Exact solution for cavitated bifurcation for compressible hyperelastic materials

International Journal of Engineering Science 39 (2001) 1101±1117

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Exact solution for cavitated bifurcation for compressible hyperelastic materials Shang Xin-Chun a, Cheng Chang-Jun b,c,* a

Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, People's Republic of China b Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, People's Republic of China c Department of Mechanics, Shanghai University, Shanghai 200072, People's Republic of China Received 16 February 2000; received in revised form 16 June 2000; accepted 28 July 2000

Abstract In this paper, a new exact analytic solution for spherical cavitated bifurcation is presented for a class of compressible hyperelastic materials. The strain energy density of the materials is assumed to be a linear function of three strain invariants, which may be regarded as a ®rst-order approximation to the general strain energy density near the reference con®guration, and also may satisfy certain constitutive inequalities of hyperelastic materials. An explicit formula for the critical stretch for the cavity nucleation and a simple bifurcation solution for the deformed cavity radius which describes the cavity growth are obtained. The potential energy associated with the cavitated deformation is examined. It is always lower than that associated with the homogeneous deformation, thus the state of cavitated deformation is relatively stable. On the basis of the presented analytic solutions for the stretches and stresses, the catastrophic transition of deformation and the jumping of stresses for the cavitation are discussed in detail. The boundary layers of the displacements, the strain energy distribution and stresses near the formed cavity wall are observed. These investigations illustrate that cavitation re¯ects a local behaviour of materials. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Cavitation; Hyperelastic materials; Bifurcation; Exact solution

1. Introduction Cavitation phenomenon, the sudden formation of voids in solid materials, has long attracted much attention in the mechanics, applied mathematics, material science and engineering circles, *

Corresponding author. Tel.: +86-21-5638-0560; fax: +86-21-5638-0560. E-mail address: [email protected] (C.-J. Cheng).

0020-7225/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 0 ) 0 0 0 9 0 - 2

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because such phenomenon relates to the fundamental mechanisms of local failure and fracture of materials. The occurrence of cavities in the interior of rubber materials under tension loading has been observed by Gent and Lindley [1] in earlier experimental work. The nonlinear theoretical investigation in a solid mechanics framework for cavitation problems started with the work of Ball [2], in which cavitation is modelled as a class of bifurcation problems with discontinuous radial symmetric solutions in ®nite elasticity. An alternative interpretation for cavitation is the sudden rapid growth of a pre-existing microvoid [3,4]. For general incompressible hyperelastic materials, an explicit formula to determine the critical stress has been given by Ball [2], and in the special case of a neo-Hookean material the theoretical critical stress thus determined has good agreement with the experimental one by Gent and Lindley [1]. However, for compressible hyperelastic materials, the study of cavitated bifurcation is considerably harder due to the inherent nonlinearity of the problems. The qualitative analyses involved in existence, uniqueness and stability of cavitation solutions have been established by Ball [2], Stuart [5], Podio-Guidugli et al. [6] and Sivaloganathan [4,7]. On the other hand, for all compressible hyperelastic materials cavitation solutions do not always exist. For example, the analysis by Haughton [8] for membranes composed of a special compressible material has shown that cavitation cannot occur. The criteria for nonexistence of cavitation have been discussed by Meynard [9]. These rigorous analyses have shown that the bifurcation behaviour of cavitation is strongly dependent on the form of strain energy density. The other approach is to search for explicit solutions for cavitated bifurcation problems. As pointed out in literature (see e.g., [10]), even for radial symmertric cavitation including the cases of spherical symmetry and axisymmetry in plane strain, it is not possible in general to determine analytically, solutions describing cavitation for compressible materials. The diculty in obtaining of the solutions analytically comes from the integrability of a nonlinear second-order ordinary di€erential equation governing the radial symmetric deformation. In the particular case of a BlatzKo material, an explicitly analytic solution of the cavitation problem has been obtained by Horgan and Abeyaratne [3]. So far, only a very few solutions have been found by di€erent authors for related compressible hyperelastic materials by di€erent authors (e.g., [8,10±16]). Other aspects of the cavitation problems that have been investigated include the e€ects of material anisotropy [17,18], material inhomogeneity [19] and ®nite strain plasticity [20]. In addition, dynamic cavitation [21,22] and asymmetric deformation of cavitation [23,24] have been examined. A recent review on the cavitation problems has been provided by Horgan and Polignone [25], in which a more extensive list of references can be found. Recently, the dynamic cavitation problem in viscoplastic solids has been investigated by Badea and Predeleanu [26]. The nonmonotonic cavity growth has been discussed by Murphy and Biwa [27]). The purpose of this paper is to seek an exact analytical solution describing spherical symmetric cavitation for a class of compressible hyperelastic materials. Of course, obtaining of an exact analytical solution for a nonlinear problem is always dicult. The strain energy density of the materials is assumed to be a linear function of three strain invariants, as a ®rst-order approximation to the general strain energy density near the reference con®guration. Also, this expression for the strain energy density must satisfy several constitutive inequalities imposed to ensure that physical behaviour of the material is realistic. A parameter transformation approach (see [3,28,29]) is employed to obtain an exact analytical solution for the principal stretches and stresses. An explicit formula to determine the critical stretch and the bifurcation solution for the

