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 ${\sigma }_{zz}=P/\left(2\pi at\right)$ while the torque induces a shear stress ${\sigma }_{z\theta }=Q/\left(2\pi {a}^{2}t\right)$.   The shear strains are related to the twist per unit length of the tube by ${\epsilon }_{z\theta }=\varphi /2L$, while the axial strains are related to the extension of the tube by ${\epsilon }_{zz}=\Delta /L$.

3.3.1.1.            Calculate a relationship between the axial load P and the extension $\Delta$ for a tube subjected only to axial loading

3.3.1.2.            Calculate a relationship between the torque Q and the twist $\varphi$ for a tube subjected only to torsional loading

3.3.1.3.            Calculate a relationship between P, Q and $\Delta$, $\varphi$ 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 ${C}_{ijkl}=\partial {\sigma }_{ij}/\partial {\epsilon }_{kl}$.  Express your answer in  matrix form by finding the matrix $\left[C\right]$ such that stress increment $\underset{_}{d\sigma }={\left[d{\sigma }_{11},d{\sigma }_{22},d{\sigma }_{33},d{\sigma }_{23},d{\sigma }_{13},d{\sigma }_{12}\right]}^{T}$ and strain increment $\underset{_}{d\epsilon }={\left[d{\epsilon }_{11},d{\epsilon }_{22},d{\epsilon }_{33},d{\epsilon }_{23},d{\epsilon }_{13},d{\epsilon }_{12}\right]}^{T}$ are related by $\underset{_}{d\sigma }=\left[C\right]\underset{_}{d\epsilon }$.  Find the eigenvalues of $\left[C\right]$ for the particular case $\left[\epsilon \right]=\left[{\epsilon }_{11},{\epsilon }_{22},{\epsilon }_{33},0,0,0\right]$.   Hence, show that the material is stable in the sense of Drucker as long as $K>0$, ${\sigma }_{0}>0$.