Cavitation in finite elasticity with surface energy effects

International Journal of Non-Linear Mechanics 41 (2006) 1084 – 1094 www.elsevier.com/locate/nlm

Cavitation in finite elasticity with surface energy effects S. Biwa ∗ Graduate School of Energy Science, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan Received 26 July 2006; received in revised form 2 November 2006; accepted 2 November 2006

Abstract Cavity formation in incompressible as well as compressible isotropic hyperelastic materials under spherically symmetric loading is examined by accounting for the effect of surface energy. Equilibrium solutions describing cavity formation in an initially intact sphere are obtained explicitly for incompressible as well as slightly compressible neo-Hookean solids. The cavitating response is shown to depend on the asymptotic value of surface energy at unbounded cavity surface stretch. The energetically favorable equilibrium is identified for an incompressible neoHookean sphere in the case of prescribed dead-load traction, and for a slightly compressible neo-Hookean sphere in the case of prescribed surface displacement as well as prescribed dead-load traction. In the presence of surface energy effects, it becomes possible that the energetically favorable equilibrium jumps from an intact state to a cavitated state with a finite cavity radius, as the prescribed loading parameter passes a critical level. Such discontinuous cavitation characteristics are found to be highly dependent on the relative magnitude of the surface energy to the bulk strain energy. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Cavitation; Finite elasticity; Surface energy; Neo-Hookean material; Slightly compressible neo-Hookean material; Variational principle

1. Introduction A class of bifurcation problems in finite elasticity, associated with the formation of a cavity in an initially intact elastic solid under certain loading conditions, has been studied by many authors since the pioneering mathematical treatment by Ball [1]. In physical terms, the bifurcation represents a phenomenon that at sufficiently large applied traction or stretch, an intact solid can reduce its associated potential energy by opening a cavity inside, depending on the specific constitutive properties of the solid. Besides rigorous mathematical aspects of the bifurcation, equilibrium solutions describing cavity formation have been explicitly explored for incompressible hyperelastic solids [2–4] as well as for various compressible hyperelastic solids [5–9] and for elastoplastic solids [10–12]. Further references can be found in a review by Horgan and Polignone [13] and, among others, recent articles by Shang and Cheng [14], Yuan et al. [15] and the references therein.

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E-mail address: [email protected]. 0020-7462/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2006.11.001

Such theoretical pursuits definitely have counterparts in the study of the mechanical behavior of materials. Cavity appearance in rubbery materials under hydrostatic tension was reported by Gent and Lindley [16] and later studied from a fracture-mechanics point of view [17]. Indeed, the predictions of the above bifurcation theory were correlated to experimental observations for rubbery materials [18]. Not only bulk rubbers but also small rubber particles dispersed in polymer matrix show cavitation, which has been a topic of interest regarding the toughness improvement of brittle polymers [19]. In addition, cavity formation in ductile metals has been found in experiments [20] and studied in the context of the cavitation instability, that is, the unbounded growth of an initially finite cavity when a certain loading level is approached [21–23]. When analyzing cavity formation, it is more realistic to account for the surface energy of the formed cavity. Such effects have been considered in the analysis of expansion as well as unbounded growth of a cavity with finite initial radius in nonlinear elastic and elastoplastic media [24–28]. In the context of cavitation of an initially intact solid, however, it was remarked by Ball [1] that the presence of surface energy would hinder

S. Biwa / International Journal of Non-Linear Mechanics 41 (2006) 1084 – 1094

the cavitating bifurcation: since then the topic seems to have been less often revisited. Many of the above-mentioned works [24–28] assume the surface energy defined per unit deformed area which is a material constant. Studies of the surface energy for solids have, however, revealed that the surface energy depends on the in-plane strain state of the surface [29]. Such stretch dependence of the surface energy is yet to be explored in the context of growth of a finite cavity or cavity formation in an intact solid. It is the purpose of this paper to examine the effect of surface energy on cavity formation at finite spherically symmetric deformation of isotropic hyperelastic solids. In order to clarify the implication of the stretch dependence of the surface energy, the formulation is first laid down for expansion of a hollow hyperelastic sphere. Consequently, the influence of the surface energy on the radial deformation of an initially intact sphere is examined, where it turns out that a characteristic cavitation phenomenon occurs instead of the bifurcation, in a manner similar to the snap-through buckling found in thin-walled structures. Namely, a jump from an intact equilibrium to a cavitated equilibrium is expected at a certain loading level in order for the sphere to have less potential energy. Such snap modes have been obtained in foregoing studies of cavitation in hyperelastic solids with material inhomogeneity [30] and anisotropy [4]. The problem of cavity formation has been recently addressed by Dollhofer et al. [27] accounting for the influence of surface tension. These authors examine equilibrium states of an incompressible neo-Hookean hollow sphere and cavity formation in an initially intact sphere. The present paper also deals with cavitation in an incompressible neo-Hookean sphere though in further detail: in addition, the analysis is extended to encompass cavitation in a slightly compressible neo-Hookean sphere. The structure of this paper is as follows. After outlining the basic kinematical relations for radial deformations in Section 2, expansion of an incompressible hyperelastic hollow sphere under tensile dead loading is considered in Section 3, where the expansion behavior is shown to depend on the functional form of the stretch-dependent surface energy. In Section 4, cavity formation in an initially intact incompressible neo-Hookean sphere is analyzed, and the energetically favorable equilibrium configuration is explored for the prescribed dead-load traction. In Sections 5 and 6, the analysis is extended to incorporate a slight compressibility in the constitutive model, and an approximate equilibrium solution describing cavitation is obtained. The energetically favorable equilibrium state is identified for both prescribed displacement and prescribed dead-load traction conditions. The paper is concluded in Section 7 with some pertinent discussions. 2. Basic kinematical relations In this paper, the spherically symmetric deformation of a sphere with radius A having a concentric spherical cavity with radius Rc is considered. This includes the case of Rc → 0, which means that the sphere is initially intact. The deformation is expressed as a mapping from the radial coordinate R in the initial configuration to the radial coordinate r in the deformed

