A nonlinear constitutive model for a two preferred direction electro-elastic body with residual stresses

Journal Pre-proof A nonlinear constitutive model for a two preferred direction electro-elastic body with residual stresses M.H.B.M. Shariff, R. Bustamante, J. Merodio

PII: DOI: Reference:

S0020-7462(19)30385-3 https://doi.org/10.1016/j.ijnonlinmec.2019.103352 NLM 103352

To appear in:

International Journal of Non-Linear Mechanics

Received date : 31 May 2019 Revised date : 3 November 2019 Accepted date : 7 November 2019 Please cite this article as: M.H.B.M. Shariff, R. Bustamante and J. Merodio, A nonlinear constitutive model for a two preferred direction electro-elastic body with residual stresses, International Journal of Non-Linear Mechanics (2019), doi: https://doi.org/10.1016/j.ijnonlinmec.2019.103352. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

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A nonlinear constitutive model for a two preferred direction electro-elastic body with residual stresses 1

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M.H.B.M. Shariff1 , R. Bustamante2 , J. Merodio3 Department of Mathematics

Khalifa University of Science and Technology, UAE. 2

Departamento de Ingenier´ıa Mec´anica, Universidad de Chile Beauchef 851, Santiago Centro, Santiago, Chile.

3

Department of Continuum Mechanics and Structures, Escuela de Ingenieros de Caminos

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Universidad Politecnica de Madrid, 28040 Madrid, Spain

Abstract

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A nonlinear spectral formulation, which is more general than the traditional classical-invariant formulation, is used to describe the mechanical behaviour of a residually stressed electro-elastic body with two preferred directions (ESTPD); the generality of the spectral formulation could facilitate the quest for good constitutive equations for ESTPDs. The strain energy is a function of spectral invariants (each with a clear physical meaning) that depend on the right stretch tensor, the residual stress tensor, two preferreddirection-structural tensors and one of the electric variables; clear meaningful physical invariants are useful in aiding the design of a rigourous experiment to construct a specific form of constitutive equation. Separable finite strain constitutive equations containing general single-variable functions, which depend only on a principal stretch or the electric field, are proposed and, in view of this, the corresponding infinitesimal strain energy functions can be easily constructed. A specific form for the strain energy function is generally easier to obtain from the general strain energy function via experiment if it is expressed in terms of general single-variable functions. In the case when ESTPDs are specialized to soft tissues the proposed constitutive equations can be easily converted to allow the mechanical influence of compressed fibres to be excluded or partially excluded and to model fibre dispersion in collagenous soft tissues. With the aid of spectral invariants, we easily prove that at most 15 of the 98 classical invariants in the corresponding minimal integrity basis are independent; this proof cannot be found in the literature. A simple tension boundary value problem with cylindrical symmetry is studied, where the residual stress is assumed to depend only on the radial position.

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Keywords: Nonlinear electro-elasticity, Separable Constitutive model, Residual stress, Two preferred directions, Spectral formulations, Physical invariants, Independent invariants.

Introduction

The existence of residual stresses in two-preferred-direction materials has been the subject of many publications, see for example, [1, 39]. The consequences of residual stresses from a mechanics point of view, 1

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are yet to be fully appreciated. This has created considerable interest during the last few years and many valuable publications have resulted from attempts to understand the influence of residual stresses on the mechanical behaviour of solid materials [14, 13]. However, the mechanical response of a residually stressed electro-sensitive material with and without preferred directions was only recently investigated [33, 34]. Electro-active materials deform under the application of an electric field, and they have recently attracted growing interest because of its potential for use in actuators, artificial muscles in robotics and biomedical applications in prostheses [2]. 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 materials characterization. Hence, constitutive equations using variables (invariants) with a clear physical meaning are particularly useful for experimental design [21]. With this in mind, and following the recent work of Shariff and co-workers [29, 33, 34], we develop constitutive models using spectral invariants (instead of classical invariants [35]) with a clear physical interpretation. The advantages of spectral invariants over classical invariants are described, for example, in references [21, 33, 34], hence we will not describe them here. 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 [29, 33, 34]. Recently, Shariff [28, 23, 32] showed that separable constitutive equations are able to satisfactorily model the mechanical behaviour of anisotropic solids. We note that the separable Valanis-Landel form [38] is also able to satisfactorily describe the mechanical behaviour of isotropic elastic solids [15, 20]. Due to lack of (possibly non-existent) experimental data obtained from rigourous experiments (experiments that require to vary one invariant while keeping the remaining invariants fixed (we called this R-experiment) [8, 21].), we are forced to assume a specific general form, and in view of the recent success of separable constitutive equations, we propose spectral constitutive equations that are expressed in terms of separable forms that contain general single-variable functions that depend on a principal stretch or the electric field. Some materials exclude or partially exclude the contribution of compressed fibres in their strain energies and in some collagenous soft tissues the fibres are dispersed. Hence, one of our aims is to construct constitutive equations so that they can be easily modified to exclude or partially exclude the mechanical influence of compressed fibres and to model fibre dispersion in collagenous soft tissues. If a constitutive model is developed via the traditional classical invariant method, it generally considers 98 classical invariants (see [35] and Appendix A) in the associated minimal integrity basis to describe its mechanical behaviour, and it is not clear in the literature how many of the numerous 98, generally nonphysical, classical invariants are independent. In view of of their non-physical nature it is not clear which of the numerous invariants are candidates for the set of unknown number of independent invariants. Hence, knowing the number of independent invariants facilitates a rigorous construction of a constitutive equation via experiments (see, for example [21, 33, 34]) and, hence, an additional aim of this paper is to evaluate the number of independent invariants from the given set of invariants in the corresponding minimal integrity basis [35].

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The paper is divided in 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 develop a spectral constitutive equation using a total energy function (see [4, 5]), that depends on the deformation, the two-preferred directions, the residual stresses and the 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 number of independent invariants based on the spectral formulation developed by Shariff and co-workers [21, 22, 24, 26, 29] and in addition to this, in Appendix A, we give relations between classical invariants in the corresponding minimal integrity basis. In Section 4, several 2

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Governing Equations

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constitutive equations are developed that lead to a development of a simple form of the total energy function. In Appendix C the proposed total energy is simply modified to exclude or partially exclude the mechanical influence of compressed fibres and to model fibre dispersion in collagenous soft tissues. A special form of the proposed total energy function for soft tissues is used in Section 5 to study a simple tension boundary value problem with cylindrical symmetry. Finally, in Section 6 we give some final remarks.

2.1 Preliminaries

In this paper, summation convention is not used and all subscripts i,j and k take the values 1, 2, 3, unless stated otherwise. Only quasi-static 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 [37].

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∂x and C = F T F = ∂X U 2 , respectively, where U is the right stretch tensor, 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.

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The deformation gradient and the right Cauchy-Green tensor are denoted by F =

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, DivT R = 0 in Br

(1)

T R N = 0 on ∂Br ,

(2)

and the boundary condition

where ∂Br is the boundary of Br and N is unit outward normal to ∂Br . In view of (1) and (2), the residual stress is non-homogeneous with the mean value Z T R dV = 0 . (3) Br

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2.3 Electrostatics

If there is no interaction with magnetic fields and there is no distribution of free charges, then the simplified forms of the Maxwell equations are curl(e) = 0 , div(d) = 0 , 3

(4)

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where e is the electric field and 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 = ε0 e ,

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(5)

where εo = 8.85 × 10−12 F/m is the electric permittivity in vacuum. For a condensed matter an extra field is needed, which is called the electric polarization p, where

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d = ε0 e + p .

(6)

In the present communication we use the concept of total Cauchy stress T defined in [4, 5]. In the absence of surface electric charges d, e and T must satisfy the continuity equations n · [[d]] = 0,

n × [[e]] = 0,

T n = ˆt + T M n,

(7)

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where · and × denote the dot product and cross product, respectively, between vectors, n is the unit outward normal vector to ∂Bt , ˆt 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 1 T M = d ⊗ e − (d · e)I. 2

(8)

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More details about electrostatics and continuum mechanics can be obtained, from, for example, in [10, 16].

Constitutive Equation

3.1 Two-preferred-direction electro-elastic body with residual stresses Following the work of Bustamante and Shariff [3] and Shariff et.al. [29, 33, 34], the total energy Ω for the problem is assumed to depend on [4, 5, 16]: U , T R , A = a ⊗ a, , D = b ⊗ b , H = f ⊗ f , e ,

(9)

where the unit vectors a and b are the preferred directions, ⊗ denotes the dyadic product, f=

1 e(L) , e = |e(L) | > 0. e

(10)

and the Lagrangian electric field e(L) is defined as e(L) = F T e[4, 5]. Hence, we can write

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Ω = W(α) (U , T R , A, D, H, e) = W(a) (U , T R , a, b, f , e) .

