Plate theories: A‐priori fulfillment of the local conditions by the consistent‐approximation approach

Classical plates theories, like Kirchhoff's plate theory [1], are based on kinematical a‐priori assumptions. Avoiding these assumptions, we derive from the three‐dimensional theory of linear elasticity by means of Taylor‐series expansions the quasi two‐dimensional problem. This problem consists of infinitely many partial‐differential equations (PDEs) written in infinitely many displacement coefficients. With the consistent approximation approach we arrive at solvable hierarchical plate theories. By using the modular structure of the displacements coefficients (modularity), we obtain from these generic‐plate theories the complete‐plate theories, whose results fulfill the strong form of the local equilibrium conditions and the Neumann boundary conditions on the upper and lower face of the plate (local conditions) a‐priori. Furthermore, we show that every variable of the complete‐plate theories can be calculated.


Introduction
We assume a homogeneous, linear elastic, cuboid solid for which the thickness h is much smaller than the in-plane dimensions a and b (cf. fig.1). The midplane of this plate continuum is the (x 1 , x 2 )-plane (with x 3 = 0). So, it is parallel to the upper and lower faces of the solid and perpendicular to the x 3 -direction. We allow arbitrary surface loads g − i = g i (x 1 , x 2 , − h 2 ) and on the upper and lower faces of the plate as well as volume loads f i . For the plate continuum at hand we can establish the elastic and dual potentials E pot and E dual of the three-dimensional theory of linear elasticity. 2 Generic-plate theories By using Taylor-series expansions in thickness-direction for all variables and integrating over the thickness, we derive from both potentials the quasi two-dimensional problem. For isotropic and transverse-isotropic material, this problem can be split into a disc and a plate problem. In the following, we only deal with the plate problem. Because it is written in infinitely many PDEs with infinitely many displacement coefficients, we use the consistent-approximation approach to arrive at solvable plate theories. The main idea of this approach is to generate theories that contain only terms up to a certain energetic magnitude [3]. Because it decreases very fast for thin plates, this magnitude is characterized by the plate parameter c = h / ( √ 12a). Following the consistent-approximation approach, a generic N th-order plate theory is given by considering all summands with c n , n ≤ 2N .

Complete-plate theories
For the plate problem, the Taylor-series of the displacements u i are given by u α = a 1 u α ξ 3 + 3 u α ξ 3 3 + 5 u α ξ 5 3 + .... and   [3], these displacement coefficients itself can be split into the infinite sum Here, with j u n i we introduced the displacement-coefficient parts (dcps) of magnitude c n . While the displacement coefficients change among the different orders of generic-plate theories, we can show that the dcps are unchangeable. We call this characteristic the modularity of the displacement coefficients. By using the modularity we can set up plate theories where all variables can be calculated (cf. next section). We call them complete-plate theories. A schema for the construction of them is provided.

Reducibility
Starting from the PDE systems of the complete-plate theories, the pseudo-reduction method leads to one main PDE, which is entirely written in the main variable ( 0 u 0 3 ), and several reduction PDEs that express the non-main variables in terms of the main variable. If there is a reduction PDE for each of the non-main variables, the PDE system is called fully reducible. Applying the pseudo-reduction method is related to solving a linear algebraic equation system. The determinant of the coefficient matrix K N of the N th-order PDE system has to be unequal to zero for a fully reducible system. With ∆ as the Laplace operator it reads .
Here, we easily see that the determinant cannot be zero and thus the complete-plate theories are fully reducible. Note that we used the determinant of the Hilbert-like matrix H ij = 1 /(2(i+j)+1) with 0 ≤ i, j ≤ N and N ≥ 0 for the derivation of Det K N .

Results
The results of the complete first and second-order plate theories coincide with the classical theories of Kirchhoff [1] and Reissner [4], respectively. Moreover the resulting stress field fulfills a-priori the local conditions. Here, we have to insert the modularity and compare only terms with the same magnitude c 2n (Neumann conditions) or the same combinations ξ m 3 c 2n (local equilibrium conditions).

Conclusion
In the talk we show that the displacement coefficients have a modular structure (modularity). Based on this, we build completeplate theories. The proof that these theories are fully reducible is provided by using the determinants of the coefficient matrices of the corresponding PDE systems. It turns out that the results of the complete-plate theories are in good accordance with those of the classical theories and that they fulfill a-priori the local conditions. Note that this is the first time a plate theory satisfies these conditions without enforcing the conditions during the derivation of the theory.