1085

configuration, r = r(R), Rc R A. The principal stretches of this deformation are given by 1 =

dr , dR

2 = 3 =

r , R

(2.1)

and its Jacobian J is given by J=

dr  r 2 . dR R

(2.2)

If the outer radius of the deformed sphere is denoted by A = r(A), the parameter  represents the expansion ratio of the sphere. Also, the deformed radius of the cavity is denoted by rc , i.e. rc = r(Rc ). Then the in-plane principal stretch of the cavity surface c is given by c = rc /Rc .

(2.3)

If the sphere is made of an incompressible material, an internal constraint is imposed on the deformation in the form J = 1. This implies that  r −2 dr = dR R

(2.4)

from Eq. (2.2). With the condition r(Rc )=rc , it is readily solved to yield  1/3 r(R) = R 3 − Rc3 + rc3

(2.5)

which gives 1/3  r(A) rc3 − Rc3 = = 1+ A A3 3 3 rc = {Rc + ( − 1)A3 }1/3 .

or (2.6)

3. Expansion of an incompressible hyperelastic hollow sphere In this section, it is assumed that the sphere consists of an incompressible isotropic hyperelastic solid that is characterized by the stored energy function W (1 , 2 , 3 ), where W is a symmetric function of the principal stretches. It is also assumed that the inner surface of the sphere is associated with the surface energy  per unit current (deformed) area. Presently,  is allowed to depend on the stretch ratio of the inner surface, namely,  = (c ), where the cavity wall stretch c is given by Eq. (2.3). For the sake of simplicity, no surface energy is considered to be associated with the outer surface of the sphere, though it is not an essential restriction. It is noted that the hollow sphere is associated with certain amount of surface energy even in its initial configuration, which is then not a stress-free state as usually understood implicitly in finite elasticity. For an incompressible sphere, the deformation mapping is solely determined by a single parameter  or rc from Eqs. (2.5) and (2.6). When the sphere is subjected to a dead-load radial tensile traction P at its outer surface, the potential energy of

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S. Biwa / International Journal of Non-Linear Mechanics 41 (2006) 1084 – 1094

the sphere is given by  A 4R 2 W (R 2 /r 2 , r/R, r/R) dR I=

incompressible sphere. In this case, the equilibrium condition in Eq. (3.6) is clearly satisfied for a given dead-load traction P by the configuration

Rc

− 4A3 ( − 1)P + 4rc2 (c ),

(3.1)

where the first term represents the strain energy stored in the sphere, the second term the work done by the dead-load traction P and the third term the surface energy of the cavity wall. The first term in the right-hand side of Eq. (3.1) is rewritten by the transformation t = r(R)/R, and partially integrated to give  A 4R 2 W (R 2 /r 2 , r/R, r/R) dR Rc

= 4(rc3

− Rc3 )

 c 2 ˆ t W (t)

(3.2)

(3.3)

Then the potential energy of the sphere is rewritten as  c ˆ 4 3 ˆ dW /dt 4 3 4 3 ˆ 3 I= A W () − R W (c ) + (r − Rc ) 3 3 c 3 c t3 − 1  3 2 × dt − 4A ( − 1)P + 4rc (c ). (3.4) Since  and c are given in terms of rc by Eqs. (2.3) and (2.6), I is a function of rc . Then the equilibrium is achieved when dI = 0. drc

implying no cavity expansion, since rc = r(0) = 0. This is a solution that keeps the sphere intact. The main interest here is an alternative equilibrium solution which has a non-zero cavity radius rc > 0. Letting Rc → 0 in Eq. (3.7) yields the desired solution. In order to find a solution to a finite dead-load traction P, however, the limit (∞) +

1 2

lim (c  (c ))

c →∞

lim (c  (c )) = 0.

where Wˆ (t) ≡ W (t −2 , t, t).

(4.1)

should have a finite value. This gives a restriction for the stretch dependence of the surface energy in order to allow the cavity formation from an intact state. This leads to a more restrictive condition

dt

(t 3 − 1)2 Wˆ () Wˆ (c ) = 4(rc3 − Rc3 ) − 3 3( − 1) 3(3c − 1)   1 c dWˆ /dt dt , + 3  t3 − 1 

r(R) = R

(3.5)

By carrying out the differentiation of Eq. (3.5) for Eq. (3.4), one obtains  c ˆ  rc2 dW /dt 2 2  (c ) rc dt − P + 2r ( ) + r = 0, (3.6) c c c Rc t3 − 1 2  where  =d/dc . When rc  = 0, the above equation is reduced to

 c ˆ dW /dt  (c ) 2 2(c ) P = 2 dt +  (3.7) + rc Rc t3 − 1  which gives the equilibrium response of the sphere for the prescribed dead-load traction P. 4. Cavity formation in an incompressible neo-Hookean sphere 4.1. Equilibrium response In this section, the limiting behavior is considered where the initial radius vanishes, i.e. Rc → 0 implying an initially intact

c →∞

(4.2)

