On the cavitation of a swollen compressible sphere in finite elasticity

International Journal of Non-Linear Mechanics 40 (2005) 307 – 321 www.elsevier.com/locate/nlm

On the cavitation of a swollen compressible sphere in finite elasticity夡 Thomas J. Pence, Hungyu Tsai∗ Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA Received 15 June 2004; accepted 16 June 2004

Abstract We consider a constitutive model for the treatment of swelling in the context of compressible hyperelasticity. It is developed as an extension of the conventional compressible theory by an additional dependence of the stored energy function on the local natural free volume due to swelling. For such a material model, we study the cavitation problem in spherical symmetry. A closed-form solution for cavitation is obtained for a class of materials characterized by two constitutive parameters (one for shear stiffness and the other for bulk stiffness). It is shown that the incompressible description of cavitation and swelling for an elastic sphere is then obtained in the limit wherein the bulk stiffness goes to infinity. In the absence of swelling this limit retrieves a neo-Hookean description for the materials under consideration. If the bulk stiffness is relatively large but finite, then a description for nearly incompressible cavitation and swelling is obtained. 䉷 2004 Elsevier Ltd. All rights reserved. MSC: 37K50; 74B20; 74E99; 74G05; 74A45 Keywords: Cavitation; Compressible elastic solids; Swelling; Bifurcation; Exact solutions

1. Introduction Cavitation of hyperelastic spheres has been an area of interest for a variety of mathematical and practical reasons as reviewed by Horgan and Polignone [1]. The mathematical issues of interest often relate to direct 夡 Dedicated to Cornelius O. Horgan on the occasion of his 60th birthday. ∗ Corresponding author. Tel.: +1-517-432-2181; fax: +1-517353-1750. E-mail addresses: [email protected] (T.J. Pence), [email protected] (H. Tsai).

0020-7462/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2004.06.004

methods in the calculus of variations for constructing minimizers to boundary value problems that require delicate arguments with respect to appropriately selected solution spaces and convergence norms [2,3]. The practical interest is due to the extent that void nucleation and growth is implicated in material failure [4]. This raises the possibility of analytical treatments that can predict locations conducive to fracture [5]. It is the purpose of this paper to consider certain basic issues with respect to cavitation phenomena in hyperelastic spheres that are also subject to swelling. Our use of the term swelling is meant to include any

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additional process leading to localized volume change. This includes not only the case of an absorbent solid in a liquid bath, which is perhaps the most common usage of the terminology, but also cases of solid–solid interactions, including those within more complicated structures such as biological tissue wherein a continuum patch is taken to span more than one biological cell. In a previous paper [6], we have examined cavitation phenomena of an otherwise incompressible hyperelastic sphere when swelling is present. The amount of ∗ swelling was denoted as v and was taken as prescribed. The deformation gradient F was then subject to the ∗ condition det F= v. The resulting treatment reduces to the conventional incompressible theory1 for the spe∗ cial case in which v is identically one. In the present paper, we discuss certain aspects of the corresponding generalization for an otherwise compressible hyperelastic material. The generalizations discussed here are straight forward in that more involved treatments can ∗ be envisioned, both with regard to the role of v in the mechanical constitutive response and with regard to ∗ the coupling of v to other physical processes. In order to make contact with the previous incompressible study we utilize a procedure by which an energy density suitable for the conventional incompressible theory can be appropriately generalized into one suitable for the conventional compressible theory. Further generalization of each framework to account for swelling is then given. With these links in place, it is possible to study the effect of swelling in both a compressible and an incompressible framework, in a unified fashion. These connections are described in Section 2. As in [6], attention is restricted to boundary value problems that preserve spherical symmetry. Boundary value problems describing cavitation of a solid sphere subject to swelling are formulated in Section 3. In order to treat this boundary value problem, a change of variable scheme considered by Horgan and Abeyaratne [7] in the conventional compressible treatment is utilized. This further restricts the form of the energy density whereupon a specific generalization of 1 We use the term “conventional” to denote the standard large

deformation theory of hyperelasticity, both compressible and incompressible. In particular, the term “conventional” does not refer to the infinitesimal strain theory of linear elasticity.

the neo-Hookean form is adopted for the subsequent study. Under the assumption of constant swelling, a closed form solution to the problem of cavitation is obtained in Section 4, thus clarifying the effect of constant swelling on cavitation. In Section 5, we study the role of material compressibility on the cavitation behavior with particular attention to the nearly incompressible regime. It is shown that the cavitation results for the otherwise incompressible treatment are retrieved by properly taking the limit wherein the bulk stiffness parameter tends to infinity. 2. Basic framework for the swelling treatment The framework considered here for the treatment of swelling is a basic extension of the framework for conventional hyperelasticity. Let X be a generic position vector in a reference configuration
X that is regarded as the state of an unloaded body prior to swelling. The load is described in the conventional way in terms of boundary tractions and body forces. For our purposes here, swelling is a generic notion ∗ of free volume change described here by a field v(X) that in the present treatment is regarded as given. Together, swelling and the application of load give rise to an invertible deformation y(X) that maps
X to the configuration
y . The total energy associated with both swelling and loading is taken to be IE = IEstore − IEload ,

(1)

where IEstore is the energy storage due to both swelling and elastic deformation, and IEload is the work due to any applied external tractions. The deformation y(X) is determined on the basis of minimizing (1) subject to conditions on applied boundary displacement. Let *
X be the boundary of
X , and let *
tX denote the subset of the boundary on which the traction vector t is specified. Then
IEload = t · y dAX . (2) *
tX

The stored energy IEstore is given in terms of a stored ∗ energy density W (F, v, X) as
IEstore = W dVX , (3)
X

T.J. Pence, H. Tsai / International Journal of Non-Linear Mechanics 40 (2005) 307 – 321 ∗

where F is the deformation gradient, F= *y/*X, and v ∗ is the local amount of swelling. In the present study v ∗ ∗ is taken to be a prescribed quantity, i.e. v = v(X). Also, we restrict attention to materials that are homogeneous so that there is no explicit dependence of W upon X, ∗ i.e. W = W (F, v). Minimization of IE with respect to suitably smooth fields y yields the field equation on
X ,

