Föppl–von Kármán equations

The Föppl–von Kármán equations, named after August Föppl and Theodore von Kármán, are a set of nonlinear partial differential equations describing the large deflections of thin flat plates. With applications ranging from the design of submarine hulls to the mechanical properties of cell wall, the equations are notoriously difficult to solve, and take the following form: ( 1 ) E h 3 12 ( 1 − ν 2 ) ∇ 4 w − h ∂ ∂ x β ( σ α β ∂ w ∂ x α ) = P ( 2 ) ∂ σ α β ∂ x β = 0 {\displaystyle {\begin{aligned}(1)\qquad &{\frac {Eh^{3}}{12(1-\nu ^{2})}}\nabla ^{4}w-h{\frac {\partial }{\partial x_{\beta }}}\left(\sigma _{\alpha \beta }{\frac {\partial w}{\partial x_{\alpha }}}\right)=P\\(2)\qquad &{\frac {\partial \sigma _{\alpha \beta }}{\partial x_{\beta }}}=0\end{aligned}}} where E is the Young's modulus of the plate material (assumed homogeneous and isotropic), υ is the Poisson's ratio, h is the thickness of the plate, w is the out–of–plane deflection of the plate, P is the external normal force per unit area of the plate, σαβ is the Cauchy stress tensor, and α, β are indices that take values of 1 and 2 (the two orthogonal in-plane directions).

Source: Wikipedia — Föppl–von Kármán equations (CC BY-SA 4.0)