plane deformations in incompressible nonlinear elasticity

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Journal of Elasticity 52: 129–158, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands. 129 Plane Deformations in Incompressible Nonlinear Elasticity DEBRA POLIGNONE WARNE 1 and PAUL G. WARNE 2 1 Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A. 2 Division of Mathematics and Computer Science, Maryville College, Maryville, TN 37804, U.S.A. Received 25 June 1998; in revised form 23 November 1998 Abstract. The substantially general class of plane deformation fields, whose only restriction requires that the angular deformation not vary radially, is considered in the context of isotropic incompressible nonlinear elasticity. Analysis to determine the types of deformations possible, that is, solutions of the governing systems of nonlinear partial differential equations and constraint of incompressibility, is developed in general. The Mooney–Rivlin material model is then considered as an example and all possible solutions to the equations of equilibrium are determined. One of these is interpreted in the context of nonradially symmetric cavitation, i.e., deformation of an intact cylinder to one with a double-cylindrical cavity. Results for general incompressible hyperelastic materials are then discussed. The novel approach taken here requires the derivation and use of a material formulation of the governing equations; the traditional approach employing a spatial formulation in which the governing equations hold on an unknown region of space is not conducive to the study of deform- ation fields containing more than one independent variable. The derivation of the cylindrical polar coordinate form of the equilibrium equations for the nominal stress tensor (material formulation) for a general hyperelastic solid and a fully arbitrary cylindrical deformation field is also given. Mathematics Subject Classifications (1991): 73G05, 73C50, 73V99, 35Q72. Key words: nonsymmetric, inhomogeneous, equilibrium, plane hyperelasticity, incompressible, nom- inal stress, material formulation of cylindrical problems, cavitation. 1. Introduction Substantial theoretical and analytical results in the area of incompressible nonlinear elasticity have been obtained since its inception in the 1940’s. However, due to the accompanying mathematical complexities, there has been little progress in our ability to analyze deformations which lie outside the realm of those which are either universal or involve simplifying assumptions on the deformation such as radial or axial symmetry. Consequently, there is little knowledge of what, if any, solutions exist which involve the independent variables (undeformed coordinates) in more complex ways. For a nice treatment of the evolution of Ericksen’s problem, that of determining all universal solutions (static deformation fields producible in every isotropic homogeneous hyperelastic material by application of surface tractions alone) and the five known families for incompressible materials, see Beatty [1]. 200702.tex; 17/03/1999; 10:20; p.1 INTERPRINT: J.N.B. elas2090 (elaskap:mathfam) v.1.15

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Page 1: Plane Deformations in Incompressible Nonlinear Elasticity

Journal of Elasticity52: 129–158, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

129

Plane Deformations in IncompressibleNonlinear Elasticity

DEBRA POLIGNONE WARNE1 and PAUL G. WARNE21Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.2Division of Mathematics and Computer Science, Maryville College, Maryville, TN 37804, U.S.A.

Received 25 June 1998; in revised form 23 November 1998

Abstract. The substantially general class of plane deformation fields, whose only restriction requiresthat the angular deformation not vary radially, is considered in the context of isotropic incompressiblenonlinear elasticity. Analysis to determine the types of deformations possible, that is, solutions of thegoverning systems of nonlinear partial differential equations and constraint of incompressibility, isdeveloped in general. The Mooney–Rivlin material model is then considered as an example andall possible solutions to the equations of equilibrium are determined. One of these is interpretedin the context of nonradially symmetric cavitation, i.e., deformation of an intact cylinder to onewith a double-cylindrical cavity. Results for general incompressible hyperelastic materials are thendiscussed. The novel approach taken here requires the derivation and use of a material formulationof the governing equations; the traditional approach employing a spatial formulation in which thegoverning equations hold on an unknown region of space is not conducive to the study of deform-ation fields containing more than one independent variable. The derivation of the cylindrical polarcoordinate form of the equilibrium equations for the nominal stress tensor (material formulation) fora general hyperelastic solid and a fully arbitrary cylindrical deformation field is also given.

Mathematics Subject Classifications (1991):73G05, 73C50, 73V99, 35Q72.

Key words: nonsymmetric, inhomogeneous, equilibrium, plane hyperelasticity, incompressible, nom-inal stress, material formulation of cylindrical problems, cavitation.

1. Introduction

Substantial theoretical and analytical results in the area of incompressible nonlinearelasticity have been obtained since its inception in the 1940’s. However, due tothe accompanying mathematical complexities, there has been little progress in ourability to analyze deformations which lie outside the realm of those which are eitheruniversal or involve simplifying assumptions on the deformation such as radial oraxial symmetry. Consequently, there is little knowledge of what, if any, solutionsexist which involve the independent variables (undeformed coordinates) in morecomplex ways. For a nice treatment of the evolution of Ericksen’s problem, thatof determining all universal solutions (static deformation fields producible in everyisotropic homogeneous hyperelastic material by application of surface tractionsalone) and the five known families for incompressible materials, see Beatty [1].

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130 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

Any other universal solutions in incompressible finite elasticity must be fully 3-dimensional in character. Fosdick and Schuler [2] obtained the complete solutionof Ericksen’s problem for plane deformations with uniform transverse stretch, i.e,

x = x(X, Y ), y = y(X, Y ), z = λZ,while Fosdick [3] closed the completeness question of Ericksen’s problem for theclass of radially (or axially) symmetric deformations

r = r(R,Z), θ = 2, z = z(R,Z).In the above and subsequently, lower (upper) case letters denote coordinates of apoint in the deformed (undeformed) configurations of a body in the appropriatecoordinate system. Discovery of other nonuniversal solutions to the equations ofnonlinear elastostatics requires specification of the strain-energy density functionto consider deformations for a particular class of materials.

In a series of three papers, Hill [4–6] determined several solutions to the equa-tions of equilibrium for neo-Hookean and Mooney–Rivlin materials involvingBessel functions of order zero, with subsequent applications to specific boundary-value problems. In [4], Hill considered plane deformation fields

x = f (R), y = y(R,2), z = λZwith y = g(R)2 + h(R) as a consequence of incompressibility, and obtainedsolutions for general Mooney–Rivlin materials. (We remark that in plane strain,there is no distinction between neo-Hookean and Mooney–Rivlin behavior as theprincipal strain invariantsI1 andI2 are equal). Radially symmetric deformations

r = f (R), θ = 2, z = z(R,Z),with, from incompressibility,z = g(R)Z + h(R), were treated in [5], while [6]considered

x = f (R), y = g(R) cos2+ h(R),z = l(R)8+m(R) cos2+ n(R),

and solutions for neo-Hookean materials and (extreme) Mooney materials (thosedepending solely on the second principal strain invariant) were determined in each.Also, Hill and Lee [7] considered the 3-dimensional deformation

x = f (R), y = g(R)2, z = h(R)Zfor general Mooney–Rivlin materials and determined several special solutions ofthe resulting system of coupled ordinary differential equations (ODEs). We note

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 131

that an explicit determination of the functionf (R) in [4–7] was hindered as thedifferential equation forf following from the incompressibility constraint involvedthe resulting Bessel function solutions obtained for the other unknown functionsfrom the equilibrium equations.

