Föppl–von Kármán equations
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The Föppl–von Kármán equations, named after August Föppl[1] and Theodore von Kármán,[2] are a set of nonlinear partial differential equations describing the large deflections of thin flat plates.[3] With applications ranging from the design of submarine hulls to the mechanical properties of cell wall,[4] the equations are notoriously difficult to solve, and take the following form: [5]
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). The 2-dimensional biharmonic operator is defined as[6]
Equation (1) above can be derived from kinematic assumptions and the constitutive relations for the plate. Equations (2) are the two equations for the conservation of linear momentum in two dimensions where it is assumed that the out–of–plane stresses (σ33,σ13,σ23) are zero.