Set of nonlinear partial differential equations
TheFöppl–von Kármán equations , named afterAugust Föppl [ 1] Theodore von Kármán ,[ 2] partial differential equations describing the large deflections of thin flat plates.[ 3] submarine hulls to the mechanical properties of cell wall,[ 4] [ 5]
( 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}}} whereE Young's modulus of the plate material (assumed homogeneous and isotropic),υ Poisson's ratio ,h w P normal force per unit area of the plate,σ αβ Cauchy stress tensor , andα ,β indices that take values of 1 and 2 (the two orthogonal in-plane directions). The 2-dimensionalbiharmonic operator is defined as[ 6]
∇ 4 w := ∂ 2 ∂ x α ∂ x α [ ∂ 2 w ∂ x β ∂ x β ] = ∂ 4 w ∂ x 1 4 + ∂ 4 w ∂ x 2 4 + 2 ∂ 4 w ∂ x 1 2 ∂ x 2 2 . {\displaystyle \nabla ^{4}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 fromkinematic assumptions and theconstitutive 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
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] [ 8] 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 [ edit ] The three Föppl–von Kármán equations can be reduced to two by introducing theAiry stress function φ {\displaystyle \varphi }
σ 11 = ∂ 2 φ ∂ x 2 2 , σ 22 = ∂ 2 φ ∂ x 1 2 , σ 12 = − ∂ 2 φ ∂ x 1 ∂ x 2 . {\displaystyle \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}}}\,.} Equation (1) becomes[ 5]
E h 3 12 ( 1 − ν 2 ) Δ 2 w − h ( ∂ 2 φ ∂ x 2 2 ∂ 2 w ∂ x 1 2 + ∂ 2 φ ∂ x 1 2 ∂ 2 w ∂ x 2 2 − 2 ∂ 2 φ ∂ x 1 ∂ x 2 ∂ 2 w ∂ x 1 ∂ x 2 ) = P {\displaystyle {\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} while theAiry function satisfies, by construction the forcebalance equation (2). An equation forφ {\displaystyle \varphi } [ 5]
Δ 2 φ + E { ∂ 2 w ∂ x 1 2 ∂ 2 w ∂ x 2 2 − ( ∂ 2 w ∂ x 1 ∂ x 2 ) 2 } = 0 . {\displaystyle \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\,.} For thepure bending of thin plates the equation of equilibrium isD Δ 2 w = P {\displaystyle D\Delta ^{2}\ w=P}
D := E h 3 12 ( 1 − ν 2 ) {\displaystyle D:={\frac {Eh^{3}}{12(1-\nu ^{2})}}} is calledflexural orcylindrical rigidity of the plate.[ 5]
[ edit ] In the derivation of the Föppl–von Kármán equations the main kinematic assumption (also known as theKirchhoff hypothesis ) is thatsurface normals to the plane of the plate remain perpendicular to the plate after deformation. It is also assumed that the in-plane (membrane) displacements are small and the change in thickness of the plate is negligible. These assumptions imply that the displacement fieldu [ 9]
