# 2D Linear Elastostatics with Displacement Boundary Conditions

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

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

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.