Abstract
Experiments on elastomers have shown that sufficiently large triaxial tensions induce the material to exhibit holes that were not previously evident. In this paper, conditions are presented that allow one to use the direct method of the calculus of variations to deduce the existence of hole creating deformations that are global minimizers of a nonlinear, purely elastic energy. The crucial physical assumption used is that there are a finite (possibly large) number of material points in the undeformed body that constitute the only points at which cavities can form. Each such point may be viewed as a preexisting flaw or an infinitesimal microvoid in the material.
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References
J.M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics. In: R.J. Knops (ed.), Nonlinear Analysis and Mechanics, Vol. I. Pitman (1977) pp. 187–241.
J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337–403.
J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. London A 306 (1982) 557–611.
J.M. Ball, Minimizers and the Euler-Lagrange equations. In: P.G. Ciarlet and M. Roseau (eds), Trends and Applications of Pure Mathematics to Mechanics. Springer, Berlin (1984) pp. 1–4.
J.M. Ball, J.C. Currie and P.J. Olver, Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Functional Anal. 41 (1981) 135–174.
P. Bauman, N.C. Owen and D. Phillips, Maximum principles and a priori estimates for a class of problems from nonlinear elasticity. Analyse non linéaire 8 (1991) 119–157.
K. Cho and A.N. Gent, Cavitation in model elastomeric composites. J. Materials Sci. 235 (1988) 141–144.
P.G. Ciarlet, Mathematical Elasticity, Vol. 1. Elsevier, Amsterdam (1988).
N. Dunford and J.T. Schwartz, Linear Operators, part. 1. Interscience, New York (1958).
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992).
I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications. Oxford Univ. Press, Oxford (1995).
A.N. Gent, Cavitation in rubber: A cautionary tale. Rubber Chem. Technology 63 (1990) G49–G53.
A.N. Gent and P.B. Lindley, Internal rupture of bonded rubber cylinders in tension. Proc. Roy. Soc. London A 249 (1985) 195–205.
M. Giaquinta, G. Modica and J. Souček, Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 106 (1989) 97–159.
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer, Berlin (1983).
C.O. Horgan and R. Abeyaratne, A bifurcation problem for a compressible nonlinearly elastic medium: Growth of a microvoid. J. Elasticity 16 (1986) 189–200.
C.O. Horgan and D.A. Polignone, Cavitation in nonlinearly elastic solids: A review. J. Appl. Mech. 48 (1995) 471–485.
R.D. James and S.J. Spector, The formation of filamentary voids in solids. J. Mech. Phys. Solids 39 (1991) 783–813.
R.D. James and S.J. Spector, Remarks on W 1,p — quasiconvexity, interpenetration of matter, and function spaces for elasticity. Anal Non-Linéaire 9 (1992) 263–280.
M. Marcus and V.J. Mizel, Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems. Bull. Am. Math. Soc. 79 (1973) 790–795.
P.A. Meyer, Probability and Potentials. Blaisdell (1966).
C.B. Morrey, Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966).
S. Müller, Det=det — A remark on the distributional determinant. C. R. Acad. Sci. Paris Sen I 311 (1990) 13–17.
S. Müller and S.J. Spector, An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal. 131 (1995) 1–66.
J.T. Schwartz, Nonlinear Functional Analysis. Gordon and Breach (1969).
J. Sivaloganathan, Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Rational Mech. Anal. 96 (1986) 97–136.
J. Sivaloganathan, Singular minimizers in the calculus of variations: A degenerate form of cavitation. Anal. Non-Linéaire 9 (1992) 657–681.
V. Šverák, Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100 (1988) 105–127.
Q. Tang, Almost-everywhere injectivity in nonlinear elasticity. Proc. Roy. Soc. Edinburgh A 109 (1988) 79–95.
E. Zeidler, Nonlinear Functional Analysis and its Applications. Springer, Berlin (1986).
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Sivaloganathan, J., Spector, S.J. (2000). On the Existence of Minimizers with Prescribed Singular Points in Nonlinear Elasticity. In: Carlson, D.E., Chen, YC. (eds) Advances in Continuum Mechanics and Thermodynamics of Material Behavior. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0728-3_8
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DOI: https://doi.org/10.1007/978-94-010-0728-3_8
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