u 1 ( x 1 , x 2 , x 3 ) = v 1 ( x 1 , x 2 ) − x 3 ∂ w ∂ x 1 , u 2 ( x 1 , x 2 , x 3 ) = v 2 ( x 1 , x 2 ) − x 3 ∂ w ∂ x 2 , u 3 ( x 1 , x 2 , x 3 ) = w ( x 1 , x 2 ) {\displaystyle u_{1}(x_{1},x_{2},x_{3})=v_{1}(x_{1},x_{2})-x_{3}\,{\frac {\partial w}{\partial x_{1}}}~,~~u_{2}(x_{1},x_{2},x_{3})=v_{2}(x_{1},x_{2})-x_{3}\,{\frac {\partial w}{\partial x_{2}}}~,~~u_{3}(x_{1},x_{2},x_{3})=w(x_{1},x_{2})} in whichv
[ edit ] The components of the three-dimensional LagrangianGreen strain tensor are defined as
E i j := 1 2 [ ∂ u i ∂ x j + ∂ u j ∂ x i + ∂ u k ∂ x i ∂ u k ∂ x j ] . {\displaystyle 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
E 11 = ∂ u 1 ∂ x 1 + 1 2 [ ( ∂ u 1 ∂ x 1 ) 2 + ( ∂ u 2 ∂ x 1 ) 2 + ( ∂ u 3 ∂ x 1 ) 2 ] = ∂ v 1 ∂ x 1 − x 3 ∂ 2 w ∂ x 1 2 + 1 2 [ x 3 2 ( ∂ 2 w ∂ x 1 2 ) 2 + x 3 2 ( ∂ 2 w ∂ x 1 ∂ x 2 ) 2 + ( ∂ w ∂ x 1 ) 2 ] E 22 = ∂ u 2 ∂ x 2 + 1 2 [ ( ∂ u 1 ∂ x 2 ) 2 + ( ∂ u 2 ∂ x 2 ) 2 + ( ∂ u 3 ∂ x 2 ) 2 ] = ∂ v 2 ∂ x 2 − x 3 ∂ 2 w ∂ x 2 2 + 1 2 [ x 3 2 ( ∂ 2 w ∂ x 1 ∂ x 2 ) 2 + x 3 2 ( ∂ 2 w ∂ x 2 2 ) 2 + ( ∂ w ∂ x 2 ) 2 ] E 33 = ∂ u 3 ∂ x 3 + 1 2 [ ( ∂ u 1 ∂ x 3 ) 2 + ( ∂ u 2 ∂ x 3 ) 2 + ( ∂ u 3 ∂ x 3 ) 2 ] = 1 2 [ ( ∂ w ∂ x 1 ) 2 + ( ∂ w ∂ x 2 ) 2 ] E 12 = 1 2 [ ∂ u 1 ∂ x 2 + ∂ u 2 ∂ x 1 + ∂ u 1 ∂ x 1 ∂ u 1 ∂ x 2 + ∂ u 2 ∂ x 1 ∂ u 2 ∂ x 2 + ∂ u 3 ∂ x 1 ∂ u 3 ∂ x 2 ] = 1 2 ∂ v 1 ∂ x 2 + 1 2 ∂ v 2 ∂ x 1 − x 3 ∂ 2 w ∂ x 1 ∂ x 2 + 1 2 [ x 3 2 ( ∂ 2 w ∂ x 1 2 ) ( ∂ 2 w ∂ x 1 ∂ x 2 ) + x 3 2 ( ∂ 2 w ∂ x 1 ∂ x 2 ) ( ∂ 2 w ∂ x 2 2 ) + ∂ w ∂ x 1 ∂ w ∂ x 2 ] E 23 = 1 2 [ ∂ u 2 ∂ x 3 + ∂ u 3 ∂ x 2 + ∂ u 1 ∂ x 2 ∂ u 1 ∂ x 3 + ∂ u 2 ∂ x 2 ∂ u 2 ∂ x 3 + ∂ u 3 ∂ x 2 ∂ u 3 ∂ x 3 ] = 1 2 [ x 3 ( ∂ 2 w ∂ x 1 ∂ x 2 ) ( ∂ w ∂ x 1 ) + x 3 ( ∂ 2 w ∂ x 2 2 ) ( ∂ w ∂ x 2 ) ] E 31 = 1 2 [ ∂ u 3 ∂ x 1 + ∂ u 1 ∂ x 3 + ∂ u 1 ∂ x 3 ∂ u 1 ∂ x 1 + ∂ u 2 ∂ x 3 ∂ u 2 ∂ x 1 + ∂ u 3 ∂ x 3 ∂ u 3 ∂ x 1 ] = 1 2 [ x 3 ( ∂ w ∂ x 1 ) ( ∂ 2 w ∂ x 1 2 ) + x 3 ( ∂ w ∂ x 2 ) ( ∂ 2 w ∂ x 1 ∂ x 2 ) ] {\displaystyle {\begin{aligned}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]\\&={\frac {\partial v_{1}}{\partial x_{1}}}-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]\\&={\frac {\partial v_{2}}{\partial x_{2}}}-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]\\&={\frac {1}{2}}{\frac {\partial v_{1}}{\partial x_{2}}}+{\frac {1}{2}}{\frac {\partial v_{2}}{\partial x_{1}}}-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{aligned}}} For small strains butmoderate rotations , the higher order terms that cannot be neglected are
