Mathematical Elasticity: Volume II: Theory of Plates (ISSN Book 27)

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 Two-dimensional 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 Three-dimensional Displacements, Once Properly Scaled, Converge In H1 Towards A Limit That Satisfies The Well-known Two-dimensional Equations Of The Linear Kirchhoff-love 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 Three-dimensional Solution Satisfies Well-known Two-dimensional Equations, Such As Those Of The Nonlinear Kirchhoff-love Theory, Or The Von Kármán Equations. Special Attention Is Also Given To The First Convergence Result Obtained In This Case, Which Leads To Two-dimensional Large Deformation, Frame-indifferent, Nonlinear Membrane Theories. It Is Also Demonstrated That Asymptotic Methods Can Likewise Be Used For Justifying Other Lower-dimensional Equations Of Elastic Shallow Shells, And The Coupled Pluri-dimensional Equations Of Elastic Multi-structures, 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.