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Problems for Chapter 3
Constitutive
Models: Relations between Stress and Strain
3.3. Hypoelasticity
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 .
3.3.1.1.
Calculate a
relationship between the axial load P and
the extension for a tube subjected only to axial loading
3.3.1.2.
Calculate a
relationship between the torque Q
and the twist for a tube subjected only to torsional
loading
3.3.1.3.
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 ,
.
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