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Abstract

This paper examines nonhomogeneous deformations of homogeneous, isotropic, compressible, nonlinearly elastic solids. It is assumed that the deformation is the gradient of a scalar field, and then determine the equation satisfied by this field from the equation of equilibrium. The strain energy function assumed is a generalization of the harmonic material and the generalized Varga material. For the special case of the generalized Varga material, a similarity solution is found for the scalar field. This solution describes the deformation of regions bounded by a family of surfaces which includes circles, ellipses, and hyperbolas in two and three dimensions. The solution is then partially extended to the more general elastic material.

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[1] Ericksen, J. L. : Deformations possible in every compressible, isotropic, perfectly elastic material. J. Math. Phys., 34, 126-128 (1955).

[2] Blatz, P. J. and Ko, W. L. : Application of finite elasticity to the deformation of rubbery materials. Trans. Soc. Rheol., 6, 223-251 (1962).

[3] Carroll, M. M. and Horgan, C. O. : Finite strain solutions for a compressible elastic solid. Quart. Appl. Math., 48, 767-780 (1990).

[4] John, F. : Plane strain solutions for a perfectly elastic material of harmonic type. Commun. Pure Appl. Math., 13, 239-296 (1960).

[5] Haughton, D. M. : Inflation and bifurcation of thick-walled compressible elastic spherical shells. IMA J. Appl. Math., 39, 259-272 (1987).

[6] Carroll, M. M. : Finite strain solutions in compressible isotropic elasticity. J. Elasticity, 20, 65-92 (1988).

[7] Ogden, R. W. and Isherwood, D. A. : Solution of some finite plane strain problems for compressible elastic solids. Q. J. Mech. Appl. Math., 31, 219-249 (1978).

[8] Jafari, A. H. , Abeyaratne, R. , and Horgan, C. O. : The finite deformation of a pressurized circular tube for a class of compressible materials. ZAMP, 35, 227-246 (1984).

[9] Currie, P. K. and Hayes, M. : On non-universal finite elastic deformations, in Finite Elasticity, Proc. IUTAM Symp., pp. 143-150, ed., D. E. Carlson & R. T. Shield , Martinus Nijhoff, The Hague, 1982.

10.

[10] Polignone, D. A. and Horgan, C. O. : Pure torsion of compressible nonlinearly elastic circular cylinders. Quart. Appl. Math., 49, 591-607 (1991).

11.

[11] Polignone, D. A. and Horgan, C. O. : Axisymmetric finite anti-plane shear of compressible nonlinearly elastic circular tubes. Quart. Appl. Math., 50, 232-341 (1992).

12.

[12] Polignone, D. A. and Horgan, C. O. : Pure azimuthal shear of compressible nonlinearly elastic circular tubes. Quart. Appl. Math., 52, 113-131 (1994).

13.

[13] Horgan, C. O. : On axisymmetric solutions for compressible nonlinearly elastic solids. ZAMP, 46, S107-S125 (1995).

14.

[14] Varley, E. and Cumberbatch, E. : Finite deformations of elastic materials surrounding cylindrical holes. J. Elasticity, 10, 341-405 (1980).

15.

[15] Wheeler, L. T. : Finite deformations of a harmonic elastic medium containing an ellipsoidal cavity. Int. J. Solids Struct., 21, 799-804 (1985).

16.

[16] Carroll, M. M. : Controllable deformations for special classes of compressible elastic solids. Stability Appl. Anal. Continuous Media, 1, 309-323 (1991).

17.

[17] Murphy, J. G. : A family of solutions describing spherical inflation in finite, compressible elasticity. Quart. J. Mech. Appl. Math., 50, 35-45 (1997).

Irrotational Deformations in Finite Compressible Elasticity

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