User:Caan L/sandbox/William's Singularity

Williams' Singularity represents a solution developed by M. L. Williams to bi-material interface cracking. Theoretical interfaces of the interface crack problem date back to the late fifties. The solution to specific problems of cracks lying along bi-material interfaces of elastic isotropic media was begun by Williams in 1959, followed by many. Williams performed an asymptotic analysis of the elastic fields at the tip of an open interface crack. He found that the stresses and displacements behaved in an oscillatory manner. Solutions following retained this feature.^[1]

In 1959, Williams determined the characteristic oscillating stress singularity solutions for the elastic interface crack problem. Since then, significant research has been performed to apply Williams’ solutions to specific fracture mechanics problems and to identify its limitations. In a study by Rice, it was found that the full form of the near tip field was not specified by Williams for interface cracks. In other words, Williams’ eigenvalues were of the form λ = n(integer) - ½ + iϵ for plane strains and stresses, but Rice determined that integer eigenvalues of the form λ = n were possible as well. Specifically, Rice solved for the complete inner and outer eigenfunction expansions of the Williams’ solution for the elastic interface crack and argued that the stress intensity factor associated with the Williams’ singular fields is valid parameter for the crack tip state with ‘small-scale’ contact zones.^[2]

The elastic interfacial crack solutions discussed by Rice predict interpenetration of the crack walls near the tip. This provides an additional characterizing parameter for the near tip state if the interpenetration zone is significantly smaller than the crack length. The physical implications of interpenetration led other researchers, like Comninou, to propose solutions predicting crack closure. Additional studies (Aravas and Sharma ) have indicated that both Williams’ and Comninou’s solutions are relevant to several current problems on elastic interface crack fields.

Williams' solution has important implications for the stress fields at different engineering attachments. In particular, it provides insight into how the morphology of an attachment governs its stress field. With some simplifications, the tendon-to-bone attachment can be modeled to relate to the Williams free-edge singularity solution. In this model, the tendon is linear elastic with elastic modulus E and Poisson ratio ν. The insertion angle θ_T of the tendon is shown here as being greater than 90°, although the graphic suggests that nature is in fact smarter than that.

Williams' solutions involve a thin, isotropic material (modeled as the tendon) attached to a rigid substratum (the bone). The tendon is loaded far from the site of attachment to bone. This model is simplified, such that the tendon connects straight to the bone without any toughening mechanisms, such as a graded interface or tendon/bone interdigitation. However, the model is still relevant to surgical practice, where the interfacial tissue is resected and the tendon is sutured directly to the bone.

One other simplification involves the anisotropy of the tendon: in this system, the tendon was modeled as isotropic. In reality, tendons are anisotropic - and this anisotropy exacerbates the free edge singularity. For example, while the isotropic tendon in the model loses the free edge singularity for insertion angles below 60°, actual anisotropic tendons present free edge singularities at angles as small as 15°. However, despite this quantitative difference, the model still yields important qualitative insight on the mechanics of attachment and the mechanical properties of healing at the tendon-to-bone insertion site.

The Free Edge Problem Williams considered can be modeled as a tendon-to-bone attachment.^[3] The asymptotic stress field in the cylindrical coordinate system of this model is represented by the following equations:

${\displaystyle \sigma _{rr}=r^{\lambda -1}(\Phi '')(\theta ;\lambda )+(\lambda +1)\Phi (\theta ;\lambda )}$

${\displaystyle \sigma _{\theta \theta }=r^{\lambda -1}(\lambda (\lambda +1)\Phi (\theta ;\lambda ))}$

${\displaystyle \sigma _{r\theta }=-r^{\lambda -1}\Phi '(\theta ;\lambda )}$

Where ${\displaystyle \Phi (\theta ;\lambda )}$ 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.^[3]

The unknown constants can be used to find the displacements by using the both the strain tensor and the delta function:

${\displaystyle \sigma _{ij}={\frac {E}{1+v}}\epsilon _{ij}+{\frac {E}{(1+v)(1-2v)}}\delta _{ij}\epsilon _{kk}}$

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

${\displaystyle \epsilon _{ij}={\frac {1}{2}}(u_{i,j}+u_{j,i})}$

where ${\displaystyle \delta _{ij}}$ is the Kronecker's delta function that equals 1 when ${\displaystyle i=j}$ and 0 otherwise, and ${\displaystyle \epsilon _{kk}=\epsilon _{11}+\epsilon _{22}+\epsilon _{33}}$.^[3]

Displacements can be written in terms of the constraints:

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

${\displaystyle u_{0}=r^{\lambda }\left({\frac {1+v}{E}}\right)(-\Phi '(\theta ;\lambda )+(1-n)(\lambda -1)\Psi (\theta ;\lambda ))}$

where ${\displaystyle n-v/(1-v)}$, and ${\displaystyle \Psi (\theta ;\lambda )}$.^[3]

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:

${\displaystyle sin^{2}(\lambda \theta _{T})=\left({\frac {1}{3-4n}}\right)(4(1-n)^{2}-(\lambda \theta _{T})^{2}sin^{2}\theta _{T})}$^[3]

The stress field near the free edge of the body becomes singular at a critical insertion angle ${\displaystyle \theta _{T}}$ where ${\displaystyle \lambda }$ reaches its lowest positive value.^[3]

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