( ∂ w ∂ x 1 ) 2 , ( ∂ w ∂ x 2 ) 2 , ∂ w ∂ x 1 ∂ w ∂ x 2 . {\displaystyle \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 thevon Kármán strains
E 11 = ∂ v 1 ∂ x 1 + 1 2 ( ∂ w ∂ x 1 ) 2 − x 3 ∂ 2 w ∂ x 1 2 E 22 = ∂ v 2 ∂ x 2 + 1 2 ( ∂ w ∂ x 2 ) 2 − x 3 ∂ 2 w ∂ x 2 2 E 12 = 1 2 ( ∂ v 1 ∂ x 2 + ∂ v 2 ∂ x 1 ) + 1 2 ∂ w ∂ x 1 ∂ w ∂ x 2 − x 3 ∂ 2 w ∂ x 1 ∂ x 2 E 33 = 0 , E 23 = 0 , E 13 = 0 . {\displaystyle {\begin{aligned}E_{11}&={\frac {\partial v_{1}}{\partial x_{1}}}+{\frac {1}{2}}\left({\frac {\partial w}{\partial x_{1}}}\right)^{2}-x_{3}\,{\frac {\partial ^{2}w}{\partial x_{1}^{2}}}\\E_{22}&={\frac {\partial v_{2}}{\partial x_{2}}}+{\frac {1}{2}}\left({\frac {\partial w}{\partial x_{2}}}\right)^{2}-x_{3}\,{\frac {\partial ^{2}w}{\partial x_{2}^{2}}}\\E_{12}&={\frac {1}{2}}\left({\frac {\partial v_{1}}{\partial x_{2}}}+{\frac {\partial v_{2}}{\partial x_{1}}}\right)+{\frac {1}{2}}\,{\frac {\partial w}{\partial x_{1}}}\,{\frac {\partial w}{\partial x_{2}}}-x_{3}{\frac {\partial ^{2}w}{\partial x_{1}\partial x_{2}}}\\E_{33}&=0~,~~E_{23}=0~,~~E_{13}=0\,.\end{aligned}}} The first terms are the usual small-strains, for the mid-surface. The second terms, involving squares of displacement gradients, are non-linear, and need to be considered when the plate bending is fairly large (when the rotations are about 10 – 15 degrees). These first two terms together are called themembrane strains . The last terms, involving second derivatives, are theflexural (bending) strains . They involve the curvatures. These zero terms are due to the assumptions of the classical plate theory, which assume elements normal to the mid-plane remain inextensible and line elements perpendicular to the mid-plane remain normal to the mid-plane after deformation.
[ edit ] If we assume that theCauchy stress tensor components are linearly related to the von Kármán strains byHooke's law , the plate is isotropic and homogeneous, and that the plate is under aplane stress condition,[ 10] σ 33 σ 13 σ 23
[ σ 11 σ 22 σ 12 ] = E ( 1 − ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 − ν ] [ E 11 E 22 E 12 ] {\displaystyle {\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