(11)

For an incompressible body, the total Cauchy stress is [4, 5] T =F

∂Ω T ∂Ω − pI = 2F F − pI , ∂F ∂C 4

(12)

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d = −F

∂Ω . ∂e(L)

Using the relations

we obtain the Lagrangian electric displacement [4, 5] d(L) = −

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∂f 1 = (I − f ⊗ f ) , ∂e(L) e

∂e =f, ∂e(L)

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where p is Lagrange multiplier evaluated at the current configuration and the Eulerian electric displacement is

∂W(a) ∂W(a) 1 ∂Ω =− f + (I − H)T , ∂e(L) ∂e e ∂f

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where d(L) = F −1 d and I is the identity tensor. We also have,

Curl(e(L) ) = 0 and Div(d(L) ) = 0 ,

(13)

(14)

(15)

(16)

where Div and Curl are, respectively, the divergence and curl operators with respect to X. In view of (16)1 we could express

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e(L) = Gradφ ,

(17)

where φ is a scalar potential and Grad is the gradient operator with respect to X. The total nominal stress is given by S=

∂Ω − pF −1 . ∂F

(18)

The variable x(X) and φ can be obtained from solving the equilibrium equation Div S = 0

(19)

and equation (16)2 with the appropriate boundary conditions.

By the definition of T R in Section 2.2, and in view of (11) and (12), the constitutive equation must satisfy TR =

∂W(a) (I, T R , a, b, f , 0) − po I ∂F

(20)

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at the reference configuration in the absence of an electric field, where p0 is the Lagrange multiplier eval∂W(a) uated at the reference configuration. Although the vector f appears in (20), should be independent ∂F of f .

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The total energy must be invariant with respect to the rotation Q [35], hence

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

W(a) (U , T R , a, b, f , e) = W(a) (QU QT , QT R QT , Qa, Qb, Qf , e) .

(21)

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The required symmetry (21) reduces W(a) to depend on the isotropic invariants of the set S = {U , T R , a, b, f }. To obtain these isotropic invariants, we simply express the components of the elements of S using the basis {u1 , u2 , u3 }, where ui is an eigenvector of U . Hence, we can express Ω in terms of the spectral component invariants e , λi = ui · U ui , tij = ui · T R uj , ai = a · ui , bi = b · ui , fi = f · ui . Alternatively, we could use the six independent invariants

(23)

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ui · T R ui , ui · T 2R ui

(22)

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instead of the invariants tij . To facilitate the construction of P -property constitutive equations [28], we rather use the invariants in (23) in our constitutive equations instead of the invariants tij . We must emphasize that the components of the vectors and tensors in the set S, with respect to an arbitrary basis, are not, in general, invariants. However, the components given in (22) are invariants as described in Appendix A. We note that although ai , bi and fi are invariants with respect to the rotation Q, they are not invariants of the of the set of tensors given in (9) that are arguments of the energy function W(α) described in (11). For example, the invariants a2i , b2i , fi2 , (a · b)ai bi , (a · f )(ai fi ) and (b · f )(bi fi ) are subset of the set of invariants for the set of the tensors in (9) for the energy function W(α) . Since a, b and f are unit vectors, we have,

a23 = 1 − a21 − a22 , b23 = 1 − b21 − b22 , f32 = 1 − f12 − f22 .

(24)

In view of (24), it is clear that, after omitting e, at most 15 of the invariants in (22) are independent. Note that in view of the incompressibility constraint λ1 λ2 λ3 = 1 only 14 of the 15 invariants are independent. The relations of the above invariants with the classical invariants in the corresponding minimal integrity basis developed by Spencer [35] are given in Appendix A. We note that a total energy function that is described by the above spectral invariants must satisfy the P -property as described in [28]. The physical meaning of the invariants in (22) are easily interpreted. The majority of the corresponding classical invariants in the corresponding minimal integrity basis do not have a clear physical meaning. Since, the classical invariants can be written explicitly in terms of the spectral invariants (22) (see Appendix A), we can claim that at most 15 (omitting e) of the 98 classical invariants in the corresponding minimal integrity basis are independent and in Appendix A, we strengthen our claim by showing there are 83 relations amongst the 98 classical invariants.

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In this paper, for simplicity, we shall not use the invariants ui · T 2R ui , hence we can express Ω = W(b) (λ1 , λ2 , λ3 , a1 , a2 , a3 , b1 , b2 , b3 , ζ1 , ζ2 , ζ3 , e) ,

(25)

where ζi = ui · T R ui . We strongly emphasize that the function W(b) must satisfy the P -property described in reference [28]. 6

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∂Ω with respect to the spectral Lagrangean ∂C basis [u1 , u2 , u3 ] and the total Cauchy stress tensor T with respect to the Eulerian basis {v1 , v 2 , v 3 }, where v i = Rui and R = F U −1 . Therefore, we have   ∂Ω 1 ∂W(b) = (26) ∂C ii 2λi ∂λi

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The spectral formulation requires the tensor components of



∂Ω ∂C



=

ij

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and for the shear components ∂W(b) ∂W(b) · uj − · ui ∂ui ∂uj , 2(λ2i − λ2j )

i 6= j .

(27)

The Eulerian spectral components of the total Cauchy stress T for an incompressible body are =

τij

=

∂W(b) −p, ∂λi   ∂Ω , i 6= j . 2λi λj ∂C ij λi

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τii

(28) (29)

The Lagrangian spectral components for the electric displacement d are:

where

3 X

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d(L)k

d(L)k uk ,

(30)

k=1

  ∂W(b) 1 T ∂W(b) (I − H) · uk . (f · uk ) + = ∂e e ∂f

(31)

The electric field in the deformed configuration can simply be expressed by d=−

4

3 X

λk d(L)k v k .

(32)

k=1

Energy functions

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Before we construct a total energy function for a ESTPD, we first construct constitutive equations for two preferred direction elastic solids in the absence of the residual stress and the electric field. There is only a few nonlinear spectral constitutive equations for two-preferred-direction solids that exist in the literature [22, 25, 28, 31]. In view of this, the authors feel that proposing general nonlinear spectral twopreferred-direction strain energy functions for different types of materials may be beneficial to the scientific community. We note that due to non-existent R-experiment data, we are not able to rigourously construct finite-strain constitutive equations for specific types for this class of materials. However, finite strain constitutive equations must be consistent with their infinitesimal strain counterparts, and in addition to this, we 7

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note that an infinitesimal strain energy function, which automatically satisfies the P -property, can be useful in facilitating the construction of a finite strain energy function, which satisfy the P -property. Hence, to aid the construction of a finite strain constitutive equation in the absence of R-experiment data, we initially construct an infinitesimal strain energy function, and using the spectral approach, we clearly show that the corresponding finite strain energy function can be easily constructed. Since the construction of the finite strain energy function is derived from the infinitesimal strain energy function, both energy functions are easily converted to each other as shown below. Two additional advantages of this ”infinitesimal strain” approach are that (a) the finite strain energy function contains separable single-variable functions; taking note that single-variable functions are easier to analyse than multivalued functions and (b) the strain energy function can be simply modified to exclude or partially exclude the mechanical influence of compressed fibres and to model fibre dispersion in collagenous soft tissues; these simple modifications are given in Appendix C.

4.1 Infinitesimal strain: Strain energy of two fibre elastic solids

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Following the work presented in [26, 36], the most general quadratic form for an incompressible twopreferred-direction infinitesimal strain energy function has the expression W(T ) = n(1) K1 + n(2) K2 + n(3) K3 + n(4) K4 + n(5) K52 + n(6) K62 +n(7) K5 K6 + n(8) K5 K7 + n(9) K6 K7 + n(10) K72 ,

K1 =

3 X i=1

K4 =

3 X i=1

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where νi2 = tr E 2 , K2 =

3 X i=1

a ¯2i νi2 = a · E 2 a, K3 =

χ ¯i νi2 = tr (ADE 2 ) , K5 =

3 X i=1

K7 =

3 X i=1

a ¯2i νi = a · Ea , K6 =

3 X

¯b2 ν 2 = b · E 2 b , i i 3 X i=1

χ ˜ i νi ,

¯b2 νi = b · Eb , i

(33)

(34)

(35)

(36)

i=1

where

χ ˜i = (a · b)(a · si )(b · si ) , a ¯i = si · a , ¯bi = si · b ,

(37)

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si is an eigenvector of the infinitesimal strain E, νi is an eigenvalue of E and n(1) , n(2) , . . . , n(10) are ground-state material constants.

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4.2 Finite strain energy function for two fibre elastic solids

W(T ) =

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Using the infinitesimal strain energy function (33), we can easily propose a finite-strain energy function of the form 3 3 X X   αi q5 (λi ))2 + n(1) q1 (λi ) + n(2) αi q2 (λi ) + n(3) βi q3 (λi ) + n(4) χi q4 (λi ) + n(5) ( i=1 3 X

i=1

n(6) (

βi q6 (λi ))2 + n(7)

3 X

αi q7 (λi )

βi q8 (λi ) + n(8)

n(9)

3 X

βi q11 (λi )

3 X

αi q9 (λi )

i=1 3 X

i=1

i=1

i=1

i=1

3 X

3 X

χi q12 (λi ) + n(10) (

χi q13 (λi ))2 , (38)

i=1

i=1

i=1

χi q10 (λi ) +

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3 X

where αi = a2i , βi = b2i , χi = (a · b)ai bi . To be consistent with the theory of infinitesimal strain, we impose the restrictions [32]

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qs (1) = 0 , s = 1, 2, . . . , 13 , q1′ (1) = q2′ (1) = q3′ (1) = q4′ (1) = 0 , q1′′ (1) = q2′′ (1) = q3′′ (1) = q4′′ (1) = 2 , qs′ (1) = 1 , s = 5, 6, . . . , 13 .

