Dissemination of IT for the Promotion of Materials Science (DoITPoMS)


Derivation of expressions

The force balance can in this case (see above figure) be written as

\[\frac{1}{2}\left[ {\left( {{\sigma _{\mathop{\rm Y}\nolimits} } - {\sigma _{{\rm{s,\;res}}}}} \right) + \left( {{\sigma _{\mathop{\rm Y}\nolimits} } - {\sigma _{{\rm{e,\;res}}}}} \right)} \right]\left( {{y_{\rm{s}}} - {y_{\rm{e}}}} \right) + \frac{1}{2}\left[ {\left( {{\sigma _{\mathop{\rm Y}\nolimits} } - {\sigma _{{\rm{e,\;res}}}}} \right) + {\sigma _{\mathop{\rm Y}\nolimits} }} \right]{y_{\rm{e}}} = {\sigma _{\mathop{\rm Y}\nolimits} }{y_{\rm{s}}}\]

which simplifies to

\[{\sigma _{{\rm{s,\;res}}}} = - {\sigma _{{\rm{e,\;res}}}}\left( {\frac{{{y_{\rm{s}}}}}{{{y_{\rm{s}}} - {y_{\rm{e}}}}}} \right)\]

Elastic unloading of the surface region gives

\[\begin{array}{l} \left( {{\sigma _{\mathop{\rm Y}\nolimits} } - {\sigma _{{\rm{s,\;res}}}}} \right) = E\left( {{\varepsilon _{\rm{s}}} - {\varepsilon _{{\rm{s, res}}}}} \right) = E\kappa {y_{\rm{s}}} - E{\kappa _{{\rm{res}}}}{y_{\rm{s}}}\\ {\rm{thus}}\; {\sigma _{{\rm{s,\;res}}}} = {\sigma _{\mathop{\rm Y}\nolimits} } - E{y_{\rm{s}}}\left( {\kappa - {\kappa _{{\rm{res}}}}} \right) \end{array}\]

while the same condition applied at the limit of the elastic core leads to

\[\begin{array}{l} \left( {{\sigma _{\mathop{\rm Y}\nolimits} } - {\sigma _{{\rm{e,\;res}}}}} \right) = E\left( {{\varepsilon _{\rm{Y}}} - {\varepsilon _{{\rm{e, res}}}}} \right) = E{\varepsilon _{\rm{Y}}} - E{\kappa _{{\rm{res}}}}{y_{\rm{e}}}\\ {\rm{thus}}\;{\sigma _{{\rm{e,\;res}}}} = {\sigma _{\mathop{\rm Y}\nolimits} } - E{y_{\rm{e}}}\left( {\kappa - {\kappa _{{\rm{res}}}}} \right) \end{array}\]

Substitution of these two expressions for the stresses into the above equation obtained from the force balance then gives the equation

\[{\sigma _{\mathop{\rm Y}\nolimits} } - E{y_{\rm{s}}}\left( {\kappa - {\kappa _{{\rm{res}}}}} \right) = - \left[{\sigma _{\mathop{\rm Y}\nolimits} } - E{y_{\rm{e}}}\left( {\kappa - {\kappa _{{\rm{res}}}}} \right)\right]\left( {\frac{{{y_{\rm{s}}}}}{{{y_{\rm{s}}} - {y_{\rm{e}}}}}} \right)\]

which can be simplified to

\[{\kappa _{{\rm{res}}}} = \kappa {\left( {1 - \frac{{{y_{\rm{e}}}}}{{{y_{\rm{s}}}}}} \right)^2}\]