σ 11 = E ( 1 − ν 2 ) [ ( ∂ v 1 ∂ x 1 + 1 2 ( ∂ w ∂ x 1 ) 2 − x 3 ∂ 2 w ∂ x 1 2 ) + ν ( ∂ v 2 ∂ x 2 + 1 2 ( ∂ w ∂ x 2 ) 2 − x 3 ∂ 2 w ∂ x 2 2 ) ] σ 22 = E ( 1 − ν 2 ) [ ν ( ∂ v 1 ∂ x 1 + 1 2 ( ∂ w ∂ x 1 ) 2 − x 3 ∂ 2 w ∂ x 1 2 ) + ( ∂ v 2 ∂ x 2 + 1 2 ( ∂ w ∂ x 2 ) 2 − x 3 ∂ 2 w ∂ x 2 2 ) ] σ 12 = E ( 1 + ν ) [ 1 2 ( ∂ v 1 ∂ x 2 + ∂ v 2 ∂ x 1 ) + 1 2 ∂ w ∂ x 1 ∂ w ∂ x 2 − x 3 ∂ 2 w ∂ x 1 ∂ x 2 ] . {\displaystyle {\begin{aligned}\sigma _{11}&={\cfrac {E}{(1-\nu ^{2})}}\left[\left({\frac {\partial v_{1}}{\partial x_{1}}}+{\frac {1}{2}}\left({\frac {\partial w}{\partial x_{1}}}\right)^{2}-x_{3}\,{\frac {\partial ^{2}w}{\partial x_{1}^{2}}}\right)+\nu \left({\frac {\partial v_{2}}{\partial x_{2}}}+{\frac {1}{2}}\left({\frac {\partial w}{\partial x_{2}}}\right)^{2}-x_{3}\,{\frac {\partial ^{2}w}{\partial x_{2}^{2}}}\right)\right]\\\sigma _{22}&={\cfrac {E}{(1-\nu ^{2})}}\left[\nu \left({\frac {\partial v_{1}}{\partial x_{1}}}+{\frac {1}{2}}\left({\frac {\partial w}{\partial x_{1}}}\right)^{2}-x_{3}\,{\frac {\partial ^{2}w}{\partial x_{1}^{2}}}\right)+\left({\frac {\partial v_{2}}{\partial x_{2}}}+{\frac {1}{2}}\left({\frac {\partial w}{\partial x_{2}}}\right)^{2}-x_{3}\,{\frac {\partial ^{2}w}{\partial x_{2}^{2}}}\right)\right]\\\sigma _{12}&={\cfrac {E}{(1+\nu )}}\left[{\frac {1}{2}}\left({\frac {\partial v_{1}}{\partial x_{2}}}+{\frac {\partial v_{2}}{\partial x_{1}}}\right)+{\frac {1}{2}}\,{\frac {\partial w}{\partial x_{1}}}\,{\frac {\partial w}{\partial x_{2}}}-x_{3}{\frac {\partial ^{2}w}{\partial x_{1}\partial x_{2}}}\right]\,.\end{aligned}}} Thestress resultants in the plate are defined as
N α β := ∫ − h / 2 h / 2 σ α β d x 3 , M α β := ∫ − h / 2 h / 2 x 3 σ α β d x 3 . {\displaystyle N_{\alpha \beta }:=\int _{-h/2}^{h/2}\sigma _{\alpha \beta }\,dx_{3}~,~~M_{\alpha \beta }:=\int _{-h/2}^{h/2}x_{3}\,\sigma _{\alpha \beta }\,dx_{3}\,.} Therefore,
N 11 = E h 2 ( 1 − ν 2 ) [ 2 ∂ v 1 ∂ x 1 + ( ∂ w ∂ x 1 ) 2 + 2 ν ∂ v 2 ∂ x 2 + ν ( ∂ w ∂ x 2 ) 2 ] N 22 = E h 2 ( 1 − ν 2 ) [ 2 ν ∂ v 1 ∂ x 1 + ν ( ∂ w ∂ x 1 ) 2 + 2 ∂ v 2 ∂ x 2 + ( ∂ w ∂ x 2 ) 2 ] N 12 = E h 2 ( 1 + ν ) [ ∂ v 1 ∂ x 2 + ∂ v 2 ∂ x 1 + ∂ w ∂ x 1 ∂ w ∂ x 2 ] {\displaystyle {\begin{aligned}N_{11}&={\cfrac {Eh}{2(1-\nu ^{2})}}\left[2{\frac {\partial v_{1}}{\partial x_{1}}}+\left({\frac {\partial w}{\partial x_{1}}}\right)^{2}+2\nu {\frac {\partial v_{2}}{\partial x_{2}}}+\nu \left({\frac {\partial w}{\partial x_{2}}}\right)^{2}\right]\\N_{22}&={\cfrac {Eh}{2(1-\nu ^{2})}}\left[2\nu {\frac {\partial v_{1}}{\partial x_{1}}}+\nu \left({\frac {\partial w}{\partial x_{1}}}\right)^{2}+2{\frac {\partial v_{2}}{\partial x_{2}}}+\left({\frac {\partial w}{\partial x_{2}}}\right)^{2}\right]\\N_{12}&={\cfrac {Eh}{2(1+\nu )}}\left[{\frac {\partial v_{1}}{\partial x_{2}}}+{\frac {\partial v_{2}}{\partial x_{1}}}+{\frac {\partial w}{\partial x_{1}}}\,{\frac {\partial w}{\partial x_{2}}}\right]\end{aligned}}} the elimination of the in-plane displacements leads to