(39)

Alternatively, due to the incompressibility constraint, we could also impose q1′ (1) = q1′′ (1) = 1. In the case when a and b are orthogonal we have only the six constant strain energy function [25] 3 X  i=1

3 X  αi q5 (λi ))2 + n(1) q1 (λi ) + n(2) αi q2 (λi ) + n(3) βi q3 (λi ) + n(5) (

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W(T ) =

3 X

n(6) (

βi q6 (λi ))2 + n(7)

i=1

i=1

3 X i=1

αi q7 (λi )

3 X

βi q8 (λi ) .

(40)

i=1

4.3 Infinitesimal strain: Mechanically equivalent incompressible solid For a mechanically equivalent (with respect to a and b) incompressible solid, the most general quadratic form for the infinitesimal strain energy function takes the form W(T M) = m(1) K1 + m(2) (K2 + K3 ) + m(3) K4 + m(4) (K52 + K62 ) + m(5) K72 + m(6) K5 K6 + m(7) (K5 + K6 )K7 , (41) where m(1) , m(2) , . . . , m(7) are ground-state material constants.

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4.4 Finite strain energy function: Mechanically equivalent incompressible solid Using (41) we propose the following strain energy function for finite deformations: W(T M) =

3 X  i=1

 m(1) s1 (λi ) + m(2) (αi + βi )s2 (λi ) + m(3) χi s3 (λi ) + 9

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i=1

i=1

m(6)

3 X

αi s6 (λi )

3 X

βi s6 (λi ) + m(7)

+ m(5)

!2

χi s5 (λi )

i=1

!

(αi + βi )s7 (λi )

i=1

i=1

i=1

3 X

3 X

+ !

3 X

χi s8 (λi )

i=1

.

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m(4)

!

of

3 3 X X βi s4 (λi ))2 αi s4 (λi ))2 + ( (

(42)

To be consistent with the theory of infinitesimal strain, we impose the restrictions [32] sr (1) = 0 , r = 1, 2, . . . , 8 , s′1 (1) = s′2 (1) = s′3 (1) = 0 ,

s′4 (1) = s′5 (1) = s′6 (1) = s′7 (1) = s′8 (1) = 1 ,

s′′1 (1) = s′′2 (1) = s′′3 (1) = 2 .

(43)

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Due to the incompressibility constraint, we could also impose s′1 (1) = s′′1 (1) = 1.

In Appendix C, we simply modify the constitutive equations proposed in this Section to exclude or partially exclude the contribution of compressed fibres in the constitutive equation and to model fibre dispersion in collagenous soft tissues.

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4.5 Total Energy: Specific form

Until now there is not enough experimental data available in the literature to propose some specific expressions for the total energy, 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, we propose an expression for Ω, which depends on a number of single-variable functions. We note that a more general constitutive equation can be constructed based on the work of Shariff et. al. [29] but the form will be too cumbersome for any practical use. For simplicity, in this section, we propose two simple total energy function prototypes for a two-preferred-direction electro-elastic residually stressed body: Ω = W(T ) + c1

3 X

r1 (λi )ζi + c2 (e)

3 X i=1

i

γi r2 (λi ) − ε0 e2

3 X

3

γi

i=1

X 1 ζi (λi − 1) + 2 2λi i=1

(44)

for non-mechanically equivalent materials and Ω = W(T M) + c1

3 X

r1 (λi )ζi + c2 (e)

i

3 X i=1

γi r2 (λi ) − ε0 e2

3 X i=1

3

γi

X 1 ζi (λi − 1) + 2 2λi i=1

(45)

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for mechanically equivalent materials. We impose c2 (0) = 0 so that in the absence of an electric field the constitutive equations (44) and (45) revert to that for residually stress material with two-preferred directions, To be consistent with the theory of infinitesimal strain, we require the restrictions r1 (1) = r2 (1) = r1′ (1) = r2′ (1) = 0 , r1′′ (1) = r2′′ (1) = 2 . 10

(46)

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Note that (44) and (45) satisfy the P -properties as described in Shariff [28].

d = −F

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In view of (44), (45) and (13), the Eulerian electric displacement and polarization then simply take the forms ∂ℵ ∂ℵ + ε0 e , p = −F , ∂e(L) ∂e(L)

3 X

ℵ=

c2 (e)γi r2 (λi ) .

i=1

The components d(L)k in (31) simply take the form

where ℵa = ℵ − ε0

i=1

γi

e2 . 2λ2i

Boundary value problem

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5

P3

3

∂ℵa X ∂ℵa − γi ∂γk ∂γi i=1

∂ℵa 2 = (f · uk ) + ∂e e

!#

,

(48)

(49)

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d(L)k

"

pro

where

(47)

Here, to illustrate our theory, we only consider a specific constitutive equation for a mechanically equivalent material given by the constitutive equation (45) with c2 (e) = c0 e2 and, for simplicity, W(T M) =

3 h X i=1

i m(1) s(1) (λi ) + m(2) (ˆ αi + βˆi )s(2) (λi ) ,

(50)

where α ˆ i and βˆi are given in (C4). We assume that fibre compression does not contribute towards the strain energy function (50). Hence, if both fibres are compressed then (50) represents a strain energy function for an isotropic material and if one of the fibre is compressed and the other is in tension then (50) represents a strain energy function of a transversely isotropic material. So in this Section W(b) =

3 h X i=1

+c1

3 X

i m(1) s(1) (λi ) + m(2) (ˆ αi + βˆi )s(2) (λi )

r1 (λi )ζi + c0 e2

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i=1

3 X i=1

γi r2 (λi ) − ε0 e2

3 X i=1

γi

The spectral components of (26) and (27) then take the form   ∂Ω 1 ∂W(b) = (i not summed) ∂C ii 2λi ∂λi

11

3

X 1 ζi (λi − 1) . + 2 2λi i=1

(51)

(52)

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and for the shear components       ∂W(b) ∂W(b) ∂Ω 1 (2) (2) = t + m s (λ ) − s (λ ) (ˆ ai a ˆj + ˆbiˆbj ) − ij i j (2) ∂C ij (λ2i − λ2j ) ∂ζi ∂ζj    ∂W(b) ∂W(b) (53) fi fj , i 6= j . − + ∂γi ∂γj

pro

For the case of s(1)′ (1) = s(1)′′ (1) = 1 at the reference configuration, in the absence of an electric field, Equation (20) becomes T R = (m(1) − p0 )I + T R .

(54)

m(1) = p0 .

(55)

The above requires

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In the case where the conditions s(1)′ (1) = 0 , s(1)′′ (1) = 2 are imposed at the reference configuration, in the absence of an electric field, Equation (20) takes the form T R = −p0 I + T R , which implies that p0 = 0.

(56)

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We only discuss results for simple tension problem with cylindrical symmetry, which could be important from the experimental point of view, and in this Section we only consider simple tension on soft tissues. Results on deformations for non-soft tissue materials such as polymers will be obtained in the near future when there are, hopefully, more experimental results. To simulate a two-preferred-direction fibrous soft tissue (in the absence of residual stress and an electric field) realistically, we use the functions Z x Z x 4 4 √ erf −1 (κ1 ln(y)) dy , s(2) (x) = √ erf −1 (κ2 ln(y)) dy , s(1) (x) = (57) 1 κ1 π 1 κ2 π where in this case s(1)′ (1) = 0 , s(1)′′ (1) = 2, κ1 , κ2 6= 0 are dimensionless material parameters. Note that the above functions have been used by Shariff [32] to fit the mitral valve anterior leaflet biaxial experimental data of Weinberg and Kaazempur-Mofrad [40]; they also able to predict their experiment data. We note that the numerical values for the ground state constants given in Section 5.2 satisfy the strong ellipticity inequalities given in Appendix B. The results for simple tension require residual stresses in a circular cylindrical tube which is given in the next section. All tensor and vector components in sections (5.1) to (5.2) are cylindrical polar coordinate components.

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5.1 Residual stress for cylindrical problems

In this section, we give a specific form of residual stresses in a reference configuration defined by A ≤ R ≤ B,

0 ≤ Θ ≤ 2π,

where R, Θ and Z are reference polar coordinates. 12

0 ≤ Z ≤ L,

(58)

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Following [12] we consider a residual stress of the form T R = t1 (R)q 1 ⊗ q 1 + t2 (R)q 2 ⊗ q 2 ,

(59)

of

where q 1 = E R and q 2 = E Θ are cylindrical polar coordinate vectors in the reference configuration. The stress tensor (59) must satisfy the equilibrium equation

The component t1 must satisfy the boundary conditions t1 (A) = 0,

pro

d(Rt1 ) = t2 . dR t1 (B) = 0.

(60)

(61)

In [12] it has been shown that for the above T R , t2 (R) has both positive and negative values for A ≤ R ≤ B. A simple example of t1 is t1 = c¯(R − A)(R − B) , (62)

5.2 Uniform extension of a cylinder

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where, for illustrative purposes, we use the value c¯ = 1000kP a/m2.

