6.2 Linear elastic
6.2.1 Anisotropic behavior
Fibrous composites exhibit a more complex constitutive response
Consider the following material description: \[\begin{equation*} \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{zz}}\cr {\gamma_{yz}}\cr {\gamma_{xz}}\cr {\gamma_{xy}}\cr \end{array} \right\} = \left[ \begin{array}{cccccc} S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16} \cr S_{21} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26} \cr S_{31} & S_{32} & S_{33} & S_{34} & S_{35} & S_{36} \cr S_{41} & S_{42} & S_{43} & S_{44} & S_{45} & S_{46} \cr S_{51} & S_{52} & S_{53} & S_{54} & S_{55} & S_{56} \cr S_{61} & S_{62} & S_{63} & S_{64} & S_{65} & S_{66} \cr \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{zz}}\cr {\sigma_{yz}}\cr {\sigma_{xz}}\cr {\sigma_{xy}}\cr \end{array} \right\} \end{equation*}\] This is generalized Hooke’s law (applicable to any linear elastic material–called anisotropic).
Note the use of engineering shear strain in this matrix to allow the symmetry of the tensorial elastic compliance matrix to be appreciated in the 6x6 compliance matrix.
- For simplicity, we can write generalized Hooke’s law as: \[\begin{equation*} \{{\varepsilon}\} = [S] \{\sigma\} \end{equation*}\]
- the values of \([S]\) are called [“elastic compliances”]
- What are the compliances for an isotropic material?
\[\begin{equation*} \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{zz}}\cr {\varepsilon_{yz}}\cr {\varepsilon_{xz}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} = \left[ \begin{array}{cccccc} \frac{1}{E} & \frac{-\nu}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \cr \frac{-\nu}{E} & \frac{1}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \cr \frac{-\nu}{E} & \frac{-\nu}{E} & \frac{1}{E} & 0 & 0 & 0 \cr 0 & 0 & 0 & \frac{1}{2 G} & 0 & 0 \cr 0 & 0 & 0 & 0 & \frac{1}{2 G} & 0 \cr 0 & 0 & 0 & 0 & 0 & \frac{1}{2 G} \cr \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{zz}}\cr {\sigma_{yz}}\cr {\sigma_{xz}}\cr {\sigma_{xy}}\cr \end{array} \right\} \end{equation*}\]
- We have a material description for the relationship between stress and strain (called a constitutive relationship) for an isotropic material
- It is important to be able to consider the inversion of this system:
\[\begin{align} \{{\varepsilon}\} =& [S] \{\sigma\} \cr \{\sigma\} =& [S]^-1 \{{\varepsilon}\} \cr \{\sigma\} =& [C] \{{\varepsilon}\} \cr \end{align}\]
- The values of \([C]\) are called [Elastic constants]
- \([C]\) and \([S]\) are fully populated for an anisotropic material
- For an isotropic material: \[\begin{equation*} \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{zz}}\cr {\sigma_{yz}}\cr {\sigma_{xz}}\cr {\sigma_{xy}}\cr \end{array} \right\} = \frac{E}{(1+\nu)(1-2\nu)} \left[ \begin{array}{cccccc} 1-\nu & \nu & \nu & 0 & 0 & 0 \cr \nu & 1-\nu & \nu & 0 & 0 & 0 \cr \nu & \nu & 1-\nu & 0 & 0 & 0 \cr 0 & 0 & 0 & 1-2\nu & 0 & 0 \cr 0 & 0 & 0 & 0 & 1-2\nu & 0 \cr 0 & 0 & 0 & 0 & 0 & 1-2\nu \cr \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{zz}}\cr {\varepsilon_{yz}}\cr {\varepsilon_{xz}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} \end{equation*}\]
\[\begin{equation*} \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{zz}}\cr {\sigma_{yz}}\cr {\sigma_{xz}}\cr {\sigma_{xy}}\cr \end{array} \right\} = \left[ \begin{array}{cccccc} 2 \mu + \lambda & \lambda & \lambda & 0 & 0 & 0 \cr \lambda & 2 \mu + \lambda & \lambda & 0 & 0 & 0 \cr \lambda & \lambda &2 \mu + \lambda & 0 & 0 & 0 \cr 0 & 0 & 0 & 2 \mu & 0 & 0 \cr 0 & 0 & 0 & 0 & 2 \mu & 0 \cr 0 & 0 & 0 & 0 & 0 & 2 \mu \cr \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{zz}}\cr {\varepsilon_{yz}}\cr {\varepsilon_{xz}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} \end{equation*}\]
Where: \[\begin{align} \mu =& \frac{E}{2 (1-\nu)} \cr \lambda =& \frac{\nu E}{(1+\nu)(1-2 \nu)} \cr \end{align}\]
6.2.2 Shear modulus
Note also: the shear modulus
\[\begin{equation*} G = \frac{E}{2 (1+\nu)} \end{equation*}\]
6.2.3 Other material descriptions
- There are materials that fit between total anisotropy (21 constants) and isotropic (2 constants).
