Chapter 3

Constitutive Models: Relations between Stress and Strain

3.2.  Linear Elastic Constitutive Equations

3.2.1.      Using the table of values given in Section 3.2.4, find values of bulk modulus, Lame modulus, and shear modulus for steel, aluminum and rubber.

3.2.2.      A specimen of an isotropic, linear elastic material with Young’s modulus E and is placed inside a rigid box that prevents the material from stretching in any direction.   This means that the strains in the specimen are zero.  The specimen is then heated to increase its temperature by $\Delta T$.  Find a formula for the stress in the specimen.  Find a formula for the strain energy density.  How much strain energy would be stored in a $1c{m}^{3}$ sample of steel if its temperature were increased by 100C?  Compare the strain energy with the heat required to change the temperature by 100C $–$ the specific heat capacity of steel is about 470 J/(kg-C)

3.2.3.      A specimen of an isotropic, linear elastic solid is free of stress, and is heated to increase its temperature by $\Delta T$.  Find expressions for the strain and displacement fields in the solid.

3.2.4.      A thin isotropic, linear elastic thin film with Young’s modulus E, Poisson’s ratio $\nu$ and thermal expansion coefficient $\alpha$ is bonded to a stiff substrate.  The film is stress free at some initial temperature, and then heated to increase its temperature by $T$.  The substrate prevents the film from stretching in its own plane, so that ${\epsilon }_{11}={\epsilon }_{22}={\epsilon }_{12}=0$, while the surface is traction free, so that the film deforms in a state of plane stress. Calculate the stresses in the film in terms of material properties and temperature, and deduce an expression for the strain energy density in the film.

3.2.5.      A cubic material may be characterized either by its moduli as

$\left[\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{12}\\ {\sigma }_{13}\\ {\sigma }_{23}\end{array}\right]=\left[\begin{array}{cccccc}{c}_{11}& {c}_{12}& {c}_{12}& 0& 0& 0\\ & {c}_{11}& {c}_{12}& 0& 0& 0\\ & & {c}_{11}& 0& 0& 0\\ & sym& & {c}_{44}& 0& 0\\ & & & 0& {c}_{44}& 0\\ & & & 0& 0& {c}_{44}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ 2{\epsilon }_{12}\\ 2{\epsilon }_{13}\\ 2{\epsilon }_{23}\end{array}\right]$

or by the engineering constants

$\left[\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ 2{\epsilon }_{12}\\ 2{\epsilon }_{13}\\ 2{\epsilon }_{23}\end{array}\right]=\left[\begin{array}{cccccc}1/E& -\nu /E& -\nu /E& 0& 0& 0\\ -\nu /E& 1/E& -\nu /E& 0& 0& 0\\ -\nu /E& -\nu /E& 1/E& 0& 0& 0\\ 0& 0& 0& 1/\mu & 0& 0\\ 0& 0& 0& 0& 1/\mu & 0\\ 0& 0& 0& 0& 0& 1/\mu \end{array}\right]\left[\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{12}\\ {\sigma }_{13}\\ {\sigma }_{23}\end{array}\right]$

Calculate formulas relating ${c}_{ij}$ to $E,\nu$ and $\mu$, and deduce an expression for the anisotropy factor $A=2\mu \left(1+\nu \right)/E$ in terms of ${c}_{ij}$.

3.2.6.      Suppose that the stress-strain relation for a linear elastic solid is expressed in matrix form as $\left[\sigma \right]=\left[C\right]\left[\epsilon \right]$, where  $\left[\sigma \right]$, $\left[\epsilon \right]$ and $\left[C\right]$ represent the stress and strain vectors and the matrix of elastic constants defined in Section 3.2.8.  Show that the material has a positive definite strain energy density ( $U>0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall \left[\epsilon \right]\ne 0$ ) if and only if the eigenvalues of $\left[C\right]$ are all positive.

3.2.7.      Write down an expression for the increment in stress resulting from an increment in strain applied to a linear elastic material, in terms of the matrix of elastic constants $\left[C\right]$.  Hence show that, for a linear elastic material to be stable in the sense of Drucker, the eigenvalues of the matrix of elastic constants $\left[C\right]$ must all be positive or zero.

3.2.8.      Let $\left[\sigma \right]$, $\left[\epsilon \right]$ and $\left[C\right]$ represent the stress and strain vectors and the matrix of elastic constants in the isotropic linear elastic constitutive equation

$\left[\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{23}\\ {\sigma }_{13}\\ {\sigma }_{12}\end{array}\right]=\frac{E}{\left(1+\nu \right)\left(1-2\nu \right)}\left[\begin{array}{cccccc}1-\nu & \nu & \nu & 0& 0& 0\\ \nu & 1-\nu & \nu & 0& 0& 0\\ \nu & \nu & 1-\nu & 0& 0& 0\\ 0& 0& 0& \frac{\left(1-2\nu \right)}{2}& 0& 0\\ 0& 0& 0& 0& \frac{\left(1-2\nu \right)}{2}& 0\\ 0& 0& 0& 0& 0& \frac{\left(1-2\nu \right)}{2}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ 2{\epsilon }_{23}\\ 2{\epsilon }_{13}\\ 2{\epsilon }_{12}\end{array}\right]-\frac{E\alpha \Delta T}{1-2\nu }\left[\begin{array}{c}1\\ 1\\ 1\\ 0\\ 0\\ 0\end{array}\right]$

3.2.8.1.            Calculate the eigenvalues of the stiffness matrix $\left[C\right]$ for an isotropic solid in terms of Young’s modulus and Poisson’s ratio.  Hence, show that the eigenvalues are positive (a necessary requirement for the material to be stable $–$ see problem ??) if and only if $-1<\nu <1/2$ and $E>0$.