Here, we consider a cylinder with a configuration defined in (58) and the deformation described by

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1 r = √ R , θ = Θ , z = λz Z , λz

where (r, θ, z) is the polar coordinate in the deformed configuration. It follows that   1 1 F ≡ diag √ , √ , λz . λz λz

(63)

(64)

1 1 We have that λ1 = λr = √ , λ2 = λr = √ , λ3 = λz . Hence, q 1 = u1 ≡ [1, 0, 0]T , q 2 = u2 ≡ λz λz [0, 1, 0]T and u3 ≡ [0, 0, 1]T . Consider the case when the preferred directions are [1]

a = cE Θ + sE Z , b = cE Θ − sE Z , (65) π where c = cos(Ψ), s = sin(Ψ) and > Ψ > 0 is the angle each fibre makes with the circumferential 2 direction [1]. We simply have the spectral components a1 = b1 = 0 , a2 = b2 = c , a3 = −b3 = s , b1 = 0 , ζ1 = t1 , ζ2 = t2 , ζ3 = 0 , tij = 0 , i 6= j .

(66)

c2 + s2 λ2z . From Fig. 1, for 1 ≤ λz ≤ 1.5, we see that λz o the fibres are in tension when Ψ = 45 , compression and tension when Ψ = 30o and compression when Ψ = 15o .

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We note that the invariant I = a · Ca = b · Cb =

In this section, we give preliminary results for three different cases for e(L) . The results depicted in Figures 2 to 7 are briefly explained in Section 5.3. 13

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Figure 1: I = a · Ca = b · Ca vs λz

5.2.1 Case 1: e(L) ≡ [0, 0, e]T

For this case we have, from (17), φ = e0 Z, where e = e0 is constant and f ≡ [0, 0, 1]T . Since e(L) ≡ [0, 0, e0 ]T = Gradφ the condition Curl(e(L) ) = 0 is automatically satisfied and we have f1 = f2 = 0 and f3 = 1. It is clear from (53) that



∂Ω ∂C



ij

= 0 , i 6= j

(67)

and hence the total stress is coaxial with the right stretch tensor U and, we have,

Jo

T = τrr u1 ⊗ u1 + τθθ u2 ⊗ u2 + τzz u3 ⊗ u3 ,

(68)

where τrr , τθθ , τzz are physical cylindrical components of T . The Lagrangian components of the electric displacement is simplified to d(L)1 = d(L)2 = 0 , d(L)3 = 14

∂ℵa . ∂e

(69)

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Figure 2: (a) e(L) ≡ [0, 0, e0 ]T (b) e(L) ≡ [

e0 e0 , 0, 0]T (c) e(L) ≡ [0, , 0]T , (d)e(L) ≡ [0, 0, 0]T . R R

In view of (49), d(L)3 is independent of X and hence Div(d(L) ) = 0 is automatically satisfied. The electric field is



e0 e ≡ 0, 0, λz

T

.

(70)

The non-zero Maxwell stress components in vacuo are (T M )zz =

ε0 e20 = −(T M )rr = −(T M )θθ . 2λ2z

(71)

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

ε0 e20 . 2λ2z

15

(72)

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pro

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Figure 3: (a) e(L) ≡ [0, 0, e0 ]T (b) e(L) ≡ [

e0 e0 , 0, 0]T (c) e(L) ≡ [0, , 0]T . R R

The non-zero components of the total stress in the body are: τrr = λr

∂W(b) ∂W(b) ∂W(b) − p , τθθ = λr − p , τzz = λz −p. ∂λ1 ∂λ2 ∂λ3

(73)

We note that the total Cauchy stress is inhomogeneous. Since, the total energy function depends on r (or in view of (63), equivalently on R) and taking note that, in view of (53) and (66), the shear stresses are zero and the Cauchy stress must satisfy the equilibrium equation r

dτrr = τθθ − τrr , dr

Jo

which can be integrated to give  Z r ∂W(b) dr ∂W(b) ε0 e20 − λr − λr τrr = ∂λ2 ∂λ1 r 2λ2z b i ε e2 1 h 0 0 = √ 2m ¯ (2) c2 s(2)′ (λ2 )ln(R/B) + (c1 r1′ (λ2 ) + 1)t1 (R) − , 2λ2z λz 16

(74)

(75)

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Figure 4: (a) e(L) ≡ [0, 0, e0 ]T (b) e(L) ≡ [

e0 e0 , 0, 0]T (c) e(L) ≡ [0, , 0]T , (d)e(L) ≡ [0, 0, 0]T . R R

where b is the current radial value of the reference radial value B and m ¯ (2) = m(2) q(p) (I)2 . The axial stress is given by τzz

= =

∂W(b) ∂W(b) − λr + τrr ∂λ3 ∂λ1   ε0 e20 ¯ (2) s2 s(2)′ (λz ) + c0 e20 r2′ (λz ) + λz m(1) s(1)′ (λz ) + 2m − λ1 m(1) s′1 (λ1 ) 2λ3z λz

+

2m ¯ (2) c2 s(2)′ (λ2 )ln(R/B) √ . λz

(76)

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5.2.2 Case 2: e(L) ≡ [e(R), 0, 0]T

e0 e0 , 0, 0]T = Gradφ and e(R) = . R R Hence Curl(e(L) ) = 0 and f1 = 1, f2 = f3 = 0. Similar to Section 5.2.1, the total stress is coaxial with Here we have taken φ = e0 ln(R), where e0 = Be(B). Hence e(L) ≡ [

17

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Figure 5: (a) e(L) ≡ [0, 0, e0 ]T (b) e(L) ≡ [

e0 e0 , 0, 0]T (c) e(L) ≡ [0, , 0]T R R

the right stretch tensor U . The non-zero Maxwell stress components in vacuo are (T M )rr =

ε0 λz e2 = −(T M )zz = −(T M )θθ . 2

(77)

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

ε0 λz e(R)2 . 2

(78)

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Following the method in Section 5.2.1, we have   1 1 1 e2 τrr = √ − ) )( 2c2 m ¯ (2) s(2)′ (λ2 )ln(R/B) + (c1 r1′ (λ2 ) + 1)t1 (R) + 0 (c0 r2′ (λ1 ) + ε0 λ−3 1 2 R2 B2 λz +

ε0 λz e20 . 2R2 18

(79)

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τzz

e0 e0 , 0, 0]T (c) e(L) ≡ [0, , 0]T , (d)e(L) ≡ [0, 0, 0]T . R R

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Figure 6: (a) e(L) ≡ [0, 0, e0 ]T (b) e(L) ≡ [

  e2 1 m(1) s(1)′ (λ1 ) + 02 (c0 r2′ (λ1 ) + ε0 λ−3 ) = λz [m(1) s(1)′ (λz ) + 2m ¯ (2) s2 s(2)′ (λz )] − √ 1 R λz   2 2 e 1 ε0 λz e0 1 1 2c2 m ¯ (2) s(2)′ (λ2 )ln(R/B) + 0 (c0 r2′ (λ1 ) + ε0 λ−3 ) + . (80) + √ 1 )( 2 − 2 2 R B 2R2 λz

The Lagrangian components of the electric displacement is simplified to e0 d(L)2 = d(L)3 = 0 , d(L)1 = (2c0 r2 (λ1 ) − ε0 λ−2 1 ). R In view of (81), clearly Div(d(L) ) = 0.

(81)

5.2.3 Case 3: e(L) ≡ [0, e(R), 0]T

e0 e0 T , 0] , where e(R) = and e0 = Be(B). Hence R R Curl(e(L) ) = 0, f1 = f3 = 0 and f2 = 1. Similar to Section 5.2.1, the total stress is coaxial with the right stretch tensor U . The non-zero Maxwell stress components in vacuo are

Jo

If we use φ = e0 Θ, we have e(L) = Gradφ ≡ [0,

(T M )rr = −

ε0 λz e2 = (T M )zz = −(T M )θθ . 2 19

(82)

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e0 e0 , 0, 0]T (c) e(L) ≡ [0, , 0]T . R R

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Figure 7: (a) e(L) ≡ [0, 0, e0 ]T (b) e(L) ≡ [

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

ε0 λz e(R)2 . 2

(83)

Following the method in Section 5.2.1, we have,   e20 1 1 1 −3 2 (2)′ ′ ′ √ 2c m ¯ (2) s (λ2 )ln(R/B) + (c1 r1 (λ2 ) + 1)t1 (R) + (c0 r2 (λ2 ) + ε0 λ2 )( 2 − 2 ) τrr = 2 B R λz

τzz

=

ε0 λz e20 . 2R2

Jo

−

(84)

1 n λz [m(1) s(1)′ (λz ) + 2m ¯ (2) s2 s(2)′ (λz )] + √ −m(1) s(1)′ (λ1 ) + 2c2 m ¯ (2) s(2)′ (λ2 )ln(R/B) λz  1 ε0 λz e20 1 e20 − ) − . (85) (c0 r2′ (λ1 ) + ε0 λ−3 )( 1 2 2 2 B R 2R2 20

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The Lagrangian components of the electric displacement is simplified to e0 (2c0 r2 (λ2 ) − ε0 λ−2 2 ). R

(86)

of

d(L)1 = d(L)3 = 0 , d(L)2 = In view of (86), clearly Div(d(L) ) = 0.

