$ , which is defined in terms of the other field vectors by the standard relation (for details, see Kovetz [14])

p = d - ε_{0} e .

(6)

In this paper, we use the concept of total Cauchy stress $T$ defined by Dorfmann and Ogden [7, 8]. In the absence of surface electric charges, $d$ , $e$ and $T$ must satisfy the continuity equations

n \cdot [[d]] = 0, n \times [[e]] = 0, Tn = \hat{t} + T_{M} n,

(7)

where · and × denote the dot product and cross product, respectively, between vectors, $n$ is the unit outward normal vector to $\partial B_{t}$ , $\hat{t}$ is the external mechanical traction, $[[]]$ denotes the difference of a quantity from outside and inside a body and $T_{M}$ is the Maxwell stress tensor outside the body in vacuum defined as

T_{M} = d \otimes e - \frac{1}{2} (d \cdot e) I,

(8)

where ⊗ denotes the dyadic product.

More details about electrostatics and continuum mechanics can be obtained, from, for example, Kovetz [14]; as well as this, a complete up-to-date review on the mechanics and electrodynamics of magneto- and electro-elastic materials can be found, for example, in Ogden and Steigmann [15].

3. Constitutive equation

3.1. A preferred direction electro-elastic body with residual stresses

Following the work of Bustamante and Shariff [16] and Shariff et al. [12], the total energy $Ω$ for the problem depends on (see also Dorfmann and Ogden [7, 8] and Ogden and Steigmann [15]):

F, T_{R}, a \otimes a, f \otimes f, e,

(9)

where the unit vector $a$ is the preferred direction,

f = \frac{1}{e} e_{(L)}, e = | e_{(L)} | > 0,

(10)

and the Lagrangian electric field $e_{(L)}$ is defined as $e_{(L)} = F^{T} e$ (see, for example, Dorfmann and Ogden [7, 8]). Hence, we can write

Ω = W_{(a)} (F, T_{R}, a, f, e) .

(11)

For an incompressible body, the total Cauchy stress is [7, 8]:

T = F \frac{\partial Ω}{\partial F} - p I = 2 F \frac{\partial Ω}{\partial C} F^{T} - p I

(12)

and the Eulerian electric displacement is

d = - F \frac{\partial Ω}{\partial e_{(L)}} .

(13)

Using the relations

\frac{\partial e}{\partial e_{(L)}} = f, \frac{\partial f}{\partial e_{(L)}} = \frac{1}{e} (I - f \otimes f),

(14)

we obtain the Lagrangian electric displacement [7, 8]:

d_{(L)} = - \frac{\partial Ω}{\partial e_{(L)}} = - \frac{\partial W_{(a)}}{\partial e} f - \frac{1}{e} (I - H)^{T} \frac{\partial W_{(a)}}{\partial f},

(15)

where $d_{(L)} = F^{- 1} d$ , $I$ is the identity tensor and $H = f \otimes f$ . We also have

Curl (e_{(L)}) = 0 and Div (d_{(L)}) = 0,

(16)

where $Curl$ is the curl operator with respect to $X$ . In view of equation (16), we could write

e_{(L)} = Grad ϕ,

(17)

where $ϕ$ is a scalar field and $Grad$ is the gradient operator with respect to $X$ . The total nominal stress tensor is given by [16]:

S = \frac{\partial Ω}{\partial F} - p F^{- 1} .

(18)

The variable $x (X)$ and $ϕ$ can be obtained by solving the equilibrium equation

Div S = 0

(19)

and equation (16) with the appropriate boundary conditions (see, for example, Section 5).

By the definition of $T_{R}$ in Section 2.2, and in view of equations (11) and (12), the constitutive equation must satisfy

T_{R} = \frac{\partial W_{(a)}}{\partial F} (I, T_{R}, a, f, 0) - p_{0} I

(20)

at the reference configuration in the absence of an electric field, where the Lagrange multiplier $p_{0}$ is evaluated at the reference configuration. Although the vector $f$ appears in equation (20), $\partial W_{(a)} / \partial F$ in equation (20) should be independent of $f$ .

3.2. Spectral representation: a preferred direction electro-elastic body with residual stresses

To satisfy objectivity, $Ω$ can be expressed in terms of

Ω = W_{(b)} (U, T_{R}, a, f, e) .

(21)

The total energy must be invariant with respect to the rotation $Q$ ; hence,

W_{(b)} (U, T_{R}, a, f, e) = W_{(b)} (QU Q^{T}, Q T_{R} Q^{T}, Qa, Qf, e) .

(22)

The required symmetry (equation (22)) reduces $W_{(b)}$ to depend on the isotropic invariants of the set vectors and tensors $S = {U, T_{R}, a, f}$ . To obtain these isotropic invariants, we simply express the components (we later show that they are also invariants) of the elements of $S$ using the basis ${u_{1}, u_{2}, u_{3}}$ of the principal vectors of $U$ . Hence, we can express $Ω$ in terms the spectral ‘components’

λ_{i} = u_{i} \cdot U u_{i}, t_{ij} = t_{ji} = u_{i} \cdot T_{R} u_{j}, a_{i} = u_{i} \cdot a, f_{i} = u_{i} \cdot f,

(23)

taking note that $u_{i} \cdot U u_{j} = 0$ for $i \neq j$ . The components in equation (23) are isotropic invariants with respect to rotation $Q$ since

λ_{i} = Q u_{i} \cdot QU Q^{T} Q u_{i}, t_{ij} = Q u_{i} \cdot Q T_{R} Q^{T} Q u_{j}, a_{i} = Q u_{i} \cdot Qa, f_{i} = Q u_{i} \cdot Qf .

(24)

Since $a$ and $f$ are unit vectors, we have

a_{1}^{2} + a_{2}^{2} + a_{3}^{2} = 1, f_{1}^{2} + f_{2}^{2} + f_{3}^{2} = 1

(25)

and hence only $13$ of the $15$ spectral invariants in equation (23) are independent. It is important to note that the total energy $Ω$ in must satisfy the $P$ -property, as described in Shariff [17]. To facilitate the satisfaction of the $P$ -property and to consider that $Ω$ depends on $a \otimes a$ and $f \otimes f$ , we express

Ω = W_{(c)} (λ_{123}, ζ_{123}^{(1)}, ζ_{123}^{(2)}, ζ_{123}^{(3)}, ζ_{123}^{(4)}, e),

(26)

where the notation $λ_{123}$ represents $λ_{1}, λ_{2}, λ_{3}$ and

ζ_{i}^{(1)} = a_{i}^{2}, ζ_{i}^{(2)} = f_{i}^{2}, ζ_{i}^{(3)} = t_{ii} = u_{i} \cdot T_{R} u_{i}, ζ_{i}^{(4)} = u_{i} \cdot T_{R}^{2} u_{i} = \sum_{j = 1}^{3} t_{ij} t_{ji} .

(27)

We could also include the invariants $(a \cdot f) a_{i} f_{i}$ in equation (26); however, for simplicity, we will not use these invariants and the invariants $ζ_{i}^{(4)}$ in this communication. In view of equation (25), only $14$ of the $16$ invariants in $W_{(c)}$ are independent; also, note that, in view of the incompressibility constraint $λ_{1} λ_{2} λ_{3} = 1$ , only $13$ of the invariants are independent.

The relations of these invariants with the classical invariants developed by Spencer [6] are given in the appendix. The physical meaning of the spectral invariants in equation (26) are easily interpreted. The majority of the corresponding $37$ classical invariants (see the appendix) in the minimal integrity basis do not have a clear physical meaning. Since the classical invariants can be written explicitly in terms of the spectral invariants in equation (26), it is clear that, if $e$ is omitted, only $13$ of the $37$ classical invariants are independent.

The spectral formulation requires the tensor components of $\partial Ω / \partial C$ with respect to the spectral Lagrangean basis $[u_{1}, u_{2}, u_{3}]$ , i.e.,

{(\frac{\partial Ω}{\partial C})}_{ii} = \frac{1}{2 λ_{i}} \frac{\partial W_{(c)}}{\partial λ_{i}} (i not summed)

(28)

and for the shear components

{(\frac{\partial Ω}{\partial C})}_{ij} = \frac{\frac{\partial W_{(c)}}{\partial u_{i}} \cdot u_{j} - \frac{\partial W_{(c)}}{\partial u_{j}} \cdot u_{i}}{2 (λ_{i}^{2} - λ_{j}^{2})}, i \neq j .

