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Onset of cavitation in compressible, isotropic, hyperelastic solids. (English) Zbl 1160.74317

Summary: In this work, we derive a \(closed-form\) criterion for the onset of cavitation in compressible, isotropic, hyperelastic solids subjected to non-symmetric loading conditions. The criterion is based on the solution of a boundary value problem where a hyperelastic solid, which is infinite in extent and contains a single vacuous inhomogeneity, is subjected to uniform displacement boundary conditions. By making use of the “linear-comparison” variational procedure of O. Lopez-Pamies and P. Ponte Castañeda [J. Mech. Phys. Solids 54, No. 4, 807–830 (2006; Zbl 1120.74726)], we solve this problem approximately and generate variational estimates for the critical stretches applied on the boundary at which the cavity suddenly starts growing. The accuracy of the proposed analytical result is assessed by comparisons with exact solutions available from the literature for radially symmetric cavitation, as well as with finite element simulations. In addition, applications are presented for a variety of materials of practical and theoretical interest, including the harmonic, Blatz-Ko, and compressible Neo-Hookean materials.

MSC:

74B20 Nonlinear elasticity
74G65 Energy minimization in equilibrium problems in solid mechanics

Citations:

Zbl 1120.74726
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References:

[1] Lopez-Pamies, O., Ponte Castañeda, P.: On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations. I–Theory. J. Mech. Phys. Solids 54, 807–830 (2006) · Zbl 1120.74726 · doi:10.1016/j.jmps.2005.10.006
[2] Busse, W.F.: Physics of rubber as related to the automobile. J. Appl. Phys. 9, 438–451 (1938) · doi:10.1063/1.1710439
[3] Yerzley, F.L.: Adhesion of neoprene to metal. Ind. Eng. Chem. 31, 950–956 (1939) · doi:10.1021/ie50356a007
[4] Gent, A.N., Lindley, P.B.: Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. Lond. A 249, 195–205 (1958)
[5] Lindsey, G.H.: Triaxial fracture studies. J. Appl. Phys. 38, 4843–4852 (1967) · doi:10.1063/1.1709232
[6] Gent, A.N., Park, B.: Failure processes in elastomers at or near a rigid spherical inclusion. J. Mater. Sci. 19, 1947–1956 (1984) · doi:10.1007/BF00550265
[7] Cho, K., Gent, A.N.: Cavitation in model elastomeric composites. J. Mater. Sci. 23, 141–144 (1988) · doi:10.1007/BF01174045
[8] Donald, A.M., Kramer, E.J.: Plastic deformation mechanisms in poly(acrylonitrile-butadiene styrene) [ABS]. J. Mater. Sci. 17, 1765–1772 (1982) · doi:10.1007/BF00540805
[9] Schirrer, R., Fond, C., Lobbrecht, A.: Volume change and light scattering during mechanical damage in polymethylmethacrylate toughened with core-shell rubber particles. J. Mater. Sci. 31, 6409–6422 (1996) · doi:10.1007/BF00356243
[10] Fond, C.: Cavitation criterion for rubber materials: A review of void-growth models. J. Polym. Sci. Part B 39, 2081–2096 (2001) · doi:10.1002/polb.1183
[11] Green, A.E., Zerna, W.: Theoretical Elasticity. Oxford University Press, London (1954) · Zbl 0056.18205
[12] Ball, J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. R. Soc. A 306, 557–611 (1982) · Zbl 0513.73020 · doi:10.1098/rsta.1982.0095
[13] Stuart, C.A.: Radially symmetric cavitation for hyperelastic materials. Ann. Inst. Henri Poincaré Anal. non Linéaire 2, 33–66 (1985) · Zbl 0588.73021
[14] Sivaloganathan, J.: Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Rat. Mech. Anal. 96, 97–136 (1986) · Zbl 0628.73018 · doi:10.1007/BF00251407
[15] Sivaloganathan, J., Spector, S.J., Tilakraj, V.: The convergence of regularized minimizers for cavitation problems in nonlinear elasticity. SIAM J. Appl. Math. 66, 736–757 (2006) · Zbl 1104.74016 · doi:10.1137/040618965
