Mathematical Elasticity, Volume 2: Theory Of Plates (Studies In Mathematics And Its Application),Used

The objective of Volume II is to show how asymptotic methods, with the thickness as the small parameter, indeed provide a powerful means of justifying twodimensional plate theories. More specifically, without any recourse to any a priori assumptions of a geometrical or mechanical nature, it is shown that in the linear case, the threedimensional displacements, once properly scaled, converge in H1 towards a limit that satisfies the wellknown twodimensional equations of the linear KirchhoffLove theory; the convergence of stress is also established.In the nonlinear case, again after ad hoc scalings have been performed, it is shown that the leading term of a formal asymptotic expansion of the threedimensional solution satisfies wellknown twodimensional equations, such as those of the nonlinear KirchhoffLove theory, or the von Krmn equations. Special attention is also given to the first convergence result obtained in this case, which leads to twodimensional large deformation, frameindifferent, nonlinear membrane theories. It is also demonstrated that asymptotic methods can likewise be used for justifying other lowerdimensional equations of elastic shallow shells, and the coupled pluridimensional equations of elastic multistructures, i.e., structures with junctions. In each case, the existence, uniqueness or multiplicity, and regularity of solutions to the limit equations obtained in this fashion are also studied.