On compressible materials capable of sustaining axisymmetric shear deformations. Part 4: Helical shear of anisotropic hyperelastic materials

Conditions on the form of the strain energy function in order that homogene ous, compressible and isotropic hyperelastic materials may sustain controll able static, axisymmetric anti-plane shear, azimuthal shear, and helical sh ear deformations of a hollow, circular cylinder have been explored in sever al recent papers. Here we study conditions on the strain energy function fo r homogeneous and compressible, anisotropic hyperelastic materials necessar y and sufficient to sustain controllable, axisymmetric helical shear deform ations of the tube. Similar results for separate axisymmetric anti-plane sh ear deformations and rotational shear deformations are then obtained from t he principal theorem for helical shear deformations. The three theorems are illustrated for general compressible transversely isotropic materials for which the isotropy axis coincides with the cylinder axis. Previously known necessary and sufficient conditions on the strain energy for compressible a nd isotropic hyperelastic materials in order that the three classes of axis ymmetric shear deformations may be possible follow by specialization of the anisotropic case. It is shown that the required monotonicity condition for the isotropic case is much simpler and less restrictive. Restrictions nece ssary and sufficient for anti-plane and rotational shear deformations to be possible in compressible hyperelastic materials having a helical axis of t ransverse isotropy that winds at a constant angle around the tube axis are derived. Results for the previous case and for a circular axis of transvers e isotropy are included as degenerate helices. All of the conditions derive d here have essentially algebraic structure and are easy to apply. The gene ral rules are applied in several examples for specific strain energy functi ons of compressible and homogeneous transversely isotropic materials having straight, circular, and helical axes of material symmetry.