Recently, a new approach and methodology for analyzing problems involvingunknown deformation functions depending arbitrarily on more than one independ-ent variable was developed and used by Polignone Warne and Warne [8, 9] todetermine, within a general class of deformations, all solutions of the governing3-dimensional equations of equilibrium DivS = 0, whereS is the nominal stresstensor, for a neo-Hookean incompressible nonlinearly elastic body. It was shownin [8, 9] that within the broad class of deformations

r = f (R,2)R, θ = h(R,2), ϕ = 8, (1.1)

there are only three special types which satisfy the equilibrium equations and in-compressibility constraint for a neo-Hookean solid. In (1.1),f andh are arbitrarysufficiently smooth functions of their arguments with∂h/∂2 > 0. These threetypes of deformations are spherically symmetric, given by

r = r(R) (= f (R)R) , θ = 2, ϕ = 8 (1.2)

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

r(R) =(

1+ β3

R3

)1/3

R, (1.3)

whereβ is a constant to be determined, the trivial solution (β = 0 in (1.2) and(1.3)), and the nonsymmetric, nonhomogeneous universal deformation (specialcase of Family 5, referred to spherical polar coordinates),

r = k−1/31 R, θ = k12, ϕ = 8, (1.4)

wherek1 is an arbitrary (positive) constant withk1 6= 1. Thus in particular, thereexist no (smooth) nonsymmetric, nonhomogeneous, nonuniversal deformations ofa neo-Hookean solid within the class (1.1). Subsequently, any new solutions mustlie outside this considerably large class of deformations.

In this paper, we wish to determine all possible solutions to the equations ofequilibrium within a general class of plane deformations in incompressible nonlin-ear elastostatics. Before proceeding, we try to present a summary of other relevantwork. Investigations into nonhomogeneous, nonradially symmetric plane deform-ations of the form

r = Rf (2), θ = g(2), z = Z (1.5)

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132 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

were carried out by a number of authors. This problem was first considered by Blatzand Ko [10] for Mooney–Rivlin (or equivalently, neo-Hookean) materials. In [10],nonhomogeneous implicit solutions within the class (1.5) were obtained and thecorresponding boundary-value problem modeling tangential spreading of a radiallycracked infinitely long incompressible rubber log by a radially rigid bonded wedgewas treated, with subsequent analysis of the stress-displacement field at the cracktip. Additionally, section 5 of Hill [11] recovered the solution of [10] for neo-Hookean materials and determined a nonhomogeneous parametric solution withinthe class (1.5) for Varga materials. Fu, et. al [12], Rajagopal and Carroll [13], andRajagopal and Tao [14] have also studied (1.5) for an isotropic hyperelastic wedge.In [12], a wedge of Mooney–Rivlin material undergoing (1.5) is considered, and theequilibrium equations with conservative body forces are reduced to a single first-order highly nonlinear ODE forρ(2) ≡ 1/f (2) (see (4.11b)); incompressibilitythen determinesg asdg/d2 = f −2. The closed-form solution corresponding tohomogeneous deformations (and a bounded pressure field) is identified (both hereand in [11]), and this solution is then analyzed for various initial wedge angles. Inaddition, evidence of the existence of nonhomogeneous solutions (correspondingto unbounded pressure fields at the origin) follows in [12] from numerical integ-ration of an appropriate initial value problem. In [13], analysis of the equationsof equilibrium in the absence of body force is initially carried out for (1.5) for anarbitrary hyperelastic material. The form of the pressure field is determined forarbitrary incompressible elastic solids to be

p(R,2) = K lnR +9(2),whereK is constant; note thatK 6= 0 corresponds to an unbounded pressurefield at the origin. An expression for the pressure corresponding to the general-ized neo-Hookean power-law model is then given. For compressible elastic solids,a necessary condition on the stored-energy function so that (1.5) is possible byapplication of surface traction alone is obtained, and then applied to the specialBlatz–Ko material and the class of Hadamard materials. In [14], the equilibriumequations in the absence of body forces are considered for deformations (1.5) ofa wedge of generalized neo-Hookean power-law material, and are reduced to asingle second-order highly nonlinear ODE forρ(2). When the power-law ex-ponentn = 1, this material reduces to the neo-Hookean material and thus thework of [12] applies. Forn 6= 1, the inhomogeneous solution for whichρ(2) =γ, g(2) = γ 22, and γ is constant, a special case of the family of universalsolutions known as Family 5, can be obtained as one solution to the governingequation where the associated pressure is unbounded at the origin. Correspondingto a bounded pressure field, it is then shown that the homogeneous deformationis the only possible solution whenn > 1

2, while for n < 12, the range ofn for

which it is well-known that the equilibrium equations for the power-law model maylose ellipticity in plane strain, a boundary-layer type solution can be constructedfor which the deformation is homogeneous in the core region of the wedge and

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 133

nonhomogeneous adjacent to the boundaries, subject to appropriate boundary con-ditions. Finally in [14], the governing nonlinear ODE forρ(2) (corresponding toan unbounded pressure field) is solved numerically, indicating that solutions whichare everywhere nonhomogeneous (and nonuniversal) exist.

In Hill [15], deformations of the form

r =√aR2+ f (2), θ = b2+ c, z = Z, (1.6)

with a, b, andc constants are considered. The explicit solution to the governingequations for a Mooney–Rivlin solid, given by

f (2) = d[1+ sin(22+ e)] and a = ±1 according as b = ±1, (1.7)

with d, e constants is determined and interpreted for the boundary-value problemof a long half-cylindrical tube compressed between parallel plates. The boundaryconditions could be satisfied only in an integral sense, and thus an approximateload-deflection relation was determined. Hill [11] also considered deformations ofthe form

r = Rf (a2+ b logR), θ = ±2+ g(a2+ b logR), z = Z, (1.8)

and determined implicit solutions to the equations of equilibrium and incompress-ibility constraint for both neo-Hookean and Varga materials.

Further investigations of deformations in incompressible finite hyperelasticityhave been carried out in a recent series of papers by Hill and Arrigo [l6–18] forthe Varga and modified Varga strain-energy functions (the mathematical analoguesof the neo-Hookean and Mooney–Rivlin materials employing the first and secondprincipal strain invariants,i1, i2 of the left or right stretch tensorsV or U as op-posed to the principal invariantsI1, I2 of the strain tensorsB = V2 or C = U2;similarly, i1 = i2 in isochoric plane strain deformations). In [16], a number ofexplicit solutions to the governing partial differential equations (PDEs) of equilib-rium for Varga-type materials are obtained for both plane and axially symmetricdeformations. As our interest in this paper is limited to the former, our subsequentdiscussion is so focussed. Among the nonhomogeneous exact solutions derived in[16] for Varga materials are a similarity solution of the form

x = Yf (ξ), y = Xg(ξ), ξ = Xm

Y

(see (5.13) of [16]), a solution of the form

r =√R2G(2)+H(2), θ = F(2) (1.9)

(see (5.28) of [16]), and a solution of the form

y = f ′(ξ), ξ = Y

X

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134 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

(see (5.35) of [16]). In [17], it is shown that plane deformations,

r = r(R,2), θ = θ(R,2), z = Z,

for incompressible Varga type materials can be given parametrically by

r = (ω2+ ω22)

1/2, θ = −2− tan−1(ω2/ω), R = ωλwith ωλ > 0 andω(λ,2) any solution of the linear PDE

ωλλ + ω22 + ω = 0,

where in the above subscripts denote partial differentiation. The analysis of [17]also leads to the discovery of a new nonsymmetric explicit solution of the govern-ing equations of equilibrium for Varga materials when deformations of the form(1.9) are considered (see (7.8) and (7.5) of [17]). Hill and Arrigo [18] similarlytreats plane stress and axially symmetric deformations of Varga-type materials.

