independent of the boundary conditions on the lateral surface(s). In Sect. 11.2.3 we impose the condition that the outer lateral surface be stress-free and use the radial equilibrium equation to complete the determination of the full stress state. Section 11.2.4 contains a brief discussion of more general classes of torsional deformations that may be more appropriate for annular specimens. General background for the material in this section can be found in the books by Truesdell and Noll [12], Gurtin [13], and Batra [14]. 11.2.1 Incompressible Isotropic Elastic Materials The Cauchy stress tensor T can decomposed into a pressure (or hydrostatic stress) p and a deviatoric stress tensor S (i.e., tr S ¼0): T ¼ pI þS; p 1 3 tr T; S ¼devT T 1 3ð tr TÞI ; (11.1) where I denotes the identity tensor. The standard sign convention for the Cauchy stress tensor is used in this paper: normal stress components are taken positive in tension. For a compressible (i.e., unconstrained) material, constitutive relations must be provided for p and S. However, for an incompressible material the Cauchy stress tensor Tis given by: T ¼ ^pI þ^S; (11.2) where ^p is an indeterminate scalar and ^S is the determinate part of the stress, which need not be deviatoric. The indeterminacy of ^p means that it is not subject to any constitutive relation. However, the value of ^p at each point must be such that the boundary conditions and momentum balance (or in the quasi-static case, the equilibrium equations) are satisfied. Equations (11.1) and (11.2) imply that p ¼^p 1 3 tr ^S; S ¼dev^S: (11.3) For an incompressible material the pressure p is also indeterminate, but ^p and p are generally not equal; nevertheless, some authors refer to ^p as the “pressure”. For an incompressible elastic material the determinate stress ^Sis, by definition, a function of the deformation gradient F. If the material is also isotropic, then ^S is an isotropic function of the left Cauchy-Green deformation tensor B ¼ FFT. A general representation for such functions is given by: ^S ¼μ ωB ð1 ωÞB 1 ¼2C10B 2C01B 1; (11.4) where the elastic moduli C10, C01 andμand the dimensionless parameter ωare scalar-valued functions of trBand tr(B 1). Note that C10 ¼ 1 2 μωand C01 ¼ 1 2 μð1 ωÞ. It has been found that the inequalities C10 >0 and C01 0 imply physically reasonable response and are consistent with experimental data on nearly incompressible isotropic elastic solids; equivalently, μ >0 and 0 < ω 1. These inequalities are assumed throughout the paper. 11.2.2 Pure Torsion Torsion of a solid cylinder or cylindrical tube of an incompressible isotropic elastic material is commonly covered in text books on continuum mechanics (cf. [12, 14, 15]). Let (r, θ, z) and (R, Θ, Z) denote the deformed and reference coordinates of a material point relative to a cylindrical coordinate system. In the undeformed reference configuration, the cylinder has length L(along the Z-axis) and (outer) radius Ro; for an annular specimen the inner radius is Ri. The end of the cylinder at Z ¼0 is fixed while the other end (Z ¼L) is rotated by an angle Ψ(t), where t denotes time. This results in an angle of twist per unit length ψðtÞ¼ΨðtÞ=L. Apure torsional deformation is given by: r ¼R; θ ¼ΘþψðtÞZ; z ¼Z; (11.5) 11 Analysis and Simulations of Quasi-static Torsion Tests on Nearly Incompressible Soft Materials 91
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