On the Mathematical Structure of Helical Shear in Anelastic Continua

We study helical shear of a thick-walled tube made of an incompressible, isotropic hyper-anelastic solid carrying a radially symmetric distribution of axial screw dislocations. Through the Bilby–Kröner–Lee decomposition, the dislocations are encoded in a Weitzenböck material manifold, whose underlying Riemannian metric alone enters the elastic response. Equilibrium reduces at each radius to an algebraic system for the azimuthal and axial shear strains, whose cross terms are proportional to the dislocation strength and to the dependence of the strain energy on the second invariant; in every material, the two shear modes therefore couple only in the presence of dislocations. Exact solutions are obtained for neo-Hookean, Fung–Demiray, Gent, Knowles, and Mooney–Rivlin materials, and numerical ones for Gent–Gent materials. The universal relations of Horgan and Saccomandi are extended to dislocated generalized neo-Hookean tubes. Dislocations stiffen the tube when the strain energy stiffens with the first invariant, couple the two shear modes chirally when it depends on the second, and, in some cases, restrict admissibility: in Gent and Gent–Gent materials they cap the dislocation strength, and in Knowles materials they contract the admissible traction region when the power-law index is below the critical exponent n = 1/2. A weakly nonlinear analysis, to third order in the applied tractions and dislocation strength, shows that, at first order, i.e. within the linear theory of elasticity, the dislocations leave the shear response unaffected while carrying a residual stress. The chiral coupling enters only at second order, through the dependence of the strain energy on the second invariant, so that, in the presence of axial screw dislocations, every material couples the modes as the Mooney–Rivlin solid with the same second- and third-order elastic moduli. At third order, no further coupling arises, but the modes interact through stiffness corrections involving the fourth-order elastic modulus, with the dislocations further correcting each mode independently of the other.