Div T = 0

(4)

and boundary condition Tn = t

on *
tX ,

(5)

where *W (6) T= *F is the Piola–Kirchhoff stress tensor and n is the unit normal to the boundary in the reference configuration. The governing equations (4)–(6) are thus formally identical to those of the conventional theory of compressible hyperelasticity. Any new effects due to ∗ swelling enter via the dependence of W upon v. In the same fashion the conventional theory of incompressible hyperelasticity can be extended so as to include the effect of swelling [6]. The constraint associated ∗ with this incompressible treatment is det F = v and a reactive pressure enters into the treatment. ∗ Thus the dependence of W upon v is a central issue in both an incompressible treatment and in a compress∗ ible treatment. Since v is the natural free volume, a basic physical idea in the compressible treatment here is that the energy minimal base state of the material should correspond to equiaxial expansion/contraction, ∗ 1/3

namely the state with F = v I. This generalizes the intuitive notion in the conventional compressible hyperelastic theory that the energy minimal base state should correspond to F = I. Introducing ∗ Fˆ = v

−1/3

F

(7)

thus allows us to express the stored energy density ∗ ˆ v). as W = W (F, The notion that equiaxial swelling provides an energy minimal base state corresponds to ∗ ∗ ˆ v) the condition W (F,
W (I, v). For isotropic material response, the dependence of W upon Fˆ can be mediated through the principal scalar invariants of ˆ = Fˆ T Fˆ to be denoted by Iˆj , j = 1, 2, 3. Clearly C

309

from (7), these are related to the scalar invariants of C = FT F, denoted by Ij , j = 1, 2, 3, via I1 I2 I3 Iˆ1 = 2/3 , Iˆ2 = 4/3 , Iˆ3 = 2 . (8) ∗ ∗ ∗ v v v Thus we may chose to further rewrite W = ∗ W (Iˆ1 , Iˆ2 , Iˆ3 , v) whereupon the Piola–Kirchhoff stress tensor is given by  *W *W ˆ 2 ˆ I+ (I1 I − C) T = 1/3 Fˆ ∗ *Iˆ1 *Iˆ2 v  *W ˆ ˆ −1 . (9) I3 C + *Iˆ3 ∗

In particular, for v =1 this type of treatment reduces to one for a conventional hyperelastic compressible material. In an otherwise incompressible setting, there is a standard generalization of any isotropic stored energy density from the conventional incompressible hyperelastic theory so as to incorporate swelling. For the purpose of this discussion let us denote by Wi and Wis the respective isotropic stored energy functions for the conventional incompressible hyperelastic theory (i) and the incompressible theory with swelling (is). Both Wi and Wis are measured with respect to the reference configuration
X which is prior to swelling. In addition, Wi can be expressed in the form Wi = Wi (I1 , I2 ) and the standard extension i → is so as ∗ to incorporate the effect of v is motivated by (8) and given by ∗

∗

∗ −2/3

Wis (I1 , I2 , v) = m(v)Wi (v

∗ −4/3

I1 , v

I2 )

(10)

∗

where m(v) > 0 with m(1) = 1. In view of (8) this i → is generalization is succinctly written as ∗ ∗ ∗ Wis (I1 , I2 , v) = m(v)Wi (Iˆ1 , Iˆ2 ). The multiplier m(v) provides the proportional change in stored energy ∗ ∗ as v varies. If m(v) is identically one, then the local swelling stores no additional energy even if the ∗ ∗ volume changes. If m(v) = v then the stored energy increases in proportion to the current local volume. ∗ ∗ ∗q A useful form for the function m(v) is m(v) = v with 0 < q < 1 in which case the energy storage lags the volume increase. The incremental elastic moduli measured from the locally swollen state with ∗ 1/3

F=v

∗

I then decrease with v, thus capturing observed

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softening behavior as described for example in [8]. In particular, the standard neo-Hookean energy density  (11) Wi (I1 , I2 ) = (I1 − 3), 2 ∗

∗ 2/3

with m(v) = v

in (10) gives

 ∗ 2/3 (I1 − 3v ), (12) 2 thus retrieving a model discussed by Treloar [9].2 Additional material models corresponding to the incompressible theory with swelling are discussed in [9–11]. Some of these models fall into the framework (10) and others do not. For the material with stored energy (12), an analysis of cavitation in solid spheres and the expansion of hollow spheres is given in [6]. Similar considerations can be invoked so as to generalize any isotropic stored energy density Wc of the conventional compressible hyperelastic theory to an isotropic stored energy density Wcs for a compressible material that includes the effect of swelling. Once again, Wc and Wcs are measured with respect to the reference configuration
X . Now Wc can be expressed in the form Wc = Wc (I1 , I2 , I3 ) and the standard extension c → cs is given by ∗

Wis (I1 , I2 , v) =

∗

Wcs (I1 , I2 , I3 , v)

∗ −2/3

∗

= m(v)Wc (v

∗ −4/3

I1 , v

∗ −2

I2 , v

I3 ),

(13)

∗

where m(v) is as in (10). In view of (8) this c → cs generalization is succinctly written as ∗ ∗ Wcs (I1 , I2 , I3 , v) = m(v)Wc (Iˆ1 , Iˆ2 , Iˆ3 ). The above discussion outlines standard procedures extending precursor stored energy densities from the conventional treatment so as to incorporate the effect of swelling in such a fashion that retrieves the conven∗ tional treatment as the special case v =1. There is also a useful procedure for extending a precursor stored energy density from the conventional incompressible theory into a stored energy density for the conventional compressible theory in a fashion that retrieves the conventional incompressible framework in an appropriate limit. This i → c extension is thus completely conventional and so independent of the notion of swelling. Again Wi = Wi (I1 , I2 ) and Wc = Wc (I1 , I2 , I3 ) and 2 Here v∗ corresponds to 1/v in Treloar [9] and his choice of 2

reference state is that of equiaxial expansion.

