Cavitation in an elastomer: Comparison of theory with experiment

Materials Science and Engineering, A 112 ( 1989) 127-131

127

Cavitation in an Elastomer: Comparison of Theory with Experiment RICHARD STRINGFELLOW and ROHAN ABEYARATNE

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 (U.S.A.) (Received October 31, 1988; in revised form December 2, 1988)

Abstract

We compare the predictions of Ball's bifurcation analysis, which provides a theoretical prediction for when a void wouM nucleate in a radially deformed solid sphere, with the experiments of Oberth and Bruenner and those of Gent and Lindley. 1. Introduction Ball [1] considered the radial deformation of a solid sphere composed of an incompressible elastic material and showed that for certain constitutive laws, when the applied stress on the boundary exceeds a critical value, it is energetically more favorable for the body to deform inhomogeneously by developing an internal cavity than for it to continue to deform homogeneously in pure dilatation. His analysis yielded an explicit formula for the critical value of stress at which a cavity is expected to nucleate. (Partial results pertaining to compressible elastic materials are given in refs. 1-6; for the influence of nonradial deformations, see ref. 7; for the influence of rate-dependence, see ref. 8.) In the present study we compare the predictions of Bali's analysis with some experimental observations. Oberth and Bruenner [9] studied the tensile deformation of a polyurethane bar which had a steel sphere embedded in it. They observed that (when the particle-matrix bonding was strong) a small cavity appeared within the matrix near a pole of the particle when the applied uniaxial stress reached a certain critical value. As argued in Section 3, the state of stress near the poles of the sphere is purely hydrostatic and consequently Bali's analysis is locally applicable there. We carried out a finite element analysis of a tensile test specimen containing an embedded spherical particle and related the applied stress to the local hydrostatic stress near 0921-5093/89/$3.50

the poles of the particle, and then used Bali's result to predict when a cavity would be expected to appear. The predictions agree well with the experimental observations. In Section 4, we re-examine the experiments of Gent and Lindley [10]; Ball [1] previously commented on this work. Gent and Lindley took a short cylinder of vulcanized rubber, bonded the plane faces of the cylinder to two metal end pieces and then pulled the metal plates apart. They observed a hole form in the center of the rubber specimen when the applied axial stress reached a critical value. They (i) assumed the state of stress at the center of the specimen to be hydrostatic, (ii) used an approximate formula to calculate this stress and (iii) used it to calculate a theoretical value for the applied stress at cavitation. Their predictions compared well with their observations. Firstly, we carry out a finite element stress analysis of Gent and Lindley's test specimen to determine the extent to which the state of stress at the center of the specimen differs from a purely hydrostatic stress state; we find that the difference is about 10% (in a certain precise sense). Secondly, we use this numerical solution to relate the mean hydrostatic stress at the center of the specimen to the applied axial stress, and couple this with Ball's result to determine the predicted value of the applied stress at cavitation.

2. Analysis details An elastic material is characterized by its strain energy density function W. When the material is isotropic, W depends on the deformation only through the principal stretches 21,42 and 23, i.e. W= W(41, 42, 43)

(1)

If the material is incompressible, 414243 = 1. The principal true stresses associated with any defor© Elsevier Sequoia/Printed in The Netherlands

128 mation of an incompressible, isotropic elastic material are given by

material the stress at cavitation is

0W ri = 2i ~-~i- q

Per(2)

for i = 1, 2, 3, where the pressure q is induced by the incompressibility constraint. Let us consider a solid sphere of such a material with undeformed outer radius b, which has a uniform nominal radial tensile stress p ( p > 0 ) applied to its surface. Because of the assumed incompressibility and spherical symmetry, as p is increased, the sphere does not deform at first. However, when p reaches a certain critical value P~r, Ball [1] has shown that it is energetically more favorable for the body to develop a spherical internal cavity. The value of the applied stress at cavitation is given by 0o

Per =

f ~l

U'(2) d2

(3)

1 where U ( 2 ) - W(2 -2, 4, 4). For certain materials, the strain energy function W is such that the integral in eqn. (3) does not converge; in this case p = oo and so such a material would never develop a void. The simplest constitutive law describing an incompressible, isotropic elastic material is the neo-Hookean model [11-13] which follows as a first approximation to the molecular theory of long-chain molecules. In this case, eqn. (1) takes on the explicit form

