Finite strain solutions in compressible isotropic elasticity

Carroll, M. M.

doi:10.1007/bf00042141

Cited by 116 publications

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“…In what follows, we consider three classes of materials of the form (7.2) for which a wide range of deformations have been examined by Carroll [24]. We adopt the terminology of [24], Class I. Here W = /(/,) + c2{i2 -3) + c3(/3 -1), (7.13) where c2, c3 are constants.…”

Section: Jr^mentioning

confidence: 99%

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Pure torsion of compressible non-linearly elastic circular cylinders

Polignone¹,

Horgan²

1991

Quart. Appl. Math.

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Abstract. The large deformation torsion problem of an elastic circular cylinder, composed of homogeneous isotropic compressible nonlinearly elastic material and subjected to twisting moments at its ends, is described. The problem is formulated as a two-point boundary-value problem for a second-order nonlinear ordinary differential equation in the radial deformation field. The class of materials for which pure torsion (i.e., a deformation with zero radial displacement) is possible is described. Specific material models are used to illustrate the results.Introduction. The finite torsion of an elastic circular cylinder due to applied twisting moments at its ends was one of the classic problems solved by Rivlin [1,2] for isotropic incompressible nonlinearly elastic materials. The nonhomogeneous deformation found by Rivlin is sustainable, in the absence of body force, in all homogeneous isotropic incompressible materials. Thus it is a universal, or controllable, deformation. Furthermore, by virtue of the constraint of zero volume change, the deformation is that of pure torsion so that there is no extension in the radial direction and the cross-section of the cylinder remains circular.The situation for compressible materials is considerably more complicated. First, by virtue of Ericksen's theorem [3], the deformation cannot be universal. Second, there will, in general, be some radial extension. The torsion problem for a circular cylinder composed of a homogeneous isotropic compressible nonlinearly elastic material has been formulated by Green [4] (see also Green and Adkins [5, p. 67]) and properties of the solution discussed. The formulation leads to a nonlinear secondorder ordinary differential equation for the radial deformation r(R) (see Eq. (2.22) of the present paper). Here R and r denote radial components of a cylindrical polar coordinate system in the undeformed and deformed configurations respectively. The deformation r(R) depends on the form of the strain-energy density. An equivalent version of the ordinary differential equation may be written in terms of the principal stretches [6] (see Eq. (7.8) of the present paper). As remarked in [4,6], an analytic solution of the governing equation, even for very special strain-energy functions, is not readily obtained and so numerical approaches are called for. Other aspects of

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Section: Jr^mentioning

confidence: 99%

“…Such alternative formulations have been widely investigated particularly for compressible materials (see, e.g., [6,24] and the references cited therein). When W = w(iy, i2, i3), (7.2) it has been shown by Carroll [24] that the analog of (2.5) is T = y0l + y1V + y_1V~1, (7.3) where V is the left stretch tensor in the polar decomposition for F = VR.…”

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Pure torsion of compressible non-linearly elastic circular cylinders

Polignone¹,

Horgan²

1991

Quart. Appl. Math.

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“…We also extend our analysis to compressible solids. We show that the stress inside an inclusion with pure dilatational eigenstrain is uniform and hydrostatic when the ball is made of compressible materials of types I, II and III according to Carroll [25]. We also consider cylindrical inclusions in both finite and infinite circular cylindrical bars made of arbitrary incompressible isotropic solids.…”

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confidence: 99%

Nonlinear elastic inclusions in isotropic solids

Yavari

Goriely

2013

Proc. R. Soc. A.

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We introduce a geometric framework to calculate the residual stress fields and deformations of nonlinear solids with inclusions and eigenstrains. Inclusions are regions in a body with different reference configurations from the body itself and can be described by distributed eigenstrains. Geometrically, the eigenstrains define a Riemannian 3-manifold in which the body is stress-free by construction. The problem of residual stress calculation is then reduced to finding a mapping from the Riemannian material manifold to the ambient Euclidean space. Using this construction, we find the residual stress fields of three model systems with spherical and cylindrical symmetries in both incompressible and compressible isotropic elastic solids. In particular, we consider a finite spherical ball with a spherical inclusion with uniform pure dilatational eigenstrain and we show that the stress in the inclusion is uniform and hydrostatic. We also show how singularities in the stress distribution emerge as a consequence of a mismatch between radial and circumferential eigenstrains at the centre of a sphere or the axis of a cylinder.

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“…Chung, Horgan, and Abeyaratne [13] obtained exact solutions of the problems of cylindrical or spherical expansion or compaction for a material with a special strain energy function proposed by Blatz and Ko [14] to model the nonlinear response of foam rubber; see also Beatty [15]. Carroll [16] recently presented several exact solutions for three classes of materials, one of which is the class of harmonic materials. Haughton [ 17] has discussed inflation of thick-walled elastic spherical shells composed of compressible materials, including harmonic materials.…”

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Finite strain solutions for a compressible elastic solid

Carroll¹,

Horgan²

1990

Quart. Appl. Math.

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Abstract. Several closed form finite strain equilibrium solutions are presented for a special compressible isotropic elastic material which was proposed as a model for foam rubber by Blatz and Ko. These solutions include bending of a cylindrical sector into another sector or a rectangular block, bending of a block into a sector, expansion, compaction or eversion of cylinders or spheres, and torsion and extension of circular cylinders or tubes.1. Introduction. Ericksen has examined the problems of finding all of the deformations which can be supported, in the absence of body force, in all homogeneous, isotropic, incompressible elastic solids [1], or in all homogeneous, isotropic, compressible elastic solids [2], The first category consists of all isochoric homogeneous deformations and five families of nonhomogeneous deformations, the so-called controllable or universal deformations ([1] and Singh and Pipkin [3]). The second category consists of homogeneous deformations only, i.e., there is no nonhomogeneous finite deformation which can be supported in every compressible isotropic elastic solid material without applying a body force.This latter result implies that nonhomogeneous deformations can be discussed, for compressible solids, only in the context of a particular strain energy function, or class of strain energy functions. Currie and Hayes [4] have suggested that Ericksen's results have had an unduly inhibiting effect on the study of nonhomogeneous finite deformations. For compressible solids, in particular, the analysis of such deformations is usually very complicated even for simple forms of the strain energy function and very few closed form solutions have been obtained.John [5] introduced the class of harmonic materials, for which the problem of finite plane strain simplifies considerably, and solutions of such problems have been presented by several authors [6][7][8][9][10], Abeyaratne and Horgan [11] obtained an exact solution of the problem of pressurization of a hollow sphere of harmonic material; see also Ogden [10]. Wheeler [12] has examined the deformation of a harmonic material containing an ellipsoidal cavity. Chung, Horgan, and Abeyaratne [13] obtained

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Finite strain solutions in compressible isotropic elasticity

Cited by 116 publications

References 15 publications

Pure torsion of compressible non-linearly elastic circular cylinders

Pure torsion of compressible non-linearly elastic circular cylinders

Nonlinear elastic inclusions in isotropic solids

Finite strain solutions for a compressible elastic solid

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