this i → c extension is given by 1/3

2/3

Wc (I1 , I2 , I3 ) = Wi (I1 /I3 , I2 /I3 ) + Kg(I3 ), (14) with constant K > 0. Here g(I3 ) provides an energy penalization in the event that I3  = 1 by requiring the following, Definition 1. A function g(z) : (0, ∞) → [0, ∞) is a penalty function for non-isochoric deformation if g(1) = 0, g (1) = 0, g (1) = 1/4, g (z) > 0 for z > 1, g (z) < 0 for 0 < z < 1, and g(z) → ∞ both as z → 0 and z → ∞. The significance of such a function is that if Wi (I1 , I2 ) is minimized at (I1 , I2 ) = (3, 3) on the attainable set of (I1 , I2 )-values for isochoric deformation (viz. [12]), then Wc (I1 , I2 , I3 ) as given by (14) is minimized at (I1 , I2 , I3 ) = (3, 3, 1) on the attainable set of (I1 , I2 , I3 )-values for more general deformations obeying I3 > 0. Further, the overall penalization for non-isochoric deformation scales with K so that the incompressible case with Wi (I1 , I2 ) is retrieved in an appropriate sense in the limit K → ∞. Finite values of K correspond to the bulk modulus of the infinitesimal strain theory by virtue of the normalization g (1) = 41 . For K ?1 the associated theory is said to be nearly incompressible. It therefore follows from the above considerations that if Wi (I1 , I2 ) is considered in the conventional incompressible theory, then the i → c → cs extension yields  ∗ ∗ 1/3 2/3 Wcs (I1 , I2 , I3 , v) = m(v) Wi (I1 /I3 , I2 /I3 )  ∗2 v +Kg(I3 / ) , (15) providing a stored energy density appropriate for the compressible theory with swelling. A conventional ∗ compressible treatment follows for v =1 and a nearly incompressible treatment is obtained for large K. In view of the fact that obtaining analytic closed form solutions to the governing equations of compressible hyperelasticity is a formidable task in general [13], one would anticipate certain challenges in treating boundary value problems in the present context for (15). Of particular interest in this regard are generalizations

T.J. Pence, H. Tsai / International Journal of Non-Linear Mechanics 40 (2005) 307 – 321

that involve the base neo-Hookean response (11). Then (15) gives   1 I1 ∗ W = m(v) −3 1/3 2 I 3

∗

∗2

+ m(v)g(I3 /v ),

(16)

where

:=

K 

(17)

is the ratio of bulk to shear modulus for the infinitesimal strain specialization in the absence of swelling. We shall adopt the form (16) for the ensuing analysis of cavitation in spherical geometries. Here and for the remainder of this paper we omit the subscripts on W since the meaning will be clear by context. If ∗

∗ 2/3

m(v) = v and  → ∞ in (16) then one anticipates results that correlate with the incompressible theory for (12) as discussed in [6]. 3. Spherical geometry formulation We consider a reference configuration given by the solid sphere
X = {(R, , ) : 0  R  Ro , 0  < 2, 0 } with associated orthonormal basis (ˆeR , eˆ  , eˆ  ) in the spherical coordinate setting. The sphere is subject to spherically symmetric ∗ ∗ swelling v = v(R), and to uniform radial tractions T in the reference configuration on the outer surface (R = Ro ). In view of the spherical symmetry of the swelling and the load, attention is restricted to spherically symmetric equilibrium deformations. Thus
y ={(r, , ) : r =r(R), ri  r(R)  ro ,  = ,  = } where ri = r(0) and ro = r(Ro ). The associated orthonormal basis is (ˆer , eˆ  , eˆ  ) and the equilibrium deformations are represented by y(X) = r(R)ˆer . The sphere remains intact in the absence of cavitation giving ri = 0. Cavitation occurs ifri > 0. The deformation gradient is dr r eˆ r ⊗ eˆ R + (ˆe ⊗ eˆ  + eˆ  ⊗ eˆ  ) dR R = eˆ r ⊗ eˆ R + (ˆe ⊗ eˆ  + eˆ  ⊗ eˆ  ),

F=

(18)

where we use the notation

=

dr , dR

=

r . R

(19)

311

Observe from (19) that deformations corresponding to constant require = and hence provide homogeneous deformation with F = I. More generally

= (R), = (R) and we let  

i : = R=0, o : = R=Ro ,   : = R=0, : = R=R , (20) i

o

o

and i and i are evaluated in the limit R → 0+ . ˆ are The principal scalar invariants of C 2 2 4 2 2 + 2 ˆ2 = 2 + , Iˆ1 = , I ∗ 2/3 ∗ 4/3 v v

4 2 Iˆ3 = 2 , (21) ∗ v and we define the reduced energy density function ∗

∗

w( , , v) = W (Iˆ1 , Iˆ2 , Iˆ3 , v), (22) with Iˆ1 , Iˆ2 , Iˆ3 as given in (21). For example, (16) gives   1 2 2/3 4/3 ∗ ∗ + 4/3 − 3 w( , , v) = m(v) 2 2/3

  4 2

∗ . (23) + m(v)g ∗2 v ∗

Returning to general w( , , v) it now follows from (7), (9) and (18) that

*w 1 *w eˆ r ⊗ eˆ R + (ˆe ⊗ eˆ  + eˆ  ⊗ eˆ  ). (24) 2 *

* The equation of equilibrium (4) reduces to the single equation     *w *w 1 d *w + = 0, (25) − 2 dR * R * *

which is in general a second order ODE for r(R). The uniform normal traction T on the outer surface R = Ro is a nominal traction, sometimes referred to as a dead load traction. The corresponding uniform normal Cauchy traction on r = ro is 1  = 2 T. (26)

o One boundary condition for (25) follows from (5) and (24) as  *w  = T. (27) * R=Ro T=

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T.J. Pence, H. Tsai / International Journal of Non-Linear Mechanics 40 (2005) 307 – 321

The other condition for the solution of (25) is sensitive to the absence or presence of a cavity. In the absence of a cavity this condition is r(0) = 0.

(28)

For the case of cavity formation, the center R=0 of the initially solid sphere is deformed to a spherical surface ri > 0 in the current configuration, so that formally i is infinite. In this case the variational argument, which had given (4)–(6) for smooth fields, now gives natural boundary condition R 2 Tn → 0 in the limit as R → 0+ . Consequently, for cavitation (28) is replaced by  lim

R→0+

1 *w

2 *

= 0.

