Math 2.0

Mechanics of deformable solids Definition

Föppl–von Kármán equations

Builds on Kirchhoff–Love plate theoryBiharmonic plate equationAiry stress functionGaussian curvature

Definition

Let Ω⊂ℝ2\Omega \subset \R^2 be the mid-plane of a thin isotropic plate of thickness hh, with Young's modulus EE, Poisson's ratio ν\nu and bending stiffness D=Eh3/(12(1−ν2))D = Eh^3/(12(1 - \nu^2)). Let qq be the transverse load per unit area, ww the deflection and FF an Airy stress function for the membrane forces, so that Nxx=∂y2F,Nyy=∂x2F,Nxy=−∂x∂yF.N_{xx} = \partial_y^2 F, \qquad N_{yy} = \partial_x^2 F, \qquad N_{xy} = -\partial_x \partial_y F. For functions f,gf, g on Ω\Omega put [f,g]=∂x2f ∂y2g+∂y2f ∂x2g−2 ∂x∂yf ∂x∂yg.[f, g] = \partial_x^2 f \, \partial_y^2 g + \partial_y^2 f \, \partial_x^2 g - 2 \, \partial_x \partial_y f \, \partial_x \partial_y g. The Föppl–von Kármán equations are D Δ2w−[F,w]=q,Δ2F=−Eh2[w,w].D \, \Delta^2 w - [F, w] = q, \qquad \Delta^2 F = -\frac{E h}{2} [w, w].

Remarks

Here [w,w]=2(∂x2w ∂y2w−(∂x∂yw)2)[w, w] = 2 (\partial_x^2 w \, \partial_y^2 w - (\partial_x \partial_y w)^2), which is twice the linearised Gaussian curvature of the deflected surface. The first equation is vertical equilibrium, in which the membrane forces act on the curvature; the second is the compatibility of the in-plane strains.

Example: if w=0w = 0, the equations reduce to Δ2F=0\Delta^2 F = 0 with q=0q = 0, the plane stress problem of a flat plate. For a cylindrical bending w=cx2/2w = c x^2/2 one has [w,w]=0[w, w] = 0, so F=0F = 0 is allowed: the plate can bend into a cylinder without stretching.

Non-example: the pair DΔ2w=qD\Delta^2 w = q and F=0F = 0 is the linear biharmonic plate equation; it fails the second equation whenever ww has non-zero Gaussian curvature, such as w=c(x2+y2)/2w = c(x^2 + y^2)/2 with [w,w]=2c2[w, w] = 2c^2.

Sign conventions for FF and [⋅,⋅][\cdot, \cdot] differ between texts.

Reading path

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Airy stress function, concept, no evidence levelAiry stress functionconceptBiharmonic plate equation, concept, no evidence levelBiharmonic plate equa…conceptGaussian curvature, concept, no evidence levelGaussian curvatureconceptKirchhoff–Love plate theory, concept, no evidence levelKirchhoff–Love plate …conceptFöppl–von Kármán equations, definition, no evidence levelFöppl–von Kármán equa…definition
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  1. Airy stress function concept
  2. Biharmonic plate equation concept
  3. Gaussian curvature concept
  4. Kirchhoff–Love plate theory concept
  5. Föppl–von Kármán equations definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Föppl–von_Kármán_equations
About this page
TagGWR4DGJR23
URIm2://GWR4DGJR23
Kinddefinition
WritingW0 stub
MSC74K20, 74B20, 74G60
DefinesFöppl–von Kármán equations
UsesKirchhoff–Love plate theory, Biharmonic plate equation, Airy stress function, Gaussian curvature
Changed3 November 2026

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