In Section 2 of this paper, we consider the deformation field

r = Rf (R,2), θ = h(2), z = Z, (1.10)

in cylindrical polar coordinates. We first refine the form of the deformation so as tocomply with the incompressibility constraint, thus recovering (1.9), and then pro-ceed with the relevant material from finite elastostatics. In Section 3, the two mainequations resulting from equilibrium (DivS = 0, whereS is the nominal stresstensor) are analyzed for a general incompressible isotropic hyperelastic material,and are reduced to a single equation ((3.5)) which highlights the role played bythe material model. In Section 4, we analyze this equation for the Mooney–Rivlinmodel, and determine the only deformations, of which there are three main types,of the form (1.10) that satisfy the governing equations of equilibrium. First, wehave (see(4.24a))

r = α√R2+ β, θ = 1

α22, z = Z, (1.11)

with α andβ both constants. This is a special case of the nonhomogeneous, con-trollable deformations possible in every homogeneous, incompressible, isotropic,hyperelastic wedge known as Family 3. In (1.11),α = 1 recovers radially sym-metric deformations, whileβ = 0 recovers another universal deformation (see(4.25a)) which is also a special case of Family 5. The second type of deformationthat may be possible (see (4.11)) is

r = R√h(2)

, θ = h(2), z = Z, (1.12)

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 135

which is exactly the deformation (1.5) upon incorporating the incompressibilityconstraint. Third, we determine the explicit, nonhomogeneous, nonuniversal closed-form solution (see (4.39)) described by

r =√R2+ k3[1± sin(22+ 2k4)], θ = 2, z = Z, (1.13)

wherek3 > 0 andk4 are arbitrary constants, and obtain the corresponding pressureand nominal stress fields. The solution (1.13) is equivalent to (1.6) and (1.7) withb = 1, derived for the Mooney–Rivlin material from the starting point (1.6) by Hill[15]. Thus, in this paper, we begin with a significantly broader class of deforma-tions, and show that within this class that theonly isochoric deformations whichcan occur for the Mooney–Rivlin material are those given above. It is worth notingthat (1.13) could give rise to deformations describing nonsymmetric cavitation of aMooney–Rivlin solid under appropriate loading. This aspect of the solution (1.13)has never before been considered and thus we discuss (1.13) in this context aswell. Interestingly, (1.13) gives rise to a deformation predicting the formation of adouble cylindrical cavity, i.e., the origin of a typical cross-section of the cylinderis mapped to two circular holes which touch tangentially at a single point. Section5 presents some concluding remarks and returns to discussion and results for thearbitrary isotropic incompressible hyperelastic material.

Critical in our analysis of nonradially symmetric deformation fields with morethan one independent variable (such as (1.1) and (1.10)) are the material formu-lations of the equilibrium equations in spherical and cylindrical polar coordinates.For such deformation fields, this approach does not suffer the difficulties inherentin a Cauchy stress (spatial) formulation which (i) requires differentiations withrespect to dependent variables which may assume a highly general unknown form,and (ii) produces equations which hold on an unknown region of space. Theseequations for the nominal stress are derived in [8] and the present paper for fullyarbitrary three-dimensional deformations

r = r(R,2,8), θ = θ(R,2,8), ϕ = ϕ(R,2,8) (1.14)

and

r = r(R,2,Z), θ = θ(R,2,Z), z = z(R,2,Z) (1.15)

respectively, of a general nonlinearly elastic solid. These rather tedious albeit notterribly engaging calculations could not be found elsewhere in the literature; theirutility however has proven quite valuable. Such a material formulation of the equi-librium equations in spherical and cylindrical polar coordinates allows for consid-eration of nonradially symmetric deformation fields containing more than one inde-pendent variable. Prior to these, there existed no straightforward general frameworkof this sort from which to pursue such problems for nonlinearly elastic solids, andthus overcoming the associated mathematical difficulties was the underlying factor

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136 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

behind the types of deformations which were analyzed. We remark that a repres-entation for the equations of motion in component form for a general curvilinearcoordinate system in terms of Christoffel symbols and total covariant derivatives isgiven in Section 44 of [20], while [4], [6], and [19], derived reduced equilibriumequations in terms of metric tensors for general curvilinear coordinates, the asso-ciated Christoffel symbols, and total covariant derivatives for plane deformationswith uniform transverse stretch as well as 3-dimensional deformations. Indeed, oneattractive feature of our approach is that for cylindrical and spherical problems, oneimmediately obtains the appropriate differential equations which are free from theabove mentioned hindrances. Previous studies have encountered and circumventedsuch difficulties in several problem-specific ways. A Cartesian coordinate approachimplementing complex variable methods is used in [2] to analyze the equations,with an eventual conversion to cylindrical polar coordinates, while [10], [15], and[12–14] pose their problems in cylindrical polar coordinates and convert derivativeswith respect to the dependent variables into derivatives with respect to independentones, a straightforward transformation as the deformation (1.5) involves unknownfunctions depending only on a single independent variable. Similarly, [11] and[7] take advantage of the particular forms of the deformation fields consideredto convert their equations to ODEs containing derivatives with respect to materialcoordinates. In [11], the equilibrium equations for plane deformations of a generalincompressible hyperelastic material are first reduced to two equations giving thepartial derivatives of the unknown pressure fieldp(r, θ). Transformations based onthe deformation field (1.8) are then employed to determine equations for the partialderivatives of the pressure with respect to material instead of spatial coordinates,which then led to the derivation of new, implicit solutions for neo-Hookean andVarga materials. The analysis of [7] originates with a material formulation usingPiola–Kirchhoff stresses for the Mooney–Rivlin material. While [7] and [11] re-markably determine solutions of their resulting systems of equations, in both casesthese are special solutions in that they follow from a particular way of satisfyingthe resulting equilibrium equations. In contrast, the approach adopted here and in[8, 9] allows for determination ofall possible solutions of the governing equationswithin a general class of deformations. In addition, the present development al-lows for conclusions regarding general incompressible hyperelastic materials. Incontrast, the substantial studies of [16–18] do not extend though beyond Vargatype materials. In particular, [17] shows that for Varga materials, exact solutions tothe equations of equilibrium corresponding to plane deformations, and referred toCartesian coordinates, can be determined from a second-order system. They even-tually arrive at their results upon utilizing complex variable methods, transformingto a Monge–Ampère equation, converting to cylindrical polar coordinates, andmaking use of conversions between derivatives with respect to spatial and materialcoordinates.

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 137

2. Preliminaries: Kinematics and Stress

We wish to study the equations governing nonradially symmetric plane deform-ations in incompressible nonlinearly elastostatics. We employ the semi-inversemethod (see, e.g., [20, Sect. 59]) and consider a class of deformations of the form

r = Rf (R,2), θ = h(2), z = Z (2.1)

where, for convenience, interior points in the undeformed configuration are de-scribed by the cylindrical polar coordinates(R,2,Z) with 0 6 A 6 R < 1,06 C < 2 < D < 2π ,−∞ < Z <∞. Points in the deformed configuration arethen described by the cylindrical polar coordinates(r, θ, z) given by (2.1) wheref andh are as yet arbitrary sufficiently smooth functions of their arguments, withdh/d2 ≡ h > 0.

We now proceed with the relevant material from finite elastostatics. More de-tail can be found, for example, in [1, 20, 21]. The deformation gradient tensorFassociated with (2.1), referred to cylindrical polar coordinates, is given by

F =

f + Rf ′ f 0

0 f h 0

0 0 1

, (2.2)

where, here and elsewhere, the superposed′ and· denote (partial or ordinary) dif-ferentiations with respect toR and2 respectively. Incompressibility then requiresthatJ ≡ detF = 1, which results in the first-order quasi-linear PDE forf (R,2)

f h(f ′R + f ) = 1. (2.3)

Upon lettingu = f 2, equation (2.3) is readily integrated with respect toR, yieldingthe general solution

f (R,2) = 1

R√h(2)

√R2+ h(2)g(2), (2.4)

whereg(2) is, as yet, an arbitrary function of integration. Thus, for the radialdeformation, it is only the dependence off on2 that remains unknown.

