Cavitation in nonlinear elastodynamics for neo-Hookean materials

OlT20-7225/89$3.00+ 0.00 Copyright @ 1989Maxwell Pergamon Macmillan plc

ht. 1. Engng Sci. Vol. 21, No. 8, pp. 961-973, 1989

Printed in Great Britain. All rights reserved

CAVITATION IN NONLINEAR ELASTODYNAMICS FOR NEO-HOOKEAN MATERIALS M.-S. (OLIVIA) College of Engineering,

CHOU-WANG

Michigan State University, East Lansing, MI 48824, U.S.A.

C. 0. HORGAN? Department

of Applied Mathematics,

University of Virginia, Charlottesville,

VA 22903, U.S.A

(Communicated by E. S. !jUHUBi) Ah&act-In this paper, we consider the problem of an incompressible elastic solid sphere, composed of a neo-Hookean material, which is set into motion by a suddenly applied uniform radial tensile dead-load pa on its boundary. One solution, for all values of pa, corresponds to a trivial homogeneous static state in which the sphere remains undeformed while stressed. However, for sufficiently large values of pe, another radially symmetric solution exists involving a traction-free spherical cavity at the center of the sphere. The applied load at which such “cavitation” can occur in the dynamic problem is shown to coincide with that for the corresponding static problem.

1. INTRODUCTION Void nucleation and growth in solids have long been of concern because of the fundamental role such phenomena play in fracture and other failure mechanisms. (See e.g. [l] for a discussion of cavity nucleation in metals.) The phenomenon of sudden void formation (“cavitation”) has also been observed experimentally in vulcanized rubber by Gent and Lindley [2]. Nonlinear theories of solid mechanics have been used recently to account for such phenomena. The impetus for much of the recent theoretical developments has been supplied by the work of Ball [3]. In [3], Ball has made an extensive study of a class of bifurcation problems for the equations of nonlinear elasticity which model the appearance of a cavity in the interior of an apparently solid homogeneous isotropic elastic body once a critical load has been attained. An alternative interpretation for such problems in terms of the growth of a pre-existing micro-void is given in [4]. Further investigations of such bifurcation problems have been carried out in [5-121. It is worth noting that cavitation can be shown to occur only when finite strain measures are taken into account (see e.g. [4,9]). The corresponding problems in linearized elasticity or in the infinitesimal strain theory of plasticity do not exhibit such bifurcations. In [3], Ball provides a detailed comprehensive bifurcation treatment, within the theory of finite elastostatics, of the problem of a solid sphere subjected to a uniform radial tensile traction on its boundary. One solution to this problem, for all values of load, is that which corresponds to a homogeneous deformation in which the sphere expands uniformly. However, for certain materials, there is a critical value of the applied load at which the homogeneous solution becomes unstable and a non-homogeneous deformation bifurcates from the homogeneous one. In this case, the body contains a traction-free spherical cavity at its center. The critical load at which such “cavitation” occurs is automatically determined by the bifurcation analysis. It is shown in [4] that this critical load may also be interpreted as the load required to produce sudden rapid growth of a pre-existing micro-void. The purpose of the present paper is to examine the bifurcation approach to void nucleation within the context of nonlinear elastodynamics. In particular, our interest is in assessing the effects of inertia on the critical load at which bifurcation may occur. We consider the radially symmetric motion of an isotropic incompressible elastic solid sphere which is set into motion at time t = 0 by a suddenly applied uniform radial tensile dead-load p,, on its boundary. For materials for which static cavitation occurs, one might conjecture that cavitation should also occur in the dynamic problem and that the critical loads in both cases would coincide. For t Author to whom correspondenceshould be addressed. 967