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5.3 Analysis

re-

It is clear from (76), (80) and (85) that the total axial stress τzz is independent of residual stress in all of the three cases. To investigate the behaviour of stress strain relationships we plot several graphs using c1 = 0 and c0 = 10−10 F/m. To simulate the mechanical behaviour of soft tissue, and for simplicity, we let s(1) = s(2) with κ1 = κ2 = 6 [32], r2 (λi ) = (λi − 1)2 , m(1) = 5.0kP a and m(2) = 10kP a. In Fig. 2, the average axial force Z b 2 Fz = 2 τzz r dr , (87) b − a2 a

6

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where a is the current radial value of the referential radial value A, is plotted for all the three cases. Fig. 2 shows that the magnitude of Fz increases as the value of electric field e0 increases. Fig. 3 indicates that magnitude of Fz decreases as the fibre angle Ψ decreases from 45o to 15o ; in both figures the magnitude of Fz for Case 1 is greater than Case 3 which in turn is greater than Case 2. The radial stress τrr is depicted in Figs. 4 and 5. Fig. 4 depicts that the magnitude of τrr increases as e0 increases. The presence of residual stress seems to lower the values of the radial stress τrr as shown in Fig. 5. The change of the value of Ψ does not significantly affect the values of τrr and hence we omit plotting the dependence of τrr on Ψ. In the case of the axial stress τzz , its behaviour due to the three cases and changes in the value of Ψ is depicted in Figs. 6 and 7.

Conclusion

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We have summarized the spectral equations to model nonlinear residual stressed electro-elastic materials with two preferred directions following the work Shariff et. al. [29, 33, 34]. In Appendix C, we show that the proposed separable form of constitutive equation can be easily converted to take account of the problem of fibre compression and to model fibre dispersion in collagenous soft tissues. To illustrate the effects of the preferred directions and electro-elastic interactions of residually stressed materials that are capable of large deformations, we give results on a simple tension deformation of a hollow cylindrical solid for soft tissues. We also show that very few of the numerous classical invariants proposed by Rivlin and Spencer [35] in the minimal integrity basis are independent; hence this may aid in reducing the complexity of modelling the type of materials discussed in this paper. The general single-variable functions are expected to be used in the future to obtain specific forms of the strain energy function for different types of ESTPDs. However, in order to do this, we require experimental data for ESTPDs. There is a pressing need for comprehensive sets of experimental data and for the assessment of large deformation material response against a body of such data that catalogues the dependence of the mechanical response on the electric field, the residual stress and the two preferred directions for specific geometries, since data of the required kind are not currently available. 21

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Appendix A: Relations between classical invariants

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In this Appendix, to further justify our claim, we how that at most 15 of the of 98 classical invariants in the minimal integrity basis are independent. We show this via constructing relations among the classical invariants. We strongly emphasize that we are not concerned with the number of classical invariants that should be in the total energy function. However, examples of discussions on how many invariants that should be in an energy function can be found, for example in Rubin [17] and Shariff and Bustamante [26].

pro

Consider the 15 independent invariants

λi , tij , aα , bα , fα , i, j = 1, 2, 3 , α = 1, 2 .

(A1)

Note that the tensor components tij and vector components ai , bi and fi are invariants with respect to the rotation tensor Q since ui · T R uj = Qui · QT R QT Quj , a · ui = Qa · Qui , b · ui = Qb · Qui and f · ui = Qf · Qui .

def

J1 = tr(C)

=

3 X

λ2i , J2 = def

i=1

def

J3 = det(C) X

J4 = tr T R =

def

tii , J5 = tr (T R C) =

X

λ2i tii , J6 = tr (T R C 2 ) = def

J7 = tr (T 2R ) =

X

λ4i tii ,

λ21 (t211 + t212 + t213 ) + λ22 (t221 + t222 + t223 ) + λ23 (t231 + t232 + t233 ) ,

J9 = tr (T 2R C 2 ) =

λ41 (t211 + t212 + t213 ) + λ42 (t221 + t222 + t223 ) + λ43 (t231 + t232 + t233 ) ,

def

(A3)

t211 + t222 + t233 + 2(t212 + t213 + t223 ) ,

J8 = tr (T 2R C) = def

(A2)

i=1

i=1

i=1

def


1 (trC)2 − trC 2 = λ21 λ22 + λ21 λ23 + λ22 λ23 , 2

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def

= λ21 λ22 λ23 .

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The following subset of the classical invariants in the minimal integrity basis can be expressed explicitly in terms of the invariants (A1), i.e.,

def

J10 = a · Ca def

J12 = b · Cb def

3 X i=1

=

=

3 X

i=1 3 X i=1

λ2i a2i , J11 = a · C 2 a = def

λ2i b2i , J13 = b · C 2 b = def

λ2i fi2 , J15 = b · C 2 b = def

Jo

J14 = f · Cf

=

3 X

(A4)

λ4i a2i

i=1

3 X

i=1 3 X

λ4i b2i λ4i fi2 .

(A5)

i=1

From (A2) the principal stretches λj can be expressed in terms of J1 , J2 and J3 via the relations [9]

1 λi = √ 3

  12 p 1 J1 + 2 (J1 )2 − 3J2 cos [θ + 2π(i − 1)] , 3 22

(A6)

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where 2(J1 )3 − 9J1 J2 + 27J3 3

2[(J1 )2 − 3J2 ] 2



, (J1 )2 − 3J2 6= 0 ,

(A7)

of

θ = arccos



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In the case when (J1 )2 − 3J2 = 0, we have r √ 2 2 1 1 1 λi = J1 + (27J3 − (J1 )3 ] 3 [cos( πi) + −1 sin( πi)] . 3 3 3 3

(A8)

The three equations in (A3) are linear in t11 , t22 and t33 . In view of (A6) (or (A8)), and assuming that the values of λi are distinct, tii can be explicitly expressed in terms of J1 , J2 , . . . J6 by solving the tii -linear equations (A3). We note that if λi s are not distinct, it can be easily shown that the number of independent invariants is less than 15 as exemplified at the end of this Appendix.

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The three equations in (A4) are linear in the invariants t212 , t213 and t223 and can be solved to express these invariants explicitly in terms of J1 , J2 , . . . J9 . From (24) and (A5), we have 3 linear equations in a2i , 3 linear equations in b2i and 3 linear equations in fi2 , hence the vector invariants a2i ,b2i and fi2 can be expressed explicitly in terms of J1 , J2 , . . . J15 . The corresponding minimal integrity basis [35] contain the following 98 invariants.

J16 = tr (T 3R ) =

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J1 , J2 , . . . , J15 , 3 X

J19 = a · CT R a =

J21 = a · C 2 T R a =

3 X

bi tij bj ,

i,j=1

Jo

J26 = b · CT 2R b =

ai tij aj ,

J18 = a·T 2R a =

i,j=1

i,j,k=1

J23 = b · T R b =

3 X

tij tjk tki , J17 = a·T R a =

ai λ2i tij aj , J20 = a · CT 2R a =

3 X

ai λ4i tij aj , J22 = a · C 2 T 2R a =

i,j=1

J24 = b · T 2R b = 3 X

3 X

i,j,k=1

bi λ2i tik tkj bj ,

i,j,k=1

J28 = b · C 2 T 2R b =

3 X

i,j,k=1

bi tik tkj bj

3 X

ai λ2i tik tkj aj ,

i,j,k=1 3 X

ai λ4i tik tkj aj ,

i,j,k=1

J25 = b · CT R b =

J27 = b · C 2 T R b =

bi λ4i tik tkj bj , J29 = f · T R f =

23

ai tik tkj aj ,

i,j,k=1

3 X

i,j=1

3 X

3 X

i,j=1

i,j=1

bi λ4i tij bj ,

i,j=1 3 X

3 X

fi tij fj ,

bi λ2i tij bj

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J34 = f · C 2 T 2R f =

i,j,k=1

3 X

fi tik tkj fj J31 = f · CT R f =

i,j,k=1

fi λ2i tij fj

i,j=1

J33 = f · C 2 T R f =

fi λ2i tik tkj fj ,

i,j,k=1

3 X

3 X

3 X

fi λ4i tij fj ,

i,j=1

3 3 X X ai tij bj , ai b i ) fi λ4i tik tkj fj , J35 = (a · b)(a · T R b) = (

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J32 = f · CT 2R f =

3 X

of

J30 = f · T 2R f =

i,j=1

i=1

3 X

J36 = (a · b)(a · Cb) = (

ai b i )

ai bi λ2i ,

i=1

i=1

3 X

ai b i )

3 X

ai b i )

3 X

ai λ2i tij bj ,

i,j=1

i=1

3 X

re-

J37 = (a · b)(a · CT R b) = (

3 X

J38 = (a · b)(a · T R Cb) = (

ai bj tij λ2j ,

i,j=1

i=1

3 3 X X ai λ4i tij bj , ai b i ) J39 = (a · b)(a · C 2 T R b) = ( i,j=1

i=1

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3 3 X X ai b i ) ai tik tkj λ2j bj , J40 = (a · b)(a · T 2R Cb) = ( i=1