(29)

The Eulerian spectral components of the total Cauchy stress $T$ with respect to the spectral Eulerian basis $[v_{1}, v_{2}, v_{3}]$ of the left stretch tensor $V$ for an incompressible body are

τ_{ii} = λ_{i} \frac{\partial W_{(c)}}{\partial λ_{i}} - p,

(30)

τ_{ij} = 2 λ_{i} λ_{j} {(\frac{\partial Ω}{\partial C})}_{ij}, i \neq j .

(31)

The Lagrangian spectral components for the electric displacement $d$ are

d_{(L)} = - \sum_{k = 1}^{3} d_{{(L)}_{k}} u_{k},

(32)

where

d_{{(L)}_{k}} = \frac{\partial W_{(c)}}{\partial e} (f \cdot u_{k}) + \frac{1}{e} [{(I - H)}^{T} \frac{\partial W_{(c)}}{\partial f}] \cdot u_{k} .

(33)

The electric displacement in the deformed configuration can simply be expressed by

d = - \sum_{k = 1}^{3} λ_{k} d_{{(L)}_{k}} v_{k} .

(34)

3.3. Remark: spectral invariants versus classical invariants

In the literature, a constitutive equation of an anisotropic solid generally contains invariants of tensors that depend explicitly on the right Cauchy–Green tensor $C$ (or left Cauchy–Green tensor $B$ ). For example, the strain energy for a transversely isotropic elastic solid with the preferred direction $a$ in the reference configuration is a function of the independent invariants

I_{1} = tr C, I_{2} = \frac{1}{2} ({(tr C)}^{2} - tr C^{2}), I_{3} = det C, {\bar{I}}_{4} = a \cdot Ca, {\bar{I}}_{5} = a \cdot C^{2} a,

(35)

where $tr$ denotes the trace of a tensor. Any attempt for a modeller to construct a good constitutive equation using the invariants in equation (35) is hampered by the restriction that the invariants in equation (35) depend explicitly on $C$ and the strain energy function generally depends explicitly on these invariants: This does not allow the modeller to use more general invariants (that cannot be explicitly expressed in terms of $C$ ), which could facilitate the construction of a good constitutive equation. In addition to this, except for $I_{3}$ and ${\bar{I}}_{4}$ , the invariants in equation (35) do not have a direct physical interpretation; hence, they are not experimentally attractive. For example, there are fairly good functions for the strain energy $W_{(e)}$ of an incompressible isotropic elastic solid that uses the invariants $I_{1}$ and $I_{2}$ ; however, in 1967, Valanis and Landel [18] 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; the functions $\partial W_{(e)} / \partial I_{1}$ and $\partial W_{(e)} / \partial I_{2}$ might be very complex and it is not easy to design experiments in which $I_{1}$ and $I_{2}$ are not interrelated. Valanis and Landel [18] hence proposed the well-known Valanis–Landel strain energy function

W_{(e)} = \sum_{i = 1}^{3} \bar{r} (λ_{i}),

(36)

which depends complicatedly on $C$ (see equation (140) in the appendix). Ogden [19] and Shariff [20] have successfully used equation (36) in constructing specific forms for $W_{(e)}$ . One of the main reasons that the form of equation (36) is attractive is that $W_{(e)}$ depends explicitly on the spectral invariants (principal stretches), which have an immediate physical interpretation; in view of this, an appropriate experiment was constructed [21].

As for the case of a transversely isotropic solid, except for $I_{3}$ and ${\bar{I}}_{4}$ , the invariants in equation (35) do not have a direct physical interpretation; hence, they are not suitable for a rigorous experiment that requires one to vary one invariant while keeping the remaining invariants fixed (we called this an R-experiment) [22]. However, using the spectral invariants $λ_{i}$ and $a_{i}$ , developed for transversely isotropic solids, Shariff [5] has shown that it is possible to construct an R-experiment. Hence, it is obvious that for an anisotropic solid that requires a large number of classical invariants² to characterize its strain energy function, it is far more difficult to construct an R-experiment using classical invariants with no physical meaning than to construct one using spectral invariants each with a clear physical meaning.

In addition, we strongly emphasize that spectral formulations are more general in the sense that, since the classical invariants depend explicitly on spectral invariants, classical invariant formulations can be easily and explicitly converted into spectral formulations but not vice versa.

4. A specific spectral energy function and a total or partial exclusion method for compressed fibres

To date, there is not enough experimental data available in the literature to propose some specific expressions for $Ω$ , but with the purpose of starting a discussion on this important issue, and also with the idea of studying some boundary value problems in the following section, here we propose an expression for $Ω$ , which depends on a number of single- and multivalued functions. Let us assume that the free energy $Ω$ does not depend on the coupling between the electric field and the residual stress. We note that a more general constitutive equation can be constructed based on the work of Shariff et al. [12] but the form will be too cumbersome for any practical use. For simplicity, in this section, we propose a simple prototype energy function. We start, by first considering a strain energy function $W_{T}$ of a transversely isotropic elastic solid [17], i.e.,

W_{T} = μ_{T} \sum_{i = 1}^{3} r_{1} (λ_{i}) + 2 μ_{1} \sum_{i = 1}^{3} ζ_{i}^{(1)} r_{2} (λ_{i}) + \frac{β}{2} {(\sum_{i = 1}^{3} ζ_{i}^{(1)} r_{3} (λ_{i}))}^{2},

(37)

where $μ_{1} = μ_{L} - μ_{T}$ . To be consistent with the theory of infinitesimal strain, we require the following restrictions to be satisfied [23]:

r_{1} (1) = r'_{1} (1) = r_{2} (1) = r'_{2} (1) = r_{3} (1) = 0, {r ″}_{1} (1) = {r ″}_{2} (1) = 2, r'_{3} (1) = 1 .

(38)

It is also possible to restrict $r'_{1} (1) = {r ″}_{1} (1) = 1$ .

The form of equation (37) has been used [23] to successfully model soft tissue, where

r_{1} (x) = \int_{1}^{x} \frac{4}{α_{1} \sqrt{π}} er f^{- 1} (α_{1} \ln (y)) d y, r_{2} (x) = \int_{1}^{x} \frac{4}{α_{2} \sqrt{π}} er f^{- 1} (α_{2} \ln (y)) d y

(39)

r_{3} (x) = \frac{2}{α_{3} \sqrt{π}} er f^{- 1} (α_{3} \ln (x)),

(40)

where $α_{1 - 3} \neq 0$ are dimensionless material parameters and $er f^{- 1} (x)$ is the inverse error function.

Using equation (37), we propose the following simple prototype total energy function for a preferred direction electro-elastic residually stressed solid body:

\begin{matrix} Ω & = W_{T} + c_{1} \sum_{i = 1}^{3} r_{4} (λ_{i}) ζ_{i}^{(3)} + c_{2} (e) \sum_{i = 1}^{3} r_{5} (λ_{i}) ζ_{i}^{(2)} - \frac{ε_{0} e^{2}}{2} \sum_{i = 1}^{3} ζ_{i}^{(2)} \frac{1}{λ_{i}^{2}} + \sum_{i = 1}^{3} ζ_{i}^{(3)} (λ_{i} - 1) \\ = W_{(d)} (λ_{123}, ζ_{123}^{(1)}, ζ_{123}^{(2)}, ζ_{123}^{(3)}, e) . \end{matrix}

(41)

The material constants in the proposed constitutive equation may depend on $t_{i}$ and we impose $c_{2} (0) = 0$ , so that in the absence of an electric field the constitutive equation reverts to that for a residually stressed elastic body with a preferred direction. To be consistent with the small strain elasticity [12], we impose the restrictions

r_{4} (1) = r_{5} (1) = r'_{4} (1) = r'_{5} (1) = 0, {r ″}_{4} (1) = {r ″}_{5} (1) = 2 .

(42)

Note that equation (41) satisfies the $P$ -properties described in Shariff [17].

In the case of $r'_{1} (1) = {r ″}_{1} (1) = 1$ , at the reference configuration, in the absence of an electric field, we have

T = (μ - p_{0}) I + T_{R} .

(43)

Hence, $T_{R}$ is the Cauchy stress if and only if

μ = p_{0} .

(44)

In the case where the conditions $r'_{1} (1) = 0, {r ″}_{1} (1) = 2$ are implemented, at the reference configuration, in the absence of an electric field, we obtain

T = - p_{0} I + T_{R}

(45)

and hence the Cauchy stress is the residual stress if and only if the Lagrange multiplier $p_{0} = 0$ .

