Föppl–von Kármán equations

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]

\begin{align} (1) \qquad & \frac{Eh^3}{12(1-\nu^2)}\Delta^2 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{align}

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, $σ αβ$ 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^[6]

\Delta^2 w := \frac{\partial^2}{\partial x_\alpha \partial x_\alpha}\left[\frac{\partial^2 w}{\partial x_\beta \partial x_\beta}\right] = \frac{\partial^4 w}{\partial x_1^4} + \frac{\partial^4 w}{\partial x_2^4} + 2\frac{\partial^4 w}{\partial x_1^2 \partial x_2^2} \,.

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 ( $σ 33, σ 13, σ 23$ ) are zero.

Validity of the Föppl–von Kármán equations

While the Föppl–von Kármán equations are of interest from a purely mathematical point of view, the physical validity of these equations is questionable.^[7] Ciarlet^[8] 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

the theory depends on an approximate geometry which is not clearly defined
a given variation of stress over a cross-section is assumed arbitrarily
a linear constitutive relation is used that does not correspond to a known relation between well defined measures of stress and strain
some components of strain are arbitrarily ignored
there is a confusion between reference and deformed configurations which makes the theory inapplicable to the large deformations for which it was apparently devised.

Conditions under which these equations are actually applicable and will give reasonable results when solved are discussed in Ciarlet.^[8]^[9]

Equations in terms of Airy stress function

The three Föppl–von Kármán equations can be reduced to two by introducing the Airy stress function $\varphi$ where

\sigma_{11} = \frac{\partial^2 \varphi}{\partial x_2^2} ~,~~ \sigma_{22} = \frac{\partial^2 \varphi}{\partial x_1^2} ~,~~ \sigma_{12} = - \frac{\partial^2 \varphi}{\partial x_1 \partial x_2} \,.

Then the above equations become^[5]

\frac{Eh^3}{12(1-\nu^2)}\Delta^2 w-h\left(\frac{\partial^2\varphi}{\partial x_2^2}\frac{\partial^2 w}{\partial x_1^2}+\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2 w}{\partial x_2^2}-2\frac{\partial^2\varphi}{\partial x_1 \, \partial x_2}\frac{\partial^2 w}{\partial x_1 \, \partial x_2}\right)=P

\Delta^2\varphi+E\left\{\frac{\partial^2 w}{\partial x_1^2}\frac{\partial^2 w}{\partial x_2^2}-\left(\frac{\partial^2 w}{\partial x_1 \, \partial x_2}\right)^2\right\}=0 \,.

Pure bending

For the pure bending of thin plates the equation of equilibrium is $D\Delta^2\ w=P$ , where

D :=\frac{Eh^3}{12(1-\nu^2)}

is called flexural or cylindrical rigidity of the plate.^[5]

Kinematic assumptions (Kirchhoff hypothesis)

In the derivation of the Föppl–von Kármán equations the main kinematic assumption (also known as the Kirchhoff hypothesis) is that surface normals to the plane of the plate remain perpendicular to the plate after deformation. It is also assumed that the in-plane (membrane) displacements and the change in thickness of the plate are negligible. These assumptions imply that the displacement field $u$ in the plate can be expressed as

u_1(x_1,x_2,x_3) = -x_3\,\frac{\partial w}{\partial x_1} ~,~~ u_2(x_1,x_2,x_3) = -x_3\,\frac{\partial w}{\partial x_2} ~,~~ u_3(x_1, x_2, x_3) = w(x_1,x_2)

This form of the displacement field implicitly assumes that the amount of rotation of the plate is small.

Strain-displacement relations (von Kármán strains)

The components of the three-dimensional Lagrangian Green strain tensor are defined as

E_{ij} := \frac{1}{2}\left[\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} + \frac{\partial u_k}{\partial x_i}\,\frac{\partial u_k}{\partial x_j}\right] \,.

