5.6 Displacement Compatibility

Undeformed, compatible, and incompatible deformation

  • Displacements are not arbitrary, they must conform to certain limits.
  • For our purposes, we limit ourselves to recognizing that displacements must be single values functions. (Points A’ and B’ must be perfectly aligned under strain.)
  • This means that the displacement functions must be continuous within the body. There can be no cracks nor voids.
  • If cracks or void form, new boundary conditions exist and the elasticity problem must be reformulated accounting for the change.

Note that there are only 3 displacements, but 6 strains.

  • To get strain from displacement, we must differentiate. Differentiation loses information, so is not difficult to stomach
  • To get displacement from strain, we must integrate. While we could get 6 integration constants, we don’t have enough unique displacements to establish them. (Note, these are pointwise equations, so at any one point we do not have 6 displacements.
    • We are either overdetermined (not physically possible for a real system), or we have redundancy in the equations.
    • Consequently, there must be additional constraints on strain:

5.6.1 Compatibility relationships

Consider first 2D. We have 2 displacements and 3 strains. \[\begin{equation} \begin{split} {\varepsilon_{xx}}= \frac{\partial u}{\partial x} \\ {\varepsilon_{yy}}= \frac{\partial v}{\partial y} \\ 2 {\varepsilon_{xy}}= \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \\ \end{split} \end{equation}\]

  • Taking derivatives of the third equations (with respect to \(x\), then \(y\)): \[\begin{equation} \begin{split} \frac{2 \, \partial {\varepsilon_{xy}}}{\partial x} = \frac{\partial}{\partial x} \frac{\partial u}{\partial y} + \frac{\partial}{\partial x} \frac{\partial v}{\partial x} \\ \frac{2 \, \partial^2 {\varepsilon_{xy}}}{\partial x \partial y} = \frac{\partial^2 \partial u}{\partial y^2 \partial x} + \frac{\partial^2 \partial v}{\partial x^2 \partial y} \\ \end{split} \end{equation}\]

Thus: \[\begin{equation} \begin{split} \frac{2 \, \partial^2 {\varepsilon_{xy}}}{\partial y \partial x} = \frac{\partial {\varepsilon_{xx}}}{\partial y^2} + \frac{\partial {\varepsilon_{yy}}}{\partial x^2} \\ \end{split} \end{equation}\]

  • This establishes a relationship between the three equations such that they are not independent.