3 X

J41 = (a · b)(a · C 2 T 2R b) = (

ai b i )

i=1

3 X

J42 = (a · b)(a · T 2R C 2 b) = (

i,j,k=1

ai b i )

i=1

3 X

ai λ4i tik tkj bj ,

i,j,k 3 X

ai tik tkj λ4j bj ,

i,j,k=1

3 3 X X ai tij λ4j bj , ai b i ) J43 = (a · b)(a · T R C 2 b) = ( i=1

3 X

J44 = (a · b)(a · CT 2R b) = (

ai b i )

i=1

3 X

ai λ2i tik tkj bj ,

i,j,k=1

Jo

3 3 X X ai tij fj , ai f i ) J46 = (a · f )(a · T R f ) = ( i=1

i,j=1

i,j=1

3 X ai bi )2 , J45 = (a · b)2 = ( i=1

3 3 X X ai fi λ2i , ai f i ) J47 = (a · f )(a · Cf ) = ( i=1

3 3 X X ai λ2i tij fj , ai f i ) J48 = (a · f )(a · CT R f ) = ( i=1

24

i,j=1

i=1

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3 3 X X ai fj tij λ2j , ai f i ) J49 = (a · f )(a · T R Cf ) = ( i,j=1

i=1

i,j=1

i=1

J51 = (a · f )(a · T 2R Cf ) = (

3 X

ai f i )

i=1

ai tik tkj λ2j fj ,

i,j,k=1

pro

3 X

of

3 3 X X ai λ4i tij fj , ai f i ) J50 = (a · f )(a · C 2 T R f ) = (

3 3 X X ai f i ) ai λ4i tik tkj fj , J52 = (a · f )(a · C 2 T 2R f ) = ( i=1

3 X

J53 = (a · f )(a · T 2R C 2 f ) = (

i,j,k

ai f i )

i=1

3 X

ai tik tkj λ4j fj ,

i,j,k=1

3 X

re-

3 X

J54 = (a · f )(a · T R C 2 f ) = (

ai f i )

i,j=1

i=1

3 3 X X ai f i ) J55 = (a · f )(a · CT 2R f ) = ( ai λ2i tik tkj fj , i=1

3 X ai fi )2 , J56 = (a · f )2 = (

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i,j,k=1

i=1

3 3 X X bi fi λ2i , bi fi ) J58 = (b · f )(b · Cf ) = (

3 3 X X bi tij fj , bi fi ) J57 = (b · f )(b · T R f ) = ( i=1

ai tij λ4j fj ,

i,j=1

i=1

3 3 X X bi λ2i tij fj , bi fi ) J59 = (b · f )(b · CT R f ) = ( i,j=1

i=1

3 3 X X bi fj tij λ2j , bi fi ) J60 = (b · f )(b · T R Cf ) = ( i,j=1

i=1

3 3 X X bi λ4i tij fj , bi fi ) J61 = (b · f )(b · C 2 T R f ) = ( i,j=1

i=1

3 X

J62 = (b · f )(b · T 2R Cf ) = (

bi fi )

i=1

3 X

bi tik tkj λ2j fj ,

i,j,k=1

Jo

3 3 X X bi fi ) bi λ4i tik tkj fj , J63 = (b · f )(b · C 2 T 2R f ) = ( i=1

3 X

J64 = (b · f )(b · T 2R C 2 f ) = (

i=1

25

bi fi )

i,j,k 3 X

i,j,k=1

bi tik tkj λ4j fj ,

i=1

Journal Pre-proof

3 3 X X bi tij λ4j fj , bi fi ) J65 = (b · f )(b · T R C 2 f ) = ( i,j=1

i=1

i=1

3 X bi fi )2 , J67 = (b · f )2 = (

of

3 3 X X bi fi ) bi λ2i tik tkj fj , J66 = (b · f )(b · CT 2R f ) = (

i=1

i,j,k=1

pro

3 3 3 X X X ai f i ) , bi tij fj )( ai bi )( J68 = (a · b)(b · T R f )(a · f ) = ( i=1

i,j=1

i=1

3 3 3 X X X ai f i ) , bi fi λ2i )( ai bi )( J69 = (a · b)(b · Cf )(a · f ) = ( i=1

i=1

i=1

3 X

3 X

3 X bi λ2i tij fj )( ai f i ) ,

3 X

3 X ai f i ) , bi fj tij λ2j )(

ai bi )(

i=1

i,j=1

i=1

re-

J70 = (a · b)(b · CT R f )(a · f ) = (

3 X

J71 = (a · b)(b · T R Cf )(a · f ) = (

ai bi )(

i,j=1

i=1

i=1

3 3 3 X X X ai f i ) , bi λ4i tij fj )( ai bi )( J72 = (a · b)(b · C 2 T R f )(a · f ) = (

urn al P

J73 = (a · b)(b · T 2R Cf )(a · f ) = (

3 X

ai bi )(

i=1

i=1

i,j=1

i=1

3 X

i,j,k=1

3 X ai f i ) , bi tik tkj λ2j fj )( i=1

3 3 3 X X X ai f i ) , ai bi )( bi λ4i tik tkj fj )( J74 = (a · b)(b · C 2 T 2R f )(a · f ) = ( i=1

3 X

J75 = (a · b)(b · T 2R C 2 f )(a · f ) = (

ai bi )(

i=1

i=1

i,j,k 3 X

i,j,k=1

3 X ai f i ) , bi tik tkj λ4j fj )( i=1

3 3 3 X X X ai f i ) , bi tij λ4j fj )( ai bi )( J76 = (a · b)(b · T R C 2 f )(a · f ) = ( i=1

3 X

J77 = (a · b)(b · CT 2R f )(a · f ) = (

ai bi )(

i=1

i=1

i,j=1

3 X

i,j,k=1

3 X ai f i ) , bi λ2i tik tkj fj )( i=1

Jo

3 3 3 X X X bi fi ) , ai tij fj )( ai bi )( J78 = (a · b)(a · T R f )(b · f ) = ( i=1

i,j=1

i=1

3 3 3 X X X J79 = (a · b)(a · Cf )(b · f ) = ( ai bi )( ai fi λ2i )( bi fi ) , i=1

26

i=1

i=1

Journal Pre-proof

3 3 3 X X X bi fi ) , ai λ2i tij fj )( ai bi )( J80 = (a · b)(a · CT R f )(b · f ) = ( i=1

i,j=1

i=1

i=1

i,j=1

i=1

3 X

ai bi )(

3 X bi fi ) , ai λ4i tij fj )(

3 X

i=1

i,j=1

i=1

pro

J82 = (a · b)(a · C 2 T R f )(b · f ) = (

of

3 3 3 X X X bi fi ) , ai fj tij λ2j )( ai bi )( J81 = (a · b)(a · T R Cf )(b · f ) = (

3 3 3 X X X bi fi ) , ai bi )( ai tik tkj λ2j fj )( J83 = (a · b)(a · T 2R Cf )(b · f ) = ( i=1

J84 = (a · b)(a · C 2 T 2R f )(b · f ) = (

i=1

i,j,k=1

3 X

ai bi )(

i=1

3 X

i,j,k

3 X bi fi ) , ai λ4i tik tkj fj )( i=1

re-

3 3 3 X X X bi fi ) , ai bi )( ai tik tkj λ4j fj )( J85 = (a · b)(a · T 2R C 2 f )(b · f ) = ( i=1

i=1

i,j,k=1

3 3 3 X X X bi fi ) , ai tij λ4j fj )( ai bi )( J86 = (a · b)(a · T R C 2 f )(b · f ) = ( i=1

i,j=1

i=1

urn al P

3 3 3 X X X bi fi ) , ai bi )( ai λ2i tik tkj fj )( J87 = (a · b)(a · CT 2R f )(b · f ) = ( i=1

i=1

i,j,k=1

3 3 3 X X X bi fi ) , ai tij bj )( ai fi )( J88 = (a · f )(a · T R b)(b · f ) = ( i=1

i,j=1

i=1

3 3 3 X X X bi fi ) , ai bi λ2i )( ai fi )( J89 = (a · f )(a · Cb)(b · f ) = ( i=1

i=1

i=1

3 X i=1

3 X

i,j=1

3 X bi fi ) , ai λ2i tij bj )(

3 X

3 X

3 X bi fi ) , ai bj tij λ2j )(

J90 = (a · f )(a · CT R b)(b · f ) = (

J91 = (a · f )(a · T R Cb)(b · f ) = (

ai fi )(

ai fi )(

i=1

i,j=1

i=1

i=1

Jo

3 3 3 X X X bi fi ) , ai λ4i tij bj )( ai fi )( J92 = (a · f )(a · C 2 T R b)(b · f ) = ( i,j=1

i=1

3 X

J93 = (a · f )(a · T 2R Cb)(b · f ) = (

i=1

27

ai fi )(

3 X

i,j,k=1

i=1

3 X bi fi ) , ai tik tkj λ2j bj )( i=1

Journal Pre-proof

3 3 3 X X X bi fi ) , ai fi )( ai λ4i tik tkj bj )( J94 = (a · f )(a · C 2 T 2R b)(b · f ) = ( i=1

i=1

i,j,k

i=1

of

3 3 3 X X X bi fi ) , ai fi )( ai tik tkj λ4j bj )( J95 = (a · f )(a · T 2R C 2 b)(b · f ) = ( i=1

i,j,k=1

3 3 3 X X X bi fi ) , ai tij λ4j bj )( ai fi )( J96 = (a · f )(a · T R C 2 b)(b · f ) = ( i=1

i,j=1

pro

i=1

3 3 3 X X X bi fi ) , ai fi )( ai λ2i tik tkj bj )( J97 = (a · f )(a · CT 2R b)(b · f ) = ( i=1

i=1

i,j,k=1

(A9)

re-

3 3 3 X X X ai f i ) . bi fi )( ai bi )( J98 = (a · b)(b · f )(a · f ) = ( i=1

i=1

i=1

urn al P

The remaining 83 classical invariants in the minimal integrity basis can be written explicitly in terms of λi , tij , ai , bi , fi and hence they all depend explicitly on the classical invariants J1 , J2 , . . . J15 . Note that care must be taken in assigning the positive or negative values of t12 , t13 , t23 , ai , bi and fi in the expressions for the remaining classical invariants. Hence, it is clear that only 15 of the 98 classical invariants in the minimal integrity basis for two symmetric tensors and three preferred unit vectors are independent. In the case of two or more principal stretches having the same values, the number of independent invariants is far less than 15 as exemplified below. Here, for simplicity we give the result only for λ1 = λ2 = λ3 = λ. The result for the case of double coalescence of principal stretches can also be done in a similar fashion, but we shall omit it here. In the case of λ1 = λ2 = λ3 = λ the eigenvectors ui are arbitrary and we let them coincide with the eigenvectors of T R . Hence TR =

