on non-symmetric deformations of an incompressible nonlinearly elastic isotropic sphere

Journal of Elasticity 47: 85–100, 1997. 85c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

On Non-Symmetric Deformations of anIncompressible Nonlinearly Elastic Isotropic Sphere

DEBRA POLIGNONE WARNE1 and PAUL G. WARNE21Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA2Division of Mathematics and Computer Science, Maryville College, Maryville, TN 37804, USA

Received 20 September 1996; in revised form 23 April 1997

Abstract. A class of non-symmetric deformations of a neo-Hookean incompressible nonlinearlyelastic sphere are investigated. It is found via the semi-inverse method that, to satisfy the governingthree-dimensional equations of equilibrium and the incompressibility constraint, only three specialcases of the class of deformation fields are possible. One of these is the trivial solution, one the solutiondescribing radially symmetric deformation, and the other a (non-symmetric, non-homogeneous)deformation describing inflation and stretching. The implications of these results for cavitationphenomena are also discussed. In the course of this work, we also present explicitly the spherical polarcoordinate form of the equilibrium equations for the nominal stress tensor for a general hyperelasticsolid. These are more complicated than their counterparts for Cauchy stresses due to the mixed bases(both reference and deformed) associated with the nominal (or Piola-Kirchhoff) stress tensor, butmore useful in considering general deformation fields.

Key words: Incompressible nonlinear elasticity, non-symmetric deformations, governing equations,material formulation, cavitation.

1. Introduction

Radially symmetric deformations describing the nucleation of a spherical cavity inthe center of a ball of nonlinearly elastic isotropic material were treated analyticallyin the groundbreaking paper of Ball [1] for both incompressible and compressiblematerials. That work sparked considerable interest in cavitation phenomena andextensive investigations by a variety of authors followed. (We shall cite here onlythe literature most directly related to the present study, and refer the reader to [2]for a recent review on cavitation and a comprehensive list of references). Nearlyall of the subsequent studies consider radially symmetric cavitation in a variety ofcontexts and as a result this problem has become rather well understood.

Non-symmetric cavitation, however, has received little attention due to the obvi-ous analytical difficulties involved in relaxing the assumption of spherical symme-try. In addition, as remarked in [3], linear theories are of little value in understand-ing cavitation, as it is an inherently nonlinear phenomenon. The limited literatureaddressing non-radially symmetric cavitation problems includes numerical studiesfor elastic-perfectly plastic solids [4] and for power-law hardening elastic-plasticsolids [5], as well as for two-dimensional compressible finite elasticity [6]. In

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86 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

three-dimensions, it has been shown analytically that for a class of compressiblenonlinearly elastic materials, the spherically symmetric radial deformation describ-ing cavitation is unstable relative to a non-symmetric deformation which opens, justoutside the cavity, a long, thin “filamentary void” pointing in the radial direction[7]. There has also been some recent progress in further understanding the questionat hand for incompressible hyperelastic materials. The relationship between theradially symmetric cavitation deformation and a non-symmetric (non-cavitated)deformation carrying the sphere into an ellipsoid under symmetric dead-leadingwas examined in [8]. There it was established that whenever the former exists, thennecessarily, so does the latter, with the non-symmetric deformation being that withless energy. These authors have also considered non-symmetric cavitation defor-mations carrying an incompressible elastic or elastic-plastic sphere into a solid orcavitated ellipsoid (ellipsoid with an ellipsoidal cavity at its center) under non-symmetric loading conditions, while an energetics argument is employed in [10]for symmetric dead-loading of a neo-Hookean sphere to further examine the occur-rence of the spherically symmetric cavitated deformation relative to asymmetricdeformations of the types considered in [8, 9].

The purpose of this paper, motivated in part by [8–10], is to begin to systemat-ically investigate non-spherically symmetric deformations possible for an incom-pressible nonlinearly elastic material and their relation to cavitation. We employ thesemi-inverse method, and, as a starting point, consider here a class of deformationsof a neo-Hookean sphere sufficiently general to include spherical symmetry as aspecial case (see (2.1)). Neither the asymmetric deformations examined in [8–10]nor those considered here are contained in the other; in addition, both reduce to theradial deformation as a special case.