In fact, if the left-hand side of Eq. (4.2) were a non-zero value, M say, it would imply that  ≈ M/c for c ?1 and lead to  ≈ M ln c which contradict the above requirement. In the sequel, ∞ = (∞) is assumed as a positive finite constant. In this circumstance, the response curve for an initially intact incompressible hyperelastic sphere is given by  ∞ dW ˆ /dt 2∞ 2 dt + . (4.3) P = rc t3 − 1  As a special case of incompressible hyperelastic solids, the constitutive law for the neo-Hookean solid is considered, which reads  (4.4) W (1 , 2 , 3 ) = (21 + 22 + 23 − 3), 2 where  is a constitutive parameter representing the shear modulus for infinitesimal deformations. With this particular form for W , Eq. (4.3) reduces to  

1 2∞ 1 P 2 = 2 + +   44 rc  1  22 = 2 + 2 + ∞ 3 (4.5) A ( − 1)1/3 2 using Eqs. (2.6). Therefore, the response depends only on the ratio of the asymptotic value ∞ to A, irrespective of the functional form of (c ). This response is shown in Fig. 1 for different values of the normalized parameter ∞ /(A), which represents the relative magnitude of the surface energy to the bulk strain energy. From Eq. (4.5), it is clear that P → ∞ as rc → 0 unless ∞ = 0. So the cavitating bifurcation, in the sense of continuous transition of the solution path from an intact configuration to a cavitated equilibrium configuration as the parameter P passes a critical point, does not occur when ∞  = 0, as mentioned already by Ball [1].

S. Biwa / International Journal of Non-Linear Mechanics 41 (2006) 1084 – 1094

10

0.2

∞/(A) = 1 0.5

∞/(A) = 1

8

P/

0.1 4

0 G ()

0.5

(1+51/2)/2

0.1

0

6

1087

-0.2

0 2

-0.4

0 1

1.2

1.4

1.6

1

1.8

1.2

Fig. 1. Surface traction–stretch relation of an initially intact incompressible neo-Hookean sphere, for different normalized surface energies. Solid circles represent the points where the intact and the cavitated states have the same potential energy.

4.2. Energetically favorable equilibrium for a prescribed dead-load traction The results in Fig. 1 show that for the dead-load traction P above a certain level, there are multiple equilibrium solutions, namely, the intact configuration and two cavitating configurations. Here, the associated energies of the two types of solutions are compared to each other to determine the energetically favorable equilibrium solution for a given P. The potential energy of the cavitated equilibrium solution is given by substituting Rc = 0 and Eq. (4.4) in Eq. (3.4). Noting that dWˆ /dt = 2(t −2 + t −5 ) t3 − 1

(4.6)

for the neo-Hookean incompressible sphere, Eq. (3.4) reduces to 4 3 Icav = A G(), (4.7) 3 where

G1 () = −

∞ G2 (), A

3( − 1)2 (22 − 1) 2

2

1.6

1.8





G() ≡ G1 () +

1.4

(4.8) ,

(4.9)

Fig. 2. Variation of the normalized potential energy of an incompressible neo-Hookean sphere with the sphere stretch, for different normalized surface energies.

When ∞ = 0, G() = G1 () < 0 for  > 1, so the cavitated solution is always preferred to the intact solution. In this case, as the cavitated solution exists for P > ( 25 ) from Eq. (4.5), the cavitated solution bifurcates from the intact solution path at P = ( 25 ) as P is increased from 0, which is a common finding in the foregoing bifurcation analysis of cavitation [1,2]. Next, the case when ∞ > 0 is considered. For  greater than but sufficiently close to 1, G() can be written as G≈−

32/3 ∞ 3( − 1)2 + ( − 1)2/3 2 A

for 0 <  − 1>1 (4.11)

which indicates that the intact solution is preferred to the cavitated state. However, the following observations can be made: d 2 G2 d2

<0

for  > 1,

(4.12)

dG2 >0 d

for 1 <  < 21/3 ,

(4.13)

dG2 <0 d

for  > 21/3 ,

(4.14)

G2 () < 0

√ 1+ 5 . for  > 2

(4.15)

2

G2 () =

3( − 1)2/3 (− +  + 1) (2 +  + 1)1/3

,

(4.10)

where  is given by Eqs. (2.6) for the prescribed P. The potential energy of the intact solution is zero, since there are no stored and surface energies in the sphere, and the applied traction makes no work. Therefore, when the intact and cavitated equilibrium solutions coexist, the former is an energetically preferred equilibrium if G() > 0, while the latter is favorable if G() < 0.

Therefore, as  increases from 1, G2 () first increases but attains its maximum at  = 21/3 , and thereafter it monotonically decreases, √becoming negative at a certain value of  smaller than (1 + 5)/2. In Fig. 2, G() is numerically calculated for different values of ∞ /(A). It is then seen that there is a value of , denoted by cr , at√which G() turns from positive to negative and 1 < cr < (1 + 5)/2. For 1 <  < cr , the intact configuration is favored while for  > cr , the cavitated solution is favorable.

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S. Biwa / International Journal of Non-Linear Mechanics 41 (2006) 1084 – 1094

0.2

2 ∞/(A)= 1

0.1

0

0.5

0.5

0.1

∞/(A) = 1

2

4 0

6

8

10

rc/A

G

0.1 0

1

P/

-0.1

-0.2

0

Fig. 3. Variation of the normalized potential energy of an incompressible neo-Hookean sphere with the surface dead-load traction, for different normalized surface energies.

0

2

4

6

8

10

P/ Fig. 5. The relation between the cavity radius in an incompressible neo-Hookean sphere and the applied dead-load traction, for different normalized surface energies. Solid circles represent the points where the intact and the cavitated states have the same potential energy and cavitation is expected to occur in the P-controlled process.