968

M.-S. (OLIVIA) CHOU-WANGand C. 0. MORGAN

simplicity of presentation, we will consider the special case of a neo-Hookean which the critical load in the static problem is known to be (see e.g. [3, IO])

material

for

PO= 5P/2,

(1.1) where p> 0 denotes the shear modulus for infinitesimal deformations of the material. For the incompressible material at hand, the effect of the tensile load is felt immediately throughout the medium and the response takes the form of a nonlinear oscillation. Oscillation problems for incompressible materials were first investigated by Knowles [13,14] for hollow circular cylinders and have received considerable attention since then (see e.g. [15-181 and the references cited therein). We will show that one solution to the dynamic problem described above, for all values of pot corresponds to a trivial homogeneous static state in which the sphere remains undeformed while stressed. However, for sufficiently large values of po, one has in addition another possible radially symmetric motion involving an internal traction-free cavity. A relationship between the applied load p. and cavity radius c(t) at time t is obtained in the form of a second-order nonlinear ordinary differential equation [see equation (3.11)]. By adapting the techniques of Knowles [13], [14], we show that periodic oscillations can occur if and only if the applied tensile dead-load p. is such that p. 2 5~12.

(1.2)

As po+ 5~/2+, the deformed cavity radius c(t) +O+. It is thus shown that the value of the “cri&cal load” at which an internal void may be initiated in the dynumic problem coincides with that for the static problem. Thus following application of a load p. > 5~12 at time t = 0, the void would expand until its radius reaches a maximum value given by equation (4.8), then would contract to zero and repeat the cycle. It should be noted that the stability of equilibrium solutions with respect to radial motions satisfying the equations of elastodynamics for a wide class of incompressible materials was discussed in [3]. An investigation of cavitation in nonlinear elastodynamics for compressible materials has also been carried out recently [19]. For compressible materials, the wave propagation character of the motion requires an entirely different approach to that described in the present paper. 2. FORMULATION We consider the radially symmetric motion of an isotropic incompressible elastic solid sphere composed of a neo-Hookean material. The undeformed sphere has radius b, and it is set into motion at time t = 0 by a suddenly applied uniform radial tensile dead-load po. In this incompressible case, the effect of the tensile load is felt immediately throughout the medium, and the response takes the form of a nonlinear oscillation. Large amplitude oscillations of hollow incompressible elastic cylinders were considered by Knowles [13,14]. Methods similar to those used in [13,14] have been applied to the case of symmetric motions of a hollow thick-walled incompressible elastic sphere in [15], and an unbounded incompressible elastic medium containing a spherical cavity has been treated in [XI]. See [17] for a review of some of this work. See also the recent paper f18] for a treatment, using phase-plane arguments, of radial motion of thick spherical shells composed of in~mpressible materials. The emphasis in ]13-161 is on the characteristics of the motion, such as the period and amplitude, and on conditions which will ensure the existence of periodic motions. Here we use the techniques developed in 113-161 to explore the possibility of dynamic cavitation. Let Do = {(r, 8, Cp)/ 0 C=r c b, 0 < 8 5 23r, 0 5 CpI; n} denote the interior of the sphere in its undeformed configuration. A point which at time t has spherical coordinates (R, 0, aZ) is assumed to have been at the point (r, 8, @) in the undeformed state. The motion is thus described by R=R(r,t)>O,O
Since the material is assumed to be incompressible,

the

969

Cavitation in nonlinear eIastoclynamics

deformation gradient F obeys det F = 1, t 10. For the motion (2.1), this implies R2 dR/& which when integrated gives R = R(r, t) = [r3 + c3(t)]‘“, c(t) s 0, t I, 0,

= r2, (2.2)

where c(t) is to be determined. The motion is completely dete~ined once c(t) is known. If it is found that c(r) = 0 for t 2 0, (2.2) implies that the body remains a solid sphere in the current configuration. On the other hand if c(t) > 0 (i.e. R(O+, t) > 0), there is a cavity of radius c(t) centered at the origin in the current configuration. In this event, the cavity surface is assumed to be traction-free. For the neo-Hookean material, the strain energy density per unit undeformed volume is given by W(AI,

A2,

j13)

=

f

(A:

+

A$

+

At

3),

-

A,h2A3=

1,

(2.3)

where hi (i = 1, 2, 3) are the principal stretches, and y > 0 is the shear modulus for infinitesimal deform&ons. For the radially symmetric motion (2.1), the principal stretches are given by il, = aR(r, t)/dr, a, = A, = R(r, t)/r. The principal components of the Cauchy stress tensor t are given by

aw tij = Ai x -

p (no sum on i),

(2.4)

