Problems for Chapter 3
Constitutive Models: Relations between Stress and Strain
3.3.1. A thin-walled tube can be idealized using the hypoelastic constitutive equation described in Section 3.3. You may assume that the axial load induces a uniaxial stress while the torque induces a shear stress . The shear strains are related to the twist per unit length of the tube by , while the axial strains are related to the extension of the tube by .
184.108.40.206. Calculate a relationship between the axial load P and the extension for a tube subjected only to axial loading
220.127.116.11. Calculate a relationship between the torque Q and the twist for a tube subjected only to torsional loading
18.104.22.168. Calculate a relationship between P, Q and , for a tube subjected to combined axial and torsional loading.
3.3.2. Consider a material with the hypoelastic constitutive equation described in Section 3.3. Calculate an expression for the tangent stiffness . Express your answer in matrix form by finding the matrix such that stress increment and strain increment are related by . Find the eigenvalues of for the particular case . Hence, show that the material is stable in the sense of Drucker as long as , .
(c) A.F. Bower, 2008