Substitution of the expressions for the displacement field into the above gives

\begin{align} E_{11} & = \frac{\partial u_1}{\partial x_1} + \frac{1}{2}\left[\left(\frac{\partial u_1}{\partial x_1}\right)^2 + \left(\frac{\partial u_2}{\partial x_1}\right)^2 + \left(\frac{\partial u_3}{\partial x_1}\right)^2\right]\\ &= -x_3\,\frac{\partial^2 w}{\partial x_1^2} + \frac{1}{2}\left[x_3^2\left(\frac{\partial^2 w}{\partial x_1^2}\right)^2 + x_3^2\left(\frac{\partial^2 w}{\partial x_1 \partial x_2}\right)^2 + \left(\frac{\partial w}{\partial x_1}\right)^2\right]\\ E_{22} & = \frac{\partial u_2}{\partial x_2} + \frac{1}{2}\left[\left(\frac{\partial u_1}{\partial x_2}\right)^2 + \left(\frac{\partial u_2}{\partial x_2}\right)^2 + \left(\frac{\partial u_3}{\partial x_2}\right)^2\right]\\ &= -x_3\,\frac{\partial^2 w}{\partial x_2^2} + \frac{1}{2}\left[x_3^2\left(\frac{\partial^2 w}{\partial x_1 \partial x_2}\right)^2 + x_3^2\left(\frac{\partial^2 w}{\partial x_2^2}\right)^2 + \left(\frac{\partial w}{\partial x_2}\right)^2\right]\\ E_{33} & = \frac{\partial u_3}{\partial x_3} + \frac{1}{2}\left[\left(\frac{\partial u_1}{\partial x_3}\right)^2 + \left(\frac{\partial u_2}{\partial x_3}\right)^2 + \left(\frac{\partial u_3}{\partial x_3}\right)^2\right]\\ &= \frac{1}{2}\left[\left(\frac{\partial w}{\partial x_1}\right)^2 + \left(\frac{\partial w}{\partial x_2}\right)^2 \right]\\ E_{12} & = \frac{1}{2}\left[\frac{\partial u_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1} + \frac{\partial u_1}{\partial x_1}\,\frac{\partial u_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1}\,\frac{\partial u_2}{\partial x_2} + \frac{\partial u_3}{\partial x_1}\,\frac{\partial u_3}{\partial x_2}\right]\\ & = -x_3\frac{\partial^2 w}{\partial x_1 \partial x_2} + \frac{1}{2}\left[x_3^2\left(\frac{\partial^2 w}{\partial x_1^2}\right)\left(\frac{\partial^2 w}{\partial x_1\partial x_2}\right) + x_3^2\left(\frac{\partial^2 w}{\partial x_1 \partial x_2}\right)\left(\frac{\partial^2 w}{\partial x_2^2}\right) + \frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2}\right]\\ E_{23} & = \frac{1}{2}\left[\frac{\partial u_2}{\partial x_3} + \frac{\partial u_3}{\partial x_2} + \frac{\partial u_1}{\partial x_2}\,\frac{\partial u_1}{\partial x_3} + \frac{\partial u_2}{\partial x_2}\,\frac{\partial u_2}{\partial x_3} + \frac{\partial u_3}{\partial x_2}\,\frac{\partial u_3}{\partial x_3}\right]\\ & = \frac{1}{2}\left[x_3\left(\frac{\partial^2 w}{\partial x_1\partial x_2}\right)\left(\frac{\partial w}{\partial x_1}\right) + x_3\left(\frac{\partial^2 w}{\partial x_2^2}\right)\left(\frac{\partial w}{\partial x_2}\right) \right]\\ E_{31} & = \frac{1}{2}\left[\frac{\partial u_3}{\partial x_1} + \frac{\partial u_1}{\partial x_3} + \frac{\partial u_1}{\partial x_3}\,\frac{\partial u_1}{\partial x_1} + \frac{\partial u_2}{\partial x_3}\,\frac{\partial u_2}{\partial x_1} + \frac{\partial u_3}{\partial x_3}\,\frac{\partial u_3}{\partial x_1}\right] \\ & = \frac{1}{2}\left[x_3\left(\frac{\partial w}{\partial x_1}\right)\left(\frac{\partial^2 w}{\partial x_1^2}\right) + x_3\left(\frac{\partial w}{\partial x_2}\right)\left(\frac{\partial^2 w}{\partial x_1 \partial x_2}\right) \right] \end{align}

For small strains but moderate rotations, the higher order terms that cannot be neglected are

\left(\frac{\partial w}{\partial x_1}\right)^2 ~,~~ \left(\frac{\partial w}{\partial x_2}\right)^2 ~,~~ \frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2} \,.

