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I am new to this type of problem and feel as if I am going in circle with regards to boundary conditions.

I am interested in finding an analytic solution for: $\mu\nabla^{2}\underline{u}+(\lambda+\mu)\nabla\nabla\cdot\underline{u}=0$,

which I think in this case boils down to solving for $\underline{u}$ such that:

$(\lambda + \mu)\frac{\partial}{\partial x}(\frac{\partial u_{x}}{\partial x}+ \frac{\partial u_{y}}{\partial y})+\mu(\frac{\partial^{2}u_{x}}{\partial x^{2}}+ \frac{\partial^{2} u_{x}}{\partial y^{2}})=0$

and

$(\lambda + \mu)\frac{\partial}{\partial y}(\frac{\partial u_{x}}{\partial x}+ \frac{\partial u_{y}}{\partial y})+\mu(\frac{\partial^{2}u_{y}}{\partial x^{2}}+ \frac{\partial^{2} u_{y}}{\partial y^{2}})=0.$

Problem Setup

Is the following process to solve for $\underline{u}$ sensible/valid:

  1. Set the traction ($t_{i}^{b}$) to zero along all four sides. [Admittedly not sure if this is sensible given displacement conditions]
  2. Solve for the stresses along the boundary via $\sigma_{ij}n_{j}=t_{i}^{b}=0$, ignoring corners as normal to a point is not well-defined.
  3. Use $\sigma$ along the boundaries as boundary conditions for the biharmonic equation ($\nabla^{4}\phi=0$) to determine a suitable Airy Stress function, $\phi(x,y)$
  4. Use $\phi(x,y)$ to solve for the internal stresses, strains and displacement field $\underline{u}$
  5. Verify that $\underline{u}$ honors boundary conditions by evaluating function along the boundaries

Or am I going about this whole problem incorrectly? In which case any help would be appreciated.

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