Using equations (13) and (41), the Eulerian electric displacement and polarization then simply take the forms

d = - F \frac{\partial ℵ}{\partial e_{(L)}} + ε_{0} e, p = - F \frac{\partial ℵ}{\partial e_{(L)}},

(46)

where

ℵ = \sum_{i = 1}^{3} c_{2} (e) ζ_{i}^{(2)} r_{5} (λ_{i}) .

(47)

The components $d_{{(L)}_{k}}$ in equation (33) simply take the form

d_{{(L)}_{k}} = (f • u_{k}) [\frac{\partial ℵ_{a}}{\partial e} + \frac{2}{e} (\frac{\partial ℵ_{a}}{\partial ζ_{k}^{(2)}} - \sum_{i = 1}^{3} \frac{\partial ℵ_{a}}{\partial ζ_{i}^{(2)}} ζ_{i}^{(2)})],

(48)

where

ℵ_{a} = ℵ - ε_{0} \sum_{i = 1}^{3} ζ_{i}^{(2)} \frac{e^{2}}{2 λ_{i}^{2}} .

We emphasize that our spectral formulation can deal with a classical invariant formulation but not vice versa; in general, the spectral constitutive equation (41) cannot be converted explicitly into a classical invariant constitutive equation.

In view of equation (41), the spectral components of equations (28) and (29) take the form

{(\frac{\partial Ω}{\partial C})}_{ii} = \frac{1}{2 λ_{i}} \frac{\partial W_{(d)}}{\partial λ_{i}} (i not summed)

(49)

and the shear components take the form

\begin{matrix} {(\frac{\partial Ω}{\partial C})}_{ij} = \frac{1}{(λ_{i}^{2} - λ_{j}^{2})} {(\frac{\partial W_{(d)}}{\partial ζ_{i}^{(3)}} - \frac{\partial W_{(d)}}{\partial ζ_{j}^{(3)}}) t_{ij} + (\frac{\partial W_{(d)}}{\partial ζ_{i}^{(2)}} - \frac{\partial W_{(d)}}{\partial ζ_{j}^{(2)}}) f_{i} f_{j} + (\frac{\partial W_{(d)}}{\partial ζ_{i}^{(1)}} - \frac{\partial W_{(d)}}{\partial ζ_{j}^{(1)}}) a_{i} a_{j}}, \\ i \neq j . \end{matrix}

(50)

4.1. Restrictions on the ground state constants

The strong ellipticity condition is a mathematical restriction on a constitutive function; it guarantees that the governing partial differential equations of equilibrium are elliptic in character, and hence, in particular, certain types of nonphysical singularity, which could otherwise occur and lead to serious numerical problems, are absent. In addition to this, strong ellipticity ensures that the speeds of infinitesimal plane waves propagating through the material are real. The material constants in equation (41) can be restricted using the strong ellipticity condition in the reference configuration $(F = I)$ . Mathematically, the strong ellipticity condition for an incompressible materials (see, for example, Shams et al. [24]) requires

m \cdot [Q (n) m] > 0, m \cdot n = 0,

(51)

where $m$ and $n$ are unit vectors, and where, in Cartesian components, we have

{(Q (n))}_{ij} = \sum_{p, q = 1}^{3} {(\frac{\partial^{2} Ω}{\partial F^{2}})}_{piqj} n_{p} n_{q},

(52)

where $n_{i}$ is a Cartesian component of $n$ . The ellipticity condition in the reference configuration is obtained in a similar manner as in the work of Shariff et al. [12], where

Q (n) = Q_{1} (n) + Q_{2} (n) + Q_{3} (n) + Q_{4} (n) + Q_{5} (n) + Q_{6} (n) + Q_{7} (n),

(53)

where

Q_{1} (n) = μ_{T} (I + n \otimes n), Q_{2} (n) = μ_{1} (An \otimes n + n \otimes An + (n • An) I + A)

(54)

Q_{3} (n) = β (An \otimes An), Q_{4} (n) = \frac{c_{1}}{2} [(T_{R} n) \otimes n + n \otimes (T_{R} n) + (n \cdot T_{R} n)) I + T_{R}],

(55)

Q_{5} (n) = \frac{c_{2} (e)}{2} [Hn \otimes n + n \otimes Hn + (n \cdot Hn)) I + H],

(56)

Q_{6} (n) = - ϵ_{0} e^{2} (n \otimes Hn + Hn \otimes n + H),

(57)

Q_{7} (n) = \frac{3}{4} [(T_{R} n) \cdot n] I - \frac{1}{4} [n \otimes (T_{R} n) + (T_{R} n) \otimes n + T_{R}] .

(58)

Since in Section 5.1 we deal with problems that can be considered two-dimensional, we only consider the case for $m$ and $n$ in a plane and assume that $T_{R} = t_{1} q_{1} \otimes q_{1} + t_{2} q_{2} \otimes q_{2}$ in that plane. When $F = I$ , the basis $[u_{1}, u_{2}, u_{3}]$ is arbitrary and, for simplicity, we let $u_{1} = q_{1}$ and $u_{2} = q_{2}$ . In the case when $β = 0$ , the necessary and sufficient condition for equation (51) is

b_{1} > 0 and 4 b_{1} b_{2} > b_{3},

(59)

where

b_{1} = μ_{T} + μ_{1} (a_{1}^{2} + a_{2}^{2}) + \frac{c_{1}}{2} (t_{1} + t_{2}) + \frac{1}{4} (3 t_{1} - t_{2}) + \frac{c_{2} (e)}{2} (f_{1}^{2} + f_{2}^{2}) - ϵ_{0} e^{2} f_{2}^{2},

(60)

b_{2} = μ_{T} + μ_{1} (a_{1}^{2} + a_{2}^{2}) + \frac{c}{2} (t_{1} + t_{2}) + \frac{1}{4} (3 t_{2} - t_{1}) + \frac{c_{2} (e)}{2} (f_{1}^{2} + f_{2}^{2}) - ϵ_{0} e^{2} f_{1}^{2},

(61)

b_{3} = 2 ϵ_{0} e^{2} f_{1} f_{2} .

(62)

In the case when $β \neq 0$ , equation (59) and $β > 0$ are sufficient conditions for equation (51) .

4.2. Fibre compression

In the literature, to take into account the fact that fibre compression does not contribute (or partially contribute) towards the strain energy function, binary functional forms (see, for example, Holzapfel and Ogden [25]) that depend on $I_{4}$ were proposed. These binary models are similar to our proposed discrete ground state constants. However, the discrete ‘functions’ $μ_{L}$ and $β$ are difficult to implement, practically. For example, if at the onset we know the numerical value of $I_{4}$ , then we can easily impose the appropriate numerical values on $μ_{L}$ and $β$ . However, in a general boundary value problem, the value of $I_{4}$ depends on the constitutive equation and hence cannot be used as a hypothesis to decide which values of $μ_{L}$ and $β$ are required in a boundary value problem solving process. To overcome this ‘vicious circle’, we approximate our proposed discrete constants using the following continuous functions:

q_{(p)} (x) = \frac{(1 + \erf (a (x - 1))}{2}, q_{(n)} (x) = \frac{(1 + \erf (a (1 - x))}{2},

(63)

where erf is the error function and $a$ is a very large positive number. We then define

μ_{L} (I_{4}) = l_{p} q_{(p)} (I_{4}) + l_{n} q_{(n)} (I_{4}), β (I_{4}) = m_{p} q_{(p)} (I_{4}) + m_{n} q_{(n)} (I_{4}),

(64)

where the constants $l_{p}$ and $l_{n}$ are nonnegative and the constants $m_{p}$ and $m_{n}$ can be negative or nonnegative but must satisfy the ellipticity condition (equation (51)). Details of the functions in equations (63) and (64) can be found in Shams et al. [23]. From Figure 1, it is clear that the continuous approximation of the discrete function is not accurate when the argument $x$ (say) of the function is very close to unity. The accuracy of the continuous function could be improved by using a value of $a$ larger that $1000$ , but in this paper the value $a = 1000$ is sufficient for our purpose. Its important to note that when $x$ is very close to unity the calculated stress is very close to zero and hence the calculated ground state constants do not affect the value of stress significantly for $x$ close to unity. For practical purposes, we let the value of the derivative of all the ground state constants be zero.

