= Torsion and tilt stiffness of a bonded elastic disk
{title2=$K_V=\pi Ga^4/(2h),\quad K_H=\pi Ma^4/(4h)$}
A circular elastic layer of thickness $h\ll a$ bonded between rigid faces develops <shear strain> $r\phi/h$ under a relative twist $\phi$. Integrating the <shear stress> times its moment arm gives $K_V=GJ_p/h$. A small relative tilt produces longitudinal <strain> $y\theta/h$; when lateral deformation is constrained, the corresponding normal traction gives $K_H=MI/h$. The <polar second moment of area> is $J_p=\pi a^4/2$ and the diametric <second moment of area> is $I=\pi a^4/4$. Edge relaxation limits this thin-layer approximation, especially in a nearly incompressible material.
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