R dr = . r dR

(30)

2

*w *w dv * w +2 − = 0, dR * v∗ * * *

2

=−

  2 2 * w * w dt + ( − ) + R dR

* * * 2

∗

+

dt + t (t − 1) dR

(29)

Applying this technique to the present situation in which swelling is present, it is found that (25) gives ( − )

∗

d * d * dv * d = + + , dR dR * dR * dR * v∗ 1 1 d

dt d = ( − ), =

+ ( − ). (32) dR R dR dR R

We shall henceforth restrict attention to swelling fields ∗ v(R) that are constant. Such fields and the more gen∗ eral case of piecewise constant v have been studied in the context of the incompressible theory in [6]. One may now rewrite (31) as R



The physical interpretation of (29) is that the Cauchy traction (but not necessarily the nominal traction) vanishes at the cavity surface. It is to be remarked that the field equation (25) is appropriate for spherical symmetry in general and so also governs boundary value problems for spherically symmetric deformations of hollow spheres, in which case R ∈ [Ri , Ro ] with Ri > 0. Spherically symmetric deformations of hollow and solid spheres in the conventional isotropic compressible theory involving a strain energy density W (I1 , I2 , I3 ) have been the object of much previous study with respect to inflation and cavitation. In addition to the work summarized by Horgan and Polignone [1], later work includes Lei and Chang [14], Murphy and Biwa [15], Shang and Cheng [16], Pericak-Spector et al. [17], and Murphy [18]. In general, closed form integration techniques are easily obtained only for special forms of the stored energy density W (I1 , I2 , I3 ). One such technique [7,19] involves the substitution t :=

where use has been made of the connections

(31)

* w w ( − ) * * + 2 ** − ** w 2

*

w

.

(33)

* 2

The right-hand side of (33) is a function of the field ∗ variables and , and of the now constant v. This is equivalent to being a function of the field variables ∗

and t, and the constant v. If the dependence of the right-hand side on and is only through t then (33) is separable, giving 1 1 dR = ∗ dt, R f (t, v)

(34)

where ∗

f (t, v) = − t (t − 1) 2

−

* w w ( − ) * * + 2 ** − ** w 2

*

w

*

.

(35)

2

It is in this separable circumstance that the substitution (30) allows for direct integration. Specifically R and r are related by parameter t in terms of functions R(t) and r(t) given by
t 1 R(t) = R¯ exp ∗ ds, ¯t f (s, v)
t s r(t) = r¯ exp (36) ∗ ds, t¯ f (s, v) ¯ r¯ obeying R¯ > 0 and with some constants t¯, R, r¯ > 0. Here (36)2 follows from the connection dr/r = t dR/R. Hence t also parametrizes the

T.J. Pence, H. Tsai / International Journal of Non-Linear Mechanics 40 (2005) 307 – 321

associated and via functions (t) and (t) given by
t s−1

(t) = ¯ exp (t) = t (t), (37) ∗ ds, t¯ f (s, v) ¯ where ¯ = r¯ /R. We now turn to consider the extent to which w of the form (23) admits to this treatment based upon the substitution (30). Such issues have been considered by Lei and Chang [14] for other strain energy forms in the setting of conventional hyperelasticity. To this end the right-hand side of (33) is required to be a function of t, say q(t). This requirement gives, after elimination ∗ √ of using 3 = v z/t and algebraic rearrangement, that g and q must obey 2z2 g (z) + zg (z) =

1 Q(t), 

(38)

with Q(t) =

t 4 − 10t 3 + 5t 2 + 4t − (t 2 + 5)q(t) . 9t 2/3 (2t 2 − 2t + q(t))

limit  → ∞ and this incompressible limit coincides with the model of Treloar [9] under the additional ∗

∗ 2/3

specialization m(v) = v . For stored energy density (42) the reduced energy density function (23) is  1 2 2/3 4/3 2 2 ∗ ∗ w( , , v) = m(v) + + ∗ 2 2/3

4/3 v    4 2

− 3 − 2 , (43) − ln ∗2 v whereupon it follows from (35) that ∗

f (t, v) = 9t (1 − t)

2(t + 1) + 3t 2/3 , 2(t 2 + 5) + 9t 2/3

(44)

which, as required, consolidates dependence on and into t = / . Further, we observe that this f is also ∗ independent of v so that we may write f (t) in place ∗ of f (t, v).

(39) 4. Cavitation for constant swelling

Since z and t are independent variables, it follows from (38) that 2z2 g (z) + zg (z) = A,

313

(40)

for some constant A. Integrating (40) and invoking g(1) = 0, g (1) = 0, g (1) = 1/4 gives A = 1/2 and √ √ g(z) = − ln z + z − 1, (41) whereupon it is verified that the above g satisfies the remaining conditions to be a penalty function: g (z) > 0 for z > 1, g (z) < 0 for 0 < z < 1, and g(z) → ∞ both as z → 0 and z → ∞. Accordingly, on the basis of (14) and (41) we henceforth consider   1 I1 ∗ W = m(v) −3 1/3 2 I3     1/2 I3 I3 1 ∗ + m(v) (42) ∗ − 2 ln ∗ 2 − 1 , v v as a stored energy density for a compressible neoHookean material that is generalized so as to include the effect of swelling. The incompressible neo-Hookean material, generalized so as to include the effect of swelling, will then be recovered in the

We henceforth confine attention to the case in which ∗ v is constant for all R and seek to determine the exterior loads T, if any, that permit cavitation solutions for material (42). For any such solutions we also seek to determine the relation between the cavity radius and the external load. In this regard we are guided by the methodology of Horgan and Abeyaratne [7] who considered similar issues for a Blatz-Ko material in conventional compressible hyperelasticity. In the present case we also seek to correlate the incompressible limit  → ∞ with results from the investigation of Pence and Tsai [6]. Homogeneous deformations are considered in the absence of cavitation. This gives = whereupon (25) and (28) are satisfied identically. The relation between external traction T and stretch as provided by (27) is then  

2 1 ∗ T = Thom ( ), Thom ( ) := m(v) ∗ −

v (45) ∗ 1/3

( ) > 0 and it is noted that Thom (v ) = 0 and Thom where the prime denotes derivative with respect to the

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T.J. Pence, H. Tsai / International Journal of Non-Linear Mechanics 40 (2005) 307 – 321

argument. Thus no external load is needed to sustain homogeneous deformation at the natural free volume, and positive (negative) traction on the outer surface is required for homogeneous deformation with local ∗ volume that is greater (lesser) than v. Further, as anticipated,  → ∞ gives the incompressible limit of ∗ 1/3

= v for all T. Cavitation solutions are described parametrically by (36) and (44) in terms of t on an interval ti  t  to provided that the following conditions are met:

Observe that the substitution t = u3 gives 
t 1/3  6u + 10u2 − 2u5 H (t) = du, 6 + 9u2 + 6u3 1

whereupon it can be shown that H (0) is bounded and that 1 2/3 5 t + t 2 9 + O(t 4/3 ) as t → 0.

eH (t)−H (0) = 1 +

R(ti ) = 0, r(ti ) > 0, R(to ) = Ro , R (t) > 0, r (t) > 0, the cavity surface supports no Cauchy traction, i.e. condition (29) holds, (v) condition (27) gives finite T.