1 E h [ 2 ( 1 + ν ) ∂ 2 N 12 ∂ x 1 ∂ x 2 − ∂ 2 N 22 ∂ x 1 2 + ν ∂ 2 N 11 ∂ x 1 2 − ∂ 2 N 11 ∂ x 2 2 + ν ∂ 2 N 22 ∂ x 2 2 ] = [ ∂ 2 w ∂ x 1 2 ∂ 2 w ∂ x 2 2 − ( ∂ 2 w ∂ x 1 ∂ x 2 ) 2 ] {\displaystyle {\begin{aligned}{\frac {1}{Eh}}\left[2(1+\nu ){\frac {\partial ^{2}N_{12}}{\partial x_{1}\partial x_{2}}}-{\frac {\partial ^{2}N_{22}}{\partial x_{1}^{2}}}+\nu {\frac {\partial ^{2}N_{11}}{\partial x_{1}^{2}}}-{\frac {\partial ^{2}N_{11}}{\partial x_{2}^{2}}}+\nu {\frac {\partial ^{2}N_{22}}{\partial x_{2}^{2}}}\right]=\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]\end{aligned}}}
and
M 11 = − E h 3 12 ( 1 − ν 2 ) [ ∂ 2 w ∂ x 1 2 + ν ∂ 2 w ∂ x 2 2 ] M 22 = − E h 3 12 ( 1 − ν 2 ) [ ν ∂ 2 w ∂ x 1 2 + ∂ 2 w ∂ x 2 2 ] M 12 = − E h 3 12 ( 1 + ν ) ∂ 2 w ∂ x 1 ∂ x 2 . {\displaystyle {\begin{aligned}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{aligned}}} Solutions are easier to find when the governing equations are expressed in terms of stress resultants rather than the in-plane stresses.
Equations of equilibrium [ edit ] The weak form of the Kirchhoff plate is
∫ Ω ∫ − h / 2 h / 2 ρ u ¨ i δ u i d Ω d x 3 + ∫ Ω ∫ − h / 2 h / 2 σ i j δ E i j d Ω d x 3 + ∫ Ω ∫ − h / 2 h / 2 p i δ u i d Ω d x 3 = 0 {\displaystyle \int _{\Omega }\int _{-h/2}^{h/2}\rho {\ddot {u}}_{i}\delta u_{i}\,d\Omega dx_{3}+\int _{\Omega }\int _{-h/2}^{h/2}\sigma _{ij}\delta E_{ij}\,d\Omega dx_{3}+\int _{\Omega }\int _{-h/2}^{h/2}p_{i}\delta u_{i}\,d\Omega dx_{3}=0} here Ω denotes the mid-plane. The weak form leads to
∫ Ω ρ h v ¨ 1 δ v 1 d Ω + ∫ Ω N 11 ∂ δ v 1 ∂ x 1 + N 12 ∂ δ v 1 ∂ x 2 d Ω = − ∫ Ω p 1 δ v 1 d Ω ∫ Ω ρ h v ¨ 2 δ v 2 d Ω + ∫ Ω N 22 ∂ δ v 2 ∂ x 2 + N 12 ∂ δ v 2 ∂ x 1 d Ω = − ∫ Ω p 2 δ v 2 d Ω ∫ Ω ρ h w ¨ δ w d Ω + ∫ Ω N 11 ∂ w ∂ x 1 ∂ δ w ∂ x 1 − M 11 ∂ 2 δ w ∂ 2 x 1 d Ω + ∫ Ω N 22 ∂ w ∂ x 2 ∂ δ w ∂ x 2 − M 22 ∂ 2 δ w ∂ 2 x 2 d Ω + ∫ Ω N 12 ( ∂ δ w ∂ x 1 ∂ δ w ∂ x 2 + ∂ w ∂ x 1 ∂ δ w ∂ x 2 ) − 2 M 12 ∂ 2 δ w ∂ x 1 ∂ x 2 d Ω = − ∫ Ω p 3 δ w d Ω {\displaystyle {\begin{aligned}\int _{\Omega }\rho h{\ddot {v}}_{1}\delta v_{1}\,d\Omega &+\int _{\Omega }N_{11}{\frac {\partial \delta v_{1}}{\partial x_{1}}}+N_{12}{\frac {\partial \delta v_{1}}{\partial x_{2}}}\,d\Omega =-\int _{\Omega }p_{1}\delta v_{1}\,d\Omega \\\int _{\Omega }\rho h{\ddot {v}}_{2}\delta v_{2}\,d\Omega &+\int _{\Omega }N_{22}{\frac {\partial \delta v_{2}}{\partial x_{2}}}+N_{12}{\frac {\partial \delta v_{2}}{\partial