[16] Sivaloganathan, J.: A field theory approach to stability of radial equilibria in nonlinear elasticity. Math. Proc. Camb. Philos. Soc. 99, 589–604 (1986) · Zbl 0612.73013 · doi:10.1017/S0305004100064513
[17] Horgan, C.O., Abeyaratne, R.: A bifurcation problem for a compressible nonlinearly elastic medium: growth of a micro-void. J. Elast. 16, 189–200 (1986) · Zbl 0585.73017 · doi:10.1007/BF00043585
[18] Horgan, C.O.: Void nucleation and growth for compressible non-linearly elastic materials: an example. Int. J. Solids Struct. 29, 279–291 (1992) · Zbl 0755.73026 · doi:10.1016/0020-7683(92)90200-D
[19] Meynard, F.: Existence and nonexistence results on the radially symmetric cavitation problem. Q. Appl. Math. L, 201–226 (1992) · Zbl 0755.73027
[20] Antman, S.S., Negrón-Marrero, P.V.: The remarkable nature of radially symmetric equilibrium states of aleotropic nonlinearly elastic bodies. J. Elast. 18, 131–164 (1987) · Zbl 0631.73016 · doi:10.1007/BF00127554
[21] Polignone, D.A., Horgan, C.O.: Cavitation for incompressible anisotropic nonlinearly elastic spheres. J. Elast. 33, 27–65 (1993) · Zbl 0856.73024 · doi:10.1007/BF00042634
[22] Pericak-Spector, K.A., Spector, S.J.: Nonuniqueness for hyperbolic systems: cavitation in nonlinear elastodynamics. Arch. Rat. Mech. Anal. 101, 293–317 (1988) · Zbl 0651.73005 · doi:10.1007/BF00251490
[23] Horgan, C.O., Polignone, D.A.: Cavitation in nonlinearly elastic solids: a review. Appl. Mech. Rev. 48, 471–485 (1995) · doi:10.1115/1.3005108
[24] James, R.D., Spector, S.J.: The formation of filamentary voids in solids. J. Mech. Phys. Solids 39, 783–813 (1991) · Zbl 0761.73020 · doi:10.1016/0022-5096(91)90025-J
[25] Hou, H.-S., Abeyaratne, R.: Cavitation in elastic and elastic-plastic solids. J. Mech. Phys. Solids 40, 571–592 (1992) · Zbl 0825.73102 · doi:10.1016/0022-5096(92)80004-A
[26] Steenbrink, A.C., Van der Giessen, E.: On cavitation, post-cavitation and yield in amorphous polymer-rubber blends. J. Mech. Phys. Solids 47, 843–876 (1999) · Zbl 0977.74575 · doi:10.1016/S0022-5096(98)00075-1
[27] Huang, Y., Hutchinson, J.W., Tvergaard, V.: Cavitation instabilities in elastic-plastic solids. J. Mech. Phys. Solids 39, 223–241 (1991) · doi:10.1016/0022-5096(91)90004-8
[28] Tvergaard, V., Hutchinson, J.W.: Effect of void shape on the occurrence of cavitation instabilities in elastic-plastic solids. J. Appl. Mech. 60, 807–812 (1993) · Zbl 0800.73093 · doi:10.1115/1.2900987
[29] Sivaloganathan, J., Spector, S.J.: On the existence of minimisers with prescribed singular points in nonlinear elasticity. J. Elast. 59, 83–13 (2000) · Zbl 0987.74016 · doi:10.1023/A:1011001113641
[30] Sivaloganathan, J., Spector, S.J.: On cavitation, configurational forces and implications for fracture in nonlinear elasticity. J. Elast. 67, 25–49 (2002) · Zbl 1089.74627 · doi:10.1023/A:1022594705279
[31] Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. A 241, 376–396 (1957) · Zbl 0079.39606 · doi:10.1098/rspa.1957.0133
[32] Lopez-Pamies, O.: On the effective behavior, microstructure evolution, and macroscopic stability of elastomeric composites. Ph.D. Dissertation, University of Pennsylvania, USA (2006) · Zbl 1120.74727
[33] Ponte Castañeda, P.: Second-order homogenization estimates for nonlinear composites incorporating field fluctuations. I. Theory. J. Mech. Phys. Solids 50, 737–757 (2002) · Zbl 1116.74412 · doi:10.1016/S0022-5096(01)00099-0
[34] Lopez-Pamies, O., Ponte Castañeda, P.: On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations. II–Application to cylindrical fibers. J. Mech. Phys. Solids 54, 831–863 (2006) · Zbl 1120.74727 · doi:10.1016/j.jmps.2005.10.010