Hence, to take into account that fibre compression does not contribute (or partially contribute) towards the strain energy function, $W_{T}$ in equation (37) is now replaced by

W_{T} = μ_{T} \sum_{i = 1}^{3} r_{1} (λ_{i}) + 2 (μ_{L} (I_{4}) - μ_{T}) \sum_{i = 1}^{3} ζ_{i}^{(1)} r_{2} (λ_{i}) + \frac{β}{2} (I_{4}) {(\sum_{i = 1}^{3} ζ_{i}^{(1)} r_{3} (λ_{i}))}^{2} .

(65)

5. Boundary value problems

To illustrate our results for some boundary value problems, a specific form of the energy function is required. To simulate fibrous soft tissue with a preferred direction (in the absence of residual stress and an electric field) realistically, we use $W_{T}$ in equation (65) with the functions given in equations (39) and (40), the values for the constants [23]

α_{1} = α_{2} = α_{3} = 6, μ_{T} = 5 kPa, μ_{1} = 10 kPa, β = 20 kPa

(66)

for fibre tension and the values

μ_{1} = 0 kPa, β = 0 kPa

(67)

for fibre compression. It is shown in Shariff [23] that the functions given in equations (39) and (40), and the numerical values of equations (66) and (67), not only fit the mitral valve anterior leaflet biaxial experimental data of Weinberg and Kaazempur-Mofrad [26] but are also able to predict their experiment data. In the case where residual stress and an electric field are present, there are no adequate quantitative data available in the literature to justify the proposed constitutive equation (equation (41)); hence, in this case, for illustrative purposes, we use the values $c_{1} = 0$ and $c_{2} (e) = c_{0} e^{2}$ with $c_{0} = 10^{- 10}$ F/m and the function

r_{5} (λ_{i}) = (λ_{i} - 1)^{2} .

(68)

We note that the numerical values given in this section satisfy the strong ellipticity inequalities of equations (59) to (62).

5.1. Cylindrical boundary value problems

Here we consider the constitutive equation presented in Section 4 to give results for some boundary value problems with cylindrical symmetry, which could be important from an experimental point of view. First, we introduce a particular residual stress in a circular cylindrical tube and use it in subsequent sections.

5.1.1. Residual stress for cylindrical problems

In the following sections, we solve some boundary value problems for a circular cylindrical tube. The reference configuration is defined by

A \leq R \leq B, 0 \leq Θ \leq 2 π, 0 \leq Z \leq L,

(69)

where $R$ , $Θ$ and $Z$ are reference polar coordinates.

Following Merodio et al. [27], we consider a residual stress of the form

T_{R} = t_{1} (R) q_{1} \otimes q_{1} + t_{2} (R) q_{2} \otimes q_{2},

(70)

where $q_{1} = E_{R}$ and $q_{2} = E_{Θ}$ are cylindrical polar coordinate vectors in the reference configuration.

The stress tensor (equation (70)) must satisfy the equilibrium equation

\frac{d (R t_{1})}{d R} = t_{2} .

(71)

The component $t_{1}$ must satisfy the boundary conditions

t_{1} (A) = 0, t_{1} (B) = 0 .

(72)

In Merodio et al. [27], it has been shown that for $T_{R}$ defined in equation (70), $t_{2} (R)$ has both positive and negative values for $A \leq R \leq B$ . A simple example of $t_{1}$ is

t_{1} = κ (R - A) (R - B),

(73)

where $κ$ has the unit kilopascal per square metre. The form of equation (73) is used in Sections 5.1.2, 5.1.3 and 5.1.4 and all tensor and vector components in these sections are cylindrical polar coordinate system components.

5.1.2. Uniform extension of a cylinder

Here, we consider a cylinder with $A = 0$ in equation (69), and the deformation

r = \frac{1}{\sqrt{λ_{z}}} R, θ = Θ, z = λ_{z} Z,

(74)

where $(r, θ, z)$ is the polar coordinate in the deformed configuration. It follows that

F \equiv diag (\frac{1}{\sqrt{λ_{z}}}, \frac{1}{\sqrt{λ_{z}}}, λ_{z}) .

(75)

We have that $λ_{1} = λ_{r} = 1 / \sqrt{λ_{z}}$ , $λ_{2} = λ_{r} = 1 / \sqrt{λ_{z}}$ and $λ_{3} = λ_{z}$ . Hence, $q_{1} = u_{1} \equiv [1, 0, 0]^{T}$ , $q_{2} = u_{2} \equiv [0, 1, 0]^{T}$ and $q_{3} = u_{3} \equiv [0, 0, 1]^{T}$ .

We consider, for example, the case $e_{(L)} \equiv [0, 0, e]^{T}$ , where $e$ is a constant and $a \equiv [0, 0, 1]^{T}$ . Hence, the condition $Curl (e_{(L)}) = 0$ is automatically satisfied and we have

ζ_{1}^{(3)} = t_{1}, ζ_{2}^{(3)} = t_{2}, ζ_{3}^{(3)} = 0, ζ_{1}^{(1)} = ζ_{2}^{(1)} = ζ_{1}^{(2)} = ζ_{2}^{(2)} = 0, ζ_{3}^{(1)} = ζ_{3}^{(2)} = 1 .

(76)

The Lagrangian components of the electric displacement is simplified to

d_{{(L)}_{1}} = d_{{(L)}_{2}} = 0, d_{{(L)}_{3}} = \frac{\partial ℵ_{a}}{\partial e} .

(77)

In view of equation (48), it is clear that $d_{{(L)}_{3}}$ is independent of $X$ and hence $Div (d_{(L)}) = 0$ is automatically satisfied.

The electric field is

e \equiv {[0, 0, \frac{e}{λ_{z}}]}^{T} .

(78)

The nonzero Maxwell stress components in vacuo are

(T_{M})_{zz} = \frac{ε_{0} e^{2}}{2 λ_{z}^{2}} = - (T_{M})_{rr} = - (T_{M})_{θ θ} .

(79)

If there is no mechanical stress on the cylindrical free surface, we have, for the total Cauchy stress at the free surface,

τ_{rr} = - \frac{ε_{0} e^{2}}{2 λ_{z}^{2}} .

(80)

The nonzero components of the total stress in the body are:

τ_{rr} = λ_{r} \frac{\partial W_{(d)}}{\partial λ_{1}} - p, τ_{θ θ} = λ_{r} \frac{\partial W_{(d)}}{\partial λ_{2}} - p, τ_{zz} = λ_{z} \frac{\partial W_{(d)}}{\partial λ_{3}} - p .

(81)

We note that the total Cauchy stress is inhomogeneous. Since the total energy function depends on $r$ (or in view of equation (74), equivalently on $R$ ) and taking note that, in view of equation (50), the shear stresses are zero, the Cauchy stress must satisfy the equilibrium equation

r \frac{d τ_{rr}}{d r} = τ_{θ θ} - τ_{rr},

(82)

which can be integrated to give

τ_{rr} = - \int_{r}^{b} (λ_{r} \frac{\partial W_{(d)}}{\partial λ_{2}} - λ_{r} \frac{\partial W_{(d)}}{\partial λ_{1}}) \frac{d r}{r} - \frac{ε_{0} e^{2}}{2 λ_{z}^{2}} = \frac{t_{1} (R)}{\sqrt{λ_{z}}} - \frac{ε_{0} e^{2}}{2 λ_{z}^{2}},

(83)

where $b = B / \sqrt{λ_{z}}$ . The axial stress is given by

\begin{matrix} τ_{zz} & = λ_{z} \frac{\partial W_{(d)}}{\partial λ_{3}} - λ_{r} \frac{\partial W_{(d)}}{\partial λ_{1}} + τ_{rr} \\ = λ_{z} [μ_{T} {r'}_{1} (λ_{z}) + 2 μ_{1} {r'}_{2} (λ_{z}) + β r_{3} (λ_{z})) {r'}_{3} (λ_{z}) + 2 c_{0} e^{2} (λ_{z} - 1)] - \frac{μ_{T}}{\sqrt{λ_{z}}} {r'}_{1} (\frac{1}{\sqrt{λ_{z}}}) + \frac{ε_{0} e^{2}}{2 λ_{z}^{2}} . \end{matrix}

(84)

It is clear from equation (83) that the value of the radial stress is zero in the absence of a residual stress and an electric field and the axial stress (equation (84)) is independent of the residual stress.