x_{1}}}\,d\Omega =-\int _{\Omega }p_{2}\delta v_{2}\,d\Omega \\\int _{\Omega }\rho h{\ddot {w}}\delta w\,d\Omega &+\int _{\Omega }N_{11}{\frac {\partial w}{\partial x_{1}}}{\frac {\partial \delta w}{\partial x_{1}}}-M_{11}{\frac {\partial ^{2}\delta w}{\partial ^{2}x_{1}}}\,d\Omega \\&+\int _{\Omega }N_{22}{\frac {\partial w}{\partial x_{2}}}{\frac {\partial \delta w}{\partial x_{2}}}-M_{22}{\frac {\partial ^{2}\delta w}{\partial ^{2}x_{2}}}\,d\Omega \\&+\int _{\Omega }N_{12}\left({\frac {\partial \delta w}{\partial x_{1}}}{\frac {\partial \delta w}{\partial x_{2}}}+{\frac {\partial w}{\partial x_{1}}}{\frac {\partial \delta w}{\partial x_{2}}}\right)-2M_{12}{\frac {\partial ^{2}\delta w}{\partial x_{1}\partial x_{2}}}\,d\Omega =-\int _{\Omega }p_{3}\delta w\,d\Omega \\\end{aligned}}} The resulting governing equations are
ρ h w ¨ − ∂ 2 M 11 ∂ x 1 2 − ∂ 2 M 22 ∂ x 2 2 − 2 ∂ 2 M 12 ∂ x 1 ∂ x 2 − ∂ ∂ x 1 ( N 11 ∂ w ∂ x 1 + N 12 ∂ w ∂ x 2 ) − ∂ ∂ x 2 ( N 12 ∂ w ∂ x 1 + N 22 ∂ w ∂ x 2 ) = − p 3 ρ h v ¨ 1 − ∂ N 11 ∂ x 1 − ∂ N 12 ∂ x 2 = − p 1 ρ h v ¨ 2 − ∂ N 21 ∂ x 1 − ∂ N 22 ∂ x 2 = − p 2 . {\displaystyle {\begin{aligned}&\rho h{\ddot {w}}-{\frac {\partial ^{2}M_{11}}{\partial x_{1}^{2}}}-{\frac {\partial ^{2}M_{22}}{\partial x_{2}^{2}}}-2{\frac {\partial ^{2}M_{12}}{\partial x_{1}\partial x_{2}}}-{\frac {\partial }{\partial x_{1}}}\left(N_{11}\,{\frac {\partial w}{\partial x_{1}}}+N_{12}\,{\frac {\partial w}{\partial x_{2}}}\right)-{\frac {\partial }{\partial x_{2}}}\left(N_{12}\,{\frac {\partial w}{\partial x_{1}}}+N_{22}\,{\frac {\partial w}{\partial x_{2}}}\right)=-p_{3}\\&\rho h{\ddot {v}}_{1}-{\frac {\partial N_{11}}{\partial x_{1}}}-{\frac {\partial N_{12}}{\partial x_{2}}}=-p_{1}\\&\rho h{\ddot {v}}_{2}-{\frac {\partial N_{21}}{\partial x_{1}}}-{\frac {\partial N_{22}}{\partial x_{2}}}=-p_{2}\,.\end{aligned}}}
In terms of stress resultants [ edit ] The Föppl–von Kármán equations are typically derived with an energy approach by consideringvariations ofinternal energy and thevirtual work done by external forces. The resulting static governing equations (equations of equilibrium) are
∂ 2 M 11 ∂ x 1 2 + ∂ 2 M 22 ∂ x 2 2 + 2 ∂ 2 M 12 ∂ x 1 ∂ x 2 + ∂ ∂ x 1 ( N 11 ∂ w ∂ x 1 + N 12 ∂ w ∂ x 2 ) + ∂ ∂ x 2 ( N 12 ∂ w ∂ x 1 + N 22 ∂ w ∂ x 2 ) = P ∂ N α β ∂ x β = 0 . {\displaystyle {\begin{aligned}&{\frac {\partial ^{2}M_{11}}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}M_{22}}{\partial x_{2}^{2}}}+2{\frac {\partial ^{2}M_{12}}{\partial x_{1}\partial x_{2}}}+{\frac {\partial }{\partial x_{1}}}\left(N_{11}\,{\frac {\partial w}{\partial x_{1}}}+N_{12}\,{\frac {\partial w}{\partial x_{2}}}\right)+{\frac {\partial }{\partial x_{2}}}\left(N_{12}\,{\frac {\partial w}{\partial x_{1}}}+N_{22}\,{\frac {\partial w}{\partial x_{2}}}\right)=P\\&{\frac {\partial N_{\alpha \beta }}{\partial x_{\beta }}}=0\,.