Homogeneous equilibrium configurations of a hyperelastic compressible cube under equitriaxial dead-load tractions. (English) Zbl 1179.74012

This interesting and well-written paper is about a classical and important problem in nonlinear elasticity. The problem has been first stated by R. S. Rivlin [Philos. Trans. Roy. Soc. London, Ser. A 240, 459–508 (1948; Zbl 0029.32605)] in the framework of homogeneous deformations of a cube made of incompressible material and subject to equitriaxial dead-load tractions. In the present paper the original analysis has been extended to compressible neo-Hookean and Mooney-Rivlin materials. The authors consider the volumetric part of strain-energy function in a form that has been proposed by P. G. Ciarlet and G. Geymonat [C. R. Acad. Sci., Paris, Sér. II 295, 423–426 (1982; Zbl 0497.73017)] on the basis of mathematical arguments. This choice allows to accompany the analysis by beautiful plots and graphs that help to understand in a clear way what is going around with respect to non-uniqueness and stability of the solutions. This paper is of sure interest for researcher working in nonlinear elasticity. and clearly the next step is to consider the same problem when more realistic forms of the strain-energy function are in force.


74B20 Nonlinear elasticity
74G60 Bifurcation and buckling
74G35 Multiplicity of solutions of equilibrium problems in solid mechanics
Full Text: DOI


[1] Rivlin, R.S.: Large elastic deformations of isotropic materials, II. Some uniqueness theorems for pure homogeneous deformation. Philos. Trans. R. Soc. Lond. A 240, 491–508 (1948) · Zbl 0029.32605
[2] Rivlin, R.S.: Stability of pure homogeneous deformations of an cube under dead loading. Q. Appl. Math. 32, 265–271 (1974) · Zbl 0324.73039
[3] Ball, J.M., Schaeffer, D.G.: Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions. Proc. Camb. Philos. Soc. 94, 315–339 (1983) · Zbl 0568.73057
[4] Ball, J.M.: Differentiability properties of symmetric and isotropic functions. Duke Math. J. 51, 699–728 (1984) · Zbl 1077.74507
[5] Chen, Y.C.: Stability of homogeneous deformations of an incompressible elastic body under dead-load surface traction. J. Elast. 17, 223–248 (1987) · Zbl 0634.73025
[6] Chen, Y.C.: Stability of homogeneous deformations in nonlinear elasticity. J. Elast. 40, 75–94 (1995) · Zbl 0834.73034
[7] Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977) · Zbl 0368.73040
[8] Ciarlet, P.G.: Mathematical Elasticity. Three-Dimensional Elasticity, vol. I. North-Holland, Amsterdam (1988) · Zbl 0648.73014
[9] Euler, L.: Nova Methodus Motum Corporum Rigidorum Determinandi. Novi Comment. Acad. Imp. Petrop. 20, 208–238 (1775)
[10] Kearsley, E.A.: Asymmetric stretching of a symmetrically loaded elastic sheet. Int. J. Solid Struct. 22, 111–119 (1986)
[11] Tarantino, A.M., Nobili, A.: Finite homogeneous deformations of symmetrically loaded compressible membranes. Z. Angew. Math. Phys. 58, 659–678 (2006, accepted for publication) · Zbl 1118.74034
[12] Ciarlet, J.M., Geymonat, G.: Sur les lois de comportement en élasticité non-linéaire compressible. C.R. Acad. Sci. Ser. II 295, 423–426 (1982) · Zbl 0497.73017
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.