Abstract
This paper points out a subtle and little known consequence of assuming rank-one convexity for an isotropic hyperelastic material [e.g., of assuming a common ordering of principal stretches and principal stresses and hence the Baker–Ericksen inequalities]. What is shown is that rank-one convexity necessarily privilages those affine deformations which are dilatations: – the stored-energy associated with a dilatation is smaller than the stored energy associated with any other affine deformation possessing the same determinant. Also pointed out are fracture related consequences of this property that arise when the stored-energy function assigns to the dilatation of determinant δ > 0 a value A(δ∈) which is not an everywhere convex function of (0,∞).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S.S. Antman, Nonlinear Problems of Elasticity(Springer-Verlag. New York 1995).
M. Aron, On a minimum property in nonlinear elasticity, Int. J. Engng. Sci. 29(1995) 1471-1478.
J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat.Mech. Anal. 63(1977) 337-403.
J.M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics. In R.J. Knops (ed.), Nonlinear Analysis and Mechanics: Heriot-Watt Symposium1. London, Pitman (1977).
J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Ph. Tr. Roy. Soc. Lond. A306(1982) 557-611.
J.M. Ball, Differentiability properties of symmetric and isotropic functions, Duke Math. J. 51(1984) 699-728.
M. Baker and J.L. Ericksen, Inequalities restricting the form of the stress-deformation relations for isotropic elastic solids and Reiner-Rivlin fluids, J. Wash. Acad. Sci. 44(1954) 33-35. Reprinted in Foundations of Elasticity Theory, C. Truesdell, (ed.) (New York, Gordon and Breach, 1965).
J.M. Ball and V.J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rat. Mech. Anal. 90(1985) 325-388.
G. Buttazzo and V.J. Mizel, Interpretation of the Lavrentiev phenomenon by relaxation, J. Funct. Anal. 110(1992) 434-460.
G. Buttazzo and V.J. Mizel, On a gap phenomenon for isoperimetrically constrained variational problems, J. Conv. Anal. 2(1995) 87-101.
P.G. Ciarlet, Mathematical Elasticity 1. North-Holland, New York. (1988).
B. Dacorogna, A relaxation theorem and its applications to the equilibrium of gases, Arch. Rat. Mech. Anal. 77(1981) 359-386.
J.L. Ericksen, Special topics in elastostatics, in Advances in Applied Mechanics 17(1977) 189-244.
R. Fosdick, Personal Communication.
A.C. Heinricher and V.J. Mizel, A stochastic control problem with different value functions for singular and absolutely continuous control, Proceedings 25th IEEE Conference on Decision and Control134-139. Athens (1986).
A.D. Ioffe and V.J. Mizel, Dilatational Traces of Isotropic Hyperelastic Materials in 2 Dimensions. (In preparation).
M. Marcus and V.J. Mizel, Transformations by functions in Sobolev spaces and lowersemicontinuity for parametric variational problems, Bull. Amer. Math. Soc. 79(1973) 790-795.
C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2(1952) 25-53.
C.B. Morrey, Multiple Integrals in the Calculus of Variations. Springer-Verlag, New York (1966).
W. Noll, On cracks in elastic bodies, in Festschrift for Rudolf Trostel(1994) 249-253.
C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, 3. Springer, New York (1965).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mizel, V.J. On the Ubiquity of Fracture in Nonlinear Elasticity. Journal of Elasticity 52, 257–266 (1998). https://doi.org/10.1023/A:1007537117202
Issue Date:
DOI: https://doi.org/10.1023/A:1007537117202