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

W = W(I1, I2,1), (2.5)

where

I1 = tr B,

I2 = 12

[(tr B)2− tr B2

],

}(2.6)

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138 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

and B is the left Cauchy–Green deformation tensorB = FFT (I3 = detB =J 2 = 1 via incompressibility). As we are interested here in isochoric plane straindeformations,I1 = I2 and thus we define, for convenience,

I ≡ I1 = I2 = 1+ (f ′R + f )2+ f 2+ 1

(f ′R + f )2 , (2.7)

wheref is given by (2.4). The corresponding response equation for the nominalstress tensorS = JF−1T (see, e.g., [21, Sect. 3.4, Sect. 4.3]), whereT is theCauchy stress tensor, is given by

S= −pF−1+ 2W1FT + 2W2FT (I11− B), (2.8)

wherep = p(R,2,Z) is the unknown hydrostatic pressure associated with theincompressibility constraint,1 is the unit tensor, andWi = ∂W

∂Ii(i = 1,2). It is

readily seen that the coefficient of 2W2 in (2.8) can be represented as

I1FT − FTB = F−1+ FT + (I1− 3)EZ ⊗ ez (2.9)

for the deformation considered here, and thus on defining

H(I) ≡ 2W1(I1, I2,1)+ 2W2(I1, I2,1),

−q(R,2,Z) ≡ −p + 2W2,(2.10)

we obtain the following alternate expression for the nominal stress response

S= −qF−1+HFT + 2W2(I − 3)EZ ⊗ ez, (2.11)

where, from (2.2) and (2.3) (withf given by (2.4)),

F−1 =

(f ′R + f )−1 −f 0

0 f ′R + f 0

0 0 1

. (2.12)

Thus, (2.2), (2.11), (2.12) give the nominal stress components corresponding to thedeformation (2.1) as

SRr = −q/(f ′R + f )+H(f ′R + f )SRθ = qfS2r = HfS2θ = −q(f ′R + f )+H/(f ′R + f )SZz = −q +H + 2W2(I − 3)

SRz = S2z = SZr = SZθ = 0.

(2.13)

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 139

The equilibrium equations, in the absence of body forces, are

Div S= 0, (2.14)

where Div denotes the divergence with respect to(R,2,Z), the undeformed co-ordinates (independent variables). As mentioned earlier, due to the generality ofthe deformation field (2.1), it is prudent to proceed with nominal stresses insteadof Cauchy stresses. The cylindrical polar coordinates form of (2.14), derived in theAppendix, give the equilibrium equations for the completely general deformationfield (1.8) as

SRr,R − θ,RSRθ − θ,ZSZθ + SZr,Z +1

R

[SRr − θ,θS2θ + S2r,2

] = 0, (2.14a)

SRθ,R + θ,RSRr + θ,ZSZr + SZθ,Z +1

R

[SRθ + θ,θS2r + S2θ,2

] = 0, (2.14b)

SRz,R + SZz,Z +1

R

[SRz + S2z,2

] = 0, (2.14c)

where, in the above and subsequently, a comma denotes differentiation.For the deformation (2.1), these general equilibrium equations, (2.14a), (2.14b),

(2.14c), reduce to

SRr,R +1

R

[SRr + S2r,2 − hS2θ

] = 0, (2.15)

SRθ,R +1

R

[SRθ + S2θ,2 − hS2r

] = 0, (2.16)

SZz,Z = 0 (2.17)

respectively onA < R < 1,C < 2 < D,−∞ < Z <∞ where the last equationin (2.13) has been used to obtain (2.15)–(2.17). Upon substitution from (2.13) andnoting that the invariantI is independent of the axial coordinateZ, equation (2.17)requires, as expected, thatq = q(R,2) only.

3. The Remaining Equations

It remains to analyze the two other equilibrium equations (2.15) and (2.16) todetermine the unknown functionsh(2), g(2), andq(R,2). Substituting first from(2.13) into (2.15) and rearranging, we obtain

− q ′

f ′R + f +[h(f ′R + f )

R−(

1

f ′R + f)′− 1

R(f ′R + f )]q

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140 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

+[H(f ′R + f )]′ + 1

R

[H(f ′R + f )+ (H f )· − hH

f ′R + f]= 0.

The coefficient ofq in the above equation, however, is identically zero due toincompressibility, or equivalently (2.4), yielding

∂q

∂R≡ q ′ = f ′R + f

R

[(RH(f ′R + f ))′ + (H f )· − hH

f ′R + f]. (3.1)

Treating equation (2.16) in a similar manner, the terms involvingq again cancelvia (2.4), resulting in

∂q

∂2≡ q = 1

f ′R + f[Rf q ′ + hH f +

(H

f ′R + f)· ]

. (3.2)

Ideally, one would like to integrate (3.1) with respect toR (creating another arbit-rary function of2), and then differentiate the result with respect to2 in order to use(3.2), along with the manner in which the variableR (explicitly) enters the equa-tion, to produce a simplified system of equations for the remaining three unknownfunctions of2, a method successfully used in [8, 9]. However, as we are yet tospecify a particular incompressible material model for this plane strain problem, in-tegration of (3.1) for an arbitraryH(I) seems unlikely. In addition, we remark thateven for one of the simplest strain-energy densities for whichH is identically con-stant, integration to determineq, while possible, is not entirely a straight-forwardcalculation. To continue developing the analysis under the assumption of arbitraryincompressible material model, we circumvent the above described difficulty, byrequiring that the mixed second partial derivatives ofq(R,2) be equal

(q)′ = (q ′)·. (3.3)

Carrying out the necessary differentiations of (3.1) and (3.2) and substituting intothe requirement (3.3), we obtain

f ′R + fR

{[RH(f ′R + f )]′· + (H f )··}+ fR

{[RH(f ′R + f )]′ + (H f )·}−f {[RH(f ′R + f )]′′ + (H f )′.} − 1

f ′R + f(

H

f ′R + f)′·

−(

1

f ′R + f)′ (

H

f ′R + f)·− 1

R(hH)

· = 0. (3.4)

Using the symbolic manipulation software Macsyma [Macsyma Inc., 1995] toperform the arduous differentiations and algebraic computations in (3.4) results inan equation (holding on the interior of our body) of the form

A1(R,2)H(I)+ A2(R,2)dH

dI(I )+ A3(R,2)

d2H

dI 2(I ) = 0, (3.5)

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 141

whereH is given by(2.10)1 and (2.5),I by (2.7), and

A1(R,2) = −(f ′R + f )′′f R − f f ′ − (f ′R + f )′f

+(f ′R + f )(f ′R + f )′· − h

R+ (f

′R + f ) ˙fR

+ f fR+ (f

′R + f )fR

+ (f′R + f )(f ′R + f )·

R

−3(f ′R + f )′(f ′R + f )·(f ′R + f )4 + (f

′R + f )′·(f ′R + f )3 , (3.6)

A2(R,2) = (f ′R + f )f[I

R− I ′′R

]

+[(f ′R + f )2− f 2− 1

(f ′R + f )2]I ′

+[(f ′R + f )(f ′R + f )· − 2(f ′R + f )′f R

−f f − (f ′R + f )f + (f ′R + f )·(f ′R + f )3

]I ′

+[(f ′R + f )2

R+ f

2

R+ 2(f ′R + f )f

R

− hR− f f ′ + (f ′R + f )(f ′R + f )′

+2(f ′R + f )′(f ′R + f )3

]I , (3.7)

and

A3(R,2) = (f ′R + f )f[I 2

R− I ′2R

]

+[(f ′R + f )2− f 2− 1

(f ′R + f )2]I ′I . (3.8)

We remark that (3.5) nicely distinguishes the roles of the deformation field andmaterial model; theAi, i = 1,2,3 depend solely on the deformation, while thematerial is described byH .