3 X i=1

J1 = 3λ2 , J2 =

3 X i=1

t2i , J8 =

X J12 J 1 J4 J3 J 2 J4 t i , J5 = , J3 = 1 , J 4 = , J6 = 1 , 3 27 3 9 i=1

(A10)

(A11)

J1 J7 J 2 J4 J1 J2 J1 J2 , J9 = 1 , J10 = , J11 = 1 , J12 = , J13 = 1 , (A12) 3 9 3 9 3 9

Jo

J7 =

ti ui ⊗ ui , ti = tii , tij = 0 , i 6= j .

J14 =

3

X J1 J2 t3i . , J15 = 1 , J16 = 3 9 i=1 28

(A13)

Journal Pre-proof

ti =

1 3

  q 1 J¯1 + 2 (J¯1 )2 − 3J¯2 cos [θ + 2π(i − 1)] , 3

where 

2(J¯1 )3 − 9J¯1 J¯2 + 27J¯3 3 2[(J¯1 )2 − 3J¯2 ] 2



, (J¯1 )2 − 3J¯2 6= 0 ,

pro

θ = arccos

of

From J4 , J7 and J16 , we have

J 2 − J7 J16 − J¯1 J7 + J¯1 J¯2 J¯1 = J4 , J¯2 = 4 , J¯3 = . 2 3 In the case when (J¯1 )2 − 3J¯2 = 0, we have

√ 1 2 1 2 1 J1 + [27J¯3 − (J¯1 )3 ] 3 [cos( πi) + −1 sin( πi)] . 3 3 3 3

Note that X

a2i =

and J17 =

X i=1

i=1

b2i =

X

ti a2i J18 =

X

t2i a2i , J23 =

i=1

X

(A15)

(A16)

(A17)

fi2 = 1

(A18)

i=1

urn al P

i=1

X

re-

ti =

(A14)

ti b2i , J24 =

X i=1

i=1

t2i b2i , J29 =

X i=1

ti fi2 , J30 =

X

t2i fi2 .

i=1

(A19)

From the linear equations in (A18) and (A19), for the case of t1 6= t2 6= t3 we have a2i , b2i and fi2 in terms of the set of ten independent invariants J1 , J4 , J7 , J16 , J17 , J18 , J23 , J24 , J29 , J30 . The remaining invariants are related to this set of invariants via λ, ti , ai , bi , fi . Hence only 10 of the 98 invariants are independent. In the case of two or more of the eigenvalues of T R are the same, we can apply the same procedure as above and the number of independent invariants is further reduced. For example, consider the case when t1 = t2 = t3 = t, we then have J42 J3 J4 J45 J1 J4 J4 J56 J4 J67 , J16 = 4 , J17 = , . . . , J37 = , . . . J46 = , . . . , J57 = ,... , 3 9 3 9 3 3

Jo

J7 =

(A20)

It is clear from (A20) that there are only 5 independent invariants and we can consider J1 , J4 , J45 , J56 and J67 to be the independent invariants.

29

Journal Pre-proof

Appendix B

m·n=0

pro

m · [Q(n)m] > 0,

of

When developing a constitutive equation it is important that we give restrictions on the material constants in order to be consistent with certain physical behaviour. In this paper, the material constants can be restricted using the strong ellipticity condition in the reference configuration (F = I). Mathematically, the strong ellipticity condition for an incompressible material [19] requires (B1)

where m and n are unit vectors, and where, in Cartesian components, we have (Q(n))ij =

3  2  X ∂ Ω np nq , 2 ∂F piqj p,q=1

(B2)

For the total energy (45) with (50), we have

re-

where ni 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. [29].

Q(n) = Q1 (n) + Q2 (n) + Q3 (n) + Q4 (n) + Q5 (n) + Q6 (n) where

Q2 (n) = Q3 (n) = Q4 (n) = Q5 (n) = Q6 (n) =

m(1) (I + n ⊗ n), 1 m(2) [(An ⊗ n + n ⊗ An + (n · An)I + A) + (Dn ⊗ n + n ⊗ Dn + (n · Dn)I + D)] 2 c1 [(T R n) ⊗ n + n ⊗ (T R n) + (n · T R n))I + T R ] 2 c2 (e) [Hn ⊗ n + n ⊗ Hn + (n · Hn))I + H] , 2 2 −ǫ0 e (n ⊗ Hn + Hn ⊗ n + H) , 3 1 ((T R n) · n)I − [n ⊗ (T R n) + (T R n) ⊗ n + T R ] . (B4) 4 4

urn al P

Q1 (n) =

(B3)

Since, in Section 5, we deal with a problem that can be considered as two dimensional, we only consider the case for m and n in a plane and assume that T R = t1 q 1 ⊗ q 1 + t2 q 2 ⊗ q 2 in that plane. Since, at F = I, ui is arbitrary, we let u1 = q 1 and u2 = q 2 . The necessary and sufficient condition for (B1) is

Jo

b1 > 0 and 4b1 b2 > b3 ,

30

(B5)

Journal Pre-proof

where

+ b2

= +

b3

=

m(2) 2 (a1 + a22 + b21 + b22 ) 2 c1 1 c2 (e) 2 (t1 + t2 ) + (3t1 − t2 ) + (f1 + f22 ) − ǫ0 e2 f22 , 2 4 2 m(2) 2 (a1 + a22 + b21 + b22 ) m(1) + 2 c1 1 c2 (e) 2 (t1 + t2 ) + (3t2 − t1 ) + (f1 + f22 ) − ǫ0 e2 f12 , 2 4 2 2ǫ0 e2 f1 f2 . m(1) +

of

=

pro

b1

Appendix C

(B7) (B8)

re-

Fibre compression

(B6)

urn al P

In this Appendix, to take into account that fibre compression does not contribute (or partially contribute) towards the strain energy, we propose that the discrete ground state constants appearing in W(T ) and W(T M) depend on I(a) = a · Ca and I(b) = b · Cb. However, these discrete constants are difficult to implement practically. For example, if on the onset we know the numerical value of I(a) and I(b) , then we can easily impose the appropriate numerical values on the discrete constants . However, in a general boundary value problem, the value of I(a) and I(b) themselves depend on the constitutive equation (for details see reference [32]) and hence they cannot be used as a hypothesis to decide what are the appropriate values of the discrete constants 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) =

(1 + erf(¯ a(1 − x)) (1 + erf(¯ a(x − 1)) , q(n) (x) = , 2 2

(C1)

where erf is the error function and a ¯ is very large positive number. We then define the ”select” function b(0) (x) = lp q(p) (x) + ln q(n) (x) ,

(C2)

Jo

where the constants lp and ln have non-negative values. This type of select function is used by Shariff [32] in modelling transversely isotropic materials and hence we shall not elaborate on it here. As described in Shariff [32], the large positive value a ¯ = 1000 is sufficient for our purpose. In the case where fibre compression partially contribute towards the strain energy, we have lp > ln > 0. For practical purposes, we let the value of the derivative of b(o) (x) to be zero.

31

Journal Pre-proof

For a non-mechanically equivalent material then (38) is simply modified to take the form: W(T ) =

3 3 h i X X α ˆ i q5 (λi ))2 + n(1) q1 (λi ) + n(2) α ˆ i q2 (λi ) + n(3) βˆi q3 (λi ) + n(4) χ ˆi q4 (λi ) + n(5) ( i=1

of

i=1

3 3 3 3 3 X X X X X χ ˆi q10 (λi ) + α ˆi q9 (λi ) βˆi q8 (λi ) + n(8) α ˆi q7 (λi ) βˆi q6 (λi ))2 + n(7) n(6) (

βˆi q11 (λi )

3 X

χ ˆi q12 (λi ) + n(10) (

i=1

i=1

i=1

χ ˆi q13 (λi ))2 , (C3)

pro

n(9)

3 X

i=1 3 X

i=1

i=1

i=1

i=1

where

ˆi = b(o) (I(a) )b(o) (I(b) )χi . α ˆi = a ˆ2i = (b(o) (I(a) )ai )2 , βˆi = ˆb2i = (b(o) (I(b) )bi )2 , χ

(C4)

From (40), the modified strain energy function for an orthotropic material is

3 3 h i X X α ˆi q5 (λi ))2 + n(1) q1 (λi ) + n(2) α ˆ i q2 (λi ) + n(3) βˆi q3 (λi ) + n(5) (

re-

W(T ) =

i=1

i=1

3 3 3 X X X βˆi q8 (λi ) . α ˆi q7 (λi ) βˆi q6 (λi ))2 + n(7) n(6) ( i=1

i=1

(C5)

i=1

urn al P

In the case of a mechanically equivalent material, we have from (41) the modified strain energy function W(T M) =

3 h i X m(1) s1 (λi ) + m(2) (ˆ αi + βˆi )s2 (λi ) + m(3) χ ˆi s3 (λi ) + i=1

m(4)

3 3 X X βˆi s4 (λi ))2 α ˆ i s4 (λi ))2 + ( ( i=1

i=1

m(6)

3 X i=1

α ˆ i s6 (λi )

3 X i=1

βˆi s6 (λi ) + m(7)

3 X i=1

!

