Pure Shearing and Pure Distortional Deformations Are Not Equivalent
- PDF / 704,536 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 83 Downloads / 203 Views
Pure Shearing and Pure Distortional Deformations Are Not Equivalent M.B. Rubin1
Received: 24 May 2020 / Accepted: 26 September 2020 © Springer Nature B.V. 2020
Abstract This paper attempts to clarify the notions of a state of pure shear stress and pure shearing deformations. Specifically, it is shown that pure shearing deformations and pure distortional deformations are not equivalent. Attention is limited to isotropic, compressible, hyperelastic materials. Differences between the distortional deformations of pure shearing, pure shear caused by tension and compression, and plane strain extension and contraction defined as pure shear by Rivlin and Saunders (Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 243(865):251–288, 1951) are discussed. It is shown that these deformations are physically different and should not be expected to test the same features of a proposed form of the strain energy function. It is also shown that two deformations of pure shearing and two deformations of pure shear caused by tension and compression are nearly universal distortional deformations valid for all strain energy functions. Keywords Finite deformation · Isotropic hyperelasticity · Pure distortional deformation · Pure shearing deformation · Nearly universal distortional deformations Mathematics Subject Classification 74A05 · 74A10 · 74B20
1 Introduction Finite deformation pure shear and related shear deformations in isotropic, compressible, hyperelastic materials have been considered at least since the experiments conducted by [14] on rubber. Analytical work on a state of pure shear stress can be found in [12, 15]. [3] considered a fixed deformation and determined pairs of material line elements which are unsheared as well a pairs of line elements which suffer the maximum shear, and a class of universal relations was developed in [1]. A more recent discussion of universal results can be found in [18]. Restrictions on constitutive equations related to the Poynting effect were considered in [11], the response to pure shear was revisited in [6], and distortional isochoric
B M.B. Rubin
[email protected]
1
Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, 32000 Haifa, Israel
M.B. Rubin
plane strain deformations, called (pure) shear (stretch), were analyzed in [19]. These later deformations should not be confused with the three-dimensional deformations associated with a state of pure shear stress discussed in this paper. By way of background, it is recalled that [7] introduced the notion of a unimodular tensor to characterize pure distortional deformation. Specifically, the deformation gradient F from a zero-stress initial configuration is used to define the dilatation J , its unimodular part F and the unimodular part B of the left Cauchy-Green deformation tensor by J = detF ,
F = J −1/3 F ,
α1 = B · I ≥ 3 ,
B = F FT ,
detB = 1 ,
(1)
α2 = B2 · I ≥ 3 .
In these expressions, J is a pure measure of dilatation, B is a pure measure of distortional deformation, α1 and α2 are t
Data Loading...