3.2.8.2.            Find the eigenvectors of $\left[C\right]$ and sketch the deformations associated with these eigenvectors.

3.2.9.      Let $\left[\sigma \right]$, $\left[\epsilon \right]$ and $\left[C\right]$ represent the stress and strain vectors and the matrix of elastic constants in the isotropic linear elastic constitutive equation for a cubic crystal

$\left[\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{23}\\ {\sigma }_{12}\\ {\sigma }_{13}\end{array}\right]=\left[\begin{array}{cccccc}{c}_{11}& {c}_{12}& {c}_{12}& 0& 0& 0\\ & {c}_{11}& {c}_{12}& 0& 0& 0\\ & & {c}_{11}& 0& 0& 0\\ & sym& & {c}_{44}& 0& 0\\ & & & 0& {c}_{44}& 0\\ & & & 0& 0& {c}_{44}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ 2{\epsilon }_{23}\\ 2{\epsilon }_{12}\\ 2{\epsilon }_{13}\end{array}\right]$

Calculate the eigenvalues of the stiffness matrix $\left[C\right]$ and hence find expressions for the admissible ranges of ${c}_{11},{c}_{12},{c}_{44}$ for the eigenvalues to be positive.

3.2.10.  Let  ${C}_{ijkl}^{e}$ denote the components of the elasticity tensor in a basis $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$. Let  $\left\{{m}_{1},{m}_{2},{m}_{3}\right\}$ be a second basis, and define ${\Omega }_{ij}={m}_{i}\cdot {e}_{j}$.  Recall that the components of the stress and strain tensor in  $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ and $\left\{{m}_{1},{m}_{2},{m}_{3}\right\}$ are related by ${\sigma }_{ij}^{\left(m\right)}={\Omega }_{ik}{\sigma }_{kl}^{\left(e\right)}{\Omega }_{jl}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\epsilon }_{ij}^{\left(m\right)}={\Omega }_{ik}{\epsilon }_{kl}^{\left(e\right)}{\Omega }_{jl}$. Use this result, together with the elastic constitutive equation, to show that the components of the elasticity tensor in $\left\{{m}_{1},{m}_{2},{m}_{3}\right\}$ can be calculated from

${C}_{ijkl}^{\left(m\right)}={\Omega }_{ip}{\Omega }_{jq}{C}_{pqrs}^{\left(e\right)}{\Omega }_{kr}{\Omega }_{ls}$

3.2.11.  Consider a cube-shaped specimen of an anisotropic, linear elastic material. The tensor of elastic moduli and the thermal expansion coefficient for the solid (expressed as components in an arbitrary basis) are   ${C}_{ijkl}^{e}$, ${\alpha }_{ij}^{e}$.  The solid is placed inside a rigid box that prevents the material from stretching in any direction.   This means that the strains in the specimen are zero.  The specimen is then heated to increase its temperature by $\Delta T$.  Find a formula for the strain energy density, and show that the result is independent of the orientation of the material with respect to the box.

3.2.12.  The figure shows a cubic crystal.   Basis vectors  $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ are aligned perpendicular to the faces of the cubic unit cell. A tensile specimen is cut from the cube $–$ the axis of the specimen lies in the $\left\{{e}_{1},{e}_{2}\right\}$ plane and is oriented at an angle $\theta$ to the ${e}_{1}$ direction. The specimen is then loaded in uniaxial tension ${\sigma }_{nn}$ parallel to its axis.  This means that the stress components in the basis $\left\{{n}_{1},{n}_{2},{n}_{3}\right\}$ shown in the picture are

$\left[\begin{array}{ccc}{\sigma }_{nn}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

3.2.12.1.        Use the basis change formulas for tensors to calculate the components of stress in the $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ basis in terms of $\theta$.

Use the stress-strain equations in Section 3.1.16 to find the strain components in the $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ basis, in terms of the engineering constants $E,\nu ,\mu$ for the cubic crystal. You need only calculate ${\epsilon }_{11},{\epsilon }_{22},{\epsilon }_{12}$.

3.2.12.2.        Use the basis change formulas again to calculate the strain components in the $\left\{{n}_{1},{n}_{2},{n}_{3}\right\}$ basis oriented with the specimen.  Again, you need only calculate ${\epsilon }_{11},{\epsilon }_{22},{\epsilon }_{12}$.  Check your answer by setting $\mu =E/2\left(1+\nu \right)$ - this makes the crystal isotropic, and you should recover the isotropic solution.

3.2.12.3.        Define the effective axial Young’s modulus of the tensile specimen as $E\left(\theta \right)={\sigma }_{nn}/{\epsilon }_{nn}$, where ${\epsilon }_{nn}=n\cdot \epsilon \cdot n$ is the strain component parallel to the ${n}_{1}$ direction.  Find a formula for $E\left(\theta \right)$ in terms of $E,\nu ,\mu$.

3.2.12.4.        Using data for copper, plot a graph of $E\left(\theta \right)$ against $\theta$.  For copper, what is the orientation that maximizes the longitudinal stiffness of the specimen?   Which orientation minimizes the stiffness?