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deformed cavity radius describing growth of the cavity are obtained. With the appearance of a cavity, an interesting feature of the radial deformation near the deformed cavity wall is the transition from extension into compression. When the prescribed stretch exceeds the critical value, there exist both the homogeneous and cavitated deformation solutions. In order to determine which of them is preferred for the material considered, the associated potential energies are compared. The result of the energy comparison shows that the potential energy associated with the cavitated deformation is always lower than that associated with the homogeneous deformation, and thus the cavitated deformation is relatively stable. The e€ect of cavitation on the distribution of strain energy is examined. The stress concentration near the deformed cavity wall is observed. For the compressible materials considered in this paper, the circumferential Cauchy stress in the cavity wall is ®nite, this feature is di€erent from that of several materials (e.g., for anisotropic material, see Fig. 4 in [25]). An expression to determine the concentration factor of circumferential stress in the cavity wall also is provided. The catastrophe and the boundary layer of the displacements and the distribution of strain energy and stresses near the cavity wall are discussed in detail and illustrated by ®gures. All these investigations show that cavitation is a local behaviour of materials. 2. Formulation Consider ®nite deformation for a solid sphere with radius R0 . The sphere is composed of a homogeneous, isotropic and compressible hyperelastic material. It is subjected to prescribed uniform radial stretch k on its surface. The points in the undeformed and the deformed con®gurations of the sphere are denoted by the spherical coordinates (R; H; U) and (r; h; /), respectively. The deformation is assumed to be radially symmetric: r ˆ r…R†;

h ˆ H;

/ ˆ U …0 6 R 6 R0 ; 0 6 H 6 2p; 0 6 U 6 p†;

…1†

where r…R† is a function to be determined. In the present case, the deformation gradient tensor F and the right stretch tensor U become F ˆ U ˆ diag…k1 ; k2 ; k3 †

…2†

and the principal stretches are k1 ˆ r0 …R†;

k2 ˆ k3 ˆ

r ; R

…3†

where … †0 ˆ d=dR… †. Also, to ensure that the map (1) is a smooth one-to-one map in (0; R0 ], the restriction on the deformation gradient det F > 0 …0 < R 6 R0 † must be satis®ed. It implies k1 ˆ r0 …R† > 0 …0 < R 6 R0 †:

…4†

The strain energy density per unit undeformed volume for the hyperelastic material is denoted by W ˆ W …k1 ; k2 ; k3 †:

…5†

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The Cauchy stresses are given by r1 ˆ