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142 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

4. The Mooney–Rivlin Model

We reserve further discussion of this problem for arbitrary isotropic incompress-ible hyperelastic materials until the next section, and now restrict attention to theMooney–Rivlin strain-energy density function, for which (2.5) has the form

W = c1(I1− 3)+ c2(I2− 3). (4.1)

In view of (2.7) and(2.10)1, the corresponding functionH(I) is simply constant;H(I) = 2(c1 + c2) > 0. Thus the equation (3.5) reduces to

A1(R,2) = 0, (4.2)

whereA1 is given by (3.6). On defining

α(2) ≡ (h(2))−1/2 and β(2) ≡ h(2)g(2), (4.3)

we can rewrite (2.4) as

f (R,2) = α

R

√R2+ β. (4.4)

On using Macsyma to substitute (4.4) forf into (3.6), carry out the necessarydifferentiations, and organize the resulting terms in powers ofR, (4.2) results in anequation of the form

1

4α3R3(R2+ β)2[B1(2)R

6+ B2(2)R4+ B3(2)R

2+ B4(2)] = 0, (4.5)

which must hold for allA < R < 1 andC < 2 < D. However, as the variouspowers ofR appearing in the brackets in (4.5) are linearly independent functions,(4.5) can be satisfied if and only if each coefficient ofRn, n = 0,2,4,6, is zero.Thus, four equations result,

Bi(2) = 0, i = 1,2,3,4. (4.6)

Analysis of the equations (4.6) is facilitated by defining

K1(2) ≡ α[α − 1

α3+ 2αβ + αβ

2β− αβ

2

4β2

],

K2(2) ≡ α[α

2− 2αβ + αβ

4β+ αβ

2

8β2

],

K3(2) ≡ α[αβ

2+ αβ

2

].

(4.7)

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 143

We remark that while (4.7) may seem to have come ‘out of thin air’, thesequantities proved useful in the direct approach to solving this problem for theMooney–Rivlin material. That is, asH is simply constant for this material model,with some effort, it is possible to integrate (3.1) and substitute the result into (3.2)directly, without resorting to the requirement (3.3). (See the paragraph followingequation (3.2)). However, as we eventually wish to interpret the result of the equa-tion (4.2) within a much more general context (see Section 5), we proceed withthe approach following from (3.3) instead. Recapping then, the final equation (3.5)reduces to (4.2) for the material model (4.1), and (4.2) then reduces to the fourequations (4.6), which we now begin to consider.

On settingB1 = 0, one obtains

4α4˙α + 4α3αα + 8(α4+ 1)α = 0, (4.8)

which, upon using (4.7) reduces to

4α3(K1+ 2K2) = 0. (4.9)

Recalling (4.3),α 6= 0, and thusB1 = 0 if and only ifK1+2K2 is constant, whereK1 andK2 are defined above in(4.7)1,2.

Upon careful observation, it may be seen that

βB3 = 4α3β3K1+ 2B4, (4.10)

and thus as bothB3 andB4 must be zero, the first term on the right-hand side of(4.10) must also be zero, resulting in the following two cases.

Case 1.β = 0.From (4.4),β = 0 impliesf = α(2) and thus, on recalling(4.3)1, the deformationfield (2.1) reduces to

r = R√h(2)

, θ = h(2), z = Z, (4.11)

which is exactly the deformation field (1.5), upon incorporating the incompressib-ility constraint, analyzed in [10–12] for the Mooney–Rivlin material (4.1). Uponsubstitutingf = h−1/2 into (3.6) and noting thatf ′ ≡ 0, one may obtain thegoverning third-order highly nonlinear ODE forh(2) ≡ y(2) from (4.2) to be

2y2˙y − 10yyy + 9y3 + 4y2(y2+ 1)y = 0. (4.11a)

As an alternative to the above equation, a first integral to the system of equationsderived in [12] is found by the authors which reduces the system to the first-orderhighly nonlinear ODE forρ(2) given by

ρ2− 2C1ρ4+ ρ2+ ρ6+ 2σρ4 ln ρ = 0, (4.11b)

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144 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

whereC1 andσ are constants of integration and, as relates to (4.11),ρ =√h, (h ≡

y). The numerical work of [12, 14] indicates that nonhomogeneous solutions tothe governing equation of the type (4.11) exist for Mooney–Rivlin and general-ized neo-Hookean power-law materials, respectively. The analytical work of [10,11] derives complicated implicit expressions for a solution of the form (4.11) forMooney–Rivlin materials, while [11] additionally provides a parametric solutionfor Varga materials. Given the existing body of work for this case, we shall not fur-ther pursue it here, and thus the remainder of this paper deals with the alternative,K1 = 0.

Case 2.K1 = 0, (β 6= 0).This case requiresK1 to be constant, and thus in view of the result of (4.9),K2 mustalso be constant. Thus, on assumingβ 6= 0 and requiringK1 = 0, from (4.10), weneed only then enforceB4 = 0 andB2 = 0 to satisfy the equations (4.6).

We first consider (within Case 2), the equationB2 = 0. It can be seen that

B2 = 8α3β(K1+ K2)+ B4

β2+ 8α3

[2K3α

α+ K3

], (4.12)

whereK3 is defined in(4.7)3. AsK1 andK2 are each constant in this case, andB4

must be zero,B2 = 0 if and only ifB4 = 0 and

2K3α

α+ K3 = 0. (4.13)

Equation (4.13) is easily integrated to give that

K3 = c3

α2, (4.14)

wherec3 is an arbitrary constant. Thus, if nonsymmetric deformations of the type(2.1), which are not of the form (4.11), are possible (for a Mooney–Rivlin material)what remains to be satisfied are the four equations

K1 = k1,

K2 = k2,

K3 = c3/α2,

and B4 = 0,

(4.15)

whereK1,K2,K3 are defined by (4.7),k1, k2, c3 are arbitrary constants, andB4 isthe coefficient of unity in the bracket in (4.5).

We first consider(4.15)3. From(4.7)3, settingK3 = c3/α2 results in the differ-

ential equation

β

2+ β2

8β= c3

α4, (4.16)

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 145

relatingα(2) andβ(2).Considering next equation(4.15)4, we first note thatB4 can be written as

B4 = 4α3β2

[K1

αβ + β

2

)+ β

α2

], (4.17)

and thus the term in brackets in (4.17) must be zero. On using(4.15)1, this givesthe equation

β

β= − 2k1αα

2+ k1α2, (4.18)

which is readily integrated to yield the following algebraic relationship betweenα

andβ,

β(2) = c2

2+ k1α2(2), (4.19)

wherec2 6= 0 is constant. Upon substituting (4.19) into (4.16), solving forα2, andorganizing the result in powers ofα, we obtain

α2 = 2c3k1

c2− 4

k1− α2+

(12c3

c2− 4

k21

)α−2

+24c3

c2k1α−4+ 16c3

c2k21

α−6. (4.20)

However, from(4.7)1,2 and(4.15)1,2,

k1+ 2k2 = αα + α2− 1

α2,

which can be integrated on multiplying byα/α to yield the alternative expressionfor α2,

α2 = −α2− 1

α2+ c1 lnα2+ c4 (4.21)

wherec1 ≡ k1+2k2 andc4 are constants. Thus, on equating (4.20) and (4.21), weobtain an algebraic equation inα(2) which must hold for all2 in our domain. Ondefining

c5 ≡ c3

c2,

this equation becomes

c1 lnα−2+ 16c5

k21

α−6+ 24c5

k1α−4+

(1+ 12c5 − 4

k21

)α−2

+2c5k1− 4

k1− c4 = 0 (4.22)

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146 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

for all 2 in the open interval. Equation (4.22) can be satisfied if and only if the(constant) coefficients of each of the linearly independent functions ofα(2) arezero, or ifα is constant. Consider the former. Then from theα−6 andα−4 terms,c5 must be zero, and thus by the above definition,c3 = 0, which implies, using(4.15)3, thatK3 = 0, which in turn requires, from(4.7)3 that

4β2(2)+ β2(2) = 0. (4.23)

Equation (4.23) holds if and only ifβ = 0, a contradiction to Case 2 (K1 = 0, β 6=0).Thus, we conclude that for solutions withβ 6= 0, α must be identically constant.

