Lets think again about the stress potato:

• We need 18 (15 unique) equations to solve for the 18 (15 unique) unknowns at each point in the body (ie field equations for the continuous body).

General concepts of material constitutive response

Intuitive development of a constitutive matrix

Uniform rectangular block pulled on both ends

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:

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

• What about for shear?

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

Linear elastic

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: \[\begin{align} \mu =& \frac{E}{2 (1-\nu)} \cr \lambda =& \frac{\nu E}{(1+\nu)(1-2 \nu)} \cr \end{align}\]

Shear modulus

Note also: the shear modulus

\[\begin{equation*} G = \frac{E}{2 (1+\nu)} \end{equation*}\]

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 17^th century)

\[\begin{equation*} {\sigma_{ij}}= E_{ijkl} \, \varepsilon_{kl} \end{equation*}\]

Number of equations to find the 18 (15 unique) unknowns

• Note: The three equations of angular momentum were used to show the symmetry of the stress tensor

Plane stress

• Recall: \[\begin{equation*} \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{zz}}\cr {\varepsilon_{yz}}\cr {\varepsilon_{xz}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} = \left[ \begin{array}{cccccc} \frac{1}{E} & \frac{-\nu}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \cr \frac{-\nu}{E} & \frac{1}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \cr \frac{-\nu}{E} & \frac{-\nu}{E} & \frac{1}{E} & 0 & 0 & 0 \cr 0 & 0 & 0 & \frac{1}{2 G} & 0 & 0 \cr 0 & 0 & 0 & 0 & \frac{1}{2 G} & 0 \cr 0 & 0 & 0 & 0 & 0 & \frac{1}{2 G} \cr \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr 0 \cr 0 \cr 0 \cr {\sigma_{xy}}\cr \end{array} \right\} \end{equation*}\]

The result is a simplified set of governing constitutive equations: \[\begin{equation*} \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} = \left[ \begin{array}{ccc} \frac{1}{E} & \frac{-\nu}{E} & 0 \cr \frac{-\nu}{E} & \frac{1}{E} & 0 \cr 0 & 0 & \frac{1}{2 G} \cr \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{xy}}\cr \end{array} \right\} \end{equation*}\]

\[\begin{equation*} \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{xy}}\cr \end{array} \right\} = \left[ \begin{array}{ccc} \frac{E}{1-\nu^2} & \frac{\nu E}{1-\nu^2} & 0 \cr \frac{\nu E}{1-\nu^2} & \frac{E}{1-\nu^2} & 0 \cr 0 & 0 & {2 G} \cr \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} \end{equation*}\]

Plane strain

\[\begin{equation*} \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{zz}}\cr {\sigma_{yz}}\cr {\sigma_{xz}}\cr {\sigma_{xy}}\cr \end{array} \right\} = \frac{E}{(1+\nu)(1-2\nu)} \left[ \begin{array}{cccccc} 1-\nu & \nu & \nu & 0 & 0 & 0 \cr \nu & 1-\nu & \nu & 0 & 0 & 0 \cr \nu & \nu & 1-\nu & 0 & 0 & 0 \cr 0 & 0 & 0 & 1-2\nu & 0 & 0 \cr 0 & 0 & 0 & 0 & 1-2\nu & 0 \cr 0 & 0 & 0 & 0 & 0 & 1-2\nu \cr \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr 0 \cr 0 \cr 0 \cr {\varepsilon_{xy}}\cr \end{array} \right\} \end{equation*}\]

\[\begin{equation*} \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{xy}}\cr \end{array} \right\} = \frac{E}{(1+\nu)(1-2\nu)} \left[ \begin{array}{cccccc} 1-\nu & \nu & 0 \cr \nu & 1-\nu & 0 \cr 0 & 0 & 1-2\nu \cr \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} \end{equation*}\]

\[\begin{equation*} \left\{ \begin{array}{c} {\varepsilon_{xx}}\cr {\varepsilon_{yy}}\cr {\varepsilon_{xy}}\cr \end{array} \right\} = \frac{1+\nu}{E} \left[ \begin{array}{cccccc} 1-\nu & -\nu & 0 \cr -\nu & 1-\nu & 0 \cr 0 & 0 & 1 \cr \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\cr {\sigma_{yy}}\cr {\sigma_{xy}}\cr \end{array} \right\} \end{equation*}\]

Summary of elasticity

So far in this class we have developed a method to solve a boundary value problem of linear elasticity.

We must do all of the following:

1. Find a continuous displacement field (subject to enforced displacement boundary conditions)
   • \({\vec{u}}= f(x,y,z)\)
2. where that displacement field results in internal strains
   • \[\begin{align} {\varepsilon_{xx}}=& {u {}_{,x}} + \frac{1}{2} \left[({u {}_{,x}})^2 + ({v {}_{,x}})^2 + ({w {}_{,x}})^2 \right] \cr {\varepsilon_{yy}}=& {v {}_{,y}} + \frac{1}{2} \left[({u {}_{,y}})^2 + ({v {}_{,y}})^2 + ({w {}_{,y}})^2 \right] \cr {\varepsilon_{zz}}=& {w {}_{,z}} + \frac{1}{2} \left[({u {}_{,z}})^2 + ({v {}_{,z}})^2 + ({w {}_{,z}})^2 \right] \cr {\varepsilon_{xy}}=& \frac{1}{2} \left({u {}_{,y}} + {v {}_{,x}}\right) + \frac{1}{2} \left[{u {}_{,x}} {u {}_{,y}} + {v {}_{,x}} {v {}_{,y}} + {w {}_{,x}} {w {}_{,y}} \right] \cr {\varepsilon_{xz}}=& \frac{1}{2} \left({u {}_{,z}} + {w {}_{,x}}\right) + \frac{1}{2} \left[{u {}_{,x}} {u {}_{,z}} + {v {}_{,x}} {v {}_{,z}} + {w {}_{,x}} {w {}_{,z}} \right] \cr {\varepsilon_{yz}}=& \frac{1}{2} \left({v {}_{,z}} + {w {}_{,y}}\right) + \frac{1}{2} \left[{u {}_{,y}} {u {}_{,z}} + {v {}_{,y}} {v {}_{,z}} + {w {}_{,y}} {w {}_{,z}} \right] \cr \end{align}\]

3. where those strains result in internal stresses through a constitutive relationship
   • \[\begin{align} \{{\varepsilon}\} =& [S] \{\sigma\} \cr \{\sigma\} =& [C] \{{\varepsilon}\} \cr \end{align}\]
4. Where the stresses result in internal forces that obey equilibrium
   • \[\begin{align} {{\sigma_{xx}} {}_{,x}} + {{\sigma_{yx}} {}_{,y}} + {{\sigma_{zx}} {}_{,z}} + b_x = \rho a_x \cr {{\sigma_{xy}} {}_{,x}} + {{\sigma_{yy}} {}_{,y}} + {{\sigma_{zy}} {}_{,z}} + b_y = \rho a_y \cr {{\sigma_{xz}} {}_{,x}} + {{\sigma_{yz}} {}_{,y}} + {{\sigma_{zz}} {}_{,z}} + b_z = \rho a_z \cr \end{align}\]
   • where \(b\) is a body force (such as gravity)
5. Where the stresses satisfy externally applied loads boundary conditions
   • \(\left\{T\right\} = [\sigma] \cdot \left\{n\right\}\)