Chapter 3

Constitutive Models: Relations between Stress and Strain

3.4.  Generalized Hooke’s Law $–$ Elastic Materials with Large Rotations

3.4.1.      A uniaxial tensile specimen with length L and cross-sectional area A is idealized with a constitutive law that relates the material stress ${\Sigma }_{ij}$ to the Lagrange strain ${E}_{ij}$ by

${\Sigma }_{ij}=\frac{E}{1+\nu }\left\{{E}_{ij}+\frac{\nu }{1-2\nu }{E}_{kk}{\delta }_{ij}\right\}\text{\hspace{0.17em}}$

where E and $\nu$ are elastic constants.   The specimen is subjected to a uniaxial force P which induces an extension $\delta$.  Calculate the relationship between $P$ and $\delta$, and compare the results with the predictions of a linear elastic constitutive equation.

3.4.2.      A thin walled tube with length L, radius a and wall thickness t is subjected to a torque Q .  The tube can be idealized using the constitutive equation described in the preceding problem.  Assume that during deformation plane sections of the tube remain plane, and that cross sections of the tube rotate through and angle $\theta \left(z\right)=\varphi z/L$.

3.4.2.1.            Calculate an expression for the Lagrange strain in the specimen

3.4.2.2.            Hence deduce an expression for the material stress in the tube

3.4.2.3.            Compute the Cauchy stress distribution

3.4.2.4.            Hence, deduce an expression relating the torque Q to the tube’s twist $\varphi$.  Compare the result with the predictions of a simple linear elastic constitutive equation.

3.4.3.      Check whether the constitutive equation given in problem 3.4.1 satisfies the test for objectivity described in Section 3.1.