[35] Brun, M., Lopez-Pamies, O., Ponte Castañeda, P.: Homogenization estimates for fiber-reinforced elastomers with periodic microstructures. Int. J. Solids Struct. 44, 5953–5979 (2007) · Zbl 1186.74095 · doi:10.1016/j.ijsolstr.2007.02.003
[36] Lopez-Pamies, O., Ponte Castañeda, P.: Second-order estimates for the macroscopic response and loss of ellipticity in porous rubbers at large deformations. J. Elast. 76, 247–287 (2004) · Zbl 1086.74032 · doi:10.1007/s10659-005-1405-z
[37] Michel, J.C., Lopez-Pamies, O., Ponte Castañeda, P., Triantafyllidis, N.: Microscopic and macroscopic instabilities in finitely strained porous elastomers. J. Mech. Phys. Solids 55, 900–938 (2007) · Zbl 1170.74018 · doi:10.1016/j.jmps.2006.11.006
[38] Lopez-Pamies, O., Ponte Castañeda, P.: Homogenization-based constitutive models for porous elastomers and implications for microscopic instabilities. I-Analysis. J. Mech. Phys. Solids 55, 1677–1701 (2007) · Zbl 1176.74150 · doi:10.1016/j.jmps.2007.01.007
[39] Lopez-Pamies, O., Garcia, R., Chabert, E., Cavaillé, J.-Y., Ponte Castañeda, P.: Multiscale modeling of oriented thermoplastic elastomers with lamellar morphology. J. Mech. Phys. Solids 56, 3206–3223 (2008) · Zbl 1172.74044 · doi:10.1016/j.jmps.2008.07.008
[40] Willis, J.R.: Bounds and self-consistent estimates for the overall moduli of anisotropic composites. J. Mech. Phys. Solids 25, 185–202 (1977) · Zbl 0363.73014 · doi:10.1016/0022-5096(77)90022-9
[41] Willis, J.R.: Variational and related methods for the overall properties of composites. Adv. Appl. Mech. 21, 1–78 (1981) · Zbl 0476.73053 · doi:10.1016/S0065-2156(08)70330-2
[42] Willis, J.R.: Anisotropic elastic inclusion problems. Q. J. Mech. Appl. Math. 17, 157–173 (1964) · Zbl 0119.39602 · doi:10.1093/qjmam/17.2.157
[43] Idiart, M.I., Ponte Castañeda, P.: Field statistics in nonlinear composites. I. Theory. Proc. R. Soc. Lond. A 463, 183–202 (2006) · Zbl 1129.74036
[44] Ogden, R.W.: Non-linear Elastic Deformations. Dover, New York (1997) · Zbl 0938.74014
[45] Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics, 3rd edn. Springer, Berlin (2004) · Zbl 1068.74002
[46] John, F.: Plane strain problems for a perfectly elastic material of harmonic type. Commun. Pure Appl. Math. 13, 239–296 (1960) · Zbl 0094.37001 · doi:10.1002/cpa.3160130206
[47] Varga, O.H.: Stress-Strain Behaviour of Elastic Materials. Wiley, New York (1966) · Zbl 0166.43201
[48] Steigmann, D.J.: Cavitation in elastic membranes–an example. J. Elast. 28, 277–287 (1992) · Zbl 0777.73031 · doi:10.1007/BF00132216
[49] Hill, R.: Aspects of invariance in solid mechanics. Adv. Appl. Mech. 18, 1–75 (1978) · Zbl 0475.73026 · doi:10.1016/S0065-2156(08)70264-3
[50] Storakers, B.: On material representation and constitutive branching in finite compressible elasticity. J. Mech. Phys. Solids 34, 125–145 (1986) · Zbl 0575.73050 · doi:10.1016/0022-5096(86)90033-5
[51] Blatz, P.J., Ko, W.L.: Application of finite elastic theory to the deformation of rubbery materials. Trans. Soc. Rheol. 6, 223–251 (1962) · doi:10.1122/1.548937
[52] Biwa, S.: Critical stretch for formation of a cylindrical void in a compressible hyperelastic material. Int. J. Non-Linear Mech. 30, 899–914 (1995) · Zbl 0871.73009 · doi:10.1016/0020-7462(96)80776-1
[53] Murphy, J.G., Biwa, S.: Nonmonotonic cavity growth in finite, compressible elasticity. Int. J. Solids Struct. 34, 3859–3872 (1997) · Zbl 0942.74530 · doi:10.1016/S0020-7683(96)00237-5
[54] Sivaloganathan, J.: Cavitation, the incompressible limit, and material inhomogeneity. Q. Appl. Math. XLIX, 521–541 (1991) · Zbl 0756.73040
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