In Figure 2, we present results for $τ_{rr}$ . It is necessary to notice that the radial stress $τ_{rr}$ is independent of the material constant $c_{0}$ . In Figure 3, we show results for $τ_{zz}$ . In both Figures 2 and 3, the fibres are in tension. The magnitudes of the electric field and residual stress are controlled by the variables $e$ and $κ$ , respectively. In Figures 4 to 6, the stresses are plotted against these variables. It is clear from Figures 4 to 6 and equations (83) and (84) that the plotted stresses depend linearly on $κ$ and quadratically on $e$ .

5.1.3. Azimuthal shear

Consider the problem of pure azimuthal shear of a circular cylindrical tube with cross-section in the reference configuration defined by equation (69) with $A \neq 0$ . The deformation is described by

r = R, θ = Θ + g (R), z = Z,

(85)

where $g (R)$ is a function to be determined by solving the equilibrium equations for a given constitutive equation. The deformation gradient is given by

F \equiv (\begin{matrix} 1 & 0 & 0 \\ γ & 1 & 0 \\ 0 & 0 & 1 \end{matrix}),

(86)

where $γ = rg' (r) = Rg' (R)$ . Again, for simplicity, we only consider the case $e_{(L)} \equiv [0, 0, e]^{T}$ , where $e$ is a constant and a $\equiv [0, 0, 1]^{T}$ . The residual stress is that given in equation (70). The nonzero components of the Maxwell stress in vacuo are

(T_{M})_{zz} = \frac{ε_{0} e^{2}}{2} = - (T_{M})_{rr} = - (T_{M})_{θ θ} .

(87)

The principal directions of $U$ take the form $u_{1} \equiv [s, c, 0]^{T}$ , $u_{2} \equiv [c, - s, 0]^{T}$ and $u_{3} \equiv [0, 0, - 1]^{T}$ .

The principal stretches take the values

λ_{1} = \frac{s}{c} = \frac{γ + \sqrt{γ^{2} + 4}}{2} \geq 1, λ_{2} = \frac{1}{λ_{1}} = \frac{c}{s} = \frac{\sqrt{γ^{2} + 4} - γ}{2} \leq 1, λ_{3} = 1,

(88)

where, in this case,

c = \frac{1}{\sqrt{1 + λ_{1}^{2}}}, s = \frac{λ_{1}}{\sqrt{1 + λ_{1}^{2}}}, c^{2} - s^{2} = - γ cs .

(89)

The nonstretch spectral invariants are simply

ζ_{1}^{(1)} = ζ_{2}^{(1)} = ζ_{1}^{(2)} = ζ_{2}^{(2)} = ζ_{3}^{(3)} = 0, ζ_{3}^{(1)} = ζ_{3}^{(2)} = 1, ζ_{1}^{(3)} = t_{1} s^{2} + t_{2} c^{2}, ζ_{2}^{(3)} = t_{1} c^{2} + t_{2} s^{2} .

(90)

Following the work of Shariff et al. [12], the nonzero components of the total Cauchy stress are

\begin{matrix} τ_{θ θ} = 2 [l_{1} (s^{2} (1 + γ^{2}) + γ cs) + l_{2} (c^{2} (1 + γ^{2}) - γ cs) - 2 l_{4} cs] - p, \\ τ_{r θ} = 2 [l_{1} (γ s^{2} + cs) + l_{2} (γ c^{2} - cs) + l_{4} γ cs], τ_{rr} = 2 (l_{1} s^{2} + l_{2} c^{2} + 2 l_{4} cs) - p, \\ τ_{zz} = 2 l_{3} - p, \end{matrix}

(91)

where we have defined

l_{1} = \frac{1}{2 λ_{1}} \frac{\partial W_{(d)}}{\partial λ_{1}} = \frac{1}{2 λ_{1}} (μ_{T} r_{1}' (λ_{1}) + t_{1} s^{2} + t_{2} c^{2}),

l_{2} = \frac{1}{2 λ_{2}} \frac{\partial W_{(d)}}{\partial λ_{2}} = \frac{1}{2 λ_{2}} (μ_{T} r_{1}' (λ_{2}) + t_{1} c^{2} + t_{2} s^{2}),

l_{3} = \frac{1}{2 λ_{3}} \frac{\partial W_{(d)}}{\partial λ_{3}} = \frac{ε_{0} e^{2}}{2}, l_{4} = {(\frac{\partial Ω}{\partial C})}_{12} = \frac{c^{2} s^{2} (t_{2} - t_{1})}{γ} .

(92)

Following the work of Shariff et al. [12], we have

τ_{θ θ} - τ_{rr} = γ τ_{r θ} - 2 l_{4} cs (4 + γ^{2}) .

(93)

The equilibrium equations are

\frac{d τ_{rr}}{d r} + \frac{1}{r} (τ_{rr} - τ_{θ θ}) = 0, \frac{d}{d r} (r^{2} τ_{r θ}) = 0,

(94)

from which we have

τ_{r θ} = τ_{r θ} (b) \frac{b^{2}}{r^{2}},

(95)

where $τ_{r θ} (b)$ is the shear traction acting on the outer surface $r = b$ in the deformed configuration. The values of $γ$ (which depend on $T_{R}$ and $e_{(L)}$ ) and, hence, the values of $g (r)$ can be obtained from equation (95). It is clear from equations (91) and (92) that $g (r)$ is independent of the electric field and, in the absence of residual stress, its implicit form is the same as that if the material is isotropic with the strain energy function $\sum_{i = 1}^{3} μ_{T} r_{1} (λ_{i})$ .

Using these values, we can obtain the value of $τ_{rr}$ using the relation

τ_{rr} = - \int_{r}^{b} G (y) d y - \frac{ε_{0} e^{2}}{2},

(96)

where we have used equation (87) in equation (96) and

G (r) = \frac{1}{r} [γ τ_{r θ} (b) \frac{b^{2}}{r^{2}} - 2 l_{4} cs (4 + γ^{2})] .

(97)

The hydrostatic pressure term

p = 2 (l_{1} s^{2} + l_{2} c^{2} + 2 l_{4} cs) + \int_{r}^{b} G (y) d y + \frac{ε_{0} e^{2}}{2}

(98)

can then be used to evaluate the stresses $τ_{θ θ}$ and $τ_{zz}$ .

In view of equation (90) and the facts that $λ_{3} = 1$ and $ℵ_{a}$ depends only on $e$ and is independent of $X$ , we have

d_{{(L)}_{1}} = d_{{(L)}_{2}} = 0, d_{{(L)}_{3}} = - \frac{\partial ℵ_{a}}{\partial e} .

(99)

Hence, $Div (d_{(L)}) = 0$ is automatically satisfied.

5.1.4. Extension and torsion of a solid cylinder

In this section, we consider an incompressible thick-walled circular cylindrical tube with initial geometry defined by equation (69), with $A = 0$ . The deformation is described by

r = λ^{- \frac{1}{2}} R, θ = Θ + λ τ Z, z = λ Z,

(100)

where $τ$ is the amount of torsional twist per unit deformed length and $λ$ is the axial stretch. In this formulation, $r$ , $θ$ and $z$ are cylindrical polar coordinates in the deformed configuration. The components of the deformation gradient are

F \equiv (\begin{matrix} λ^{- \frac{1}{2}} & 0 & 0 \\ 0 & λ^{- \frac{1}{2}} & λ γ \\ 0 & 0 & λ \end{matrix}),

(101)

where $γ = r τ$ and, in this paper, we only consider $λ \geq 1$ and use the radial electric field $e_{(L)} \equiv [e, 0, 0]^{T}$ , where $e$ is a constant and $a \equiv [1, 0, 0]^{T}$ . The Lagrangian principal directions have cylindrical components:

u_{1} \equiv [1, 0, 0]^{T}, u_{2} \equiv [0, c, s]^{T}, u_{3} \equiv [0, - s, c]^{T},

(102)

where

c = \cos (ϕ) = \frac{2}{\sqrt{2 ({\hat{γ}}^{2} + 4) + 2 \hat{γ} \sqrt{{\hat{γ}}^{2} + 4}}}, s = \sin (ϕ) = \frac{\hat{γ} + \sqrt{{\hat{γ}}^{2} + 4}}{\sqrt{2 ({\hat{γ}}^{2} + 4) + 2 \hat{γ} \sqrt{{\hat{γ}}^{2} + 4}}},

(103)

with

\frac{π}{4} \leq \frac{π - \overset{- 1}{\tan} (\frac{1}{\sqrt{λ^{3} - 1}})}{2} \leq ϕ < \frac{π}{2}

(104)

and

\hat{γ} = \frac{λ^{3} γ^{2} + λ^{3} - 1}{λ^{\frac{3}{2}} γ} \geq 0 .