1 oW ; k2 k3 ok1

r2 ˆ r3 ˆ

1 oW k1 k2 ok2

…6†

and the equilibrium equation for spherically symmetric deformation (no body force) becomes dr1 2 ‡ …r1 r dr

r2 † ˆ 0:

…7†

Using (3) and (6) yields the nonlinear second-order ordinary di€erential equation for r…R† as follows:
 2 r00 …R† ‡ f1 R; r; r0 r0 R

r 2 ‡ f2 …R; r; r0 † ˆ 0 R R

…0 < R < R0 †;

…8†

where o2 W f1 ˆ ok1 ok2 def



o2 W ; ok21



oW f2 ˆ ok1 def

oW ok2



o2 W : ok21

The boundary condition on the surface R ˆ R0 is r…R0 † ˆ kR0

…k P 1†:

…9†

The condition at the centre of the sphere R ˆ 0 is one of the following two cases, that is Case I. The centre of the sphere is ®xed (no cavity appears) r…0† ˆ 0 or

…10†

Case II. A cavity is formed at the centre of sphere lim r…R† ˆ d > 0;

R!0‡

…11†

where the deformed cavity radius d is a constant to be determined. The surface of the cavity is assumed to be traction-free lim‡ r1 …R† ˆ lim‡

R!0

R!0

 2   R o dr r r ; ; W ˆ 0: r ok1 dR R R

…12†

Clearly, one trivial solution to the boundary value problems (8)±(12) together with the restriction (4) is r…R† ˆ kR

…k P 1; 0 6 R 6 R0 †:

…13†

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It corresponds to a homogeneous deformation of the sphere in radial extension and satis®es the condition (10). The present problem now is to search for a solution, called the cavitation solution, to the nonlinear boundary value problems (8), (9) and (12) with the restrictions (4) and (11). In fact, the cavitation solution bifurcates from the homogeneous one when the prescribed stretch k exceeds a certain critical value kcr . Thus, there are two principal problems here: (i) how to determine a bifurcation point kcr for the cavity nucleation and (ii) how to obtain a cavitated bifurcation sodef lution r ˆ r…R; k† for k P kcr , especially, to ®nd solution d ˆ d…k†ˆ r…0; k† describing the growth of cavity. Physically, the cavitation phenomenon re¯ects a material instability. As pointed out in the literature (Horgan and Abeyaratne [3]), such instability cannot be analysed mathematically by using a linearized approach because the problem has an inherent nonlinearity. Actually, no information about the bifurcation point kcr is obtained from the linearized problem around the trivial solution (13) associated with the nonlinear boundary value problems (8), (9) and (12) [2]. Here, it is worth mentioning that the conditions at the centre R ˆ 0 for the cavitation solution (11) and (12) are totally di€erent from the condition (10) for the trivial solution. For this reason, in general, it is not possible to determine the bifurcation point (critical external stretch or load) by solving linearized problem near the trivial solution, though the linearized approach is much used in the instability of structures such as buckling of compressed rods and plates. On the other hand, for all compressible hyperelastic materials cavitation solutions do not always exist [8,9]. Thus, it is more dicult to study the above problem of cavitated bifurcation. In this paper, we seek a new explicit analytic cavitation solution for a particular class of compressible materials.

3. Strain energy density The strain energy density of the isotropic materials from (5) can be rewritten as W ˆ W …U† ˆ W …j1 ; j2 ; j3 †;

…14†

where j1 , j2 and j3 are the invariants of the stretch tensor U and they are de®ned by j1 ˆ i1 ˆ tr U ˆ k1 ‡ k2 ‡ k3 ; i2 1 1 1 j2 ˆ ˆ tr U 1 ˆ ‡ ‡ ; k1 k2 k3 i3 j3 ˆ i3 ˆ det U ˆ k1 k2 k3 in which i1 , i2 and i3 are the principal invariants of U. Consider a class of strain energy densities for compressible materials which has the form W ˆ C1 …j1

3† ‡ C2 …j2

3† ‡ C3 …j3

1†;

where C1 , C2 and C3 are material constants.