In Section 2 of this paper, we pose the problem we wish to consider and refinethe form of the deformation so as to comply with the incompressibility constraint.We specialize the (nominal) stress response equation to that for a neo-Hookeanmaterial to arrive at the associated equilibrium equations, one of which is easilyintegrated. The equilibrium equations in terms of the nominal stresses are derivedfor a fully general deformation field in spherical polar coordinates and given incomponent form in the Appendix; we were unable to find these explicitly elsewherein the literature. Due to the generality of the deformation we consider, it is prudentto employ a stress tensor measuring force per unit area in the reference configu-ration, as opposed to the deformed configuration as with the Cauchy stress. Thecorresponding equilibrium equations (in component form) are more complicatedthan their counterparts for Cauchy stresses in non-Cartesian coordinate systems.In Section 3 we analyze the remaining two equilibrium equations to establish that,aside from the trivial deformation, only two special cases of the class of deforma-tions considered can possibly satisfy these equations and incompressibility. One ofthese is the radially symmetric deformation, while the other is a non-homogeneousuniversal deformation for incompressible materials describing inflation and stretch-ing. We also determine the hydrostatic pressure in each case. Finally, in Section 4,

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DEFORMATIONS OF AN ISOTROPIC SPHERE 87

we discuss our results and their implications for cavitation phenomena. In particu-lar, we conclude that there exist no non-symmetric deformations of the type (2.1)describing cavitation for a neo-Hookean sphere.

2. Preliminaries: Kinematics and Stress

We wish to investigate the possibility of non-spherically symmetric deformations ofan incompressible nonlinearly elastic sphere. We employ the semi-inverse method(see, e.g., [11, Sect. 59]) and consider a class of deformations of the form

r = f(R;�)R; � = h(�); ' = �; (2.1)

where interior points in the undeformed configuration of the unit sphere aredescribed by the spherical polar coordinates (R;�;�) with 0 6 R < 1, 0 6

� < 2�, 0 6 � 6 �. Points in the deformed configuration are then describedby the spherical polar coordinates (r; �; ') given by (2.1) where f and h are asyet arbitrary sufficiently smooth functions of their arguments, with _h 6= 0. Thedeformation (2.1) contains, as a special case, the well-studied radially symmetricdeformation

r = r(R)�= f(R)R

�; � = �; ' = � (2.2)

for which the radial deformation r(R) is explicitly determined from the constraintof incompressibility as

r(R) =

1 +

�3

R3

!1=3

R; (2.3)

where � is a constant to be determined.As analyzed in [1] and further explained in [2] and the references cited therein,

when � > 0, for certain materials (2.2), (2.3) can be used to describe radiallysymmetric cavitation in which a spherical hole of radius �, centered at the originin the current configuration, arises due to sufficiently large tensile dead-loading onthe boundary.

The main goal of this paper is to show that, with the exception of a deformationdescribing inflation and stretching, the only deformation of the type (2.1) that aneo-Hookean sphere can sustain are those radially symmetric deformations givenby (2.2), (2.3).

We now proceed with the relevant material from finite elastostatics. More detailcan be found, for example, in [11,12]. The deformation gradient tensor F associatedwith (2.1), referred to spherical polar coordinates, is given by

F =

264 f +Rf 0 _f= sin� 00 f _h 00 0 f

375 ; (2.4)

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where f 0 � @f=@R, _f � @f=@�, _h � dh=d�, that is, the superposed 0 and �

denote differentiation with respect toR and �, respectively. Incompressibility thenrequires that J � det F = 1, which results in the first-order quasi-linear partialdifferential equation for f(R;�)

_hf 3 +R _hf 2f 0 = 1 (2.5)

with general solution

f(R;�) =

1

_h(�)��3(�)

R3

!1=3

; (2.6)

where � is, as yet, an arbitrary function of �. Thus, for the radial deformation, itis only the dependence of f on � that remains unknown.

The strain-energy density per unit undeformed volume for an incompressibleisotropic nonlinearly elastic material is given by

W =W (I1; I2); (2.7)

where

I1 = tr B;

I2 = 12 [(tr B)2 � tr B2];

)(2.8)

and B is the left Cauchy–Green deformation tensor B = FFT (I3 = det B = J2 = 1via incompressibility). The corresponding response equation for the nominal stresstensor S = JF�1T (see, e.g., [12, Sect. 3.4, Sect. 4.3]), where T is the Cauchystress tensor, is given by

S = �pF�1 + 2W1FT � 2W2F�1B�1 or

S = �pF�1 + 2W1FT + 2W2FT (I11 � B)

9=; ; (2.9)

where p = p(R;�;�) is the unknown hydrostatic pressure associated with theincompressibility constraint, 1 is the unit tensor, and Wi = @W=@Ii (i = 1; 2).

