6.1 General concepts of material constitutive response

---

6.1.1 Intuitive development of a constitutive matrix

---

6.1.1.1 Uniform rectangular block pulled on both ends

Public Domain Kerina yin 2011

---

Block subject to normal stress

What strains do you expect for \({\sigma_{xx}}\ne 0\) (all others stresses = 0)?

• We are all familiar with Hooke’s Law: \[\begin{equation*} {\sigma_{xx}}= E {\varepsilon_{xx}} \end{equation*}\]
• Rearranging: \[\begin{equation*} {\varepsilon_{xx}}= \frac{{\sigma_{xx}}}{E} \end{equation*}\]
• But what are the other strains?

---

• But what are the other strains? \[\begin{align} {\varepsilon_{yy}}=& \, -\nu {\varepsilon_{xx}}\cr {\varepsilon_{zz}}=& \, -\nu {\varepsilon_{xx}}\cr =& \, -\nu \frac{{\sigma_{xx}}}{E} \cr {\varepsilon_{ij}}=& \, 0 \hspace{5mm} \mbox{ for } i\neq j \cr \end{align}\]

---

• Similarly, we can obtain similar equations in the other directions:

For \({\sigma_{yy}}\ne 0\), all others 0? \[\begin{align} {\varepsilon_{yy}}=& \frac{{\sigma_{yy}}}{E} \cr {\varepsilon_{xx}}=& -\nu \frac{{\sigma_{yy}}}{E} \cr {\varepsilon_{zz}}=& -\nu \frac{{\sigma_{yy}}}{E} \cr {\varepsilon_{ij}}=& \; 0 \hspace{5mm} \mbox{ for } i\neq j \cr \end{align}\]

For \({\sigma_{zz}}\ne 0\), all others 0? \[\begin{align} {\varepsilon_{zz}}=& \, \frac{{\sigma_{zz}}}{E} \cr {\varepsilon_{xx}}=& \, -\nu \frac{{\sigma_{zz}}}{E} \cr {\varepsilon_{yy}}=& \, -\nu \frac{{\sigma_{zz}}}{E} \cr {\varepsilon_{ij}}=& \, 0 \hspace{5mm} \mbox{ for } i\neq j \cr \end{align}\]

• We’ve found a pattern for the normal stress-normal strain response

---

• What about for shear?

---

Block subject to shear stress

For \({\tau_{xy}}\ne 0\), all others 0? \[\begin{align} {\gamma_{xy}}=& \, \frac{{\tau_{xy}}}{G} \cr {\gamma_{xz}}=& \, 0 \cr {\gamma_{yz}}=& \, 0 \cr {\varepsilon_{ii}}=& \, 0 \hspace{5mm} \mathrm{No \; sum} \cr \end{align}\]

For \({\tau_{xz}}\ne 0\), all others 0? \[\begin{align} {\gamma_{xz}}=& \, \frac{{\tau_{xz}}}{G} \cr {\gamma_{xy}}=& \, 0 \cr {\gamma_{yz}}=& \, 0 \cr {\varepsilon_{ii}}=& \, 0 \hspace{5mm} \mathrm{No \; sum} \cr \end{align}\]

---

Block subject to shear stress

For \({\tau_{yz}}\ne 0\), all others 0? \[\begin{align} {\gamma_{yz}}=& \, \frac{{\tau_{yz}}}{G} \cr {\gamma_{xy}}=& \, 0 \cr {\gamma_{xz}}=& \, 0 \cr {\varepsilon_{ii}}=& \, 0 \hspace{5mm} \mathrm{No \; sum} \cr \end{align}\]

For multiple simultaneous stresses: use superposition

---

---