P

Cavitated configuration Pcr Pmin

Intact configuration 1

Pmin

cr



Fig. 4. A schematic representation of a possible scenario of cavity formation in an incompressible sphere in the P-controlled process.

The critical applied traction Pcr is given by

  Pcr 1 ∞ 1 2 2 . + 4 + = cr 2  cr A (3cr − 1)1/3 4cr

(4.16)

In Fig. 3, the normalized potential energy G is illustrated as a function of P, where Pcr is found as the point at which the curve intersects G = 0. In the response curves in Fig. 1, the points (cr , Pcr /) are plotted as solid circles for different ∞ /(A), where the energetically favorable state switches from the intact to the cavitated solution. Based on the above analysis, a possible scenario of cavity formation is illustrated schematically in Fig. 4, provided that the sphere can spontaneously take the equilibrium configuration with as low potential energy as possible for the prescribed boundary condition. This means that any possible resistances to the transition between equilibrium states are neglected, which should be discussed in a different framework. As P is applied

to the sphere, it first remains in an intact state, as it is the only possible equilibrium state. For P > Pmin depicted in Fig. 4, the cavitated equilibrium becomes possible, but the sphere still remains intact since it is associated with less potential energy than forming a cavity until P = Pcr at which the two configurations become energetically equivalent. For P > Pcr , it is energetically favorable to turn into a cavitated configuration. Therefore, a snap-through-type jump can be expected from an intact to a cavitated state as shown with an arrow in Fig. 4. In Fig. 4, Pcr is greater than Pmin where dP /d = 0. As a matter of fact, 

  dG 2∞ (3 − 2) 1 = − 3( − 1) 2− 3 + d A (3 − 1)4/3  d(P /) = − 3( − 1) , (4.17) d so at  = Pmin , dG/d=0 and G(Pmin ) > 0, which implies that Pmin < cr , where Pmin is defined as the stretch corresponding to Pmin as shown in Fig. 4. Therefore, although two cavitated configurations coexist for the same P when P > Pmin , any cavitated equilibrium which is energetically preferred to the intact equilibrium is one with larger  as shown in Fig. 4. It is noted that as ∞ → 0, Pmin → 1, cr → 1 and Pcr → ( 25 ): the jump distance diminishes. As ∞ → √ ∞ in the other extreme, Pmin → 21/3 1.260, cr → (1 + 5)/2 1.618 and Pcr → ∞: the cavitation becomes impossible. In Fig. 5, the variation of the cavity radius with the deadload traction P is illustrated, where the switching points defined above are plotted as solid circles for different ∞ /(A), with arrows representing the transition. When ∞ = 0, the cavity starts to grow at the critical point, starting from null radius. On the other hand, when ∞ > 0, a cavity of a finite radius appears in the sphere at the critical point, and grows continuously with increasing P.

S. Biwa / International Journal of Non-Linear Mechanics 41 (2006) 1084 – 1094

5. Expansion of a compressible hyperelastic hollow sphere 5.1. Equilibrium response In this section, the foregoing analysis is repeated for a more general situation incorporating material compressibility. For compressible solids, the equilibrium analysis becomes more complicated than the incompressible case which essentially reduces to a one degree-of-freedom system due to the internal constraint. Although the problem was considered for a prescribed dead-load traction in Sections 3 and 4, the prescribed displacement problem is considered in addition to the prescribed traction problem for a compressible sphere in the present and the subsequent sections. When a compressible isotropic hyperelastic hollow sphere with outer radius A and inner radius Rc , characterized by the stored energy function W (1 , 2 , 3 ) and the surface energy function (c ), is subjected to the surface displacement of the form r(A) = A,

(5.1)

where  > 1 denotes the applied stretch ratio of the sphere, the relevant potential energy is  A J= 4R 2 W (dr/dR, r/R, r/R) dR Rc   r(Rc ) 2 + 4{r(Rc )}  . (5.2) Rc This is a functional of r(R), and the equilibrium condition is given by a variational principle, i.e. from J =0. With r(A)=A fixed, this leads to the Euler–Lagrange equation   

 d dr r r dr r r R 2 W1 , , − 2RW 2 , , = 0, dR dR R R dR R R (5.3) and the natural condition   dr  r(Rc ) r(Rc ) 2 Rc W1 , , dR R=Rc Rc Rc     {r(Rc )}2  r(Rc ) r(Rc ) = 2r(Rc ) + ,  Rc Rc Rc

(5.4)

j W (1 , 2 , 3 ) , j

 = 1, 2.

(5.3), the natural condition (5.4) and, additionally, the following condition are obtained:    dr  r(A) r(A) , = P. (5.7) , W1 dR R=A A A In fact, in the prescribed displacement problem, the left-hand side of Eq. (5.7) gives the surface traction that is required to deform the sphere to meet Eq. (5.1). On the other hand, in the prescribed dead-load traction problem,  can be defined as r(A)/A from the solution which conforms to Eq. (5.1). Therefore, the equilibrium responses for the two cases with dead-load traction and radial displacement prescribed are considered in a unified manner until the associated potential energies in the two problems are examined separately in later sections. 5.2. Change of variable Introducing the change of variable [7,31] t=

r(R) , R

s(t) =

dr , dR

(5.8)

the Euler–Lagrange equation (5.3) reduces to a first-order ordinary differential equation

ds W1 (s, t, t)−W2 (s, t, t) W11 (s, t, t) = −2 +W12 (s, t, t) , dt s−t (5.9) where W1 (1 , 2 , 3 ) ≡

j2 W (1 , 2 , 3 ), j j1

 = 1, 2,

(5.10)

Let  = s() denote the radial principal stretches at R = A. Given  = s() as the initial condition, the differential (5.9) gives s = s(t) for  < t < r(Rc )/Rc . Here,  is a given quantity in the prescribed displacement problem, while it should satisfy Eq. (5.7) in the prescribed dead-load traction problem, i.e. W1 (, , ) = P .

