6.5 Summary of elasticity

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

“Stress potato”

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
    • \[\begin{align} {\varepsilon_{xx}}=& {u {}_{,x}} + \frac{1}{2} \left[({u {}_{,x}})^2 + ({v {}_{,x}})^2 + ({w {}_{,x}})^2 \right] \cr {\varepsilon_{yy}}=& {v {}_{,y}} + \frac{1}{2} \left[({u {}_{,y}})^2 + ({v {}_{,y}})^2 + ({w {}_{,y}})^2 \right] \cr {\varepsilon_{zz}}=& {w {}_{,z}} + \frac{1}{2} \left[({u {}_{,z}})^2 + ({v {}_{,z}})^2 + ({w {}_{,z}})^2 \right] \cr {\varepsilon_{xy}}=& \frac{1}{2} \left({u {}_{,y}} + {v {}_{,x}}\right) + \frac{1}{2} \left[{u {}_{,x}} {u {}_{,y}} + {v {}_{,x}} {v {}_{,y}} + {w {}_{,x}} {w {}_{,y}} \right] \cr {\varepsilon_{xz}}=& \frac{1}{2} \left({u {}_{,z}} + {w {}_{,x}}\right) + \frac{1}{2} \left[{u {}_{,x}} {u {}_{,z}} + {v {}_{,x}} {v {}_{,z}} + {w {}_{,x}} {w {}_{,z}} \right] \cr {\varepsilon_{yz}}=& \frac{1}{2} \left({v {}_{,z}} + {w {}_{,y}}\right) + \frac{1}{2} \left[{u {}_{,y}} {u {}_{,z}} + {v {}_{,y}} {v {}_{,z}} + {w {}_{,y}} {w {}_{,z}} \right] \cr \end{align}\]
  1. where those strains result in internal stresses through a constitutive relationship
    • \[\begin{align} \{{\varepsilon}\} =& [S] \{\sigma\} \cr \{\sigma\} =& [C] \{{\varepsilon}\} \cr \end{align}\]
  2. 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)
  3. Where the stresses satisfy externally applied loads boundary conditions
    • \(\left\{T\right\} = [\sigma] \cdot \left\{n\right\}\)