(105)

We also have the relation

c^{2} - s^{2} = - \hat{γ} cs .

(106)

In the case of simple torsion, $λ = 1$ and we have $\hat{γ} = γ$ . The principal stretches for a combined extension and torsion deformation have the forms

λ_{1} = \frac{1}{λ^{\frac{1}{2}}}, λ_{2} = \sqrt{\frac{1}{λ} + \frac{s γ \sqrt{λ}}{c}}, λ_{3} = \sqrt{\frac{1}{λ} - \frac{c γ \sqrt{λ}}{s}} .

(107)

The nonstretch invariants have the form

ζ_{2}^{(1)} = ζ_{3}^{(1)} = ζ_{2}^{(2)} = ζ_{3}^{(2)} = 0, ζ_{1}^{(1)} = ζ_{1}^{(2)} = 1, ζ_{1}^{(3)} = t_{1} (R), ζ_{2}^{(3)} = c^{2} t_{2} (R),

ζ_{3}^{(3)} = s^{2} t_{2} (R) .

(108)

The nonzero Maxwell stress components are

(T_{M})_{rr} = \frac{ε_{0} λ e^{2}}{2} = - (T_{M})_{zz} = - (T_{M})_{θ θ} .

(109)

The cylindrical components of the total stress take the form:

\begin{matrix} τ_{θ θ} = - p + 2 [\frac{l_{2} c^{2} + l_{3} s^{2} - 2 l_{6} cs}{λ} + 2 \sqrt{λ} γ ((l_{2} - l_{3}) cs + l_{6} (c^{2} - s^{2})) + λ^{2} γ^{2} (l_{2} s^{2} + l_{3} c^{2} + 2 l_{6} cs)], \\ τ_{z θ} = 2 [\sqrt{λ} ((l_{2} - l_{3}) cs + l_{6} (c^{2} - s^{2})) + λ^{2} γ (l_{2} s^{2} + l_{3} c^{2} + 2 l_{6} cs)], \\ τ_{zz} = - p + 2 λ^{2} (l_{2} s^{2} + l_{3} c^{2} + 2 l_{6} cs), τ_{r θ} = 2 [\frac{l_{4} c - l_{5} s}{λ} + \sqrt{λ} γ (l_{4} s + l_{5} c)], \\ τ_{rr} = - p + \frac{2 l_{1}}{λ}, τ_{zr} = 2 \sqrt{λ} (l_{4} s + l_{5} c), \end{matrix}

(110)

where

l_{5} = {(\frac{\partial Ω}{\partial C})}_{13}, l_{6} = {(\frac{\partial Ω}{\partial C})}_{23} .

(111)

It can be easily shown that $l_{4} = l_{5} = 0$ , which implies that $τ_{rz} = τ_{r θ} = 0$ . We also have

l_{1} = \frac{1}{2 λ_{1}} [μ_{T} {r'}_{1} (λ_{1}) + 2 μ_{1} {r'}_{2} (λ_{1}) + β r_{3} (λ_{1}) {r'}_{3} (λ_{1}) + 2 c_{0} e^{2} (λ_{1} - 1) + \frac{ε_{0} e^{2}}{λ_{1}^{3}} + t_{1} (R)],

l_{2} = \frac{1}{2 λ_{2}} [μ_{T} {r'}_{1} (λ_{2}) + c^{2} t_{2} (R)], l_{3} = \frac{1}{2 λ_{3}} [μ_{T} {r'}_{1} (λ_{3}) + s^{2} t_{2} (R)], l_{6} = - \frac{cs t_{2} (R)}{λ_{2} + λ_{3}} .

(112)

Since the deformation depends on $γ$ , $λ$ and $e$ , the total energy function can be expressed by $Ω = \bar{Ω} (λ, γ, e)$ . We can easily deduce that

τ_{z θ} = \frac{\partial \bar{Ω}}{\partial γ} .

(113)

To obtain equation (113), we require the following formulae:

\frac{\partial λ_{2}}{\partial γ} = \frac{1}{λ_{2}} (sc \sqrt{λ} + λ^{2} γ s^{2}), \frac{\partial λ_{3}}{\partial γ} = \frac{1}{λ_{3}} (- sc \sqrt{λ} + λ^{2} γ c^{2})

(114)

and

\frac{\partial s^{2}}{\partial γ} = - \frac{\partial c^{2}}{\partial γ} = \frac{sc}{λ_{2}^{2} - λ_{3}^{2}} (2 \sqrt{λ} (c^{2} - s^{2}) + 4 λ^{2} γ sc) .

(115)

The total (not the mechanical) traction $N$ and the torque $M$ applied at the ends of the cylinder are as follows:

N = 2 π \int_{0}^{b} τ_{zz} r d r, M = 2 π \int_{0}^{b} \frac{\partial \bar{Ω}}{\partial γ} r^{2} d r .

(116)

To remove the hydrostatic pressure term in equation (116), we reformulate equation (116) in the form

N = π \int_{0}^{b} (2 τ_{zz} - τ_{rr} - τ_{θ θ}) r d r + π b^{2} \frac{ε_{0} λ e^{2}}{2} .

(117)

using the relation

τ_{rr} + τ_{θ θ} = \frac{1}{r} \frac{d (r^{2} τ_{rr})}{d r}

(118)

and equation (109).

Since $d_{{(L)}_{2}} = d_{{(L)}_{3}} = 0$ and $d_{{(L)}_{1}} = \partial ℵ_{a} / \partial e$ , where

ℵ_{a} = c_{0} e^{2} (λ_{1} - 1)^{2} - \frac{ε_{0} e^{2}}{λ_{1}^{2}}

and $λ_{1} = λ^{- \frac{1}{2}}$ , it is clear that $Div ($ $d_{(L)}) = 0$ .

In Figure 7, the axial force per undeformed area $N / π B^{2}$ is plotted as a function of axial stretch $λ$ . Note that, since $I_{4} = 1 / λ < 1$ , the fibre is in compression and the values of $μ_{1} = 0$ and $β = 0$ are used. Equation (117) indicates that the total axial traction depends linearly on the residual stress and quadratically on the electric field, and this is depicted in Figures 8 and 9.

5.2. Spherically symmetric deformation of a spherical shell

Moving away from cylindrical problems, we consider a thick-walled spherical shell whose reference geometry is defined by

A \leq R \leq B, 0 \leq Θ \leq π, 0 \leq Φ \leq 2 π

(119)

in terms of spherical polar coordinates $(R, Θ, Φ)$ . The current geometry is defined by

a \leq r \leq b, θ = Θ, ϕ = Φ

(120)

in terms of spherical polar coordinates $(r, θ, ϕ)$ , and we consider a deformation described by

r = Rf (R) .

(121)

The deformation gradient in spherical polar coordinates is

F \equiv (\begin{matrix} 1 + Rf' (R) & 0 & 0 \\ 0 & f (R) \\ 0 & 0 & f (R) \end{matrix}) .

(122)

Owing to the incompressibility condition, we have

f (R) = {(1 + \frac{a^{3} - A^{3}}{R^{3}})}^{\frac{1}{3}} = \frac{r}{R} .

(123)

The principal stretches are

λ_{1} = \frac{1}{λ^{2}}, λ_{2} = λ_{3} = λ = \frac{r}{R}

(124)

and the eigenvectors of $U$ in terms of spherical polar coordinates are $u_{1} \equiv [1, 0, 0]^{T}$ , $u_{2} \equiv [0, 1, 0]^{T}$ and $u_{3} \equiv [0, 0, 1]^{T}$ .

In this section, we consider the residual stress to take the form

T_{R} = t_{1} (R) E_{R} \otimes E_{R} + t_{2} (R) E_{Θ} \otimes E_{Θ} + t_{3} (R) E_{Φ} \otimes E_{Φ},

(125)

where $t_{2} (R) = t_{3} (R)$ and ${E_{R}, E_{Θ}, E_{Φ}}$ is the reference spherical polar coordinate basis. For simplicity, we assume that

t_{1} (R) = κ (R - A) (R - B)

(126)

and $κ$ has the unit kilopascal per square metre. The equilibrium equation requires

t_{2} (R) = \frac{1}{2 R} \frac{d (R^{2} t_{1} (R))}{d R} .

(127)

It is thus clear that the free stress surface (equation (2)) is satisfied.