…15†

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Since the strain energy of material vanishes in the reference con®guration (undeformed state), that is, W …3; 3; 1† ˆ 0, the expression (15) is just the linear part of the Taylor expansion of W around the invariants j1 , j2 and j3 . Thus, the expression (15) can be regarded as a ®rst-order approximation to the general strain energy density (14) near the reference con®guration. The Cauchy stresses from (6) and (15) have the expressions  1 C1 k i ri ˆ k1 k2 k3

C2 ki

 ‡ C3

…i ˆ 1; 2; 3†:

…16†

Suppose that the undeformed state is a natural state (stress-free state). This leads to the restriction on the material constants: C1

C2 ‡ C3 ˆ 0:

…17†

Further, since in®nitesimal strain is a limiting case of the ®nite one, in the Taylor expansion of the strain energy density (15) in terms of the Green strain tensor E, retaining only the linear and second-order terms gives W ˆ …C1

1 C2 ‡ C3 †tr E ‡ C3 …tr E†2 2

1 …C1 2

3C2 ‡ 2C3 †tr E2 :

It should be identical to the strain energy density in the linear theory of elasticity [30] h m i 2 …tr E† ‡ tr E2 ; W ˆl 1 2m

…18†

…19†

where l and m are the shear modulus and Poisson ratio in the state of in®nitesimal strain, respectively. Hence, comparing (18) with (19), the material constants in the strain energy function (15) can be de®ned by C1 ˆ l

1 1

3m ; 2m

C2 ˆ l

1 m ; 1 2m

C3 ˆ l

2m : 1 2m

…20†

In general, the strain energy function W must be nonnegative for any positive stretches k1 ; k2 and k3 . Obviously, if the restriction (4) holds, then from (10) and (11), both homogeneous and cavitation solutions satisfy r…R† > 0 …0 < R 6 R0 †. Thus, all stretches are positive. For the compressible hyperelastic material (15), it is not dicult to prove that the strain energy density W …k1 ; k2 ; k3 † has a unique local minimum point (1, 1, 1) which corresponds to the natural state and it is always nonnegative if all material constants satisfy following sucient conditions: C1 > 0;

C2 > 0;

C3 > 0:

…21†

This requires l > 0;

1 0
…22†

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In addition, certain constitutive inequalities should be imposed on the strain energy density, in order to ensure that the physical behaviour of the material is realistic. For an isotropic material, a simple tensile load produces a simple extension and the Cauchy stress tensor r and stretch tensor U share the same principal axes. It is reasonable to assume that the principal stress in certain principal direction must be increased if the stretch in this direction is increased [31,32]. In mathematical terms, this means that the following tension-extension inequalities should be satis®ed ori >0 oki

…i ˆ 1; 2; 3 no sum†:

…23†

Substitution of (16) into the inequalities (23) yields C2 > 0. Other restrictions are the Baker±Ericksen inequalities …ri

rj †…ki

kj † > 0

if ki 6ˆ kj …i; j ˆ 1; 2; 3

no sum†:

…24†

They are based on the assumption that for an isotropic material the greatest (least) principal stress ri occurs in the direction corresponding to the greatest (least) stretch ki [31,32]. By introducing (16) into the inequalities (24) and using (4), it turns out that the inequalities (24) hold if C1 > 0 and C2 > 0. Moreover, for hyperelastic materials, the following restriction called the behaviour of the strain energy density for large strain [30,33] must be required W …k1 ; k2 ; k3 † ! ‡1 when ki ! 0‡ or ‡ 1;

…25†

where ki is any of k1 ; k2 ; and k3 while the others are ®xed in (0; 1). Recalling the expression for the strain energy density (15), the restriction (25) is satis®ed if the conditions (21) hold. Consequently, if the conditions (21) are imposed, then the constitutive restrictions (23)±(25) are automatically satis®ed. 4. Exact solution For the particular class of strain energy densities (15), Eq. (8) reduces to r00

