A characterisation of the boundary displacements which induce cavitation in an elastic body. (English) Zbl 1255.74008

Summary: We present numerical and theoretical results for characterising the onset of cavitation-type material instabilities in solids. To model this phenomenon we use nonlinear elasticity to allow for the large, potentially infinite, stresses and strains involved in such deformations. We give a characterisation of the set of linear displacement boundary conditions for which energy minimising deformations produce a single isolated hole inside an originally perfect elastic body, based on a notion of the derivative of the stored energy functional with respect to hole-producing deformations. We conjecture that, for many stored energy functions, the critical linear boundary conditions which cause an isolated cavity to form correspond to the zero set of this derivative. We use this characterisation to propose a numerical procedure for computing these critical boundary displacements for general stored energy functions and give numerical examples for specific materials. For a degenerate stored energy function (with spherically symmetric boundary deformations) and for an elastic fluid, we show that the vanishing of the volume derivative gives exactly the critical boundary conditions for the onset of this type of cavitation.


74B20 Nonlinear elasticity
35J50 Variational methods for elliptic systems
49K20 Optimality conditions for problems involving partial differential equations
74G65 Energy minimization in equilibrium problems in solid mechanics
74G15 Numerical approximation of solutions of equilibrium problems in solid mechanics
74G55 Qualitative behavior of solutions of equilibrium problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74G70 Stress concentrations, singularities in solid mechanics
Full Text: DOI


[1] Antman, S.S., Negrón-Marrero, P.V.: The remarkable nature of radially symmetric equilibrium states of aelotropic nonlinearly elastic bodies. J. Elast. 18, 131–164 (1987) · Zbl 0631.73016 · doi:10.1007/BF00127554
[2] Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977) · Zbl 0368.73040 · doi:10.1007/BF00279992
[3] Ball, J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. Lond. A 306, 557–611 (1982) · Zbl 0513.73020 · doi:10.1098/rsta.1982.0095
[4] Ball, J.M.: Some open problems in elasticity. In: Geometry, Mechanics, and Dynamics, pp. 3–59. Springer, New York (2002) · Zbl 1054.74008
[5] Ball, J.M., Knowles, G.: A numerical method for detecting singular minimisers. Numer. Math. 51, 181–197 (1987) · Zbl 0636.65064 · doi:10.1007/BF01396748
[6] Ball, J.M., Mizel, V.J.: One-dimensional variational problems whose minimisers do not satisfy the Euler-Lagrange equation. Arch. Ration. Mech. Anal. 90, 325–388 (1985) · Zbl 0585.49002 · doi:10.1007/BF00276295
[7] Ball, J.M., Murat, F.: W 1,p -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58, 225–253 (1984) · Zbl 0549.46019 · doi:10.1016/0022-1236(84)90041-7
[8] Conti, S., De Lellis, C.: Some remarks on the theory of elasticity for compressible Neohookean materials. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 5, 521–549 (2003) · Zbl 1114.74004
[9] Evans, L.C.: Quasiconvexity and partial regularity in the calculus of variations. In: Proceedings of the International Congress of Mathematicians, Berkeley, California, USA (1986) · Zbl 0627.49006
[10] Gent, A.N.: Cavitation in rubber: a cautionary tale. Rubber Chem. Technol. 63, G49–G53 (1990)
[11] Gent, A.N., Lindley, P.B.: Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. Lond. A 249, 195–205 (1958) · doi:10.1098/rspa.1959.0016
[12] Henao, D.: Cavitation, invertibility, and convergence of regularized minimisers in nonlinear elasticity. J. Elast. 94, 55–68 (2009) · Zbl 1159.74322 · doi:10.1007/s10659-008-9184-y