- In the aerospace world, a critical one is “orthotropic” (9 constants) \[\begin{equation*} \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{zz}}\cr {\varepsilon_{yz}}\cr {\varepsilon_{xz}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} = \left[ \begin{array}{cccccc} \frac{1}{E_{xx}} & -\frac{\nu_{yx}}{E_{yy}} & -\frac{\nu_{zx}}{E_{zz}} & 0 & 0 & 0 \cr -\frac{\nu_{xy}}{E_{xx}} & \frac{1}{E_{yy}} & -\frac{\nu_{zy}}{E_{zz}} & 0 & 0 & 0 \cr -\frac{\nu_{xz}}{E_{xx}} & -\frac{\nu_{yz}}{E_{yy}} & \frac{1}{E_{zz}} & 0 & 0 & 0 \cr 0 & 0 & 0 & \frac{1}{2 G_{yz}} & 0 & 0 \cr 0 & 0 & 0 & 0 & \frac{1}{2 G_{zx}} & 0 \cr 0 & 0 & 0 & & 0 & \frac{1}{2 G_{xy}} \cr \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{zz}}\cr {\sigma_{yz}}\cr {\sigma_{xz}}\cr {\sigma_{xy}}\cr \end{array} \right\} \end{equation*}\]
Due to symmetry: \[\begin{align} \frac{\nu_{yx}}{E_{yy}} =& \frac{\nu_{xy}}{E_{xx}} \cr \frac{\nu_{zx}}{E_{zz}} =& \frac{\nu_{xz}}{E_{xx}} \cr \frac{\nu_{zy}}{E_{zz}} =& \frac{\nu_{yz}}{E_{yy}} \cr \end{align}\]
This is often the best description of a composite ply.
- It works for bone in some cases too
- Also:
- Monoclinic (13 constants)
- Orthotropic (9 constants)
- Tetragonal (6 constants)
- Transversely isotropic (5 constants – ex: unidirectional composites, rolled steel, some bone)
- Cubic (3 constants-ex: silicon)
Finally, this entire description is referred to as “generalized Hooke’s law” (Robert Hooke, Late 17th century)
\[\begin{equation*} {\sigma_{ij}}= E_{ijkl} \, \varepsilon_{kl} \end{equation*}\]
6.2.4 Number of equations to find the 18 (15 unique) unknowns
| Law | Quantity |
|---|---|
| Linear momentum (equilibrium) | 3 |
| Angular momentum (equilibrium) | (3) |
| Constitutive law (\(\sigma\)-\({\varepsilon}\)) | 6 |
| Kinematics (\({\varepsilon}\)-displacement | 6 |
| Total | 18 (15) |
- Note: The three equations of angular momentum were used to show the symmetry of the stress tensor