1 0 Rr03 r ‡ 2 ˆ0 R r

…0 < R < R0 †:

…26†

In order to solve this equation explicitly, it is useful to introduce the variable transformation [3,28,29] R ˆ R…t†;

r ˆ r…t†;

tˆ

k1 Rr0 : ˆ k2 r

…27†

Thus, (26) can be transformed to the system of ®rst-order ordinary di€erential equations for R…t† and r…t† in parametric form:

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dR…t† R…t† ˆ ; dt t…1 t†…2 ‡ t†

dr…t† ˆ dt …1

r…t† : t†…2 ‡ t†

…28†

Also, the boundary condition (9) can be rewritten as R…t0 † ˆ R0 ;

r…t0 † ˆ kR0 ;

…29†

where t0 is a constant to be determined. Solving (28) with the initial condition (29), the solution is found to be  1=2  1=3  1=6 t 1 t0 2 ‡ t0 R…t† ˆ R0 ; t0 1 t 2‡t 1=3  1=3  1 t0 2‡t r…t† ˆ kR0 : 2 ‡ t0 1 t

…30†

Obviously, using (4) from (27) one ®nds t > 0. The solution (30) has a singularity at t ˆ 1. From (27) and (8) the case of t ˆ 1 corresponds to trivial solution (13). Also, if t > 1 then (28) results in dR=dt < 0 since R P 0, and the solution (30) implies that limt!0 R…t† ˆ 0, however, this leads to contradiction that R…t† 6 0. Thus for the cavitation solutionit is required that 0 6 t < 1. As a result, in view of 0 6 R 6 R0 and (30), the range of variable t can be determined as 0 6 t 6 t0 < 1:

…31†

Further, the principal stretches have the expressions  t 1=2  2 ‡ t 1=2 r…t† 0 ; ˆk k2 …t† ˆ R…t† 2 ‡ t0 t

k1 …t† ˆ tk2 …t†:

…32†

Substituting (32) into (16), one obtains the solution for the Cauchy stresses C1 t r1 …t† ˆ 2 k t0 C1 r2 …t† ˆ 2 k t0

 

2 ‡ t0 2‡t 2 ‡ t0 2‡t

 

C2 k4 t02 C2 t k4 t02

 

2 ‡ t0 2‡t 2 ‡ t0 2‡t

2 ‡ C3 ;

…33†

2 ‡ C3 :

At the centre of the sphere t ˆ 0, the conditions (11) and (12) become  d ˆ lim r…t† ˆ kR0 t!0

r1 …0† ˆ

1 t0 1 ‡ t0 =2

1=3 ;

 2 C2 2 1 ‡ ‡ C3 ˆ 0: t0 4k4

…34†

…35†

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Since d ˆ 0 and k ˆ kcr at the critical state (34) implies that t0 ˆ 1 and (35) leads to the critical stretch (Bifurcation point)  kcr ˆ

9C2 4C3

1=4

  9 1 ˆ 8 m

1=4 :

1

…36†

Combining (36) with (35), one ®nds that t0 …k† ˆ

2 3…k=kcr †2

1

…k P kcr †:

…37†

Finally, from (34) and (37) the solution for the growth of the cavity is obtained as  d…k† ˆ kR0 1

kcr k

2 !1=3 …k P kcr †:

…38†

In summary, for the class of strain energy densities (15), a new exact solution for cavitated bifurcation has been explicitly given by (30)±(33) together with (36)±(38). From (36) the critical stretch kcr depends only on material parameter (Poisson ratio) m. The relation between kcr and m is shown in Fig. 1. The critical stretch kcr decreases monotonically with increasing Poisson ratio m. It should be emphasised that (38) gives a closed and quite simple form of solution for the deformed cavity radius d…k†. Also, the solution (38) is slightly di€erent from Horgan's solution d…k† ˆ kR0 …1 …kcr =k†3 †1=3 …k P kcr † which is associated with a class of strain energy densities called the generalized Varga materials (see formula (29) in [10] or the formula (3.31) in [25]). The bifurcation curves for the deformed cavity radius d are shown in Fig. 2. The curves of radial

Fig. 1. Variation of the critical stretch kcr with material parameter m.

