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:
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 or 2. The 2-dimensional biharmonic operator is defined as
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.
While the Föppl–von Kármán equations are of interest from a purely mathematical point of view, the physical validity of these equations is questionable. Ciarlet states: The two-dimensional von Karman equations for plates, originally proposed by von Karman [1910], play a mythical role in applied mathematics. While they have been abundantly, and satisfactorily, studied from the mathematical standpoint, as regards notably various questions of existence, regularity, and bifurcation, of their solutions, their physical soundness has been often seriously questioned. Reasons include the facts that
Conditions under which these equations are actually applicable and will give reasonable results when solved are discussed in Ciarlet.
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