We consider a sphere composed of a neo-Hookean material, so that (2.7)becomes

W = 12�(I1 � 3); (2.10)

where � > 0 is the shear modulus for infinitesimal deformations. Defining p � �q,(2.9) simplifies, upon substitution from (2.10), to

S = �(FT � qF�1); (2.11)

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the response equation for the nominal stress for a neo-Hookean material.From (2.4), (2.5),

F�1 =

266641=(f +Rf 0) �f _f= sin� 0

0 1=(f _h) 0

0 0 1=f

37775 ; (2.12)

and thus (2.4), (2.11), (2.12) give the dimensionless nominal stresses correspondingto the deformation (2.1) as

SRr = f +Rf 0 � q=(f +Rf 0);

S�� = f _h� q=(f _h);

S�' = f � q=f;

SR� = qf _f= sin�;

S�r = _f= sin�;

SR' = S�' = S�r = S�� = 0

9>>>>>>>>>>>>=>>>>>>>>>>>>;; (2.13)

where f is given by (2.6) via incompressibility.The equilibrium equations, in the absence of body forces, are

Div S = 0; (2.14)

where Div denotes the divergence with respect to (R;�;�), the undeformed coor-dinates (independent variables). Our choice to proceed with nominal stresses,as opposed to Cauchy stresses, now becomes apparent, since the correspondingequilibrium equations div T = 0 introduce complications in differentiation w.r.t.(r; �; '), the deformed coordinates (dependent variables) given by (2.1). How-ever, due to the mixed bases associated with the nominal stresses, that is, sincethe tensor S is represented as S = SijEi ej , i; j = 1; 2; 3 (summation con-vention is implied here), where fEig and fejg are orthonormal bases in the unde-formed and deformed configurations respectively, care must be taken in calculatingDiv S, as the components of each fejg also depend on (R;�;�) via (2.1). We havebeen unable to find these equations explicitly in the literature, and thus present aderivation for spherical polar coordinates in the Appendix. (See, e.g. [11, (44.5)]for a representation of these equations in component form for a general curvilinearcoordinate system in terms of Christoffel symbols.)

For the deformation (2.1) and the neo-Hookean material (2.10) of interest here,the general equilibrium equations, r � S = 0, from (A.35), (A.34), and (A.33),reduce to

(S�' � _hS��) cot�+@

@�S�' = 0; (2.15)

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@

@RSR� +

1R

�2SR� +

_hS�r +1

sin �

@

@�S��

�= 0; (2.16)

@

@RSRr +

1R

�2SRr � _hS�� S�' +

1sin �

@

@�S�r

�= 0; (2.17)

respectively for 0 < R < 1, 0 < � < 2�, 0 < � < �, where the last equationin (2.13) has been used to obtain (2.15)–(2.17). Upon substitution from (2.13),Equation (2.15) determines the dependence of the unknown pressure on �; (2.15)becomes the simple partial differential equation for q(R;�;�)

@q

@�= f 2(1 � _h2) cot�;

which integrates to

q(R;�;�) = f2(R;�)[1 � _h2(�)] ln(sin �) + �(R;�); (2.18)

where � is an, as yet, arbitrary function of its arguments, and we recall from (2.6)that f = (_h�1 � �3(�)R�3)1=3.

It remains to analyze the two other equilibrium equations (2.16) and (2.17) todetermine the unknown functions �(�), h(�), and �(R;�). This is the focus ofthe next section.

3. Analysis of the Remaining Equations

In the analysis to follow, we exploit our explicit knowledge of the manner in whichthe variable � enters the equations. We first consider (2.16) and multiply (2.16)by sin� for convenience. Using the symbolic manipulation software Macsyma[Macsyma Inc., 1995], we substitute the expressions (2.13) for the stresses and(2.18) for the pressure q, into (2.16) and factor out the explicit � dependence.Thus we can organize (2.16) into two terms, one multiplying the term ln(sin�)and the other multiplying unity. As ln(sin�) and unity are linearly independent,an equation of the form