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Fig. 2. Bifurcation curves of the cavity radius d…k† for various values of m. def

displacement u…R; k†ˆ r…R; k† R for di€erent values k in Fig. 3 are produced by (30) and (37). In spite of forming a cavity in the critical state k ˆ kcr , no jumping of displacement occurs. In other words, when the prescribed stretch k increases and exceeds the critical value kcr , the state of homogeneous deformation continuously bifurcates into the state of cavitated deformation. However, it should be noticed that the transition for the slope of radial displacement u…R† is not continuous. As shown in Fig. 3 the cavitated displacement u…R† has positive and negative slopes in the di€erent domains of R. The domain of positive or negative slope of u…R† corresponds to one of

Fig. 3. Curves of radial displacement u…R† for various values of k.

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extension and compression, respectively. More importantly, before the cavity forms (k < kcr ) the deformation of the whole sphere is extension, while after the cavity forms (k > kcr ) the deformation near the cavity becomes compressive. Such a situation is clearly di€erent from that of the homogeneous deformation.

5. Energy analysis As shown in the discussion in the preceding sections, both the homogeneous and cavitated deformation solutions exist when the prescribed stretch k > kcr . However, for two di€erent deformations, the one which minimizes the energy would be stable. Thus, it is required to compare the potential energy for the two types of deformations. Since the displacement is prescribed on the outer surface, the total potential energy is def

E…k†ˆ

Z

Z V

W dV ˆ 4p

R0 0

R2 W dR:

…39†

For the case of spherically symmetric deformation, applying the equilibrium equation leads to [10]   d 3R W ˆ R3 W dR 2

…k1

oW k2 † ok1

 :

As proved by Sivaloganathan [7] for the cavitated solution the following limit, denoted by E0 , vanishes. In fact, for the strain energy density (15), it is directly veri®ed that    oW 3 …k1 ; k2 ; k2 † E0 ˆ lim‡ R W …k1 ; k2 ; k2 † …k1 k2 † R!0 ok1   C2 3 3 ‡ C3 k2 …t† ˆ 0: ˆ lim‡ R …t† t!0 t2 k2 …t† def

…40†

Thus, the total potential energy (39) can be rewritten as  4 3 E…k† ˆ pR W …k1 ; k2 ; k2 † 3

…k1

 oW k2 † …k1 ; k2 ; k2 † : ok1

…41†

For the cavitated solution for the material (15) the above expression reduces to  4 3 Ec …k† ˆ pR0 C1 …ac ‡ 2k 3

 3† ‡ C2

1 2 ‡ ac k

 3 ‡ C3 …ac k2

1†

…ac

k†…C1

 C2 2 ‡ C3 k † ; a2c …42†

where

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2k

def

ac …k†ˆ k1 …t0 † ˆ

3…k=kcr †2

1

6k

…k P kcr †;

while for the homogeneous solution for the material (15) the energy expression (41) becomes   4 3 2 3C2 Eh …k† ˆ pR0 …k 1† …43† ‡ C3 …k ‡ 2† : 3 k The di€erence of the energy between the cavitated and homogeneous solutions is Ec …k†

Eh …k† ˆ

C2 3pR30 k



k kcr

!2

2 1

:

…44†

Obviously, it follows since C2 > 0 that the energy corresponding to the cavitated solution is strictly less than that of homogeneous solution for the same prescribed stretch k > kcr , namely Ec …k† < Eh …k†

…k > kcr †:

…45†

This result ensures that the present cavitated solution is relatively stable and it is energetically favourable to form a spherical cavity within spherically symmetric deformation. The energy curves corresponding to the homogeneous and cavitated solutions are shown in Fig. 4. Actually, it gives bifurcation curves of total potential energy E…k†. It is easy to verify from (44) that dEc dEh ˆ dk dk

…k ˆ kcr †:

…46†

It indicates that the two curves in Fig. 4 have the same tangent at the bifurcation point k ˆ kcr .