To conclude this section, we consider the possibility of solutions withα =constant. From(4.3)1, α = constant implies that the angular deformationh(2)is simply h = 2/α2 (neglecting rigid displacements), and from (4.4), the radialdeformation isr = α√R2+ β(2).

We first comment on the special case ofα = constant andβ = constant. In thisspecial case the deformation field (2.1) reduces to

r = α√R2+ β, θ = 1

α22, z = Z. (4.24a)

The deformation field (4.24a) is a special case of the nonhomogeneous, control-lable deformations possible in every homogeneous, incompressible, isotropic, hy-perelastic wedge known as Family 3 (see e.g., [1,20]), for which

r =√AR2+ B, θ = C2+DZ, z = E2+ FZ (4.24b)

with A(CF − DE) = 1. SettingA = α2, B = α2β,C = α−2,D = E = 0, andF = 1 recovers (4.24a). Note thatα = 1 in (4.24a) describes radially symmetricdeformations. We also remark thatα = constant,β = 0 (a special case of Case 1,(see (4.11))) results in the deformations field

r = αR, θ = 1

α22, z = Z, (4.25a)

which is seen to be a special case of the class of controllable deformations knownas Family 5 (see [1,22]) given by

r = AR, θ = B logR + C2, z = Z

A2C. (4.25b)

It thus remains to determine the possiblity of solutions of the form

r = α√R2+ β(2), θ = 1

α22, z = Z, (4.26)

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 147

with α constant andβ 6= 0. Thus (4.15) becomes

K1 = k1,

K2 = k2,

K3 = k3,

and B4 = 0,

(4.27)

whereK1,K2,K3 are defined by (4.7),k1, k2, k3 are as yet arbitrary constants, andB4 is the coefficient of unity in the bracket in (4.5). We first consider(4.27)4 andon settingB4 = 0 obtain the equation

4α3β2β

(K1

2+ 1

α2

)= 0. (4.28)

As none ofα, β, β are zero, (4.28) requires

K1 = − 2

α2(= k1), (4.29)

and thus definesk1 from (4.27)1. AsK1(2) was defined in(4.7)1 in terms ofα andβ, setting(4.7)1 equal to−2/α2 for α constant results in the ODE forβ(2),

β

β− β2

2β2= − 2

α4. (4.30)

On next equating(4.7)2 and(4.27)2, we obtain,

1

2

β− β2

2β2

)= 1− 2k2

α2. (4.31)

Thus (4.30) and (4.31) imply that

k2 = α2

2(α−4+ 1). (4.32)

Finally, on equating(4.7)3 and(4.27)3 and differentiating, we obtain

α2β

4

[2+ β

β− β2

2β2

]= 0,

which, upon substitution from (4.30), requires

α2β

2

[1− 1

α4

]= 0. (4.33)

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148 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

Thus, from (4.33) andα, β 6= 0, we obtainα = ±1, and in view of (4.26), it mustbe that

α = 1. (4.34)

With α = 1 now determined, (4.29) and (4.32) givek1 = −2 andk2 = 1, respect-ively. Having satisfied(4.27)4 via (4.29), we return to(4.27)1,2,3 and (4.7) withα = 1. The equations forK1 andK2 with α = 1 result in the same ODE forβ(2),(one could also see this from (4.30) and (4.31))

1

2

β− β2

2β2

)= −1, (4.35)

while the equations forK3 with α = 1 become

1

2

(β + β2

)= k3, (4.36)

with k3 still an arbitrary constant. Differentiation of (4.36) implies that

β

4

(2+ β

β− β2

2β2

)= 0. (4.37)

However, on noting that multiplying (4.35) byβ/2 (recall,β 6= 0) results in (4.37),we need only solve (4.36) and its solution will then automatically satisfy (4.37) andthus (4.35). In addition, as (4.26) requiresR2+ β(2) > 0 for all 06 A 6 R 6 1and 06 C 6 2 6 D < 2π , we takeβ > 0, and thus (4.36) implies thatk3 > 0 aswell. (We note thatk3 = 0 is inconsistent withβ 6= 0.) Thus on solving (4.36), weobtain

β(2) = k3[1± sin(22+ 2k4)], (4.38)

wherek4 is an arbitrary constant. Thus, with (4.26), (4.34), and (4.38), we obtainfor a Mooney–Rivlin material that the only possible (smooth) deformation field ofthe form (2.1) which is not included within the class of deformations (4.11) or theclass (4.24a) is the following nonhomogeneous, nonsymmetric deformation field

r =√R2+ k3[1± sin(22+ 2k4)], θ = 2, z = Z, (4.39)

wherek3 > 0 andk4 are arbitrary constants. The deformation (4.39) recovers (1.6),(1.7) witha = b = 1, which was first determined in [15]. Here, however, we haveshown that (4.39) is theonlysolution within the very general class of deformations(2.1) other than those of either the form (4.11), or (4.24a) withα andβ constant.The invariant (2.7) corresponding to (4.39) is

I = 3+ 2k23[sin(22+ 2k4)± 1]

k3R2[sin(22+ 2k4)± 1] ± R4, (4.40)

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 149

and the unknown functionq becomes

1

Hq(R,2) = ln

(r2

R2

)− k3

r2+ k5,

wherek5 is a constant of integration,r is given by(4.39)1, and we recall that forthe Mooney–Rivlin material,H = 2(c1 + c2) > 0 is constant. Thus, the unknownpressurep is determined from the above and (2.10) as

p(R,2) = 2(c1 + c2)

[ln

(r2

R2

)− k3

r2

]+ k6, (4.41)

wherek6(= 2(c1 + c2)k5 + 2c2) is an arbitrary constant. Note that, as occurredin [10–14], the pressure field is singular at the origin. We may then determine thecorresponding nonzero nominal stress components from (2.13) as

1HSRr(R,2) = −

[ln

(r2

R2

)− k3

r2+ k5

]rR+ R

r

1HSRθ(R,2) = ±

k3 cos(22+ 2k4)[ln(r2

R2

)− k3

r2 + k5

]Rr

1HS2r(R,2) = ±k3 cos(22+ 2k4)

Rr

1HS2θ(R,2) = −

[ln

(r2

R2

)− k3

r2+ k5

]Rr+ r

R

1HSZz(R,2) = −

[ln

(r2

R2

)− k3

r2+ k5

]+ 1+ 2c2

H(I − 3)

, (4.42)

whereH = 2(c1 + c2), r is given by(4.39)1, andI by (4.40).As mentioned earlier, [15] interpreted the solution (4.39) for the boundary-

value problem of a long half-cylindrical Mooney–Rivlin tube compressed betweenparallel plates. We wish here to briefly consider a different aspect of (4.39), thatof the possibility of cavitation in a Mooney–Rivlin cylinder. (See the fundamentalwork of Ball [23] on radially symmetric cavitation in solids; for a recent reviewarticle, see Horgan and Polignone [24]). While radially symmetric cavitation forincompressible nonlinearly elastic solids has become relatively well-understood,little is known beyond radial symmetry; see [8] for a discussion of this topic, as wellas for additional references on nonsymmetric cavitation deformations. As treatedin [23] and studied by numerous authors (see [24] and references cited therein), thesolution (1.3) for spherically symmetric deformations, written alternatively as

r(R) = (R3+ β3)1/3, θ = 2, ϕ = 8,

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150 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

and the solution (1.11) withα = 1, in particular,

r =√R2+ β, θ = 2,

with β > 0 constant can be used to describe radially symmetric cavitation inwhich a spherical or circular hole of radiusβ centered at the origin in the currentconfiguration arises due to sufficiently large tensile dead-loading on the boundary.In considering cavitation here, without loss of generality we choose the plus sign in(4.39) (and similarly in (4.40) and (4.42)) ask4 is an arbitrary constant. Therefore,on defining

9 = 2+ k4 (4.43)

and making use of trigonometric identities, we obtain

r(0,2) = √k3(sin9 + cos9) when sin9 + cos9 > 0

and

r(0,2) = −√k3(sin9 + cos9) when sin9 + cos9 < 0.