Thus (48) provides the small t expansion   1 5

(t) = ¯ eH (0) t −5/9 + t 1/9 + t 4/9 2 9 7/9 + O(t ) as t → 0.

One may alternatively consider to  t  ti in which case the inequality in (iii) is reversed, but the technique under consideration here does not generate such solutions. In order to meet (i) it is found on the basis of (36) and (44) that ti = 0. This could have been anticipated from (30) since the cavitation singularity implies i → ∞. In order to meet (iii) it is required that 0 < to  1 and regardless of the value to it is convenient to take t¯ = 1 in (36). Consideration of (ii) then gives
1 1 ¯ R = Ro exp ds. (46) to f (s)

  1 *w  2 ¯ −3 −3H (0) ∗ = m(v) ∗ − e 3

2 * v 2/3 + O(t ) as t → 0,

(i) (ii) (iii) (iv)

In order to satisfy (iv) observe that the natural boundary condition (29) involves  1 *w 2 −10/3 −5/3 2 ∗ v) = m(

( − 2 ) 2 * 3

  1 +  ∗ − −2 −1 . (47) v Evaluation of (29) as R → 0 requires t → ti =0 in the present framework. Since (44) gives (t − 1)/f (t) → −5/9t as t → 0 it is convenient to write (t) as given by (37) in the form

(t) = ¯ t −5/9 eH (t) , 
t 5 s−1 H (t) = + ds. f (s) 9s 1

(48)

(49)

(50)

(51)

Using = t and substituting from (51) into (47) gives

(52)

so that (29) requires 

¯ =

∗ 1/3

2v 3

e−H (0) .

(53)

The value of r¯ now follows from (46) and (53). Consolidating results, it follows that: Cavitation solutions are described by
t 1 R(t) = Ro exp ds, f (s)   to1/3  ∗
1 v 2 1 r(t) = Ro e−H (0) exp ds  3 to f (s)
t s × exp ds, (54) f (s) 1 with 0  t  to to any 0 < to  1, provided that such to gives consistency with (v). Thus to parametrizes different potential cavitation solutions, each of which involves a radial deformation that is in turn described with the aid of parameter t ∈ [0, to ]. The different cavitation solutions involve different external dead loads T and different cavity

T.J. Pence, H. Tsai / International Journal of Non-Linear Mechanics 40 (2005) 307 – 321

radii ri . The external dead load T follows from (27) as  2 −5/3 −1 2 ∗ T = m(v)

o (to − 1) t 3 o  ∗ −1 + ( 2o v − to−1 −1 ) , (55) o where

o = ¯ exp


1

s−1 ds. f (s)

(56)

Define the normalized cavity radius a := ri /Ro . It then follows from (54) that  a=

∗ 1/3

2v 3

simple. Namely, it follows from (53)–(57) that   1/3   ∗ 1/3 ¯ = v∗ ¯    ∗ , r(R) = v r(R) ∗ , v =1

 ∗ 1/3 

o = v o  ∗

to−5/9 exp


0

to



5 1 − 9s f (s)

 ds.

(57)

It is to be remarked that both T and a can be regarded as parametrized by to , although we do not show this in the notation. Of particular interest is placing a into correspondence with either o or T by this parametrization so as to generate a graph of cavity radius a vs. either exterior stretch o or dead load T. Such graphs admit interpretation as a branch of cavitation solutions that bifurcates from the standard solution branch of homogeneous solutions a = 0 at the corresponding cavity nucleation value, say o = nuc or T = Tnuc . In the conventional compressible hyperelastic theory it is convenient to organize the discussion of cavitation around the former type of diagram. For example, in a Blatz-Ko material one obtains [7] nuc = 1.30874. In the conventional incompressible theory there is no deformation prior to cavitation, thus making nuc = 1, and so it is typical to instead organize the discussion about the latter type of diagram. For example, the neoHookean material (11) gives Tnuc = 5/2. In addition to T the other physical constants defining the problem under consideration are: , , Ro , and ∗ v. The dependence upon  is trivial since it simply provides a multiplier for T, and the dependence upon Ro is trivial since it simply provides a multiplier for r. The dependence upon  is highly non-trivial, since  is resident in f (t) (viz. (44)) and so is implicit in H (t) and in the integral expressions appearing in (53)–(57). This dependence in the large  regime corresponding to nearly incompressible behavior is described in ∗ the next section. Finally, the dependence upon v is

v =1

(58)

v =1

 ∗ m(v)  a=v , T = 1/3 T  . (59) ∗ ∗ v =1 v v =1 These simple scalings are identical to those obtained in the incompressible theory of spherically symmetric ∗ swelling for globally constant v. It is to be remarked however that the simple scalings (58) and (59) are a ∗ ∗ ∗ consequence of constant v. If instead v = v(R) then more complicated behavior will take place, as for example is suggested by the counterpart problem in the incompressible theory [6]. At this juncture it is useful to provide some brief technical commentary. The cavitation solutions are described in terms of integrals on either the full interval [0, 1] or some subinterval. The integrands are either of the general form 1/f (s) or s/f (s) where f is given by (44). Here f (t) > 0 for t ∈ (0, 1) with f (0)=f (1)=0 and f (t)=O(t) and f (t)=O(t −1) at t =0 and t =1. Thus the integrals with integrand 1/f (s) are singular for endpoints s = 0 and 1, but the respective endpoint singularities are removed if the integrand is of the respective form s/f (s) or (s − 1)/f (s). If the integrand is of the form 1/f (s) then simple closed form expressions result for the integral. For example, (57) gives  ∗ 1/3 v a= to−5/9 (1−to )1/3 × (2+2to +3to2/3 )1/3 . 3 ∗ 1/3

to

315

  a  ∗

(60) On the other hand, if the integrand is of the form s/f (s) then closed form expressions are rather complicated. Obtaining such a closed form expression is equivalent to obtaining a closed form expression for H (t). This follows by virtue of (48) and the closed form expressions for the integral with integrand 1/f (s). In order to keep the current discussion relatively unencumbered, we postpone display of such an expression until (74)–(79) at which point such presentation is central to the development. It is also useful to remark that other compressible stored energy densities in the conventional theory give rise to expressions

