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This paper deals with a classic but very difficult type of problems, i.e., pursuing analytic buckling solutions of biaxially loaded rectangular thin plates with two free adjacent edges that are characterized by having both the free edges and a free corner.The primary challenge is to find the solutions satisfying both the governing high-order partial differential equations (PDEs) and non-Lévy-type boundary constraints. Here, an up-to-date symplectic superposition method is developed for the issues, which yields the analytic solutions by converting the problems to be solved into the superposition of two elaborated subproblems that are solved by the symplectic elasticity approach. The distinctive merit of the method is that a direct rigorous derivation helps to access the analytic solutions without any assumptions/prior knowledge of the solution forms, which is attributed to the implementation in the symplectic space-based Hamiltonian system rather than in the classic Euclidean space-based Lagrangian system. As the important outputs, comprehensive new analytic results are obtained, with 1200 critical buckling loads and 100 buckling mode shapes presented, which are all well validated by the refined finite element analysis.The rapid convergence and favorable accuracy of the present method make it competent as a benchmark one for similar problems.

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In a flexible electronic heater (FEH), periodic metal wires are often encapsulated into the soft elastic substrate as heat sources. It is of great significance to develop analytic models on transient heat conduction of such an FEH in order to provide a rapid analysis and preliminary designs based on a rapid parameter analysis. In this study, an analytic model of transient heat conduction for bi-layered FEHs is proposed, which is solved by a novel symplectic superposition method (SSM). In the Laplace transform domain, the Hamiltonian system-based governing equation for transient heat conduction is introduced, and the mathematical techniques incorporating the separation of variables and symplectic eigen expansion are manipulated to yield the temperature solutions of two subproblems, which is followed by superposition for the temperature solution of the general problem. The Laplace inversion gives the eventual temperature solution in the time domain. Comprehensive time-dependent temperatures by the SSM are presented in tables and figures for benchmark use, which agree well with their counterparts by the finite element method. A parameter analysis on the influence of the thermal conductivity ratio is also studied. The exceptional merit of the SSM is on a direct rigorous derivation without any assumption/predetermination of solution forms, and thus, the method may be extended to more heat conduction problems of FEHs with more complex structures.

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