[13] Henao, D., Mora-Corral, C.: Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity. Arch. Ration. Mech. Anal. 197, 619–655 (2010) · Zbl 1248.74006 · doi:10.1007/s00205-009-0271-4
[14] Henao, D., Mora-Corral, C.: Lusin’s condition and the distributional determinant for deformations with finite energy. Preprint (2009). Advances in Calculus of Variations. ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, doi: 10.1515/ACV.2011.016
[15] Henao, D., Serfaty, S.: Energy estimates and cavity interaction for a critical-exponent cavitation model. Preprint (2011) · Zbl 1388.74082
[16] Henao, D., Xu, X.: An efficient numerical method for cavitation in nonlinear elasticity. Math. Models Methods Appl. Sci. 21, 1733–1760 (2011) · Zbl 1276.74007 · doi:10.1142/S0218202511005556
[17] Horgan, C.O.: Void nucleation and growth for compressible non-linearly elastic materials: an example. Int. J. Solids Struct. 29(3), 279–291 (1992) · Zbl 0755.73026 · doi:10.1016/0020-7683(92)90200-D
[18] Horgan, C.O., Abeyaratne, R.: A bifurcation problem for a compressible nonlinearly elastic medium: growth of a microvoid. J. Elast. 16, 189–200 (1986) · Zbl 0585.73017 · doi:10.1007/BF00043585
[19] Horgan, C.O., Polignone, D.A.: Cavitation in nonlinearly elastic solids: a review. Appl. Mech. Rev. 48(8), 471–485 (1995) · doi:10.1115/1.3005108
[20] 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
[21] James, R., Spector, S.J.: Remarks on W 1,p -quasiconvexity, interpenetration of matter, and function spaces for elasticity. J. Mech. Phys. Solids 39, 783–813 (1991) · Zbl 0761.73020 · doi:10.1016/0022-5096(91)90025-J
[22] James, R., Spector, S.J.: The formation of filamentary voids in solids. Anal. Non Linéaire 9, 263–280 (1992) · Zbl 0773.73022
[23] Kristensen, J.: On the non locality of quasiconvexity. Anal. Non Linéaire 16, 1–13 (1999) · Zbl 0932.49015 · doi:10.1016/S0294-1449(99)80006-7
[24] Lavrentiev, M.: Sur quelques problemes du calcul des variations. Ann. Mat. Pura Appl. 4, 107–124 (1926)
[25] Li, Z.: Element removal method for singular minimizers in variational problems involving Lavrentiev phenomenon. Proc. R. Soc. Lond. A 439, 131–137 (1992) · Zbl 0777.49003 · doi:10.1098/rspa.1992.0138
[26] Li, Z.: Element removal method for singular minimizers in problems of hyperelasticity. Math. Models Methods Appl. Sci. 5(3), 387–399 (1995) · Zbl 0829.73078 · doi:10.1142/S0218202595000231
[27] Lian, Y., Li, Z.: A dual-parametric finite element method for cavitation in nonlinear elasticity. J. Comput. Appl. Math. (2011). doi: 10.1016/j.cam.2011.05.020 · Zbl 1300.74051
[28] Lian, Y., Li, Z.: A numerical study on cavitations in nonlinear elasticity–defects and configurational forces. Math. Models Methods Appl. Sci. (2011). doi: 10.1142/S0218202511005830
[29] Lopez-Pamies, O., Idiart, M.I., Nakamura, T.: Cavitation in elastomeric solids: I–A defect-growth theory. J. Mech. Phys. Solids 59, 1464–1487 (2011) · Zbl 1270.74025 · doi:10.1016/j.jmps.2011.04.015
[30] Lopez-Pamies, O., Idiart, M.I., Nakamura, T.: Cavitation in elastomeric solids: II–Onset-of-cavitation surfaces for Neo-Hookean materials. J. Mech. Phys. Solids 59, 1488–1505 (2011) · Zbl 1270.74026 · doi:10.1016/j.jmps.2011.04.016
[31] Lopez-Pamies, O.: Onset of cavitation in compressible, isotropic, hyperelastic solids. J. Elast. 94, 115–145 (2009) · Zbl 1160.74317 · doi:10.1007/s10659-008-9187-8
[32] Meyers, N.G.: Quasiconvexity and lower semicontinuity of multiple integrals of any order. Trans. Am. Math. Soc. 119, 225–249 (1965) · doi:10.1090/S0002-9947-1965-0188838-3
[33] Morrey, C.B.: Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2, 25–53 (1952) · Zbl 0046.10803 · doi:10.2140/pjm.1952.2.25
