Application of Finite Elastic Theory to the Deformation of Rubbery Materials

Blatz, P. J.; Ko, William L.

doi:10.1122/1.548937

Cited by 622 publications

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“…The Blatz-Ko material. It has been shown recently by Beatty [15] and independently by Carroll and Horgan [16] that pure torsion is possible for a special compressible material proposed by Blatz and Ko [14] as a model for foam rubber material. In this section, we examine this special case in detail.…”

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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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“…In Sec. 3, we consider the special case of the Blatz-Ko material [14] for which pure torsion, in the absence of lateral surface tractions, is possible. This was recently observed by Beatty in [15], and independently by Carroll and Horgan in [16].…”

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confidence: 99%

Pure torsion of compressible non-linearly elastic circular cylinders

Polignone¹,

Horgan²

1991

Quart. Appl. Math.

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“…For the application of the analysis presented in the preceding sections, only one specific form of (6.1) 1 will be chosen. This is the particular case (σ, ν) = (0, ¼), which is referred to as the highly compressible Blatz-Ko foam polyurethane [12], which was also employed in [1]. and, in accordance with the concept of pseudo transverse isotropy (see also Appendix A), it may be expressed alternatively as…”

Section: Application: the Blatz-ko Foam Polyurethane Modelmentioning

confidence: 99%

“…Consider next the three-parameter family of isotropic strain energy functions introduced by Blatz and Ko [12]. When specialized for the present problem, this family has the form ( ) It is recalled that the first and the third of the three parameters µ 0 , σ and ν appearing in (6.1) 1 represent the shear modulus and Poisson's ratio, respectively, as defined in classical isotropic elasticity.…”

Section: Application: the Blatz-ko Foam Polyurethane Modelmentioning

confidence: 99%