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 ${\sigma }_{zz}=P/(2\pi at)$ while the torque induces a shear stress ${\sigma }_{z\theta }=Q/(2\pi a^{2}t)$.   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 $[C]$ 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 $\underset{\_}{d\sigma }=[C]\underset{\_}{d\epsilon }$.  Find the eigenvalues of $[C]$ for the particular case $[\epsilon ]=[{\epsilon }_{11},{\epsilon }_{22},{\epsilon }_{33},0,0,0]$.   Hence, show that the material is stable in the sense of Drucker as long as $K>0$, ${\sigma }_0>0$.