+ m(5)

3 X

!2

χ ˆi s5 (λi )

i=1

!

(ˆ αi + βˆi )s7 (λi )

3 X i=1

+ !

χ ˆi s8 (λi )

.

(C6)

Jo

It is important to note that when λi is very close to unity, the values and the derivative values of the functions in (C3) and (C6) are very close to zero and hence the function b(o) does not effect the value of stress significantly for λi close to unity.

Fibre dispersion

The mechanical influence of fibre dispersion in collagenous soft tissues has been studied recently in the literature [7, 11]. The collagen fibers in these tissues may be dispersed randomly in space, in a certain 32

Journal Pre-proof

of

pattern such as predominately in a particular direction [41], as a rotationally symmetric dispersion about a mean direction, or as the recently observed non-symmetric dispersion in arterial walls [18]. In this section, we show how the above model can be easily modified to take fibre dispersion into account. Macroscopically, the fibre dispersion model require the dispersion tensors H (1) = k1 I + (1 − 3k1 )A , H (2) = k2 I + (1 − 3k2 )D ,

1 is required. Using operations such as 3 ! 3 3 X X [k1 + (1 − 3k1 )ˆ αi ]q3 (λi )] , q3 (λi )ui ⊗ ui ) = (k1 I + (1 − 3k1 )b(o) (I(a) )A)(

tr

pro

where the restriction 0 ≤ k1 , k2 ≤

(C7)

(C8)

i=1

i=1

where, in (C8), H (1) is modified to take the problem of fibre compression into account, we propose, using (C3), for a non-mechanically equivalent material, the constitutive equation 3 3 X X   α ¯i q5 (λi ))2 + n(1) q1 (λi ) + n(2) α ¯ i q2 (λi ) + n(3) β¯i q3 (λi ) + n(4) χ ¯i q4 (λi ) + n(5) (

re-

W(T ) =

i=1

i=1

3 3 3 3 3 X X X X X χ ¯i q10 (λi ) + α ¯ i q9 (λi ) β¯i q8 (λi ) + n(8) α ¯ i q7 (λi ) β¯i q6 (λi ))2 + n(7) n(6) (

β¯i q11 (λi )

i=1

i=1

where

3 X

urn al P

n(9)

3 X

i=1

i=1

i=1

i=1

i=1

3 X χ ¯i q13 (λi ))2 , (C9) χ ¯i q12 (λi ) + n(10) ( i=1

χ ¯i = k1 k2 + k2 (1 − 3k1 )b(o) α ˆi + k1 (1 − 3k2 )βˆi + (1 − 3k1 )(1 − 3k2 )χ ˆi ,

(C10)

α ¯ i = k1 + (1 − 3k1 )ˆ αi and β¯i = k2 + (1 − 3k2 )βˆi .

For a mechanically equivalent material with respect to the preferred directions a and b we let k1 = k2 = k and, using the method that is similar to the above and Equation (C6), we propose the form W(T ) =

3 X  i=1

m(4)

 m(1) s1 (λi ) + m(2) (¯ αi + β¯i )s2 (λi ) + m(3) χ ¯i s3 (λi ) +

3 3 X X β¯i s4 (λi ))2 ( α ¯ i s4 (λi ))2 + ( i=1

α ¯ i s6 (λi )

3 X

Jo

m(6)

3 X

i=1

i=1

i=1

β¯i s6 (λi ) + m(7)

3 X i=1

!

+ m(5)

3 X

!2

χ ¯i s5 (λi )

i=1

+

! ! 3 X ¯ χ ¯i s8 (λi ) . (¯ αi + βi )s7 (λi ) i=1

(C11)

33

Journal Pre-proof

References

of

[1] Ahamed, T., Dorfmann, L., Ogden, R.W. (2016) Modelling of residually stressed materials with application to AAA. J. of the Mech. Behav. Biomedical Materials, 61, 221-234.

pro

[2] Y. Bar-Cohen, Y. (2002) Electro-active polymers: current capabilities and challenges. In: Y. BarCohen (ed.), Proceedings of the 4th Electroactive Polymer Actuators and Devices (EAPAD) Conference, 9th Smart Structures and Materials Symposium, Vol. 4695, San Diego, pp. 17. [3] Bustamante, R. and Shariff, M.H.B.M., (2016) New sets of invariants for an electro-elastic body with one and two families of fibres. European J. Mech. A, 58 42–53. [4] Dorfmann, A., Ogden, R.W., (2005) Nonlinear electroelasticity. Acta Mech., 174, 167–183. [5] Dorfmann, A., Ogden, R.W., (2006) Nonlinear electroelastic deformations. J. Elasticity, 82, 99–127.

re-

[6] Holzapfel, G.A., Ogden,R.W.(2010) Modelling the layer-specific three-dimensional residual stresses in arteries, with an application to the human aorta. J.R.Soc.Interface, 7, 787-799. [7] Gasser, T.C., Ogden, R.W. Holzapfel, G.A. (2006) Hyperelastic modelling of arterial layers with distributed collagen fibre orientations, J. R. Soc. Interface, 3, 15 -35. [8] Holzapfel, G.A., & Ogden, R.W. (2009). On planar biaxial tests for anisotropic nonlinearly elastic solids: a continuum mechanical framework. Math. Mech. Solids, 14, 474 - 489.

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[9] Itskov, M. : Tensor Algebra and Tensor Analysis for Engineers 3rd edn (Springer, 2013). [10] Kovetz, A., (2000) Electromagnetic Theory. University Press, Oxford. [11] Li, K., Ogden, R. W., Holzapfel, G. A. (2018) A discrete fiber dispersion method for excluding fibers under compression in the modeling of fibrous tissues, J. R. Soc. Interface, 15(138), doi: 10.1098/rsif.2017.0766. [12] Merodio, J. Ogden, R.W. & Rodriguez, J. (2013) The influence of residual stress on finite deformation elastic response. Int. J. Non-linear Mech., 56, 43-49. [13] Merodio, J. & Ogden, R.W. (2016) Extension, inflation and torsion of a residually stressed circular cylindrical tube. Cont. Mech. Thermodyn., 28 (1-2), 157-174. [14] Nam, N.T., Merodio, J., Ogden, R.W. & Vinh, P.C. (2016) The effect of initial stress on the propagation of surface waves in a layered half-space Int. J. of Solids Struct., 88, 88-100. [15] Ogden, R.W. (1972). Large deformation isotropic elasticity: on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. Soc. Lond., A 326, 565 - 584.

Jo

[16] Ogden, R.W., Steigmann, D.J., (2011) Mechanics and Electrodynamics of Magneto- and Electroelastic Materials. CISM Courses and Lectures Series, Vol. 527, Springer, Wien. [17] Rubin, M. B. (2016). Seven Invariants Are Needed to Characterize General Orthotropic Elastic Materials: A Comment on [Shariff, J. Elast., 110:237241 (2013)]. Journal of Elasticity, 123(2), 253254.

34

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[18] Schriefl, A.J., Zeindlinger, G., Pierce, D.M., Regitnig, P., Holzapfel, G.A. (2012) Determination of the layer-specific distributed collagen fiber orientations in human thoracic and abdominal aortas and common iliac arteries, J. R. Soc. Interface, 9, 1275 - 1286.

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[19] Shams, M., Destrade, M. & Ogden, R.W. (2011) Initial stresses in elastic solids: Constitutive laws and acoustoelasticity. Wave Motion, 48, 552–567.

pro

[20] Shariff, M.H.B.M.(2000). Strain energy function for filled and unfilled rubberlike material, Rubber Chem. Technol., 73, 1-21. [21] Shariff, M.H.B.M., (2008) Nonlinear transversely isotropic elastic solids: An alternative representation. Quat. J. Mech. Appl. Math., 61, 129–149. [22] Shariff, M.H.B.M., (2011) Physical invariants for nonlinear orthotropic solids. Int. J. Solids Struct., 48, 1906–1914.

re-

[23] Shariff, M.H.B.M. (2013) Physical invariant strain energy function for passive myocardium. Biomech. Model. Mechanobiol., 12(2), 215-223. [24] Shariff, M.H.B.M. (2013) Nonlinear orthotropic elasticity: Only six invariants are independent. J. Elast., 110, 237–241. [25] Shariff, M.H.B.M. (2013). Physical invariant strain energy function for passive myocardium. Biomech. Model Mechanobiol., 12(2), 215-223.

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