5E 6

Alternatively, because the infinitesimal shear modulus/~ = E/3 for an incompressible material, one can write Per 5/*/2. (This differs from ref. 1, eqn. (5.66), by a factor of two, because/~ in Bali's paper refers to one-half of the shear modulus.) =

3. Cavitation in the vicinity of a rigid inclusion

Oberth and Bruenner [9] conducted tensile tests on polyurethane bars, each of which had a 0.25 in steel sphere embedded centrally in it. By adjusting the degree of cross-linking in the polyurethane, they were able to test specimens which had different mechanical properties. They also varied the matrix-particle bond strength. When a tensile test was carried out on specimens in which the matrix-particle bonding was strong, they observed a small cavity to appear in the matrix near a pole of the embedded spherical panicle, i.e. on the central axis of the specimen very near the particle. This occurred when the applied (nominal) tensile stress S reached a critical value Scr. The test was repeated on a range of specimens whose Young's moduli varied from about 10 to 200 lb in -2. The following correlation between the applied tensile stress at cavitation Scr and the Young's modulus E was obtained experimentally S , = 0.5E +8

W(21, 22, 23) = E (212 + 222 + ~,32-- 3)

(4)

where E is the Young's modulus of the material at infinitesimal deformations. From eqn. (2), the associated stress-stretch relationship is E

for i = 1 , 2 , 3 . According to eqn. (5), the stress-stretch relationship of a neo-Hookean material in uniaxial tension is r = (E/3 )(22 _ 2 - 1) for the true stress, and 0 = ( E / 3 ) ( 2 - 2 -2) for the nominal stress. In simple shear, the shear stress is related linearly to the amount of shear. When the particular form of eqn. (4) is used in eqn. (3), one finds that for a neo-Hookean

(6)

(lb in -2)

(7)

Oberth and Breuhner found no correlation between Scr and the tensile strength of the materials. The state of stress at the poles in this specimen is precisely one of pure hydrostatic tension. This follows from the fact that the two hoop stretches at each pole are both unity (assuming the bond to be perfect and the spherical particle to be rigid), which in turn implies that the third principal stretch is also unity (assuming the matrix material to be incompressible). Thus, the three principal stretches at each pole are equal and so from eqn. (2) the stress state is purely hydrostatic. Consequently, the analysis of Section 2 is locally applicable near the poles of the panicle. Using the finite-element program ABAQUS (version 4.5-159), we performed a stress analysis of the tensile test specimen. The tensile test speci-

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mens used in the experiments had a rectangular cross-section of 0.35 in x 0.5 in. To simplify the numerical solution we studied two bars of circular cross-section with diameters of 0.35 and 0.5 in. One would expect the stresses in the rectangular bar to lie between those in the two circular bars. The non-linear geometry and hyperelastic material options of ABAQUS were used to carry out an incremental analysis. Using mid-plane and axial symmetry, a mesh of 140 elements and 1109 degrees of freedom was used to model onequarter of the test specimen. Because of the incompressible nature of the neo-Hookean constitutive law, 8-node hybrid stress-displacement elements were chosen for the analysis; these treat the pressure terms independently, thereby avoiding the numerical difficulties arising from the use of a purely displacement-based finiteelement method. The particle was modeled as being rigid. Figure 1 shows the contours of constant mean nominal stress (0.11 -t- 0"22 --b 0"33)/3, indicating that it is largest near the poles of the inclusion. (The detailed numerical results show that, in fact, the maximum is attained at a point slightly away from the poles. This phenomenon of a maximum stress occurring at an interior point of a body has been observed previously, e.g. by Abeyaratne and Horgan [14], and Wilner [15].) We now let p be the maximum mean nominal stress in the specimen. For different values of the applied stress S, we determined the associated value of p; the results are shown in Fig. 2. The linear relationships p = 1.824S

(0.5 in diameter bar)

p = 1.482S

(0.35 in diameter bar)

(0.5 in diameter)

Scr = 0.562E

(0.35 in diameter)

2

?'6

t

Fig. 1. Contours of constant mean nominal stress in a bar of diameter 0.5 in containing a particle of diameter 0.25 in. Average applied axial stress is 1.35E. 1, P/E=O.O0; 2, P/E=0.33; 3, P/E=0.67; 4, P/E= 1.00; 5, PIE = 1.33; 6, P/E= 1.67; 7, P/E=2.00. 3

