## 2 Materials and Methods
### Similarity Solutions in the Corner of an Elastic Body
#### 2.2.1 Elastostatic Expansion  
It has been known for over 50 years that static deformations (eigensolutions) in a two-dimensional corner of an elastic material can be found by use of an Airy stress function, an approach that is well understood for similar problems (see, e.g., [25]). We shall examine this briefly here, and then move on to the more-relevant elastodynamic case afterwards. Following [5], we pose an Airy stress function of the form \(\chi(\rho,\theta)=\rho^{\gamma_{m}+1}\Theta(\theta)\), where \(\gamma_{m}\in\mathbb{C}\), from which we can recover the stresses as  
\[\tau_{\rho\theta}= -\frac{\partial}{\partial\rho}\left(\frac{1}{\rho}\frac{\partial \chi}{\partial\theta}\right), \tau_{\theta\theta}= \frac{\partial^{2}\chi}{\partial\rho^{2}}. \tag{30}\]  
Then, [PERSON]'s equation is satisfied, and compatibility yields \(\
abla^{4}\chi=0\), which requires that the \(\theta\) dependence is of the form  
\[\Theta(\theta)=c_{1}\cos((\gamma_{m}+1)\theta)+c_{2}\sin((\gamma_{m}+1)\theta )+c_{3}\cos((\gamma_{m}-1)\theta)+c_{4}\sin((\gamma_{m}-1)\theta). \tag{31}\]  
We now apply the boundary conditions, and use the symmetry properties of the \(\theta\) dependence as given in (31) to simplify the problem. We do this by considering the linear combinations of conditions given by \(\tau_{\rho\theta}|_{\xi/2}\pm\tau_{\rho\theta}|_{-\xi/2}=0\) and \(\tau_{\theta\theta}|_{\xi/2}\pm\tau_{\theta\theta}|_{-\xi/2}=0\), which yield two decoupled linear systems of equations given by  
\[\begin{bmatrix}\cos((\gamma_{m}+1)\xi/2)&\cos((\gamma_{m}-1)\xi/2) \\ (\gamma_{m}+1)\sin((\gamma_{m}+1)\xi/2)&(\gamma_{m}-1)\sin((\gamma_{m}-1)\xi/2 )\end{bmatrix}\begin{bmatrix}c_{1}\\ c_{3}\end{bmatrix}= \begin{bmatrix}0\\ 0\end{bmatrix}, \tag{32}\] \[\begin{bmatrix}\sin((\gamma_{m}+1)\xi/2)&\sin((\gamma_{m}-1)\xi/ 2)\\ (\gamma_{m}+1)\cos((\gamma_{m}+1)\xi/2)&(\gamma_{m}-1)\cos((\gamma_{m}-1)\xi/2 )\end{bmatrix}\begin{bmatrix}c_{2}\\ c_{4}\end{bmatrix}= \begin{bmatrix}0\\ 0\end{bmatrix}. \tag{33}\]  
In order for these conditions to possess non-trivial solutions to the homogeneous problem, the determinants must be equal to zero, which yields constraints on the corner exponents:  
\[\sin(\gamma_{m}\xi)\pm\gamma_{m}\sin(\xi)=0. \tag{34}\]  
In (34), a positive sign corresponds to the determinant of the first system (32) and an Airy stress function symmetric around \(\theta=0\), while the negative sign corresponds to the determinant of the second system (33) and an Airy stress function antisymmetric about \(\theta=0\). In order for the solutions to be physical, we must also have finite displacements, which limits the solutions to those where \(\Re[\gamma_{m}]>0\).  
The roots of (34), which can easily be found numerically, are labelled \(\gamma_{m}\) and ordered according to the size of the real parts, and also by imaginary parts when two roots have the same real part. For each solution of the compatibility condition (34) we find a non-trivial  
Figure 4: A general traction-free elastic corner. The elastic body is the shaded area extending to infinity.  
solution to the homogeneous problem with a single unknown constant; we can therefore write a general similarity solution as the series  
\[\tau_{\rho\theta} =\sum_{m=1}^{M}\eta_{m}f_{0}(\gamma_{m},\theta)\rho^{\gamma_{m}-1}, \tag{35}\] \[\tau_{\theta\theta} =\sum_{m=1}^{M}\eta_{m}g_{0}(\gamma_{m},\theta)\rho^{\gamma_{m}-1}, \tag{36}\]  
for arbitrary number of terms \(M\). Here \(f_{0}\) and \(g_{0}\) are known functions and \(\eta_{m}\) are arbitrary unknown constants, each associated with the \(m\)th similarity solution.