[34] Müller, S., Spector, S.J.: An existence theory for nonlinear elasticity that allows for cavitation. Arch. Ration. Mech. Anal. 131, 1–66 (1995) · Zbl 0836.73025 · doi:10.1007/BF00386070
[35] Müller, S., Sivaloganathan, J., Spector, S.J.: An isoperimetric estimate and W 1,p -quasiconvexity in nonlinear elasticity. Calc. Var. 8, 159–176 (1999) · Zbl 0929.74013 · doi:10.1007/s005260050121
[36] Negrón-Marrero, P.V.: A numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity. Numer. Math. 58, 135–144 (1990) · Zbl 0706.73080 · doi:10.1007/BF01385615
[37] Negrón-Marrero, P.V., Betancourt, O.: The numerical computation of singular minimizers in two dimensional elasticity. J. Comput. Phys. 113(2), 291–303 (1994) · Zbl 0805.73077 · doi:10.1006/jcph.1994.1136
[38] Negrón-Marrero, P.V., Sivaloganathan, J.: The numerical computation of the critical boundary displacement for radial cavitation. Math. Mech. Solids 14, 696–726 (2009) · Zbl 1197.74017 · doi:10.1177/1081286508089845
[39] Negrón-Marrero, P.V., Sivaloganathan, J.: The volume derivative and its approximation in the case of radial cavitation. SIAM J. Appl. Math. (2011). doi: 10.1137/110835943 · Zbl 1448.74021
[40] Neuberger, J.W.: Sobolev Gradients and Differential Equations. Lecture Notes in Math., vol. 1670. Springer, Berlin (1997) · Zbl 0935.35002
[41] 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
[42] Sivaloganathan, J.: Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Ration. Mech. Anal. 96, 97–136 (1986) · Zbl 0628.73018 · doi:10.1007/BF00251407
[43] Sivaloganathan, J.: Singular minimisers in the calculus of variations: a degenerate form of cavitation. Anal. Non Linéaire 9, 657–681 (1992) · Zbl 0769.49030
[44] Sivaloganathan, J., Spector, S.J.: On the existence of minimisers with prescribed singular points in nonlinear elasticity. J. Elast. 59, 83–113 (2000) · Zbl 0987.74016 · doi:10.1023/A:1011001113641
[45] Sivaloganathan, J., Spector, S.J.: On the optimal location of singularities arising in variational problems in nonlinear elasticity. J. Elast. 58, 191–224 (2000) · Zbl 0977.74005 · doi:10.1023/A:1007629229174
[46] Sivaloganathan, J., Spector, S.J.: On cavitation, configurational forces and implications for fracture in a nonlinearly elastic matrial. J. Elast. 67, 25–49 (2002) · Zbl 1089.74627 · doi:10.1023/A:1022594705279
[47] Sivaloganathan, J., Spector, S.J.: Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity. Anal. Non Linéaire 25, 201–213 (2008) · Zbl 1137.74011 · doi:10.1016/j.anihpc.2006.11.013
[48] Sivaloganathan, J., Spector, S.J., Tilakraj, V.: The convergence of regularised minimisers for cavitation problems in nonlinear elasticity. SIAM J. Appl. Math. 66, 736–757 (2006) · Zbl 1104.74016 · doi:10.1137/040618965
[49] Sokolowski, J., Zochowski, A.: On topological derivative in shape optimization. SIAM J. Control Optim. 37, 1251–1272 (1999) · Zbl 0940.49026 · doi:10.1137/S0363012997323230
[50] Sokolowski, J., Zochowski, A.: Energy change due to the appearance of cavities in elastic solids. Int. J. Solids Struct. 40, 1765–1803 (2003) · Zbl 1035.74009 · doi:10.1016/S0020-7683(02)00641-8
[51] Spector, S.J.: Linear deformations as global minimizers in nonlinear elasticity. Q. Appl. Math. 32, 59–64 (1994) · Zbl 0812.73014
[52] Stuart, C.A.: Radially symmetric cavitation for hyperelastic materials. Ann. Inst. Henri Poincaré 2, 33–66 (1985) · Zbl 0588.73021
[53] Stuart, C.A.: Estimating the critical radius for radially symmetric cavitation. Q. Appl. Math. 13(2), 251–263 (1993) · Zbl 0790.73016
[54] Varvaruca, L.: Singular minimizers in the calculus of variations and nonlinear elasticity. Doctoral dissertation, University of Bath, Department of Mathematical Sciences (April 2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.