(5.11)

Furthermore, s(t) needs to satisfy Eq. (5.4), which using Eq. (2.3) becomes

where W (1 , 2 , 3 ) ≡

1089

(5.5)

If the dead-load traction P is prescribed on the surface instead of the displacement, the pertinent potential energy is, instead of Eq. (5.2),  A I= 4R 2 W (dr/dR, r/R, r/R) dR Rc

  r(A) r(Rc ) − 4A3 − 1 P + 4{r(Rc )}2  (5.6) A Rc in a similar fashion to Eq. (3.1) in the incompressible case. By the variational principle, the same Euler–Lagrange equation

W1 (s(c ), c , c ) = 2c

2(c ) + c  (c ) . r(Rc )

(5.12)

Using s=s(t) so determined, the deformation field in the sphere is in turn expressed from Eq. (5.8) by  t

s( )/ r = A exp d , s( ) −  t

1 R = A exp d  s( ) − with t as a parameter [7], where  < t < r(Rc )/Rc .

(5.13)

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S. Biwa / International Journal of Non-Linear Mechanics 41 (2006) 1084 – 1094

6. Cavity formation in a slightly compressible neo-Hookean sphere

10 ε = 0.02

 2 { + 22 + 23 − 3 − 2(1 2 3 − 1) 2 1 + (1 2 3 − 1)2 },

1

(6.1)

is a sufficiently small parameter which accounts for slight compressibility. In particular, Poisson’s ratio  for infinitesimal deformations is given by (6.3)

which approaches the incompressible limit 0.5 when ε → 0. As ε → 0, the model implies the incompressible neo-Hookean behavior as treated in the previous sections. With this particular form, the equilibrium equation (5.9) becomes (1 + t 2 )

ds = −2(1 + st 3 ), dt

(6.4)

which is integrated to give C0 − (t) s(t) =  , 1 + t 4

(6.5)

where C0 is an integration constant and  (t) = 

2 1 + t 4

.

(6.6)

Although (t) can be expressed in terms of an elliptic integral, the present interest lies in the case of slight compressibility, and a solution for ε>1 is sought for, which is given by s(t) =

 2ε 2  ε −4 −4 ( − t ) − 2 (−1 − t −1 ) 1 + 2 t2 t

to first order in ε, using  = s(). Substitution of Eq. (6.7) in Eq. (5.13)1 yields  1/3 3 − 2 r = At t 3 − 2

 4−1 + −2 − 3t −1 − 2−4 t 3 × exp −ε , 6(t 3 − 2 )

0 2

, Intact solution

(6.2)

1−ε = 2

0.1 4

0

where  is again the parameter denoting the shear modulus for small deformations, and ε≡

0.5

6 P/

In the sequel, the analysis is confined to the deformation of an initially intact sphere made of a compressible hyperelastic solid, i.e. Rc → 0. Furthermore, a specific form of the stored energy function, referred to as the slightly compressible neoHookean solid [32], is employed hereafter. For this material, W is given by W (1 , 2 , 3 ) =

∞/(A) = 1

8

6.1. Equilibrium response

(6.7)

(6.8)

1

1.2

1.4

1.6

1.8

 Fig. 6. Surface traction–stretch relation of an initially intact slightly compressible neo-Hookean sphere, for ε = 0.02 and for different normalized surface energies. Solid circles represent the points where the intact and the cavitated states have the same potential energy for the prescribed dead-load traction, and open circles represent the corresponding points for the prescribed surface displacement.

and the cavity radius is given by  3

2 1/3

rc = lim r = A( −  ) t→∞

exp

ε 34

 .

(6.9)

On the other hand, setting Rc → 0 and substituting Eqs. (6.1) and (6.7), Eq. (5.12) reduces to 

ε−4 2∞ 2 −1 (6.10) = 1 + ε 1 + 2 +  1 + 2 rc employing ∞ as in Section 4. In the prescribed displacement problem, Eqs. (6.9) and (6.10) determine  for a given , and then the function s(t) and the whole deformation field r(R) from Eqs. (6.7) and (6.8), respectively. In the prescribed deadload traction problem, the parameter  should be determined from Eq. (5.11), which reduces to

1   − 2 + (2 − 1)2 = P . (6.11) ε In the prescribed displacement problem, Eq. (6.11) gives the surface traction at R = A. Some features of the solution given by Eqs. (6.7)–(6.11) are noted here. As rc → 0, Eq. (6.9) implies that  → . In this situation, Eqs. (6.10) and (6.11) imply that  → ∞ and P → ∞ unless ∞ = 0. On the other hand, as  → ∞ another equilibrium is possible corresponding to rc → A{3 − (1 + ε)}1/3 ,

(6.12)

 → (1 + ε)−2

(6.13)

and P → ∞. In Fig. 6, the relations between  and P are calculated numerically for different values of ∞ /(A) with ε = 0.02.