On considering the polar coordinates

x = r cos9, y = r sin9

we obtain thatr(0,2) can be described by(x +√k3

2

)2

+(y +√k3

2

)2

= k3

2when sin9 + cos9 > 0

and (x −√k3

2

)2

+(y −√k3

2

)2

= k3

2when sin9 + cos9 < 0

which corresponds to the formation of two circular holes of radius√k3/2, centered

at (−√k3/2,−√k3/2) and(√k3/2,

√k3/2) respectively, which touch tangentially

at a single point. To illustrate the deformation (4.39), we consider a long circularcylinder of undeformed radius unity and plot in Figure 1 a typical cross-sectionof the deformed configuration. The outermost curve depicts the deformation ofthe boundary of a cross-section,r(1,2), the innermost curve shows the doublecircular cavity r(0,2), and the remaining nine curves display the deformationcorresponding toR = 0.1 throughR = 0.9 in intervals of 0.1. Ask4 merely effectsa rigid rotation, we takek4 to be zero for simplicity in our further consideration ofthe cavitation problem.

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 151

Figure 1. Cavitation deformation ((4.39) withk3 = k4 = 1) of a cross-section of aMooney–Rivlin cylinder.

We first calculate the dead-loading condition on the boundaryr(1,2) requiredto produce the deformation

r =√R2+ k3[1+ sin(22)], θ = 2, z = Z, (4.44)

The appropriate boundary condition (see e.g. [21]) satisfies

STN = s(X),

whereS is the nominal stress tensor,N is the unit normal to the boundary of theundeformed configuration of the body,X is a point on this boundary, ands theapplied traction vector. Thus, on defining

r(2) ≡ r(1,2) = √1+ k3[1+ sin(22)], (4.45)

we calculate the necessary applied dead-load for this problem to be

s(2) = 2(c1+ c2)

{[1

r(k3+ 1)− r(2 ln r + k5)

]ER

+k3 cos(22)

r

(2 ln r − k3

r2+ k5

)E2

}, (4.46)

whereER,E2 are the appropriate orthonormal basis vectors associated with theundeformed configuration. We remark that the total surface traction on the outerboundary,∫ 2π

0s(2)d2, (4.47)

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152 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

is zero. The choice ofk4 = π/4, as employed in Hill [15], transforms (4.39) to

r =√R2+ 2k3 cos2(2), θ = 2, z = Z, (4.48)

and the resulting integrals associated with (4.47) are more easily evaluated withthis choice than withk4 = 0. Similarly, by symmetry, there is also no resultanttraction on the surface of the double cavity. This is analogous to the traction-freecavity surface boundary condition employed for radially symmetric cavitation. Inthat problem this condition is satisfied pointwise, whereas, we can only satisfy itin an integral sense for this problem. We note that [15] also could not satisfy point-wise stress boundary conditions for the particular problem considered there forthis deformation, and instead satisfied them in an integral sense. We note that thiscavitation solution can be seen to be an infinite energy solution as the total energyassociated with this equilibrium configuration is unbounded. Ball [23] reports ananalogous infinite energy radially symmetric cavitation solution in two dimensionsfor a neo-Hookean material.

5. Concluding Remarks and General Incompressible Hyperelasticity

We remark that the deformation (4.39) is neither homogeneous nor contained withinknown universal solutions. One way to see the latter is by employing Macsymato substitute (4.4) withα = 1 andβ given by (4.38), andI given by (4.40), into(3.7) and (3.8), the equations forA2(R,2) andA3(R,2), respectively. (Recall thatA1(R,2) = 0 was the subject of Section 4; we remark that upon using Macysmato substitute the appropriate expessions corresponding to (4.39) into (3.6) and sim-plifying, we have additionally verified that the result is zero). In view of (3.5), ifA2 andA3 are identically zero for this deformation, then the deformation would beuniversal, as theAi are independent of the material model. However, this is not thecase as neitherA2 norA3 is zero for the deformation (4.39). We note that, for thedeformation (4.24a),f = 0 and soI = 0, and one can easily see that from (3.7)and (3.8) thatA2 = A3 = 0; thus, as remarked earlier, (4.24a) is universal.

Recall from Section 4 thatA1 = 0 is compatible with (4.39), (4.24a), and (4.11)provided (4.11a) or (4.11b) holds. As (4.24a) is a universal solution, we shall notconsider it further, however it is interesting to do so for (4.39). Consider first (4.39)and an arbitrary isotropic incompressible material. AsA2 andA3 are nonzero whenevaluated at (4.39), we may then determine the most general class of materials forwhich (4.39) is a solution upon considering (3.5). SinceA1 is zero when evaluatedat (4.39), (3.5) then requires thatA2 (dH/dI )(I ) + A3 (d2H/dI 2)(I ) evaluated at(4.39), (4.40) must be zero for someH(I). One solution is that each of dH/dI andd2H/dI 2 are zero, thus recovering (4.39) as a solution for Mooney–Rivlin or neo-Hookean materials. The other possibility, dH/dI and d2H/dI 2 nonzero, producesan ODE for(dH/dI )(I ) resulting in

dH

dI(I ) = c e−

∫ A2A3

dI, (5.1)

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 153

wherec is a constant of integration. The above equation would determine the mostgeneral form ofH(I) (with dH/dI and d2H/dI 2 nonzero) for which the solution(4.39) ofA1 = 0 holds, provided the quantityA2/A3 can represented as a functionof I alone. Upon employing Macsyma to calculateA2 andA3 evaluated at (4.39),(4.40), we obtain

A2

A3= ±3R2

{k3[sin(22+ 2k4)± 1] ± R2

}2k2

3[cos2(22+ 2k4)] .

On recalling (4.40) and simplifying, the above becomes

A2

A3= 3

(I − 3)[1∓ sin(22+ 2k4)] , (5.2)

which is not expressible as a function ofI alone, and thus (5.1) cannot determinesuch a function. Thus, (4.39) is a solution of the equilibrium equations and incom-pressibility constraintif and only ifthe material is Mooney–Rivlin. We note that in[15] it was remarked that this solution was not applicable to isotropic incompress-ible materials more general than Mooney–Rivlin, but no subsequent justificationwas given. The general development of Section 3 here allows for a simple proof ofthis result.

In attempting to explore the possibility of nonsymmetric, nonhomogeneous,nonuniversal solutions to the equations of incompressible elastostatics, both hereand for spherical coordinate formulations (see [8, 9]), we have found the equationsto be prohibitively overdetermined. In the spherical case for a neo-Hookean mater-ial, the simplest incompressible nonlinearly elastic model, it was found that thereexist no (smooth) solutions beyond radially symmetric and a class of materialscontained within Family 5 for the general deformation field

r = f (R,2)R, θ = h(R,2), ϕ = 8. (5.3)

As derived here and in [15], a nonsymmetric, nonhomogeneous, nonuniversal solu-tion ((4.39)) to the equations of elastostatics for a Mooney–Rivlin material is pos-sible. Thus, it has been shown in this paper that(4.39) is the only smooth deforma-tion field possible beyond deformations which are radially symmetric, special casesof Families 3 or 5, or of the form (4.11) within the class

r = f (R,2)R, θ = h(2), z = Z, (5.4)

with h > 0 for the Mooney–Rivlin material. We remark that an analogous argumentto that presented here consideringh < 0 would recover the case ofa = b = −1of (1.6), (1.7). While the results in the present paper and in [8, 9] have been shownfor particular (simple) material models, the underlying difficulty has been the pro-hibitively overdetermined systems of equations which result, in most cases forcing