Neglecting all other higher order terms, and enforcing the requirement that the plate does not change its thickness, the strain tensor components reduce to the von Kármán strains

\begin{align} E_{11} & = -x_3\,\frac{\partial^2 w}{\partial x_1^2} + \frac{1}{2}\left(\frac{\partial w}{\partial x_1}\right)^2 \\ E_{22} & = -x_3\,\frac{\partial^2 w}{\partial x_2^2} + \frac{1}{2}\left(\frac{\partial w}{\partial x_2}\right)^2 \\ E_{12} & = -x_3\frac{\partial^2 w}{\partial x_1 \partial x_2} + \frac{1}{2}\,\frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2}\\ E_{33} & = 0 ~,~~ E_{23} = 0 ~,~~ E_{31} = 0 \,. \end{align}

Stress-strain relations

If we assume that the Cauchy stress tensor components are linearly related to the von Kármán strains by Hooke's law, the plate is isotropic and homogeneous, and that the plate in under a plane stress condition,^[10] we have $σ 33$ = $σ 13$ = $σ 23$ = 0 and

\begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix} = \cfrac{E}{(1-\nu^2)} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & 1-\nu \end{bmatrix} \begin{bmatrix} E_{11} \\ E_{22} \\ E_{12} \end{bmatrix}

Expanding the terms, the three non-zero stresses are

\begin{align} \sigma_{11} &= \cfrac{E}{(1-\nu^2)}\left[\left(-x_3\,\frac{\partial^2 w}{\partial x_1^2} + \frac{1}{2}\left(\frac{\partial w}{\partial x_1}\right)^2 \right) + \nu\left(-x_3\,\frac{\partial^2 w}{\partial x_2^2} + \frac{1}{2}\left(\frac{\partial w}{\partial x_2}\right)^2 \right) \right] \\ \sigma_{22} &= \cfrac{E}{(1-\nu^2)}\left[\nu\left(-x_3\,\frac{\partial^2 w}{\partial x_1^2} + \frac{1}{2}\left(\frac{\partial w}{\partial x_1}\right)^2 \right) + \left(-x_3\,\frac{\partial^2 w}{\partial x_2^2} + \frac{1}{2}\left(\frac{\partial w}{\partial x_2}\right)^2 \right) \right] \\ \sigma_{12} &= \cfrac{E}{(1+\nu)}\left[-x_3\frac{\partial^2 w}{\partial x_1 \partial x_2} + \frac{1}{2}\,\frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2}\right] \,. \end{align}

Stress resultants

The stress resultants in the plate are defined as

N_{\alpha\beta} := \int_{-h/2}^{h/2} \sigma_{\alpha\beta}\, d x_3 ~,~~ M_{\alpha\beta} := \int_{-h/2}^{h/2} x_3\,\sigma_{\alpha\beta}\, d x_3 \,.

Therefore,

\begin{align} N_{11} &= \cfrac{Eh}{2(1-\nu^2)}\left[\left(\frac{\partial w}{\partial x_1}\right)^2 + \nu\left(\frac{\partial w}{\partial x_2}\right)^2 \right] \\ N_{22} &= \cfrac{Eh}{2(1-\nu^2)}\left[\nu\left(\frac{\partial w}{\partial x_1}\right)^2 + \left(\frac{\partial w}{\partial x_2}\right)^2 \right] \\ N_{12} &= \cfrac{Eh}{2(1+\nu)}\,\frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2} \end{align}

and

\begin{align} M_{11} &= -\cfrac{Eh^3}{12(1-\nu^2)}\left[\frac{\partial^2 w}{\partial x_1^2} +\nu \,\frac{\partial^2 w}{\partial x_2^2} \right] \\ M_{22} &= -\cfrac{Eh^3}{12(1-\nu^2)}\left[\nu \,\frac{\partial^2 w}{\partial x_1^2} +\frac{\partial^2 w}{\partial x_2^2} \right] \\ M_{12} &= -\cfrac{Eh^3}{12(1+\nu)}\,\frac{\partial^2 w}{\partial x_1 \partial x_2} \,. \end{align}

Solutions are easier to find when the governing equations are expressed in terms of stress resultants rather than the in-plane stresses.