For simplicity, we only discuss the case when $a \equiv [1, 0, 0]^{T}$ and $e_{(L)} = e (R)$ $E_{R}$ , where the function $e$ is determined using $Div (d_{(L)}) = 0$ , as described at the end of this section. Note that with this definition of $e_{(L)}$ , equation (16) is automatically satisfied.

In view of the aforementioned, the nonzero components of the Maxwell stress tensor are

{(T_{M})}_{rr} = \frac{ε_{0} e {(R)}^{2} λ^{2}}{2} = - {(T_{M})}_{θ θ} = - {(T_{M})}_{ϕ ϕ} .

(128)

The nonstretch invariants simply take the form

ζ_{1}^{(1)} = ζ_{1}^{(2)} = 1, ζ_{2}^{(1)} = ζ_{3}^{(1)} = ζ_{2}^{(2)} = ζ_{3}^{(2)} = 0, ζ_{i}^{(3)} = t_{i} (R) .

(129)

The nonzero spherical polar components of the total stress are $τ_{rr}$ , $τ_{θ θ}$ and $τ_{ϕ ϕ}$ .

Since $ζ_{2}^{(3)} = ζ_{3}^{(3)}$ , it is clear from equations (30) and (41) that

τ_{θ θ} = τ_{ϕ ϕ} .

(130)

Hence, the equilibrium equation takes the form

\frac{d τ_{rr}}{d r} + \frac{2}{r} (τ_{rr} - τ_{θ θ}) = 0 .

(131)

Integrating equation (131) and taking account of the Maxwell stress at $r = a$ , we get

τ_{rr} = \int_{a}^{r} \frac{2}{r} (λ \frac{\partial W_{(d)}}{\partial λ_{2}} - \frac{1}{λ^{2}} \frac{\partial W_{(d)}}{\partial λ_{1}}) d r + \frac{ε_{0} e {(A)}^{2} a^{2}}{2 A^{2}} .

(132)

The components $d_{{(L)}_{2}} = d_{{(L)}_{3}} = 0$ , and

d_{{(L)}_{1}} = (2 c_{0} {(λ_{1} - 1)}^{2} - \frac{ε_{0}}{λ_{1}^{2}}) e (R)

(133)

only depends on $R$ . Using equation (133) in solving $Div ($ $d_{(L)}) = 0$ , we have

e (R) = \frac{Q (A) e (A)}{Q (R)}, Q (R) = R^{2} [2 c_{0} {(λ_{1} - 1)}^{2} - \frac{ε_{0}}{λ_{1}^{2}}],

(134)

where $e (A)$ is the value of the electric field $e (R)$ at $R = A$ .

In Figure 10, the radial stress is depicted when the sphere is moved inwards, so that $a / A = 0.95$ . We only consider $B / A = 1.182$ . Although the fibre is in tension, with $(b - a) / (B - A) = 1.082$ , the radial stress is negative, taking note that $b / B = 0.970$ . It is indicated in Figure 10 that the magnitude of the radial stress $τ_{rr}$ increases in the presence of a residual stress or an electric field. However, in the case when the sphere is is moved outwards (see Figure 11), so that $a / A = 1.1$ , $b / B = 1.062787231$ , $(b - a) / (B - A) = 0.8583214670$ and hence the fibre is in compression and we use the values $μ_{1} = β = 0$ . The residual stress makes part of the radial stress negative.

Our specific constitutive equation (equation (41)) depends linearly and quadratically on the residual stress and electric field, respectively, and is indicated in Figures 12 and 13 for a sphere that is moved inwards. These behaviours are similar for a sphere that is moved outwards; hence, we omit their graphs.

6. Conclusion

In this paper, we have summarized the spectral equations to model nonlinear residual stressed electro-elastic materials with a preferred direction (RSEPD), following the work of Shariff et al. [28]. We then give results for four problems that illustrate the effects of the preferred direction and electro-elastic interactions in RSEPD materials that are capable of large deformations. We also show that fewer than half of the classical invariants proposed by Spencer [6] are independent; hence, this reduces the complexity of modelling RSEPD.

There is a pressing need for comprehensive sets of experimental data and for the assessment of large deformation material responses against a body of such data that catalogues the dependence of the mechanical response on the electric field, residual stress and preferred direction for specific geometries since data of the required kind are not currently available.

Footnotes

Appendix

The $37$ classical invariants (and their relations to our spectral invariants) in the minimal integrity basis for the set of tensors $C, T_{R}, H$ and $A$ are given as:

J_{1} = I_{1} = tr (C) = \sum_{i = 1}^{3} λ_{i}^{2}, J_{2} = I_{2} = \frac{1}{2} ({(tr C)}^{2} - tr C^{2}) = λ_{1}^{2} λ_{2}^{2} + λ_{1}^{2} λ_{3}^{2} + λ_{2}^{2} λ_{3}^{2},

\begin{matrix} J_{3} = I_{3} = det (C) = λ_{1}^{2} λ_{2}^{2} λ_{3}^{2}, \\ J_{4} = tr (T_{R}) = \sum_{i = 1}^{3} t_{ii}, J_{5} = tr (C T_{R}) = \sum_{i}^{3} λ_{i}^{2} t_{ii}, J_{6} = tr (C^{2} T_{R}) = \sum_{i}^{3} λ_{i}^{4} t_{ii}, \\ J_{7} = tr ({CT}_{R}^{2}) = \sum_{i, j = 1}^{3} λ_{i}^{2} t_{ij} t_{ji}, J_{8} = tr (C^{2} T_{R}^{2}) = \sum_{i, j = 1}^{3} λ_{i}^{4} t_{ij} t_{ji} \\ J_{9} = tr (T_{R}^{2}) = \sum_{i, j = 1}^{3} t_{ij} t_{ji}, J_{10} = tr (T_{R}^{3}) = \sum_{i, j, k = 1}^{3} t_{ij} t_{jk} t_{ki}, \\ J_{11} = a \cdot T_{R} a = \sum_{i, j = 1}^{3} a_{i} t_{ij} a_{j}, J_{12} = a \cdot T_{R}^{2} a = \sum_{i, j, k = 1}^{3} a_{i} t_{ik} t_{kj} a_{j}, \\ J_{13} = a \cdot C T_{R} a = \sum_{i, j = 1}^{3} a_{i} λ_{i}^{2} t_{ij} a_{j}, \end{matrix}

\begin{matrix} J_{14} = a \cdot {CT}_{R}^{2} a = \sum_{i, j, k = 1}^{3} a_{i} λ_{i}^{2} t_{ik} t_{kj} a_{j}, J_{15} = a \cdot C^{2} T_{R} a = \sum_{i, j = 1}^{3} a_{i} λ_{i}^{4} t_{ij} a_{j}, \\ J_{16} = a \cdot C^{2} T_{R}^{2} a = \sum_{i, j, k = 1}^{3} a_{i} λ_{i}^{4} t_{ik} t_{kj} a_{j}, J_{17} = a \cdot Ca = \sum_{i = 1}^{3} a_{i}^{2} λ_{i}^{2}, J_{18} = a \cdot C^{2} a = \sum_{i = 1}^{3} a_{i}^{4} λ_{i}^{4} . \\ J_{19} = b \cdot T_{R} b = \sum_{i, j = 1}^{3} f_{i} t_{ij} f_{j}, J_{20} = b \cdot T_{R}^{2} b = \sum_{i, j, k = 1}^{3} f_{i} t_{ik} t_{kj} f_{j}, J_{21} = b \cdot C T_{R} b = \sum_{i, j = 1}^{3} f_{i} λ_{i}^{2} t_{ij} f_{j} \\ J_{22} = b \cdot {CT}_{R}^{2} b = \sum_{i, j, k = 1}^{3} f_{i} λ_{i}^{2} t_{ik} t_{kj} f_{j}, J_{23} = b \cdot C^{2} T_{R} b = \sum_{i, j = 1}^{3} f_{i} λ_{i}^{4} t_{ij} f_{j}, \\ J_{24} = b \cdot C^{2} T_{R}^{2} b = \sum_{i, j, k = 1}^{3} f_{i} λ_{i}^{4} t_{ik} t_{kj} f_{j}, J_{25} = b \cdot Cb = \sum_{i = 1}^{3} f_{i}^{2} λ_{i}^{2}, J_{26} = b \cdot C^{2} b = \sum_{i = 1}^{3} f_{i}^{2} λ_{i}^{4}, \\ J_{27} = (a \cdot b) (a \cdot T_{R} b) = (\sum_{i = 1}^{3} a_{i} f_{i}) \sum_{i, j = 1}^{3} a_{i} t_{ij} f_{j}, J_{28} = (a \cdot b) (a \cdot Cb) = (\sum_{i = 1}^{3} a_{i} f_{i}) \sum_{i = 1}^{3} a_{i} f_{i} λ_{i}^{2}, \\ J_{29} = (a \cdot b) (a \cdot C T_{R} b) = (\sum_{i = 1}^{3} a_{i} f_{i}) \sum_{i, j = 1}^{3} a_{i} λ_{i}^{2} t_{ij} f_{j}, \\ J_{30} = (a \cdot b) (a \cdot T_{R} Cb) = (\sum_{i = 1}^{3} a_{i} f_{i}) \sum_{i, j = 1}^{3} a_{i} f_{j} t_{ij} λ_{j}^{2}, \\ J_{31} = (a \cdot b) (a \cdot C^{2} T_{R} b) = (\sum_{i = 1}^{3} a_{i} f_{i}) \sum_{i, j = 1}^{3} a_{i} λ_{i}^{4} t_{ij} f_{j}, \\ J_{32} = (a \cdot b) (a \cdot T_{R}^{2} Cb) = (\sum_{i = 1}^{3} a_{i} f_{i}) \sum_{i, j, k = 1}^{3} a_{i} t_{ik} t_{kj} λ_{j}^{2} f_{j}, \\ J_{33} = (a \cdot b) (a \cdot C^{2} T_{R}^{2} b) = (\sum_{i = 1}^{3} a_{i} f_{i}) \sum_{i, j, k}^{3} a_{i} λ_{i}^{4} t_{ik} t_{kj} f_{j}, \end{matrix}