*

where p is the hydrostatic pressure associated with the incompressibility constraint &L& = 1. For the material (2.3) and the motion (2.1), equation (2.4) can be written as r&R,

- P(R, t),

t) = p (R3 ;c’4’3 2

rcm(R,

t)

=

Q&C

4

=

F

(R3

”

c3)z3

where P(R, t) now represents the arbitrary hydrostatic It is assumed that the sphere is in an undeformed R(r, 0) = I, d(r, 0) = 0, and so from (2.2) we deduce satisfy the initial conditions c(0) = 0, d(0) =

-

P(R, t), t =-0,

(2.5)

pressure. state and at rest at time t = 0, so that that the current cavity radius c(t) must 0,

(2.6)

where the dot denotes differentiation with respect to time. A dead-load p0 is suddenly applied and maintained at the surface of the sphere so that the boundary conditions are (2.7)

where p. is a positive constant and A = R(b, t) = {b3 + c(t)3}1’3 is the deformed outer radius. In addition if c(t) > 0, then the condition for a traction-free cavity surface r&c,

t) = 0, t 220,

(2.8)

must also hold. The equations of motion, in the absence of body force, governing the radially symmetric motion of the sphere reduce to the single equation

%tR

1

-+-(2~~R-t08-Z~Qt)=p-ji, dR R

tro,

(2.9)

where p is the constant mass density of the material. Thus the problem to be solved is the following: For a prescribed value of the dead-load traction po> 0, we seek a pressure fieid P(R, t), and a time dependent function c(t) 1: 0, sucla that (2.23, (2.61, (2.9) and (2.7) are

910

M.-S. (OLIVIA) CHOU-WANG and C. 0. HORGAN

satisfied where tRR, tss, satisfied.

z*@ are given by (2.5). In addition if c(t) > 0, then (2.8) must also be

3.

SOLUTIONS

It is readily shown that one solution to the problem described in Section 2, for all values of PO, 1s

P(R, t)=p-po, This corresponds

c(t)-0,

to the trivial homogeneous

tro.

(3. I)

(static) state of deformation

R(r, t) = r, t 20,

(3.2)

with corresponding stresses rRR = too = tmQ = pO. Next we describe solutions for which c(t) > 0, corresponding to the presence traction-free cavity at the origin. On substitution from (2.5) into (2.9) we obtain

is[CR3;t3)u3-

t)] + g [ CR3 --f3)M - (R3 “:3)u3] =pa*

P(R,

of a

(3.3)

The incompressibility condition (2.2) is now used to compute the acceleration d2R/dt2 in terms of the acceleration d2c(t)/dt2 of particles on the cavity surface, so that we have d2R

-

2+c’R-2?$

= ~cR-~(R~ _ c3)

dt2 Equation (3.4) is now introduced

(3.4)

into the right hand side of (3.3) to yield

(R3 --;3)4’3 _P(R, t)] + s [(R3--~3)4’3 _(R3 !;3,_] = 2pK5(R3

- c”)

2 + ~c’R-~$.

(3.5)

Equation (3.5) is now integrated with respect to R, and so we obtain ij

cL(R3 - c3)4’3

R4

- P(R, t) + P(c, t) + 2,~

(53

=2pc

(

_

c3)2/3

1

dE

dc 2 R(53-c3)dS+pc2& & >I C C5

The integral on the left hand side of (3.6) may be simplified, on integration 5

34

(53

_

c3)2/3

Rd5 dt2.F’ I

(36) ’

by parts, to yield

1 dg

= _ p(R3 - c3)4’3+2cl

R(E3-C3)113

2R4

IC

d5 -2~

E2

R g(lj3 - c3)u3 d&

(3.7)

The first integral on the right hand side of (3.7) is also simplified on integrating by parts to yield 2p

IR(5”-E2c3)1’3 d5

=

(R3- ~~1~‘~ +

_2p

2p

Thus on combining (3.7), (3.8) and evaluating directly, we rewrite (3.6) as follows: - P(R, t) = -P(c, (R3 -R4c3)4’3

CL

(R3

t) + 2~

(3.8)

R

c

_

the integrals on the right hand side of (3.6)

c3)4’3

4R4 +2pc

(

$

+

1

(R3 - c~)I’~ R

2 c3 1 *-R+-J >[

+@$[;-;I.