316

T.J. Pence, H. Tsai / International Journal of Non-Linear Mechanics 40 (2005) 307 – 321

for R(t) and r(t) defined in terms of integrals, all of which admit to evaluation in terms of relatively simple closed form expressions [7,19]. Our interest here in considering stored energy densities that specifically allows for retrieval of the conventional neo-Hookean ∗ case (11) in a particular limit (v =1,  → ∞) thwarts comparable simplifications. We now turn to consider the issue of cavity nucleation in terms of the appearance of an originally zero radius cavity that exhibits subsequent growth. Observe from (60) that a = 0 requires to = 1. Thus (56) with to =1 shows that such nucleation corresponds to o = ¯ , so that nuc = ¯ . However, to avoid excessive notation we shall maintain usage of ¯ in the sequel. Using both

o = ¯ given by (53) and to = 1 in (55), the nucleation dead load Tnuc is found to take the form ∗



Tnuc = m(v)

3

1/3

∗

2v

× eH (0)



 2 −3H (0) − . e 3 (61)

The expression (61) for Tnuc coincides with that given by use of (53) in (45) thereby confirming that T = Tnuc locates the bifurcation point at which cavitation solutions emerge from the homogeneous branch of solutions (45). Cavitation solutions with non-zero a are subsequently obtained by taking values to < 1 with T and o given by (55) and (56), respectively. It now follows on the basis of the technical discussion in the paragraph containing (60) that T as given by (55) is finite for all 0 < to  1. This confirms that (v) holds for these to , whereupon the final non-italicized provisional phrase from (54) is rendered spurious. Subsequent cavity growth can be described in terms of the relation between o and a. Observe that (56) gives d o to − 1 <0 = o dto f (to )

for 0 < to < 1,

(62)

for 0 < to < 1,

Fig. 2. Normalized cavity size a = ri /Ro as a function of the external dead load T for the same values of  as in Fig. 1. Asymptotes for small a are shown with dashed curves indicating that bifurcation from the a = 0 axis is to the left for sufficiently small . For large a, the (T , a) curves are given by (73).

modulus ratio  so that the graph of cavitation solutions in the ( o , a)-plane is always a monotonically increasing curve that emerges from the o -axis at o = ¯ . Fig. 1 shows a variety of such graphs for different values  > 0. Notice that the cavity nucleation stretch ¯ ∗ 1/3

while (60) using (44) gives a da =− <0 dto f (to )

Fig. 1. Normalized cavity size a = ri /Ro as a function of the external stretch o = ro /Ro for various  values. These values are chosen to correspond to = 0, 1/8, 1/4, 3/8, 1/2. Asymptotes for small a are shown with dashed curves. For large a, all ( , a) curves tend to a = o .

(63)

indicating that the cavity radius grows with exterior stretch (i.e. as the parameter to decreases from 1 toward 0). This behavior is general for all values of the

becomes v in the incompressible limit ( → ∞) with subsequent increase of the nucleation value ¯ as the material becomes relatively more compressible (i.e. as  decreases). This issue is explored in more detail in Section 5. The graph of cavitation solutions in the (T , a)-plane is given in Fig. 2 for a variety of  > 0. The scaled nucleation load Tnuc / increases as the relative compressibility decreases (i.e. as  in-

T.J. Pence, H. Tsai / International Journal of Non-Linear Mechanics 40 (2005) 307 – 321

creases). Each (T , a)-graph is found to be monotonically increasing for sufficiently large . However if  is sufficiently small, specifically if  < 1.47332, then the cavity radius is not a monotonic increasing function of load. It is to be remarked that the corresponding (T , a)graph in the conventional incompressible theory for homogeneous neo-Hookean spheres always involves monotonic increase, as is suggested by the large  limit in the present analysis. However, a similar transition from supercritical bifurcation to subcritical bifurcation has been observed in the conventional incompressible theory for spheres composed of a neoHookean material for certain cases in which either the neo-Hookean modulus is radially varying (viz. [20]) or if additional reinforcing is present so as to render the material transversely isotropic about the radial direction (viz. [21]). The qualitative change in the cavitation bifurcation suggests the possibility of stability mediated abrupt transitions in cavity radius. With respect to the present analysis it is of interest to establish the leading order asymptotic relations between a, o and T for both small and large cavity radius. To establish the behavior for small a requires consideration of to near 1. Observe from (55), (56), (60), (61) that  1/3  2/3  1 2 ∗ T = Tnuc + m(v) ∗ e−2H (0) 3 3 v

 × 2 − 3e−3H (0) ( + 2) (1 − to ) + O((1 − to )2 ), 

1¯

(1 − to ) + O (1 − to )2 , 3  ∗ 1/3  1/3 v 4 + (1 − to )1/3 a=  3

 + O (1 − to )4/3 .

o = ¯ +

(64) (65)

Fig. 3. A close-up plot of the cavity size a versus (T /Tnuc − 1) for the indicated  values. The critical value  =1.47332 separates the subcritical and supercritical bifurcation.

  ∗   2m(v)  4/3 3 1/3 T − Tnuc =  3 + 4 v∗ 2   H (0) 2 −3H (0) ×e −−2 a 3 +O(a 6 ), e 3 (68) and so gives appropriate asymptotes to the graphs of Figs. 1 and 2 near the bifurcation point. In particular the sign of the bracketed term in (68) determines whether the bifurcation in Fig. 2 is to the right or to the left. This term vanishes for  = 1.47332 thus determining the previously given transitional value. A close-up plot in Fig. 3 shows clearly this transition. To establish the asymptotic behavior when the cavity is large requires the consideration of to near 0. It is found that (55), (56), (60) together lead to ∗

∗

T = m(v)(5/3)(2/3)2/3 (/ v )2/5 to−1/9 + O(to5/9 ), ∗

(66)

Elimination of (1 − to ) between these expressions indicates that the initial cavity growth after nucleation is described by  ∗ 1/3   v 4 + 3 1/3 a= ( o − ¯ )1/3 

¯

 + O ( o − ¯ )2/3 ,

317

o = (2 v /3)1/3 to−5/9 + O(to1/9 ), ∗

a = (2 v /3)1/3 to−5/9 + O(to1/9 ).