S. Biwa / International Journal of Non-Linear Mechanics 41 (2006) 1084 – 1094

The response for the case when ∞ = 0 is also illustrated in Fig. 6, which is given by 

 = −2 1 + ε 1 + 2−1 −

−4 2

 ,

  −2 −4 −2 −1 P =  2 + 1 + 2 − + ε , 2 2

(6.14)

from Eqs. (6.10) and (6.11) to first order in ε, and coincides with the results obtained earlier [33]. For this solution, the relation between  and  is monotonic, and this solution bifurcates from the path of the intact solution, as shown below, at  = cr0 , or P = Pcr0 , which are given by 3cr0 − 1

−4 1 + 2−1 cr0 − cr0 /2

= ε,

−2 Pcr0 = 3cr0 − cr0 .  2

r(R) = R,

2 (3 − 1) W1 (, , ) =   −  + ε



2

= P,

(6.16)

is depicted. It is straightforward to check that  given in Eq. (6.16) indeed satisfies Eqs. (5.3), (5.4) and (5.7). 6.2. Energetically favorable equilibrium for a prescribed displacement In the above discussion, it has been found that for a prescribed surface displacement , an intact solution is possible, and for a sufficiently large , two cavitated solutions are additionally possible. In this section, the most preferable solution is identified from the viewpoint of the potential energy in Eq. (5.2). For a prescribed , let Jintact and Jcav denote the potential energy of the intact and cavitated solutions, respectively. For the intact mode, the stored energy is uniform in the sphere and no surface energy is accompanied, so Jintact is given by Jintact =

∞/(A) = 1 0.2

0.5

0.1 0.1 0 1

1.1 

0

1.2

-0.1 Fig. 7. Variation of the normalized potential energy of a slightly compressible neo-Hookean sphere with the sphere stretch, for ε = 0.02 and for different normalized surface energies.

(6.15)

In Fig. 6, another type of equilibrium solution corresponding to the intact mode, namely,

0.3

(Jcav-Jintact)/{(4/3)πA3}

1091

4 3 A W (, , ). 3

(6.17)

For the cavitated solution, Eq. (5.2) is rewritten as [7,34]  4 Jcav = R 3 R 3 W (dr/dR, r/R, r/R) 3  

R→A r dr + − W1 (dr/dR, r/R, r/R) R dR R→0 4 A3 {W (, , ) + 4rc2 ∞ = 3 4 2 + ( − )W1 (, , )} + r  . 3 c ∞

(6.18)

Therefore, the task is to evaluate the difference 4 3 Jcav − Jintact = A W (, , ) − W (, , ) 3 + ( − )W1 (, , ) 

 2ε ∞ 2/3 4/3 . + ( − )  exp A 34

(6.19)

For the slightly compressible neo-Hookean material, W (, , ) − W (, , ) + ( − )W1 (, , )  4   = − ( − )2 1 + <0 2 ε

(6.20)

and

 Jcav − Jintact 1 4 2 = − ( − ) 1 + 2 ε (4/3)A3  +

  ∞ 2ε , ( − )2/3 4/3 exp A 34

(6.21)

from which it is found that for  sufficiently close to , Jcav − Jintact > 0. On the other hand, when  grows sufficiently large, Jcav − Jintact < 0. Therefore, as has been found in the incompressible case, a cavitated solution with a cavity small enough is not energetically favorable, but one with a sufficiently large cavity can be a preferable mode of deformation to the intact mode. The difference Jcav − Jintact is calculated numerically in Fig. 7, from which cr , the critical applied stretch corresponding to Jcav − Jintact = 0, is identified, and the corresponding traction–stretch points are shown in Fig. 6 as open circles. Furthermore, Fig. 8 schematically illustrates a possible scenario of the sphere deformation with the sphere stretch  as a controlling parameter, in the spirit mentioned in Section 4. Compared to Fig. 6, Fig. 8 shows a region around the critical point with a magnified scale. It shows that the energetically preferable equilibrium state jumps off the path of the intact deformation mode to that of the cavitated mode at the critical stretch cr . Numerically, it has been found that two equilibrium states are possible when  > min , where min is depicted

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S. Biwa / International Journal of Non-Linear Mechanics 41 (2006) 1084 – 1094

3

P

∞/(A) = 1

Intact configuration

(Icav-Iintact)/{(4/3)πA3}

2

Cavitated configuration

0.5 1 0.1 0 2

0

-1

4

6 P/

8

10

-2 -3

cr



min

Fig. 10. Variation of the normalized potential energy of a slightly compressible neo-Hookean sphere with the dead-load traction, for ε =0.02 and for different normalized surface energies.

Fig. 8. A schematic representation of a possible scenario of cavity formation in a slightly compressible sphere in the -controlled process.

Eq. (5.6) is given as Iintact = 1

4 3 A W (0 , 0 , 0 ) − 4A3 (0 − 1)P , 3

where 0 satisfies W1 (0 , 0 , 0 ) = 

rc /A

∞/(A) = 0 0.5

0 − 20

(3 − 1)20 + 0 ε

(6.22)

 = P.