(3.9)

Cavitation in nonlinear elastociynamics

971

Equation (3.9) is now introduced into the right hand side of the first of (2.5), and then the traction-free cavity surface condition (2.8) is imposed. This leads to P(c, t) = 0, t 2 0, and so we obtain

cmm0 = 2Pf Finally

(R3- c3y3 + (R3- c3y R

dR4

]+2PC(~)g~-f+~]+pEZ~[~-~].

the boundary condition (2.7) at R = A = {b’ + am} Po[ (b3 :c3J

= 2P[ 4@3 !$W3

(3.10)

is satisfied if

+ (b3 _+!c~)I/3]

c3 1 +-1 4(b3 + c~).+‘~ - (b3 + c~)*‘~ 4c

I

1 (b3 + c~)“~ ’

I

t 2 0.

(3.11)

The second-order nonlinear ordinary differenti~ equation (3.11) provides a relationship between the applied load p. and cavity radius c(t) > 0 for the neo-Hookean material. The static counterpart of this relationship may be obtained by formally replacing c(t) in (3.11) by the constant c. Thus, for the equilibrium problem, the load required to sustain a cavity of radius c is given by (cf. [3, lo]) p. = ; [(1+ c3/b3)-z3 + 4(1 + cW)“‘3].

(3.12)

As c + O+, po+ 5~/2+ from which we recover the result (1.1) for the critical load at which a cavity may be initiated in the static case.

4.

To treat the differential equation dimensionless cavity radius by

OSCILLATIONS

(3.11), we adopt the techniques

of [13]. Defining the

c(t)

x(t)=7>0,

where & is the original undeformed

(4.1)

radius of the solid sphere, we have

dc c(I)=bx(f),~=b~,~=b~.

(4.2)

On introducing the notation (4.3)

f(xf=~[411+:3)“+(1+:3~~“]’

and using (4.2), we rewrite (3.11) as PO pb2( 1 + x3)213=x

I dx2x3 +2x (- )[ dt

4(1 +-x3)4/3-(l

Since the motion starts when the sphere is undeformed (2.6), (4.2) that the initial conditions x(0)=0, must also hold.

F=o,

1

+x3)1/3 +-;]

+f(x),

tro.

(4.4)

and at rest [see (2.6)], we deduce from

(4.5)

972

M.-S. (OLIVIA) CHOU-WANG

and C. 0. HORGAN

With the notation u = dwldt, d2xldt2 = v dvldx, equation (4.4) in the form

it is possible

to write the differential

2X”Po

(4.6)

pb2( 1 + x3)=’

Using (4.3), we find that (4.6) may be integrated with respect to x over the interval from zero to x to yield x4

:--

(I+~3)~,3]v~+~[(‘+~)u3-4(~+Ix3)“ia]=~[(l+x~)~’~-l],

tro.

(4.7)

[

It is well known from the theory of vibrations that the motion x(t) is periodic if and only if the “energy curves” (4.7) are closed curves in the x-v plane with a finite period

The energy curve in the x-v plane is symmetric about the x-axis. This curve, given by (4.7), starts at the initial point x = 0, v = 0 at time t = 0. If p,, is sufficiently large to produce an internal cavity, x and v then move into the region x > 0, v > 0 as t increases from zero. If v passes through a maximum and returns to zero as x increases from zero, the curve will be closed. According to (4.7), this will happen for a given p. if there is a root x >O of (4.7) when v = 0. Setting v = 0 in (4.7) we obtain ;=

[(l +x3)=

1]-‘i(l

+x3)Z’3-2(1

:,,,,,,

-iI.

