User:Caan L/sandbox

Solution to the Williams Free Edge Problem
The Free Edge Problem Williams considered can be modeled as a tendon-to-bone attachment. The asymptotic stress field in the cylindrical coordinate system of this model is represented by the following equations:


 * $$ \sigma_{rr}=r^{\lambda-1}(\Phi'')(\theta;\lambda)+(\lambda+1)\Phi(\theta;\lambda)$$
 * $$ \sigma_{\theta\theta}=r^{\lambda-1}(\lambda(\lambda+1)\Phi(\theta;\lambda)) $$
 * $$ \sigma_{r\theta}=-r^{\lambda-1}\Phi'(\theta;\lambda) $$

Where $$ \Phi(\theta;\lambda) $$ is a function whose form, given by Williams, contains four unknown constants that must be found along with a set of eigenvalues using the boundary conditions.

The unknown constants can be used to find the displacements by using the equations below:


 * $$ \sigma_{ij}=\frac{E}{1+v}\epsilon_{ij} + \frac{E}{(1+v)(1-2v)}\delta_{ij}\epsilon_{kk} $$

where $$ \epsilon_{ij} $$ represents the 3 x 3 matrix of components of the strain tensor $$ \epsilon $$ in a particular coordinate system, $$ u_i $$ represents the three components of the displacement vector u in that coordinate system, and $$ u_{i,j}=dy/dx $$ and


 * $$ \epsilon_{ij} = \frac{1}{2}(u_{i,j}+u_{j,i}) $$

where $$ \delta_{ij} $$ is the Kronecker's delta function that equals 1 when $$ i = j $$ and 0 otherwise, and $$ \epsilon_{kk}=\epsilon_{11}+\epsilon_{22}+\epsilon_{33} $$.

Displacements can be written in terms of the constraints:


 * $$ u_r = r^\lambda \left ( \frac{1+v}{E} \right )(-(\lambda+1)\Phi(\theta;\lambda)+(1-n)\Psi'(\theta;\lambda)) $$
 * $$ u_0 = r^\lambda \left ( \frac{1+v}{E} \right )(-\Phi'(\theta;\lambda)+(1-n)(\lambda-1)\Psi(\theta;\lambda)) $$

where $$ n-v/(1-v) $$, and $$ \Psi(\theta;\lambda) $$.

The four boundary conditions used to determine the unknown constants result in a system of equations with a nontrivial solution when the equation below is satisfied:


 * $$ sin^2(\lambda\theta_T)=\left ( \frac{1}{3-4n} \right )(4(1-n)^2-(\lambda\theta_T)^2sin^2\theta_T) $$

The stress field near the free edge of the body becomes singular at a critical insertion angle $$ \theta_T $$ where $$ \lambda $$ reaches its lowest positive value.