\begin{matrix} J_{34} = (a \cdot b) (a \cdot T_{R}^{2} C^{2} b) = (\sum_{i = 1}^{3} a_{i} f_{i}) \sum_{i, j, k = 1}^{3} a_{i} t_{ik} t_{kj} λ_{j}^{4} f_{j}, \\ J_{35} = (a \cdot b) (a \cdot T_{R} C^{2} b) = (\sum_{i = 1}^{3} a_{i} f_{i}) \sum_{i, j = 1}^{3} a_{i} t_{ij} λ_{j}^{4} f_{j}, \\ J_{36} = (a \cdot b) (a \cdot {CT}_{R}^{2} b) = (\sum_{i = 1}^{3} a_{i} f_{i}) \sum_{i, j, k = 1}^{3} a_{i} λ_{i}^{2} t_{ik} t_{kj} f_{j}, J_{37} = (a \cdot b)^{2} = {(\sum_{i = 1}^{3} a_{i} f_{i})}^{2} . \end{matrix}

Since these classical invariants are expressed explicitly in terms of the proposed spectral invariants in equation (23), it is clear that only $13$ of the $37$ classical invariants are independent.

Next, based on the work of Shariff [29, 30], we show relations between the classical invariants that further validate our claim that only $13$ of the classical invariants are independent.

For convenience, we write the following subset of the classical invariants in the minimal integrity basis in the notation

(136)

\begin{matrix} I_{1} \overset{def}{=} tr (C) = \sum_{i = 1}^{3} λ_{i}^{2}, I_{2} \overset{def}{=} \frac{1}{2} ({(tr C)}^{2} - tr C^{2}) = λ_{1}^{2} λ_{2}^{2} + λ_{1}^{2} λ_{3}^{2} + λ_{2}^{2} λ_{3}^{2}, \\ I_{3} \overset{def}{=} det (C) = λ_{1}^{2} λ_{2}^{2} λ_{3}^{2} . \end{matrix}

(137)

I_{4} \overset{def}{=} tr T_{R} = \sum_{i = 1} t_{ii}, I_{5} \overset{def}{=} tr (T_{R} C) = \sum_{i = 1} λ_{i}^{2} t_{ii}, I_{6} \overset{def}{=} tr (T_{R} C^{2}) = \sum_{i = 1} λ_{i}^{4} t_{ii},

I_{7} \overset{def}{=} tr (T_{R}^{2}) = t_{11}^{2} + t_{22}^{2} + t_{33}^{2} + 2 (t_{12}^{2} + t_{13}^{2} + t_{23}^{2}),

I_{8} \overset{def}{=} tr (T_{R}^{2} C) = λ_{1}^{2} (t_{11}^{2} + t_{12}^{2}) + t_{13}^{2} + λ_{2}^{2} (t_{21}^{2} + t_{22}^{2} + t_{23}^{2}) + λ_{3}^{2} (t_{31}^{2} + t_{32}^{2} + t_{33}^{2}),

(138)

I_{9} \overset{def}{=} tr (T_{R}^{2} C^{2}) = λ_{1}^{4} (t_{11}^{2} + t_{12}^{2}) + t_{13}^{2} + λ_{2}^{4} (t_{21}^{2} + t_{22}^{2} + t_{23}^{2}) + λ_{3}^{4} (t_{31}^{2} + t_{32}^{2} + t_{33}^{2}),

(139)

\begin{matrix} I_{10} \overset{def}{=} a \cdot Ca = \sum_{i = 1}^{3} λ_{i}^{2} a_{i}^{2}, I_{11} \overset{def}{=} a \cdot C^{2} a = \sum_{i = 1}^{3} λ_{i}^{4} a_{i}^{2} \\ I_{12} \overset{def}{=} b \cdot Cb = \sum_{i = 1}^{3} λ_{i}^{2} f_{i}^{2}, I_{13} \overset{def}{=} b \cdot C^{2} b = \sum_{i = 1}^{3} λ_{i}^{4} f_{i}^{2} . \end{matrix}

From equation (136), the principal stretches $λ_{j}$ can be expressed in terms of $I_{1}$ , $I_{2}$ and $I_{3}$ via the relations [31]

(140)

λ_{i} = \frac{1}{\sqrt{3}} {I_{1} + 2 \sqrt{{(I_{1})}^{2} - 3 I_{2}} \cos (\frac{1}{3} [θ + 2 π (i - 1)])}^{\frac{1}{2}},

where

(141)

θ = \arccos [\frac{2 (I_{1})^{3} - 9 I_{1} I_{2} + 27 I_{3}}{2 [(I_{1})^{2} - 3 I_{2}]^{\frac{3}{2}}}], (I_{1})^{2} - 3 I_{2} \neq 0,

In the case when $(I_{1})^{2} - 3 I_{2} = 0$ , we have

(142)

λ_{i} = \sqrt{\frac{1}{3} I_{1} + \frac{1}{3} {(27 I_{3} - {(I_{1})}^{3}]}^{\frac{1}{3}} [\cos (\frac{2}{3} π i) + \sqrt{- 1} \sin (\frac{2}{3} π i)]} .

The three equations in equation (137) are linear in $t_{11}, t_{22}$ and $t_{33}$ . In view of equation (140) (or equation (142)), and assuming that the values of $λ_{i}$ are distinct, $t_{ii}$ can be explicitly expressed in terms of $I_{1}, I_{2}, \dots, I_{6}$ by solving the $t_{ii}$ -linear equations (equation (137)).

Equation (138) is linear in the invariants $t_{12}^{2}, t_{13}^{2}$ and $t_{23}^{2}$ and can be solved to express these invariants explicitly in terms of $I_{1}, I_{2}, \dots, I_{9}$ .

From equations (25) and (139), we have three linear equations in $a_{i}^{2}$ and three linear equations in $f_{i}^{2}$ ; hence, the vector invariants $a_{i}^{2}$ and $f_{i}^{2}$ can be expressed explicitly in terms of $I_{1}, I_{2}, \dots, I_{13}$ .

The remainder of the numerous classical invariants in the minimal integrity basis in equation (135) can be written explicitly in terms of $λ_{i}, t_{ij}, a_{i}, f_{i}$ and hence they all depend explicitly on the classical invariants $I_{1}, I_{2}, \dots, I_{13}$ . Note that care must be taken in assigning the positive or negative values of $t_{12}, t_{13}, t_{23}, a_{i}$ and $f_{i}$ in the expressions for the remaining classical invariants. Hence, it is clear that only $13$ of the $37$ classical invariants in the minimal integrity basis for two symmetric tensors and two preferred unit vectors are independent.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

ORCID iDs

MHBM Shariff

Roger Bustamante

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