(6.23)

On the other hand, for the cavitated solution, 0.1

∞/(A) = 1

0.5

Icav =

0 1

1.05

1.1

1.15

 Fig. 9. The relation between the cavity radius in a slightly compressible neo-Hookean sphere and the sphere stretch, for ε = 0.02 and for different normalized surface energies. Open circles represent the points where the intact and the cavitated states have the same potential energy and cavitation is expected to occur in the -controlled process.

in Fig. 8. The critical stretch cr is greater than min and the energetically favorable equilibrium is one with lower P of the two possible equilibrium states. The jump event can be seen in the corresponding relation between the applied stretch  and the cavity radius rc shown in Fig. 9, which demonstrates that at  = cr a cavity of a finite radius would appear in the sphere. 6.3. Energetically favorable equilibrium for a prescribed dead-load traction Next, the problem is examined for a prescribed dead-load traction on the sphere surface. In this case, it is the functional of Eq. (5.6) that is to be considered. For the intact mode,

4 3 A {W (, , ) + ( − )W1 (, , )} 3 4 2 + r  − 4A3 ( − 1)P , 3 c ∞

(6.24)

using the same formula as used in Eq. (6.18), where  and  are given in accordance with Eqs. (6.9) and (6.10). The difference is given by 4 3 A W (, , ) − W (0 , 0 , 0 ) Icav − Iintact = 3  + (30 − 2 − )P + ∞ ( − )2/3 4/3 A 

 2ε , (6.25) × exp 34 where  and  are determined as a cavitated equilibrium, and 0 as an intact equilibrium, for the same prescribed P. Although it appears difficult to draw rigorous conclusions, it is now straightforward to check the sign of Icav − Iintact in Eq. (6.25) in a numerical manner. The numerical results are shown in Fig. 10, where a similar trend to Fig. 3 of the incompressible case can be found, except a looping behavior in each curve. From these results it can be conjectured that, like the previous cases, the sphere prefers to remain intact for small P, but a transition occurs from an intact to a cavitated configuration with a finite cavity radius at a critical dead-load traction. The critical dead-load tractions and the corresponding stretches are plotted as solid circles in Fig. 6. The transition behavior is

S. Biwa / International Journal of Non-Linear Mechanics 41 (2006) 1084 – 1094

1093

shown schematically in Fig. 11. The relation between the cavity radius and the applied dead-load traction is shown in Fig. 12.

been demonstrated, which is valid for slight compressibility. The energetically favorable equilibrium has been determined for both prescribed surface displacement and prescribed deadload traction. The present analysis has shown that in the presence of surface energy effects, there is a finite jump of the energetically favorable equilibrium from an intact state to a cavitated state, as the prescribed loading parameter is increased and passes a critical level. This is in contrast to the case without surface energy, which reveals continuous bifurcation from the path of an intact state to the path of a cavitated solution. The critical dead-load traction or the critical surface displacement is dependent on the non-dimensional parameter ∞ /(A). This parameter represents the relative magnitude of the surface energy to the bulk strain energy. In the present paper, the results have been displayed for the cases ∞ /(A) = 0.1, 0.5, 1 as well as for the case ∞ /(A) = 0. For polymeric solids, the surface energy is typically in the order of 0.03 J/m2 [19] and the shear modulus is in the order of 0.5–1 MPa. Therefore, the results shown in this paper are relevant to micrometer- or submicron-size spatial scales. A point worth noting from the present analysis is that it is the surface energy at unbounded cavity surface stretch ∞ that influences cavitation from an intact state, although a full functional form (c ) and its asymptotic value are not specifiable for the time being. A recent article by Dollhofer et al. [27] discusses a practical implication of the surface energy effect on the cavitation behavior in the context of adhesive materials. In a molecular dynamics simulation for nickel, Makino et al. [35] have shown that a cavity is formed in an atomic lattice at a certain critical level of dilatational loading. Their results show that there occurs rather abrupt appearance of a cavity at a critical point. This simulation may partially support the present analysis from an atomistic point of view, since the excess energy of the initiated cavity surface is naturally incorporated in such direct simulations. In this context, it may be worth pointing out that such discontinuous cavitation characteristics found in this paper can be also examined by using the elastoplastic constitutive model instead of hyperelastic ones, referring to the foregoing cavitation analysis [10–12] for elastic–plastic solids, and the void-growth analysis [25] incorporating the surface energy effect.

7. Concluding remarks

References

P

Cavitated configuration Pcr Pmin

Intact configuration  Fig. 11. A schematic representation of a possible scenario of cavity formation in a slightly compressible sphere in the P-controlled process.

2

rc /A

∞/(A) = 1

1

∞/(A) = 1

0.5

0.1

0 0

2

4

6

8

10

P/ Fig. 12. The relation between the cavity radius in a slightly compressible neo-Hookean sphere and the dead-load traction, for ε = 0.02 and for different normalized surface energies. Solid circles represent the points where the intact and the cavitated states have the same potential energy and cavitation is expected to occur in the P-controlled process.

In this paper, cavity formation in incompressible as well as compressible isotropic hyperelastic materials has been examined by accounting for the effect of surface energy. While the expansion behavior of a hollow sphere depends on the specific surface-stretch dependence of the surface energy, the cavitation behavior from an intact state has been shown to depend only on the asymptotic value of the surface energy at unbounded surface stretch. For incompressible solids, the cavitated equilibrium solution has been illustrated in detail for a neo-Hookean model, and the energetically favorable equilibrium has been identified for the prescribed dead-load traction. For compressible solids, an approximate cavitated equilibrium solution has

[1] J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. R. Soc. London A 306 (1982) 557–611. [2] M.-S. Chou-Wang, C.O. Horgan, Void nucleation and growth for a class of incompressible nonlinearly elastic materials, Int. J. Solids Struct. 25 (1989) 1239–1254. [3] H.-S. Hou, A study of combined asymmetric and cavitated bifurcations in neo-Hookean material under symmetric dead loading, Trans. ASME J. Appl. Mech. 60 (1993) 1–7. [4] D.A. Polignone, C.O. Horgan, Cavitation for incompressible anisotropic nonlinearly elastic spheres, J. Elasticity 33 (1993) 27–65. [5] C.O. Horgan, R. Abeyaratne, A bifurcation problem for a compressible nonlinearly elastic medium: growth of a micro-void, J. Elasticity 16 (1986) 189–200.