Fig. 4. The curves of the total potential energy.

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Next, in order to analyse the distribution of strain energy in the radial direction, the true density of strain energy is introduced as def

wˆ

W W ˆ ; det F k1 k22

…47†

which denotes the strain energy density per unit deformed volume. Using solutions (30)±(32) and (15) the dimensionless true density of strain energy w=l along the dimensionless undeformed radial coordinate R=R0 is shown in Fig. 5. It should be noted that the distribution of strain energy per unit deformed volume has a catastrophic transition when the prescribed stretch k increases and exceeds the critical value kcr . Such a transition is a jump from a homogeneous distribution into a nonhomogeneous one. However, the transition of total potential energy is continuous at the bifurcation point k ˆ kcr (see Fig. 4). Also, the asymptotic expansion of solutions (30) and (32) near t ˆ 0 (R ˆ 0) shows that k1  O…R† and k2  O…1=R† when R ! 0‡ . Although the strain energy per unit undeformed volume W (nominal density of strain energy) from (14) has the singularity of O…1=R† at R ˆ 0, the true density of strain energy is ®nite since w  O…1† at R ˆ 0. In addition, Fig. 5 shows that the cavitation of material is accompanied with the concentration of strain energy, and the e€ects of cavitation on the distribution of strain energy are concentrated in a narrow layer near the surface of the cavity. The boundary layer of the distribution of strain energy illustrates that cavitation is a local behaviour of materials.

6. Catastrophe and concentration of stresses For the homogeneous deformation, recalling (3), (13) and (16), the stresses have the form

Fig. 5. The distribution of the strain energy.

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 1 r1h …R† ˆ r1h …0† ˆ r2h …R† ˆ r2h …0† ˆ 2 C1 k

C2 k2

 ‡ C3 :

…48†

For the cavitated deformation, using (33) and combining with (36) and (37), one obtains the stresses at the boundary of the cavity R ˆ 0 as r1c …0† ˆ 0;

r C3 r2c …0† ˆ C1 ‡ C3 > 0 C2

…k P kcr †:

…49†

Clearly, the radial stress r1 and circumferential stress r2 at R ˆ 0 are dependent on the prescribed stretch k for the homogeneous deformation. However, they have catastrophic jumping if k increases and exceeds the critical value kcr . The discontinuity of stresses r1 …0† and r2 …0† associated with k is exhibited in Fig. 6. In addition, for the cavitated deformation the distribution of stresses r1 …R† and r2 …R† associated with the undeformed radial coordinate R can be obtained by (30), (33) and (37), while the distribution of stresses r1 …R† and r2 …R† for the homogeneous deformation is given by (48). Thus, when a cavity is formed, as shown in Fig. 7, the stresses r1 …R† and r2 …R† have an obviously catastrophic transition from a homogeneous distribution …k 6 kcr † into nonhomogeneous one …k > kcr †. It should be noted that the circumferential stress in the deformed cavity wall r2 …0† is ®nite for the compressible materials considered here, but r2 …0† is in®nite for certain materials (e.g., for anisotropic materials, see Fig. 4 in Horgan and Polignone [25]). Simultaneously, the concentrations of both radial stress r1 …R† and circumferential stress r2 …R† appear near the cavity wall and will be reduced when k increases. Therefore, the concentration of stresses for cavitation is a local behaviour as an e€ect of boundary layer. The concentration factor of circumferential stress in the surface of cavity is introduced as

Fig. 6. The jumping of the stresses at the surface of the cavity R ˆ 0.

