View source for Elasticity/Disk with hole

== Disk with a central hole==
{|align="center"
 |[[image:elastic_disk_with_hole.png|thumb|400px|An elastic disk with a central circular hole]] 
 |}

Under general loading, for the stresses and displacements to be single-valued and continuous, they must be periodic in <math>\theta</math>, e.g.,
<math>\sigma_{11}(r,\theta) = \sigma_{11}(r,\theta + 2m\pi)</math>.

An Airy stress function appropriate from this situation is
:<math> \text{(83)} \qquad 
    \varphi = \sum^{\infty}_{n=0} f_n(r) \cos(n\theta) +
              \sum^{\infty}_{n=0} g_n(r) \sin(n\theta)
  </math>
In the absence of body forces,
:<math> \text{(84)} \qquad 
    \nabla^4{\varphi} = \nabla^2{(\nabla^2{\varphi})} = 0 ~;~~
    \nabla^2{} = \left(\cfrac{\partial^2}{\partial r^2} + \cfrac{1}{r}\cfrac{\partial}{\partial r} +
                   \cfrac{1}{r^2}\cfrac{\partial^2}{\partial \theta^2}\right)
  </math>
Plug in <math>\varphi</math>.
:<math>\begin{align}
    \nabla^2{\varphi} = &
      \sum^{\infty}_{n=0} \left[
        f^{''}_n(r) \cos(n\theta) + \cfrac{1}{r} f^{'}_n(r) \cos(n\theta)
        - \cfrac{n^2}{r^2} f_n(r) \cos(n\theta) \right] +  \\
      & \sum^{\infty}_{n=0} \left[
        g^{''}_n(r) \sin(n\theta) + \cfrac{1}{r} g^{'}_n(r) \sin(n\theta)
        - \cfrac{n^2}{r^2} g_n(r) \sin(n\theta) \right]  \qquad \text{(85)}
  \end{align}</math>
or,
:<math> \text{(86)} \qquad 
    \nabla^2{\varphi} = \sum^{\infty}_{n=0} F_n(r) \cos(n\theta) +
              \sum^{\infty}_{n=0} G_n(r) \sin(n\theta)
  </math>
Therefore,
:<math>\begin{align}
    \nabla^4{\varphi} = &
      \sum^{\infty}_{n=0} \left[
        F^{''}_n(r) \cos(n\theta) + \cfrac{1}{r} F^{'}_n(r) \cos(n\theta)
        - \cfrac{n^2}{r^2} F_n(r) \cos(n\theta) \right] +  \\
      & \sum^{\infty}_{n=0} \left[
        G^{''}_n(r) \sin(n\theta) + \cfrac{1}{r} G^{'}_n(r) \sin(n\theta)
        - \cfrac{n^2}{r^2} G_n(r) \sin(n\theta) \right]  \qquad \text{(87)}
  \end{align}</math>
To satisfy the compatibility condition <math>\nabla^4{\varphi} = 0</math>, we need
:<math>\begin{align}
    \text{(88)} \qquad F^{''}_n(r) + \cfrac{1}{r} F^{'}_n(r) - \cfrac{n^2}{r^2} F_n(r) & = 0
          \\
    \text{(89)} \qquad G^{''}_n(r) + \cfrac{1}{r} G^{'}_n(r) - \cfrac{n^2}{r^2} G_n(r) & = 0
         
  \end{align}</math>
The general solution of these Euler-Cauchy type equations is
:<math>\begin{align}
    \text{(90)} \qquad F_n(r) & = A_1 r^n + B_1 r^{-n}   \\
    \text{(91)} \qquad G_n(r) & = C_1 r^n + D_1 r^{-n}  
  \end{align}</math>
We can use either to determine <math>f_n(r)</math>.  Thus,
:<math> \text{(92)} \qquad 
    f^{''}_n(r) + \cfrac{1}{r} f^{'}_n(r) - \cfrac{n^2}{r^2} f_n(r) =
      A_1 r^n + B_1 r^{-n}
  </math>
or,
:<math> \text{(93)} \qquad 
    r^2 f^{''}_n(r) + r f^{'}_n(r) - n^2 f_n(r) = A_1 r^{n+2} + B_1 r^{-n+2}
  </math>
The homogeneous and particular solutions of this equation are
:<math>\begin{align}
    \text{(94)} \qquad f^h_n(r) & = A_2 r^n + B_2 r^{-n}   \\
    \text{(95)} \qquad f^p_n(r) & = A_1 r^{n+2} + B_1 r^{-n+2}  
  \end{align}</math>
Hence, the general solution is
:<math> \text{(96)} \qquad 
    f_n(r)  = A_1 r^{n+2} + B_1 r^{-n+2} + A_2 r^n + B_2 r^{-n}
  </math>
This form is valid for <math>n > 1</math>.  If <math>n = 0, 1</math>, alternative forms are obtained.  Thus,
:<math>\begin{align}
    \text{(97)} \qquad f_0(r) & = A_O r^2 + B_0 r^2 \ln r + C_0 + D_0 \ln r  \\
    \text{(98)} \qquad f_1(r) & = A_1 r^3 + B_1 r  + C_1 r \ln r + D_1 r^{-1}   \\
    \text{(99)} \qquad f_n(r) & = A_n r^{n+2} + B_n r^n  + C_n r^{-n+2} + D_n r^{-n} ~,~~n > 1
       
  \end{align}</math>
Terms in <math>f_n</math> are chosen according to the specific problem of interest.

=== Traction BCs===
: at <math> r = a </math>
:<math>
  \text{(100)} \qquad
  \sigma_{rr} = T_1(\theta) ~,~~ \sigma_{r\theta} = T_2(\theta)
 </math>
: at <math> r = b </math>
:<math>
  \text{(101)} \qquad
  \sigma_{rr} = T_3(\theta) ~,~~ \sigma_{r\theta} = T_4(\theta) 
 </math>
Express <math>T_i(\theta)</math> in Fourier series form.
:<math> \text{(102)} \qquad
    T_i(\theta) = \sum^{\infty}_{n=0} A_{ni}\cos(n\theta) +
                  \sum^{\infty}_{n=0} B_{ni}\sin(n\theta) ~,~~i=1,2,3,4
  </math>
Terms in <math>T_i</math> are chosen according to the specific problem of interest.

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[[Category:Elasticity]]