Linear code sequence and jump

2013-09-12T10:38:35Z

92.26.55.12: Consistency

The '''Föppl–von Kármán equations''', named after [[August Föppl]]<ref>Föppl, A., "Vorlesungen über technische Mechanik", ''B.G. Teubner'', Bd. 5., p. 132, Leipzig, Germany (1907)</ref> and [[Theodore von Kármán]],<ref>von Kármán, T., "Festigkeitsproblem im Maschinenbau," ''Encyk. D. Math. Wiss.'' '''IV''', 311–385 (1910)</ref> are a set of nonlinear [[partial differential equation]]s describing the large deflections of thin flat plates.<ref>E. Cerda and L. Mahadevan, 2003, "Geometry and Physics of Wrinkling" [http://prola.aps.org/abstract/PRL/v90/i7/e074302 Phys. Rev. Lett. 90, 074302 (2003)]</ref> With application ranging from the design of submarine hulls to the mechanical properties of cell wall,<ref>http://focus.aps.org/story/v27/st6</ref> the equations are notoriously difficult to solve, and take the following form:
<ref name="ld">"Theory of Elasticity". L. D. Landau, E. M. Lifshitz, (3rd ed. ISBN 0-7506-2633-X)</ref>

:<math>
\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}
</math>

where {{math|''E''}} is the [[Young's modulus]] of the plate material (assumed homogeneous and isotropic), {{math|''υ''}} is the [[Poisson's ratio]], {{math|''h''}} is the thickness of the plate, {{math|''w''}} is the out–of–plane deflection of the plate, {{math|''P''}} is the external normal force per unit area of the plate, {{math|''σ''''αβ''}} is the [[Cauchy stress tensor]], and {{math|''α'', ''β''}} are [[Einstein notation|indices]] that take values of 1 or 2. The 2-dimensional [[biharmonic equation|biharmonic operator]] is defined as<ref>The 2-dimensional [[Laplacian]], {{math|Δ}}, is defined as
<math> \Delta w := \frac{\partial^2 w}{\partial x_\alpha \partial x_\alpha} = \frac{\partial^2w}{\partial x_1^2} + \frac{\partial^2w}{\partial x_2^2} </math></ref>
:<math>
\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} \,.
</math>
Equation (1) above can be derived from [[kinematic]] assumptions and the [[constitutive relation]]s 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 ({{math|''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.<ref>[http://imechanica.org/node/6618 von Karman plate equations http://imechanica.org/node/6618 Accessed Tue July 30 2013 14:20.]</ref> Ciarlet<ref name=Ciarlet>{{Citation|last=Ciarlet|first=P. G.|year=1990|title=Plates and Junctions in Elastic Multi-Structures|publisher=Springer-Verlag.}}</ref> 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.<ref name=Ciarlet/><ref>{{Citation|last= Ciarlet|first= Philippe G. | title=A justification of the von Kármán equations|journal= Archive for Rational Mechanics and Analysis |volume=73|year=1980|pages= 349–389.|issue=4|bibcode= 1980ArRMA..73..349C|doi= 10.1007/BF00247674}}</ref>

== Equations in terms of Airy stress function ==
The three Föppl–von Kármán equations can be reduced to two by introducing the [[Stress functions#Airy stress function|Airy stress function]] <math>\varphi</math> where
:<math>
\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} \,.
</math>
Then the above equations become<ref name="ld"/>
:<math>
\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
</math>
:<math>
\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 \,.
</math>

== Pure bending==
For the [[pure bending]] of thin plates the equation of equilibrium is <math>D\Delta^2\ w=P</math>, where
:<math>
D :=\frac{Eh^3}{12(1-\nu^2)}
</math>
is called [[flexural rigidity|flexural]] or ''cylindrical rigidity'' of the plate.<ref name="ld"/>

== 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 normal]]s 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 {{math|'''u'''}} in the plate can be expressed as
:<math>
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)
</math>
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 [[Finite strain theory|Green strain tensor]] are defined as
:<math>
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] \,.
</math>
Substitution of the expressions for the displacement field into the above gives
:<math>
\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}
</math>
For small strains but '''moderate rotations''', the higher order terms that cannot be neglected are
:<math>
\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} \,.
</math>
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'''
:<math>
\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}
</math>

== 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,<ref>Typically, an assumption of '''zero out-of-plane stress''' is made at this point.</ref> we have {{math|''σ''33}} = {{math|''σ''13}} = {{math|''σ''23}} = 0 and
:<math>
\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}
</math>
Expanding the terms, the three non-zero stresses are
:<math>
\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}
</math>

== Stress resultants ==
The [[stress resultants]] in the plate are defined as
:<math>
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 \,.
</math>
Therefore,
:<math>
\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}
</math>
and
:<math>
\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}
</math>
Solutions are easier to find when the governing equations are expressed in terms of stress resultants rather than the in-plane stresses.

== Föppl–von_Kármán equations in terms of stress resultants ==
The Föppl–von_Kármán equations are typically derived with an energy approach by considering [[variational calculus|variation]]s of internal energy and the work done by external forces. A similar approach can be used to write these equations in terms of stress resultants. The resulting governing equations are
:<math>
\begin{align}
&\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{align}
</math>

==References==
<references/>

== See also ==
* [[Plate theory]]

{{DEFAULTSORT:Foppl-Von Karman Equations}}
[[Category:Partial differential equations]]
[[Category:Continuum mechanics]]

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