[� � �] ln(sin �) + (� � �) � 1 = 0 (3.1)

which holds for all 0 < � < � necessarily implies that the terms in [� � �] and(� � �) (which are independent of �) are identically zero. Thus, (2.16) (multipliedby sin �) has the form (3.1), and so on setting the coefficient of the ln(sin �)

term equal to zero, we obtain (after some algebraic manipulations carried out usingMacsyma)

0 = �2(�3 _h�R3)1=3

3R5 _h7=3[(2 _h2 + 1)R3�h+ 3�2 _� _h2(1 � _h2)]: (3.2)

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Note that � and h are functions of � only, so that for (3.2) to hold, the term inbrackets must be zero. Further, this term is zero, i.e., (3.2) holds throughout thebody, if and only if

(2 _h2 + 1)�h = 0 (3.3a)

and

�2 _� _h2(1 � _h2) = 0: (3.3b)

From (3.3a) we conclude that h can be at most linear in �,

h(�) = k1�+ k2; (3.4)

and from (3.3b) that at least one of the following must hold

� = 0; _� = 0; _h = 1 = k1; (3.5)

the last equality following from (3.4). From physical considerations and as we arenot interested here in deformations describing eversion, we require _h 6= 0 and take_h > 0. We may also take k2 = 0 in (3.4) since this represents a rigid angulardisplacement. Substituting from (3.4) and simplifying, one can determine that, forthe coefficient of unity in (2.16) to be zero, we must have

0 = (k1�3 �R3)1=3[k1�

2 _��0 +R2 _�]� 2k7=31 �2 _�: (3.6)

Equation (3.6) is a first-order linear partial differential equation to determine�(R;�). Note that (3.6) contains the, as yet, unknowns �(�) and k1. We willreturn to (3.6) in our subsequent consideration of (2.17).

In an analogous manner, we multiply (2.17) by sin2 � for convenience andgroup the terms as coefficients of the various explicit functions of �, with the aidof Macsyma. In this case we have three different terms which multiply the functionsunity, sin2 � ln(sin �), and sin2 �, respectively. Similarly, the coefficients of eachof these functions must be zero. Setting the coefficient of unity to zero implies,after simplification,

0 = k2=31 �[k1�

4 �� R3(� �� + 2 _�2)]: (3.7)

Thus either � = 0 or the term in brackets is zero. Consider the latter. Then, as with(3.2), this requires

k1�4 �� = 0 (3.8a)

and

� �� + 2 _�2 = 0: (3.8b)

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On assuming � 6= 0, (3.8a) implies �� = 0 which then implies, using (3.8b), that_� = 0 also. Thus, � is simply a constant. Notice also that �� = 0 ensures that (3.5),

and thus (3.3b), holds.On returning to (3.6), _� = 0 reduces this equation for �(R;�) to

(k1�3 �R3)1=3R2 _� = 0; (3.9)

which requires _� = 0. Thus, we conclude that

� = �(R): (3.10)

On collecting next the coefficients of the sin2� ln(sin �) term in (2.17), we findthat

2k2=31 �3

R5 (k21 � 1)(R3 � k1�

3)1=3 = 0: (3.11)

Again, for (3.11) to hold for all 0 < R < 1, either

� = 0 or k1 = 1: (3.12)

We now consider the above two possibilities separately.

Case 1. k1 = 1, � 6= 0, constant.

From the information gathered thus far, (2.1), (2.6), (3.4), and the comment fol-lowing (3.4) imply that the resulting deformation has the form

r =

1 �

�3

R3

!1=3

R; � = �; ' = �; (3.13)

which is exactly the radially symmetric deformation (2.2), (2.3), with � replaced by��. The final unknown function �(R) is determined from the third part of (2.17),the coefficients of sin2 � which also must equal zero. In this case, this equationsimplifies to the ordinary differential equation

�0 = �2�6(R3 � �3)�7=3: (3.14)

Note that, from (2.18) and since _h = k1 = 1 in this case, the resulting pressure q issimply the function �(R). Integrating (3.14), we thus recover the (dimensionless)pressure for radially symmetric deformations,

p(R)

�� q(R) = �(R) = �

R(3R3 � 4�3)

2(R3 � �3)4=3+ k3; (3.15)

k3 constant.

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Case 2. � = 0, (k1 6= 1).