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154 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

the general deformation fields considered to reduce to much simpler, previouslystudied, ones. Any additional solutions must be contained within even less restrict-ive deformation fields than (5.3), (5.4). We expect that this difficulty will not bealleviated in considering materials more general than those studied here and in [8,9]. In fact, more complicated material models will likely result in more restrictions,not less, following from the general equations, and thus we infer that additionalsolutions within the classes (5.3) or (5.4) for more general incompressible materialmodelsW(I1, I2) is unlikely, except perhaps for some very special forms for thestrain-energy density. Indeed for deformations (5.4) and recalling (3.5), considerthe case ofH , dH/dI , and d2H/dI 2 linearly independent functions ofI , the casefor many material models in the literature. (Recall thatH is defined by(2.10)1, andI by (2.7), and that theAi depend solely on the deformation and are independentof the material model.) Then, to satisfy (3.5) without placing what would seemto be tremendously restrictive conditions on the deformation (for example, theAior their ratios must be able to be written solely as functions of the invariantI ) itfollows that

A1 = 0, A2 = 0, A3 = 0. (5.5)

As theAi are independent of the material, if (5.5) holds for a deformation field,then that deformation field is necessarily universal. However, Fosdick and Schuler[2] guarantees that new universal solutions will not exist within the frameworkconsidered here. Thusfor materials such thatH , dH/dI , andd2H/dI 2 are linearlyindependent functions ofI and deformation fields such that theAi or their ratioscannot be written solely as functions of the invariantI , isochoric deformationsof the form (5.4) must be contained within the fields categorized by universal de-formations.Given the above, new solutions to the equations of incompressibilityand equilibrium, if they exist, must lieoutsidethe class of deformations (5.4) forany such material. That is, under the above conditions, there existno additionalsolutions within the broad class of deformations (5.4) other than those of the form(1.11) or (1.12) foranysuch material. The only exception being (1.6), (1.7) whichholds for Mooney–Rivlin solids alone. It is worth remarking of course that lessrestrictive conditions than (5.5) are possible. For example, the material model forbiological tissue,

W = µ0

2γ[eγ (I1−3) − 1]

with µ0 > 0 andγ constants (see [1]), would result in the equation

A1+ γA2+ γ 2A3 = 0.

In conclusion, the methodology begun in [8, 9] and continued here has begun topush beyond the assumption of radial symmetry in the nonlinear theory of elasti-city, and has led to the development of a framework within which one may begin to

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 155

systematically consider deformation fields containing more than one independentvariable in cylindrical or spherical coordinates. This framework naturally producessystems of partial differential equations which hold on a fixed region of space andinvolve derivatives of dependent variables with respect to independent variables, asopposed to the corresponding traditional Cauchy stress formulation. Implementa-tion of our approach has led to the determination of all possible smooth solutions tothe equations of equilibrium within the general classes of deformation fields (5.3)and (5.4) for the neo-Hookean model, and has also given insight into such solutionsfor more general incompressible hyperelastic materials.

Appendix

The goal of this appendix is to derive, in a manner similar to that carried outfor spherical coordinates in [8], the component form referred to cylindrical polarcoordinates of the expression DivS≡ ∇ ·S, whereS represents the nominal stresstensor.

Let {X1, X2, X3} represent rectangular Cartesian Lagrangean coordinates withorthonormal basis vectors{E1,E2,E3} and originO, and let{x1, x2, x3} representthe corresponding rectangular Cartesian Eulerian coordinates ofX with orthonor-mal basis vectors{e1,e2,e3} and origino. We introduce the cylindrical coordinates

{X1, X2, X3} = {R cos2,R sin2,Z} (A.1)

and

{x1, x2, x3} = {r cosθ, r sinθ, z} (A.2)

with corresponding orthonormal basis vectors

ER = cos2E1+ sin2E2, (A.3)

E2 = − sin2E1+ cos2E2, (A.4)

EZ = E3, (A.5)

and

er = cosθe1+ sinθe2, (A.6)

eθ = − sinθe1 + cosθe2, (A.7)

ez = e3, (A.8)

for X andx respectively.

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156 DEBRA POLIGNONE WARNE AND PAUL G. WARNE

The nominal stresstensorS has component formSijEi ⊗ ej with respect tothe orthonormal bases{Ei} and {ej } in the reference and current configurationsrespectively, and theSij represent the physical components of the stress. Withrespect to the orthonormal bases (A.3)–(A.5) and (A.6)–(A.8),

GradS = ∇ ⊗ S

= [SRrER ⊗ er + SRθER ⊗ eθ + SRzER ⊗ ez + S2rE2 ⊗ er

+S2θE2 ⊗ eθ + S2zE2 ⊗ ez + SZrEZ ⊗ er + SZθEZ ⊗ eθ

+SZzEZ ⊗ ez]⊗ [ ∂

∂RER + 1

R

∂2E2 + ∂

∂ZEZ

], (A.9)

where

ER,R = E2,R = EZ,R = 0, (A.10)

ER,Z = E2,Z = EZ,Z = 0, (A.11)

ER,2 = E2, E2,2 = −ER, EZ,2 = 0, (A.12)

and

er,i = θ,ieθ for i = R,2,Z, (A.13)

eθ,i = −θ,ier for i = R,2,Z, (A.14)

ez,i = 0 for i = R,2,Z, (A.15)

where in (A.10)–(A.15) and subsequently, the comma notation denotes differen-tiation. Carrying out the operations in (A.9), using (A.10)–(A.15), the third-ordergradient tensor∇ ⊗ S can be factored to the form

∇ ⊗ S= CijkEi ⊗ ej ⊗ Ek for i, k = R,2,Z and j = r, θ, z, (A.16)

where the coefficientsCijk , of the 27 different basis tensorsEi⊗ej⊗Ek, are some-what complicated expressions which depend onR,2,Z, r, θ , z, S, and derivativesof S.

As discussed in [8],∇ · T, the divergence of a second-order tensorT, is definedaccording to a contraction between∇ andT applied to∇⊗T. For∇⊗S in (A.16),this contraction defines∇ · Ssuch that

∇ · S≡ Cijk(Ei · Ek)ej for i, k = R,2,Z and j = r, θ, z, (A.17)

and as the{Ei} of (A.3)–(A.5) are orthonormal, the dependence of (A.17) onCijkis reduced to its trace componentsCiji so that

∇ · S= Cijiej for i = R,2,Z and j = r, θ, z. (A.18)

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PLANE DEFORMATIONS IN INCOMPRESSIBLE NONLINEAR ELASTICITY 157

Upon calculating, theCiji are found to be such that

∇ · S ={SRr,R − θ,RSRθ +

1

R

[SRr + S2r,2 − θ,2S2θ

]+ SZr,Z − θ,ZSZθ

}er (A.19)

+{θ,RSRr + SRθ,R +

1

R

[SRθ + θ,2S2r + S2θ,2

]+ θ,ZSZr + θ,ZSZθ

}eθ (A.20)

+{SRz,R +

1

R

[SRz + S2z,2

]SZz,Z

}ez, (A.21)

where (A.19)–(A.21) denote the three terms as indicated. Thus the equilibriumequations, DivS= ∇ ·S= 0, for the nominal stressS, referred to cylindrical polarcoordinates, are given in component form by setting the coefficients ofer , eθ , andez in (A.19)–(A.21), respectively, to zero.

Acknowledgements

The authors are grateful to Prof. Stuart S. Antman, who suggested our study ofthe plane strain problem, and are indebted to a reviewer who guided us to thereferences [4–7, 10, 11, 15, 19] which were unknown to us in the first version ofthis manuscript. The authors also thank Ryan Miller of Maryville College who, aspart of an NSF REU program at the University of Tennessee, created many graphicsrelated to this problem, including Figure 1.

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