1094

S. Biwa / International Journal of Non-Linear Mechanics 41 (2006) 1084 – 1094

[6] C.O. Horgan, Void nucleation and growth for compressible non-linearly elastic materials: an example, Int. J. Solids Struct. 29 (1992) 279–291. [7] S. Biwa, E. Matsumoto, T. Shibata, Effect of constitutive parameters on formation of a spherical void in a compressible nonlinear elastic material, Trans. ASME J. Appl. Mech. 61 (1994) 395–401. [8] S. Biwa, Critical stretch for formation of a cylindrical void in a compressible hyperelastic material, Int. J. Non-Linear Mech. 30 (1995) 899–914. [9] J.G. Murphy, S. Biwa, Nonmonotonic cavity growth in finite, compressible elasticity, Int. J. Solids Struct. 34 (1997) 3859–3872. [10] D.-T. Chung, C.O. Horgan, R. Abeyaratne, A note on a bifurcation problem in finite plasticity related to void nucleation, Int. J. Solids Struct. 23 (1987) 983–988. [11] H.-S. Hou, R. Abeyaratne, Cavitation in elastic and elastic–plastic solids, J. Mech. Phys. Solids 40 (1990) 571–592. [12] S. Biwa, Finite expansion of an infinitesimal void in elastic–plastic materials under equitriaxial stress, JSME Int. J. Ser. A 40 (1997) 23–30. [13] C.O. Horgan, D.A. Polignone, Cavitation in nonlinearly elastic solids: a review, Appl. Mech. Rev. 48 (1995) 471–485. [14] X.C. Shang, C.J. Cheng, Exact solution for cavitated bifurcation for compressible hyperelastic materials, Int. J. Eng. Sci. 39 (2001) 1101–1117. [15] X.G. Yuan, Z.Y. Zhu, C.J. Cheng, Study on cavitated bifurcation problems for spheres composed of hyperelastic materials, J. Eng. Math. 51 (2005) 15–34. [16] A.N. Gent, P.B. Lindley, Internal rupture of bonded rubber cylinders in tension, Proc. R. Soc. London A 249 (1958) 195–205. [17] A.N. Gent, C. Wang, Fracture mechanics and cavitation in rubber-like solids, J. Mater. Sci. 26 (1991) 3392–3395. [18] R. Stringfellow, R. Abeyaratne, Cavitation in an elastomer: comparison of theory with experiment, Mater. Sci. Eng. A 112 (1989) 127–131. [19] C.B. Bucknall, A. Karpodinis, X.C. Zhang, A model for particle cavitation in rubber-toughened plastics, J. Mater. Sci. 29 (1994) 3377–3383. [20] M.F. Ashby, F.J. Blunt, M. Bannister, Flow characteristics of highly constrained metal wires, Acta Metall. 37 (1989) 1847–1857. [21] Y. Huang, J.W. Hutchinson, V. Tvergaard, Cavitation instabilities in elastic–plastic solids, J. Mech. Phys. Solids 39 (1991) 223–241.

[22] V. Tvergaard, Y. Huang, J.W. Hutchinson, Cavitation instabilities in a power hardening elastic–plastic solid, Eur. J. Mech. A/Solids 11 (1992) 215–231. [23] Y. Huang, K.X. Hu, C.P. Pen, N.-Y. Li, K.C. Hwang, A model study of thermal stress-induced voiding in electronic packages, Trans. ASME J. Electron. Packag. 118 (1996) 229–234. [24] A.N. Gent, D.A. Tompkins, Surface energy effects for small holes or particles in elastomers, J. Polym. Sci. Part A2 7 (1969) 1483–1488. [25] B. Huo, Q.-S. Zheng, Y. Huang, A note on the effect of surface energy and void size to void growth, Eur. J. Mech. A/Solids 18 (1999) 987–994. [26] C. Fond, Cavitation criterion for rubber materials: a review of voidgrowth models, J. Polymer Sci. Part B 39 (2001) 2081–2096. [27] J. Dollhofer, A. Chiche, V. Muralidharan, C. Creton, C.Y. Hui, Surface energy effects for cavity growth and nucleation in an incompressible neo-Hookean material: modeling and experiment, Int. J. Solids Struct. 41 (2004) 6111–6127. [28] Z.P. Huang, J. Wang, A theory of hyperelasticity of multi-phase media with surface/interface energy effect, Acta Mech. 182 (2006) 195–210. [29] R.J. Needs, M.J. Godfrey, Surface stress of aluminum and jellium, Phys. Rev. B 42 (1990) 10933–10939. [30] C.O. Horgan, T.J. Pence, Void nucleation in tensile dead-loading of a composite incompressible nonlinearly elastic sphere, J. Elasticity 21 (1989) 61–82. [31] C.A. Stuart, Radially symmetric cavitation for hyperelastic materials, Ann. Inst. Henri Poincaré Anal. Non-Linéaire 2 (1985) 33–66. [32] M. Levinson, I.W. Burgess, A comparison of some simple constitutive relations for slightly compressible rubberlike materials, Int. J. Mech. Sci. 13 (1971) 563–572. [33] S. Biwa, Nonlinear analysis of particle cavitation and matrix yielding under equitriaxial stress, Trans. ASME J. Appl. Mech. 66 (1999) 780–785. [34] A.E. Green, On some general formulae in finite elastostatics, Arch. Ration. Mech. Anal. 50 (1973) 73–80. [35] M. Makino, T. Tsuji, N. Noda, MD simulation of atom-order void formation in Ni fcc metal, Comput. Mech. 26 (2000) 281–287.