In this case the resulting (non-spherically symmetric) deformation is one of inflationand stretching,

r = k�(1=3)1 R; � = k1�; ' = �; (3.16)

where k1 is an arbitrary (positive) constant with k1 6= 1. Here, the equation for�(R) which follows from setting the coefficients of sin2 � to zero is

�0 = k�(2=3)1 (1 � k2

1)=R: (3.17)

Integrating and using (2.18) with f = k�(1=3)1 , _h = k1, we obtain the (dimension-

less) pressure corresponding to the deformation (3.16) as

q(R;�) = k�(2=3)1 (1 � k2

1) ln(R sin�) + k4; (3.18)

k4 constant.We remark that when � = 0 and k1 = 1 we recover the trivial solution r = R,

� = �, ' = � for which the sphere remains undeformed but stressed, with aconstant hydrostatic pressure field.

4. Discussion and Conclusions

We conclude that based solely on the equations of equilibrium for a neo-Hookeansphere and the incompressibility constraint, a deformation of the type (2.1) (with_h > 0) is possible only in two special cases: the well-studied radially symmetricdeformation (2.2), (2.3), and the deformation of inflation and stretching describedby (3.16). (We remark that taking _h = k1 < 0 would result in deformations whichcan be used to describe eversion). The above deformations can each be identifiedwithin the five well-known families of nonhomogeneous static deformations thatcan be maintained in any homogeneous, incompressible, isotropic elastic materialby application of surface tractions alone. The deformation (2.2), (2.3) is recognizedas a special case of Family 4: Inflation or eversion of a sector of a spherical shelldefined by (see, e.g. [11, Sect. 57])

r = (�R3 +A)1=3; � = ��; ' = �; (4.1)

where A is an arbitrary constant. The above solution for inflation was obtainedin [13] while that for eversion in [14]. Its relevance in the context of cavitationin an incompressible initially solid sphere was treated in [1], while a differentinterpretation of cavitation was given in [3] as the sudden rapid expansion of apre-existing microvoid. The deformation (3.16) can be seen to be a special case

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of Family 5: Inflation, bending, extension, and azimuthal shearing of an annularwedge, obtained in [15] and, described, using cylindrical polar coordinates, as

r = AR; � = B log R+ C�; z =Z

A2C; (4.2)

where A, B, and C are arbitrary constants. When (4.2) is referred to sphericalpolar coordinates, it is easy to show that the deformation (3.16) is a special caseof (4.2) with A = k

�(1=3)1 , B = 0, C = k1. Since B = 0 for this special case, no

singularity in the deformation arises at the origin.The goal of this paper was to investigate deformations of the type (2.1) possible

for a neo-Hookean sphere. Our results, following from the equations of elastostaticsand incompressibility, show that no deformation of the type (2.1) can be sustained(in the absence of body forces) except for the two special cases of (2.1) ((3.13), orequivalently (2.2), (2.3), and (3.16)) for which the radial deformation r dependssolely on the radial variable R, and the deformed spherical polar angle � dependslinearly on �. We view the steps taken here as a starting point toward moregeneral results. In particular, sparked by our interest in non-spherically symmetricdeformations describing cavitation phenomena, future work will consider similarquestions for more general deformations than (2.1) and more general materialmodels than neo-Hookean. The present results show that for a neo-Hookean sphere,no new cavitation phenomena exist for deformations of the type (2.1), since onlythe two special cases (2.2), (2.3), which describes the well understood radiallysymmetric cavitation, and (3.16), which does not allow for cavitation, may occur.

In [16] it is shown that this is also the result for (sufficiently smooth) deforma-tions of the form

r = f(R;�)R; � = h(R;�); ' = �; (4.3)

and in view of the analyses here and in [16], holds for a normal, bounded, neo-Hookean solid. In particular, the argument is equally valid for the domain D �

f(R;�;�)j(0 6 A < R < B; 0 6 C < � < D 6 2�; 0 6 E < � < F 6 �g.The authors have also considered similar questions for plane deformations of theform

r = f(R;�)R; � = h(�); z = Z; (4.4)

for hyperelastic incompressible solids [17]. Critical in the analysis of such generaldeformation fields as (2.1), (4.3) and (4.4) are the material formulations of theequilibrium equations in spherical and cylindrical polar coordinates. This approachdoes not suffer the difficulties inherent in a spatial formulation employing Cauchystresses in considering deformation fields with more than one independent variable.These equations, which seem not to appear anywhere else in the literature, arederived below for deformations

r = r(R;�;�); � = �(R;�;�); ' = '(R;�;�) (4.5)

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of a general hyperelastic solid, and in [17] for deformations

r = r(R;�; Z); � = �(R;�; Z); z = z(R;�; Z): (4.6)

As opposed to the spatial formulation, the material formulation naturally gives thegoverning equations on a known, fixed region of space, with differentiations takenwith respect to the independent, and not the dependent, variables.