(70) (71)

Elimination of to between these expressions indicates that a cavity with large radius is described by a = o + O( −1/5 ), o ∗

(67)

(69)

T = m(v)

    5 2 3/5  2/5 1/5 a + O(a −1 ), ∗ 3 3 v

(72) (73)

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T.J. Pence, H. Tsai / International Journal of Non-Linear Mechanics 40 (2005) 307 – 321

Fig. 4. Radial deformation r(R) normalized by Ro for  =5/3 and hence = 1/4. The curves correspond to increasing outer stretch values o / ¯ = 1/2, 1, 1.04, 1.2, 1.4, 1.7, 2. The nucleation stretch ∗1/3 for  = 5/3. is ¯ = 1.30155v

and so would provide the large a asymptotes to the graphs of Figs. 1 and 2. Each point on the graphs of Figs. 1 and 2 correlates with a particular to and represents a cavity solution. The displacement field r(R) follows from (54) for each such to . On this basis, Fig. 4 shows the deformation field r(R) for  = 5/3 due to a sequence of increasing o . 5. The effect of
and the nearly incompressible regime It is anticipated that the incompressible description of constant swelling cavitation as described in [6] should be recovered as  → ∞. In addition, a nearly incompressible regime is given by large but finite . More generally, varying  ∈ (0, ∞) provides for a family of material models and the question arises as to how the cavitation behavior varies within this family. One qualitative change has already been noted, namely the non-monotonic dependence of a on T for  < 1.47332. Alternatively, this family of materials can be parametrized by 3 − 2 := 6 + 2

(74)

for ∈ (−1, 1/2) since d /d > 0. It is easily verified ∗ by linearization of the v =1 uniaxial stress response about the undeformed state that corresponds to Poisson’s ratio from the isotropic theory of infinitesimal

strain elasticity. This interpretation for also follows from the relation between shear modulus, bulk modulus and Poisson’s ratio in the infinitesimal strain theory. Incompressibility is obtained in the limit as Poisson’s ratio → 1/2. Such infinitesimal strain constitutive constants can be extracted for any isotropic compressible hyperelastic material, for example the standard form of the Blatz-Ko material [7] gives = 41 and  = 53 . This accounts for the choice of  in Fig. 4. On the basis of (74) the non-monotonic dependence of a on T occurs for < 0.22325. In order to further clarify the cavitation behavior within this family of materials we first turn to a more detailed consideration of the function H (t) which is central to (53)–(61). It follows from (49) that H (t) = G(t 1/3 ) − G(1)  −1 + 2t 1/3 c2 + C2 tan−1 √ −1 − 4c3 2  −1 −1 + 2c − tan √ −1 − 4c3 −c + t 1/3 + C3 log −c + 1 −c − t 1/3 + t 2/3 c2 + C4 ln , −c − 1 + c2

(75)

with 1 1 3 G(u) = − 2 u + u2 − u3 , 4 4 9

(76)

C2 =

1 + c3  −1 − 4c3 , 3c6

(77)

C3 =

1 − c3 , 3

(78)

C4 =

−1 − 3c3 + 2c6 6c6

(79)

and  √ 1 c = c() = −  + ( 3 + 2 + 2)2/3   2 √ +( 3 + 2 − 2)2/3 .

(80)

Here c is the unique real root of the cubic 6 + 9u2 + 6u3 which appears in the denominator of the kernel for (49). In particular, c < − max{3/2, 1} for  > 0.

T.J. Pence, H. Tsai / International Journal of Non-Linear Mechanics 40 (2005) 307 – 321

319

∗

We are now in a position to verify that v =1 and  → ∞ retrieves the well known cavity nucleation results of the conventional incompressible theory associated with the neo-Hookean material. For large  one finds that c=−3/2−4−2 /9+64−5 /243+O(−8 ) and that (75)–(80) lead to   2 1 (1 − t 2/3 ) + t 2/3 H (t) = ln 3 3   11−12t 1/3 +t 4 1+t 1/3 −2t 2/3 − + 18 9 3t 2/3 + 2−2t 2/3     1 3 1/3 4 2 −1 × − tan t  9 3 2    3 1 − tan−1 2 3/2 + O(−2 )

as  → ∞.

(81)

In particular,

   5 1 2 2 1 2 1 + − H (0) = ln 3 3 6 9 3 3/2 −2 + O( ).

1/3

∗ 1/3 5

+v

1 + O(−3/2 ) 6

or equivalently ∗

¯ = v

1/3

∗ 1/3 5

+v

4



1 − 2 1+ 

as

(82)

 → ∞, (83)



1 (84) → . 2 It follows from (61) that the nucleation dead load admits the series expansion  ∗ ∗ m(v) 5 m(v) 2 2  Tnuc = 1/3  − 1/3  ∗ ∗ 2 3 3 1/2 v v + O(−1 ) as  → ∞, (85) + O (1 − 2 )3/2

as

or equivalently ∗

∗



as

1 → . 2

m(v) 2 m(v) 5 Tnuc = 1/3  − 1/3 ∗ ∗ 2 3 v v + O(1 − 2 )

This confirms the results of the counterpart problem ∗ 1/3 in the incompressible theory [6] that ¯ → v and ∗

Consequently, it follows from (53) that the nucleation stretch admits the series expansion ∗

¯ = v

Fig. 5. Plot of the nucleation stretch ¯ vs. Poisson’s ratio . The nucleation value ¯ = 1.30874 for the Blatz-Ko material ( = 1/4) is marked by a dot. The dash curve describes the asymptote for → 1/2 as given by (84).

1 − 2  1+ (86)

∗ 1/3

Tnuc → 5m(v)/(2v ) for  → ∞. In particular, the conventional neo-Hookean results ¯ → 1 and Tnuc → ∗ 5/2 are recovered for v =1,  → ∞. The effect of a small amount of compressibility is described by means of the corrections given by the second terms in expansions (83)–(86). Specifically compressibility gives rise to expansion of the solid sphere prior to cavity nucleation. In addition the cavity nucleation load Tnuc is depressed below the incompressible result. Fig. 5 shows how the nucleation value ¯ increases as decreases from the incompressible value ∗ 1 v 2 . For = 1/4 and =1 it is found that the nucleation stretch ¯ is approximately 1.30155 for the material model (42) under consideration, a value that is remarkably close to the Blatz-Ko nucleation value 1.30874 (marked by a dot in Fig. 5) given by Horgan and Abeyaratne [7]. The variation of Tnuc with is shown in Fig. 6. 6. Concluding remarks We have examined cavitation phenomena for a compressible hyperelastic sphere that is also subject to swelling. A constitutive framework (13) is established for the purpose of incorporating the effect of swelling into the conventional compressible theory. Although this framework is general, a special class of materials from the conventional compressible theory (14)