(8)

were found to fit the data very well over the range 0 ~ S/E ~<1 which includes the range of S encountered by Oberth and Bruenner [9]. (In an infinite, linearly elastic medium, Goodier [16] has shown that p =2S.) Because the analysis of Section 2 predicts the appearance of a cavity near the poles of the particle when p reaches 5E/6, eqn. (8) leads to the theoretical relationships Scr = 0.457E

3

2 P/E 1

0

0.0

0.5

1.0

1.5

5/E Fig. 2. Variation in maximum mean nominal stress p with average applied stress S: (a) bar of diameter 0.5 in; (b) bar of diameter 0.35 in.

(9)

The theoretical and experimental values of Scr (eqn. (9) and eqn. (7) respectively) are seen to be in good agreement. As one might expect, the critical stress for the rectangular specimen is

essentially bounded by the corresponding results for the two circular specimens. The additive term in eqn. (7) may be explained by the fact that the analysis leading to eqn. (3) assumed the cavity to be traction-free after nucleation. Allowing for

130 internal (atmospheric) pressure introduces an additive term into eqn. (9) of the form found in eqn. (7).

Gent and Lindley [10] took short circular cylinders of vulcanized rubber, bonded the flat surfaces to two metal disks, and studied the response of the rubber as the metal disks were moved apart. They observed a distinct kink in the load-extension curve at a certain level of load and also heard a "popping" sound at that load. By cutting open the rubber disks, they concluded that the kink and popping signalled the formation of a small void in the center of the specimen. They repeated the test on a number of different vulcanized rubbers whose Young's moduli ranged from 1000 to 4000 kPa; they concluded experimentally that the critical (nominal) axial stress Scr, although not correlated to the ultimate tensile strength of the material, was related to the Young's modulus E by +49

(kPa)

a P/E zl

4. Cavitation in a circular disk

Scr--- 0 . 5 5 E

4

(10)

Gent and Lindley used an approximate stress analysis to relate the applied axial stress S to the mean nominal stress p at the center of the specimen. They then used a failure criterion based on the expansion of an infinitesimal spherical void (which is equivalent to the analysis of Section 2) to obtain the theoretical value Scr = 0.49E. The state of stress at the center of the specimen is not purely hydrostatic and so the analysis of Section 2 (or that of Gent and Lindley) does not rigorously apply to the present problem. To examine the departure from a purely hydrostatic state, we carried out a finite element stress analysis of Gent and Lindley's specimen using the program ABAQUS. The cylinder had a diameter Of 2 cm and a height of 0.3 cm. The constitutive law was again assumed to be neo-Hookean. We calculated the ratio between the axial stress and the radial stress at the center of the specimen at different values of the applied tensile stress S. It was found that this ratio was approximately 1.15 and that it remained essentially constant at this value for all values of the applied stress in the range 0 ~
0 0.0

I 0.5

I 1.0

1.5

S/E Fig. 3. Variation in maximum mean nominal stress p with average applied stress S. by Gent and Lindley. The axial stress exceeds the mean pressure by 10%, whereas the hoop and radial stresses are 5% smaller than the mean pressure. Next, we used the results of the finite element stress analysis to relate the applied axial stress S to the mean nominal stress p at the center of the specimen; a graph of S vs. p is shown in Fig. 3. For stresses in the range of Gent and Lindley's experiments, S and p were found to be related linearly by p=1.75S

(11)

The analysis of Section 2 predicts that a cavity will appear, under purely hydrostatic conditions, when p = 5E/6. It is not unreasonable to assume that, as a first.approximation, this continues to be true even in the present non-hydrostatic case. Thus, from eqns. (6) and (11) we obtain the theoretical relationship • Scr= 0.476E

(12)

which compares favorably with the experimental result of eqn. (10). 5. Conclusion

The simple formula of eqn. (3) appears to provide an accurate estimate for the stress level at which cavitation occurs in rubber-like materials under hydrostatic conditions. Acknowledgments

Computations were performed on a Data General MV-10000 computer donated to M.I.T. by the Data General Corporation. The A B A Q U S finite element program was made available under

131

academic licence from Hibbitt, Karlsson and Sorensen, Inc., Providence, RI. The results were obtained in the course of an investigation supported in part by the U.S. Army Research Office.

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