A. Appendix

The goal of this appendix is to derive the component form, referred to sphericalpolar coordinates, of the expression Div S � r�S, where S represents the nominalstress tensor.

Let fX1;X2;X3g represent rectangular Cartesian Lagrangean coordinates withorthonormal basis vectors fE1;E2;E3g and origin O, and let fx1; x2; x3g representthe corresponding rectangular Cartesian Eulerian coordinates of X with orthonor-mal basis vectors fe1; e2; e3g and origin o. We introduce the spherical coordinates

fX1;X2;X3g = fR sin � cos �; R sin � sin �; R cos �g (A.1)

and

fx1; x2; x3g = fr sin ' cos �; r sin ' sin �; r cos 'g (A.2)

with corresponding orthonormal basis vectors

ER = sin � cos �E1 + sin � sin �E2 + cos �E3; (A.3)

E� = � sin �E1 + cos �E2; (A.4)

E� = cos � cos �E1 + cos � sin �E2 � sin �E3; (A.5)

and

er = sin ' cos �e1 + sin ' sin �e2 + cos 'e3; (A.6)

e� = � sin �e1 + cos �e2; (A.7)

e' = cos ' cos �e1 + cos ' sin �e2 � sin 'e3; (A.8)

for X and x respectively.From [12, pp. 49–50, 60], the divergence of a second-order tensor field T, the

vector r � T, for the curvilinear coordinates fx1; x2; x3g is defined according to

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a contraction between, r and T, applied to its third-order gradient tensor r T.The gradient r T in component form is given by [12, p. 58]

r T =

�@Tij

@xk� �r

ikTrj � �rjkTir

�gi gj gk; (A.9)

where fgjg is the reciprocal basis of the natural basis fgig for the fxig, Tij arethe covariant components of T, i.e. T = Tijgigj , �i

jk are the Christoffel symbolsdefined by

�ijk = �gj �

@gi

@xk; (A.10)

and � are the standard dyadic and vector dot products, respectively, and summationconvention for repeated indices is employed here and throughout. The expressionfor the gradient of T in (A.9) is equivalent to

r T =@Tij

@xkgi gj gk + Tij

" @gi

@xk� gr

!gr# gj gk

+Tijgi

" @gj

@xk� gr

!gr# gk; (A.11)

and (A.11) is equivalent tohTijgi gj

i

�@

@xkgk�; (A.12)

in which the gradient operator, the term on the right in (A.12), is distributed acrossT, the term on the left in (A.12). On letting bTij and bgk represent the physicalcomponents of T and the unit vectors bgk � gk=kgkk, (A.12) becomesh bTijbgi bgji �kgkk

@

@xkbgk� : (A.13)

The nominal stress tensor S has component form SijEi ej with respect tothe orthonormal bases fEig and fejg in the reference and current configurationsrespectively, and the Sij represent the physical components of the stress, [12,p. 153]. With respect to the orthonormal bases (A.3)–(A.5) and (A.6)–(A.8), anexpression for Grad S = r S using (A.13) is

[SRrER er + SR�ER e� + SR'ER e' + S�rE� er + S��E� e�

+S�'E� e' + S�rE� er + S��E� e� + S�'E� e']

�@

@RER +

1R sin �

@

@�E� +

1R

@

@�E�

�: (A.14)

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As fSijg, fEig, and fejg can depend on fR;�;�g the following transformationsare useful when carrying out the dyadic distribution in (A.14):

ER;R= E�;

R= E�;

R= E�;

�= 0; (A.15)

ER;�= E�; E�;

�= �ER; (A.16)

ER;�= sin �E�; E�;� = cos �E�; (A.17)

