A Note on the Eigenfunction Expansion for the Elastic Strip
D. A. Spence · SIAM Journal on Applied Mathematics · 1982
The Papkovitch–Fadle expansion of the biharmonic stress function for a semi-infinite strip $| x |\leqq 1,y\geqq 0$ with free edges $x = \pm 1$ is obtained as the residue sum of a Fourier integral for the case of end data corresponding to prescribed normal stress and transverse displacement, which implies knowledge of $\Phi _{xx} ( x,0 ) \equiv f^{( 2 )} ( x )$ and $\Delta \Phi ( x,0 ) \equiv f^{( 4 )} ( x )$. It is shown that if $f = ( f^{( 2 )} ,f^{( 4 )} )$ is twice differentiable with ${\bf f}'' \in L_2 (0,1)$ and the data are compatible with the edge conditions at the corners $x = \pm 1$, $y = 0$, the derived expansions for $\Phi _{xx} ( x,y )$ and $\Delta \Phi (x,y)$ converge to $f^{(2)} (x),f^{(4)} (x)$ as $y \downarrow 0$, for x on the closed interval $[0,1]$; even when the compatibility conditions are not met, the expansions converge to the data on the open interval $(0,1)$.