On the development and mathematical justification of consistent-plate theories of any approximation order

Abstract

Plate theories are widely used to calculate the statics of two-dimensional, thin-walled and plane structures (e.g., ceilings in civil engineering). Classically, the derivation of these theories is based on kinematical a-priori assumptions (cf. Bernoulli assumptions). On the other hand, consistent-plate theories are developed in a mathematically rigorous manner from the generally accepted three-dimensional theory of linear elasticity. This systematic derivation offers many advantages for the development of plate theories for more complex materials. In Schneider & Kienzler (Schneider and Kienzler, 2020), consistent theories are derived for quasi one-dimensional continua. We adopt their argumentation for the development of consistent two-dimensional theories. In doing so, we first arrive from the elastic and dual potential of the three-dimensional theory by means of Taylor-series expansions at the “quasi-two-dimensional problem”. We show that this problem decouples into a plate and a disc problem for certain anisotropy. Moreover, we prove that the coupling relations follow from the sparsity scheme of the stiffness tensor. Using the “consistent-approximation approach”, which estimates the magnitude of the potential parts via powers of a geometric parameter, we finally obtain hierarchical “generic-(consistent) plate theories” . By applying the “pseudo-reduction method”, these systems of partial-differential equations (PDE systems) finally can be reduced to one main PDE, depending only on the main variable, and several reduction PDEs, expressing the non-main variables in dependence of the main variable. Following Kienzler & Schneider (Kienzler and Schneider, 2017), we decompose the variables (coefficients of the Taylor series of the displacements) into their energetic parts. This allows us to prove for the generic-plate theories that the displacement coefficients have a modular structure. Using this modularity, we develop the “complete-(consistent) plate theories”, by which all displacement coefficients can be calculated (proof). Moreover, we present a computer program that automatically generates complete-plate theories of arbitrary approximation orders and that reduces them to a main PDE and several reduction PDEs. It turns out that these PDEs satisfy the PDE systems of the three-dimensional theory. Based on this finding, we derive directly from the three-dimensional theory the “ original-(consistent) plate theories”. Again, we prove that all displacement coefficients can be calculated and develop a program that automatically establishes and reduces original-plate theories of arbitrary approximation orders. A comparison of the complete and original-plate theories reveals that their main and reduction PDEs coincide. Based on this observation, we establish the relations between both theories. Furthermore, we give formulas for some main and reduction PDEs that follow a pattern across all approximation orders and motivate the use of the consistent-approximation approach and modularity from a mathematical point of view. Using the main and reduction PDEs, it is finally proved that the third component of the curl of the displacement field is zero. By means of the Helmholtz decomposition, a comparison of this finding with a finite-element simulation makes it clear that our displacement approach is not complete. On the basis of additional coefficients and by changing one assumption of the consistent-approximation approach, we arrive at the “original-plate theories for the boundary layer”. We prove that all displacement coefficients of these theories can be calculated and give a analytical formula for the main PDEs of arbitrary approximation order. It turns out that the first-order main PDE agrees with the second PDE of Reissner’s plate theory except for a prefactor.