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Fig. 7. The distribution of stresses and e€ect of boundary layers for the concentration of stresses. def

K…k†ˆ

r2c …0† : r2h …0†

In the critical state of cavitation, the concentration factor K…k† has maximum value as p 9…C1 ‡ C2 C3 † p : Kcr ˆ K…kcr † ˆ 6C1 ‡ 5 C2 C3

Fig. 8. The comparison for the circumferential stress concentration factor at the surface of the cavity.

…50†

…51†

1116

X.-C. Shang, C.-J. Cheng / International Journal of Engineering Science 39 (2001) 1101±1117

Substituting from (20) into the above expression, one ®nds that the stress concentration factor Kcr in the range of Poisson's ratio 0 < m < 1=2 is shown in Fig. 8. As a comparison, according to the linear theory of elasticity [34], when the stresses in the far ®eld are homogeneous, the concentration factor of circumferential stress near an in®nitesimal cavity can be determined to be the constant 1.5 for the linear elastic materials. Therefore, as shown in Fig. 8, the nonlinear e€ect of the hyperelastic material (15) makes the stress concentration factor have a higher value. Acknowledgements This work was supported by the Scienti®c Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry and also by the National Natural Science Foundation of China (No.19802012). References [1] A.N. Gent, P.B. Lindley, Internal rupture of bounded rubber cylinders in tension, Proc. R. Soc. London 249 (1958) 195. [2] J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. R. Soc. London A 306 (1982) 557. [3] C.O. Horgan, R. Abeyaraine, A bifurcation problem for a compressible nonlinear elastic medium: growth of a micro-void, J. Elasticity 16 (1986) 189. [4] J. Sivaloganathan, Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity, Arch. Rat. Mech. Anal. 96 (1986) 97. [5] C.A. Stuart, Radially symmetric cavitation for hyperelastic materials, Ann. Inst. Henri Poincare ± Analyses non lineare 2 (1985) 33. [6] P. Podio-Guidugli, G. Vergara Ca€arelli, E.G. Virga, Discontinuous energy minimizers in nonlinearly elastostatics: an example of J. Ball revisited, J. Elasticity 16 (1986) 75. [7] J. Sivaloganathan, A ®eld theory approach to stability of radial equilibria in nonlinear elasticity, Math. Proc. Camb. Phil. Soc. 99 (1986) 589. [8] D.M. Haughon, Cavitation in compressible elastic membranes, Int. J. Eng. Sci. 28 (1990) 162. [9] F. Mernard, Existence and nonexistence results on the radially symmetric cavitation problem, Quart. Appl. Math. 50 (1992) 201. [10] C.O. Horgan, Void nucleation and growth for compressible nonlinearly elastic materials:an example, Int. J. Solids and Struct. 29 (1992) 279. [11] N. Ertan, In¯uence of compressibility and hardening on cavitation, ASCE J. Eng. Mech. 114 (1988) 1231. [12] H. Tian-hu, A theory of the appearance and growth of the micro-spherical void, Int. J. Fract. 43 (1990) R51. [13] S. Biwa, E. Matsumoto, T. Shibata, E€ect of constitutive parameters on formation of a spherical void in a compressible nonlinear elastic material, Trans. ASME J. Appl. Mech. 61 (1994) 395. [14] S. Biwa, Critical stretch for formation of a cylindrical void in a compressible hyperelastic material, Int. J. Nonlinear Mech. 30 (1995) 899. [15] H.C. Lei, H.W. Chang, Void formation and growth in a class of compressible solids, J. Eng. Math. 30 (1996) 693. [16] X.-C. Shang, C.-J. Cheng, The spherical cavitation bifurcation in hyperelastic materials, Acta Mech. Sinica 28 (1996) 751. [17] S.S. Antman, P.V. Negron-Marrero, The remarkable nature of radially symmetric equilibrium states of aeolotropic nonlinearly elastic bodies, J. Elasticity 18 (1987) 131. [18] D.A. Polignone, C.O. Horgan, Cavitation for incompressible anisotropic nonlinearly elastic spheres, J. Elasticity 33 (1993) 27.

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