E�;� = �[sin �ER + cos �E�]; (A.18)

and

er;i = �;i sin 'e� + ';ie' for i = R;�;�; (A.19)

e�;i = ��;i[sin 'er + cos 'e'] for i = R;�;�; (A.20)

e';i = �;i cos 'e� � ';ier for i = R;�;�; (A.21)

where in (A.15)–(A.21) and subsequently, the comma notation denotes differen-tiation. To illustrate how r S may be calculated from (A.14), we distributethe differential (Gradient) operator r, the right term in (A.14), to the first termSRrER er of S, the left expression in (A.14), yielding

[SRrER er]@

@RER + [SRrER er]

@

@�

1R sin �

E�

+ [SRrER er]@

@�

1R

E�; (A.22)

which, on using (A.15)–(A.21) and the product rule, is equivalent to

SRr;RER er ER +SRr�;R sin 'ER e� ER +SRr';RER e'ER(A.23)

+1

R sin �[SRr;�ER er E� + sin �SRrE� er E�

+ �;� sin'SRrER e� E� + ';�SRrER e' E�]

9>>=>>; ; (A.24)

+1R[SRr;�ER er E� + SRrE� er E�

+ �;� sin 'SRrER e� E� + ';�SRrER e' E�]

9>=>; ; (A.25)

where (A.23), (A.24), and (A.25) represent the first, second, and third terms in(A.22) respectively. The rest of r S can be calculated analogously distributing

elas1266.tex; 13/10/1997; 13:08; v.7; p.13


r to the remaining eight terms of S in (A.14). After these differentiations are takenthe resulting expression can then be factored to the form

r S = CijkEi ej Ek for i; k = R;�;� and j = r; �; '; (A.26)

where the coefficient Cijk, of the 27 different basis tensors Ei ej Ek, aresomewhat complicated expressions which depend on R, �, �, r, �, ', S, andderivatives of S.

As mentioned above,r�T, the divergence of a second-order tensor T, is definedaccording to a contraction between r and T applied to r T. Specifically, thecontraction is chosen such that

(r � T) � a = r � (Ta); (A.27)

which has the Cartesian component representation

(r � T) � a = Tij;iaj (A.28)

for all vectors a in three-dimensional Euclidean space, and Ta represents thestandard tensor-vector multiplication [12, pp. 49, 50]. Such a contraction for somethird-order tensor

R = Rijkgi gj gk; (A.29)

here with a covariant representation, is defined as the vector

c = Rijk(gi � gk)gj; (A.30)

[12, p. 38]. For r S in (A.26) the contraction in (A.30) defines r � S such that

r � S � Cijk(Ei � Ek)ej for i; k = R;�;� and j = r; �; ': (A.31)

Since the fEig of (A.3)–(A.5) are orthonormal, the dependence of (A.31) on Cijk

is reduced to its trace components Ciji so that

r � S = Cijiej for i = R;�;� and j = r; �; ': (A.32)

After some effort these Ciji are found to be such that

r � S

=

�SRr;R � sin '�;RSR� � ';RSR'

+1R[2SRr + cot�S�r + S�r;� � sin '�;�S�� ';�S�']

+1

R sin �[S�r;� � sin '�;�S�� ';�S�']

�er

9>>>>>>>>>>>=>>>>>>>>>>>;(A.33)

elas1266.tex; 13/10/1997; 13:08; v.7; p.14


+

�sin '�;RSRr + SR�;R + cos '�;RSR'

+1R[2SR� + cot�S�� + sin '�;�S�r + S��;� + cos '�;�S�']

+1

R sin �[sin '�;�S�r + S��;� + cos '�;�S�']

�e�

9>>>>>>>=>>>>>>>;(A.34)

+

�';RSRr � cos '�;RSR� + SR';R

+1R[2SR' + cot�S�' + ';�S�r � cos '�;�S�� + S�';� ]

+1

R sin �[';�S�r � cos '�;�S�� + S�';� ]

�e'

9>>>>>>>=>>>>>>>;; (A.35)

where (A.33)–(A.35) denote the three terms as indicated. Thus the equilibriumequations, Div S = r � S = 0, for the nominal stress S, referred to spherical polarcoordinates, are given in component form by setting the coefficients of er, e�, ande' in (A.33)–(A.35), respectively, to zero.

Acknowledgements

D. Polignone Warne gratefully acknowledges the support of the research via aProfessional Development Grant from the University of Tennessee, Knoxville anda Research Planning Grant (NSF DMS-9500642) from the National Science Foun-dation.

References

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on non-symmetric deformations of an incompressible nonlinearly elastic isotropic sphere

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