(4.8)

The right hand side of (4.8) is a monotone increasing function of x for x > 0. As x + 0+ in (4.8), we find, using 1’Hopital’s rule, that

PO*:+ Y

2’

For a given p. > $12, we denote by x, the non-zero right hand side of (4.8) is monotonic increasing]. The in the oscillation process. If po< 5~/2, no positive motions do not occur for this range of applied tensile

ofthe

(4.9) root of (4.8) [there is only one since the quantity x, is the maximum cavity radius root of (4.8) exists, and hence periodic loads. Thus we have shown that the value

“critical load” at which an internal cavity may be initiated in the dynamic problem coincides with that for the static problem. [Recall (l.l).] Thus following application of a pressure p. > 5~/2, an internal cavity would form and expand until it would reach the value

x,, given by the root of (4.8), then would contract to zero and repeat the cycle. It is of interest to note that Knowles and Jakub [16] found that no periodic motions exist for values of pressure above 5~/2 for the problem of an unbounded solid, composed of a neo-Hookean material, containing a spherical cavity which is set into motion by the sudden application of a spatially uniform radial pressure to the cavity wall. In fact for this problem, the deformed cavity radius tends to infinity as the applied pressure tends to the value 5~/2. A related observation was made by Gent and Lindley [2] and by Ball [3] for the corresponding static problems. Acknowledgements-This work was supported in part by the National Science Foundation under Grant MSM 85-12825 MSM 8944719. This research is part of a dissertation [20] submitted in partial fulfilment of the requirements for the Ph.D. degree at Michigan State University.

REFERENCES [l] S. H. GOODS and L. M. BROWN, The nucleation of cavities by plastic deformation. Acta met. 27, 1-15 (1979). [2] A. N. GENT and P. B. LINDLEY, Internal rupture of bonded rubber cylinders in tension. Proc. R. Sot. Land. A. 249, 195-205 (1958).

Cavitation in nonlinear ei~t~ynami~

973

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G. VERGARA CAFFARELLI and E. G. VIRGA, Discontinuous energy minimizers in nonlinear elastostatics: an example of J. Ball revisited. J. Efmticity 16, 75-96 (1986). [7] J. SIVALOGANATHAN, Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Rat. Mech. Anal. %,97-136 (1986). [8] J. SIVALOGANATHAN, A field theory approach to stability of radial equilibria in nonlinear elasticity. Math. Proc. Comb, Phif. Sot. 99,589~604 (1986). [9] D.-T. CHUNG, C. 0. HORGAN and R. ABEYARATNE, A note on a bifurcation problem in finite plasticity related to void nucleation. ht. J. Solids Sfruct. 23,983-988 (1987). [lo] C. 0. HORGAN and T. J. PENCE, Void nucleation in tensile dead-loading of a composite incompressible nonlinearly elastic sphere. J. Efasticiry 21. In Press. [ll] C. 0. HORGAN and T. J. PENCE, Cavity formation at the center of a composite incompressible nonlinearly elastic sphere. 1. appf. Mech. 56. In press. [12] S. S. ANTMAN and P. V. NEGRON-MARRERO, The remarkable nature of radially symmetric equilibrium states of aeolotropic nonlinearly elastic bodies. J. Elasticity 18, 131-164 (1987). [13] J. K. KNOWLES, Large amplitude oscillations of a tube of incompressible elastic material. (2. appf. Math. 18, 71-77 (1960). [14] J. K. KNOWLES, On a class of oscillations in the finite deformation theory of elasticity. J. appl. Mech. 29, 283-286 (1962). [15] Z. H. GUO and R. SOLECKI, Free and forced finite amplitude oscillations of an elastic thick-walled hollow sphere made of incompressible material. Arch. Mech. Stos. lS,427-433 (1963). [16] J. K. KNOWLES and M. T. JAKUB, Finite dynamic deformations of an incompressible elastic medium containing a spherical cavity. Arch. Rat. Mech. Anai. 18, 367-378 (1964). 1171 A. C. ERINGEN and E. S. SUHUBI, Ef~tody~fcs, Vol. 1. Academic Press, New York (1975). [18] C. CALDERER, The dynamical behavior of nonlinear elastic spherical shells. I: ~f~~ci~ W, 17-47 (1983). 1191 K. A. PERICAK-SPECTOR and S. J. SPECTOR, Nonuniqueness for a hyperbolic system: cavitation in nonlinear eiastodynamics. IMA Preprint 305, University of Minnesota. Arch. Rui. h4ech. Amf. 101, 293-317 (1988). [20] M.-S. (OLIVIA) CHOU-WANG, Studies on void formation and growth for incompressible nonlinearly elastic materials. Ph.D. dissertation, Michigan State University (1988). (Received 21 November

1988)