\end{aligned}}} When the deflections are small compared to the overall dimensions of the plate, and the mid-surface strains are neglected,
∂ w ∂ x 1 ≈ 0 , ∂ w ∂ x 2 ≈ 0 , v 1 ≈ 0 , v 2 ≈ 0 {\displaystyle {\begin{aligned}{\frac {\partial w}{\partial x_{1}}}\approx 0,{\frac {\partial w}{\partial x_{2}}}\approx 0,v_{1}\approx 0,v_{2}\approx 0\end{aligned}}}
The equations of equilibrium are reduced (pure bending of thin plates) to
∂ 2 M 11 ∂ x 1 2 + ∂ 2 M 22 ∂ x 2 2 + 2 ∂ 2 M 12 ∂ x 1 ∂ x 2 = P {\displaystyle {\frac {\partial ^{2}M_{11}}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}M_{22}}{\partial x_{2}^{2}}}+2{\frac {\partial ^{2}M_{12}}{\partial x_{1}\partial x_{2}}}=P} ^ Föppl, A., "Vorlesungen über technische Mechanik",B.G. Teubner , Bd. 5., p. 132, Leipzig, Germany (1907) ^ von Kármán, T., "Festigkeitsproblem im Maschinenbau,"Encyk. D. Math. Wiss. IV , 311–385 (1910) ^ Cerda, E.; Mahadevan, L. (19 February 2003). "Geometry and Physics of Wrinkling".Physical Review Letters .90 (7) 074302. American Physical Society (APS).Bibcode :2003PhRvL..90g4302C .doi :10.1103/physrevlett.90.074302 .hdl :10533/174540 ISSN 0031-9007 .PMID 12633231 . ^ David Harris (11 February 2011)."Focus: Simplifying Crumpled Paper" .Physical Review Focus . Vol. 27. Retrieved4 February 2020 . ^a b c d "Theory of Elasticity". L. D. Landau, E. M. Lifshitz, (3rd ed.ISBN 0-7506-2633-X ^ The 2-dimensionalLaplacian ,Δ , is defined asΔ w := ∂ 2 w ∂ x α ∂ x α = ∂ 2 w ∂ x 1 2 + ∂ 2 w ∂ x 2 2 {\displaystyle \Delta w:={\frac {\partial ^{2}w}{\partial x_{\alpha }\partial x_{\alpha }}}={\frac {\partial ^{2}w}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}w}{\partial x_{2}^{2}}}} ^ von Karman plate equations http://imechanica.org/node/6618 Accessed Tue July 30 2013 14:20. ^a b Ciarlet, P. G. (1990),Plates and Junctions in Elastic Multi-Structures , Springer-Verlag,ISBN 3-540-52917-9 ^a b Ciarlet, Philippe G. (1980), "A justification of the von Kármán equations",Archive for Rational Mechanics and Analysis ,73 (4):349– 389,Bibcode :1980ArRMA..73..349C ,doi :10.1007/BF00247674 ,S2CID 120433309 ^ Typically, an assumption ofzero out-of-plane stress is made at this point. 