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T.J. Pence, H. Tsai / International Journal of Non-Linear Mechanics 40 (2005) 307 – 321

Fig. 6. Plot of the cavity nucleation dead load Tnuc versus Poisson’s ratio . The critical value = 0.22325 separates the subcritical and supercritical bifurcation shown in Fig. 3. The dash curve describes the asymptote for → 1/2 as given by (86).

provide the focus for this study. Specifically, this special class of compressible materials are generalizations from the incompressible theory so as to account for material compressibility. Generalization (14) of an incompressible energy density into a compressible one appears to have the advantage of isolating the effect of volume change into a single penalty function g(I3 ). 1/3 2/3 The factoring of I1 by I3 and I2 by I3 preserves the simplicity that is typically associated with the structure of an incompressible energy density form. In addition, it only requires the precursor incompressible energy density to be defined on (I1 , I2 ) values that are actually associated with isochoric deformation. Together, these generalizations give (15) which is then applied to (11) so as to give a compressible version of the neo-Hookean material that also incorporates the effect of swelling. A physically reasonable penalty function is obtained in (41) to complete the generalization from the incompressible to the compressible framework. This particular completion allows construction of closed form solutions describing cavitation under constant swelling using the change of variable technique (30) employed by Horgan and Abeyaratne [7]. It is shown that a proper limit retrieves previously obtained cavitation results appropriate to the neo-Hookean material both with and without swelling. Highly compressible materials in the family under consideration exhibit a qualitative change in cavitation response that has been associated with a snap-cavitation phenomena in conventional hyperelasticity.

It is to be noted that the effect of swelling is rela∗ tively trivial for the case of constant v considered here, ∗ wherein v simply participates in the determination of the scale factors given in (58) and (59). It is antic∗ ∗ ipated that radially varying swelling fields v = v(R) would give rise to other interesting effects, as have recently been obtained in the incompressible treatment. One such effect is that if the extent of swelling near the outer surface R = Ro is sufficiently larger than that in the interior, then cavitation can take place with no external load. The swelling of the outer shell in such a case acts as an effective loading device on the inner core [6]. It is also to be pointed out that the effect of swelling on the growth of a pre-existing micro-void, as for example discussed by Horgan and Abeyaratne [7] in the conventional compressible theory, can be treated by an appropriate extension of the methodology presented here. In this context, the effect of swelling on the growth of a pre-existing microchannel in the incompressible setting is discussed in [22]. References [1] C.O. Horgan, D.A. Polignone, Cavitation in nonlinear elastic solids: a review, Appl. Mech. Rev. 48 (1995) 471–485. [2] J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. R. Soc. London Ser. A 306 (1982) 557–610. [3] S. Muller, S.J. Spector, An existence theory for nonlinear elasticity that allows for cavitation, Arch. Ration. Mech. Anal. 131 (1995) 1–66. [4] A. Dorfmann, S.L. Burtscher, Experimental and computational aspects of cavitation in natural rubber, Plast. Rubber Composites 29 (2000) 80–87. [5] J. Sivaloganathan, S.J. Spector, On cavitation configurational forces and implications for fracture in a nonlinearly elastic material, J. Elasticity 67 (2002) 25–49. [6] T.J. Pence, H. Tsai, Swelling induced cavitation of elastic spheres, Math. Mech. Solids (2004), to appear. [7] C.O. Horgan, R. Abeyaratne, A bifurcation problem for a compressible nonlinearly elastic medium: growth of a microvoid, J. Elasticity 16 (1986) 189–200. [8] S.K. De, N.R. Aluru, B. Johnson, W.C. Crone, D.J. Beebe, Equilibrium swelling and kinetics of pH-responsive hydrogels: models, experiments, and simulations, J. Microelectromech. S. 11 (2002) 544–555. [9] L.R.G. Treloar, The Physics of Rubber Elasticity, third ed., Clarendon Press, Oxford, UK, 1975. [10] W.H. Han, F. Horkay, G.B. McKenna, Mechanical and swelling behaviors of rubber: a comparison of some molecular models with experiment, Math. Mech. Solids 4 (1999) 139–167.

T.J. Pence, H. Tsai / International Journal of Non-Linear Mechanics 40 (2005) 307 – 321 [11] M.C. Boyce, E.M. Arruda, Swelling and mechanical stretching of elastomeric materials, Math. Mech. Solids 6 (2001) 641–659. [12] K.N. Sawyers, On the possible values of the strain invariants for isochoric deformations, J. Elasticity 7 (1977) 99–102. [13] C.O. Horgan, Equilibrium solutions for compressible nonlinearly elastic materials, in: Ogden, Fu (Eds.), Nonlinear Elasticity: Theory and Applications, Cambridge University Press, Cambridge, 2001, pp. 135–159. [14] H.C. Lei, H.W. Chang, Void formation and growth in a class of compressible solids, J. Eng. Math. 30 (1996) 693–706. [15] J.G. Murphy, S. Biwa, Nonmonotonic cavity growth in finite, compressible elasticity, Int. J. Solids Struct. 34 (1997) 3859–3872. [16] X. Shang, C. Cheng, Exact solution for cavitated bifurcation for compressible hyperelastic materials, Int. J. Eng. Sci. 39 (2001) 1101–1117.

321

[17] K.A. Pericak-Spector, J. Sivaloganathan, S.J. Spector, An explicit radial cavitation solution in nonlinear elasticity, Math. Mech. Solids 7 (2002) 87–93. [18] J.G. Murphy, Inverse radial deformations and cavitation in finite compressible elasticity, Math. Mech. Solids 8 (2003) 639–650. [19] C.O. Horgan, Some remarks on axisymmetric solutions in finite elastostatics for compressible materials, Proc. R. Ir. Acad. Sect. A 89 (1989) 185–193. [20] C.O. Horgan, T.J. Pence, Void nucleation in tensile deadloading of a composite incompressible nonlinearly elastic sphere, J. Elasticity 16 (1989) 189–200. [21] D.A. Polignone, C.O. Horgan, Cavitation for incompressible anisotropic nonlinearly elastic spheres, J. Elasticity 33 (1993) 27–65. [22] T.J. Pence, H. Tsai, Swelling induced microchannel formation in nonlinear elasticity, IMA J. Appl. Math. (2004), to appear.