Föppl–von Kármán equations
Encyclopedia
The Föppl–von Kármán equations, named after August Föppl
and Theodore von Kármán
, are a set of nonlinear partial differential equation
s describing the large deflections of thin flat plates. With application 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
Introducing the Airy stress function
the above equations become
of thin plates the equation of equilibrium is
, where
is called flexural or cylindrical rigidity of the plate.
August Föppl
August Otto Föppl was a professor of Technical Mechanics and Graphical Statics at the Technical University of Munich, Germany...
and Theodore von Kármán
Theodore von Karman
Theodore von Kármán was a Hungarian-American mathematician, aerospace engineer and physicist who was active primarily in the fields of aeronautics and astronautics. He is responsible for many key advances in aerodynamics, notably his work on supersonic and hypersonic airflow characterization...
, are a set of nonlinear partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s describing the large deflections of thin flat plates. With application 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
-
= Young's modulusYoung's modulusYoung's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...
-
= Stress tensor
-
= Thickness of the plate
-
= Out of plane deflection
-
= Poisson's ratioPoisson's ratioPoisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....
-
= External normal force per unit area of the plate
-
= 2-dimensional Laplacian
Introducing the Airy stress function
the above equations becomePure bending
For the pure bendingPure bending
Pure bending is a condition of stress where a bending moment is applied to a beam without the simultaneous application of axial, shear, or torsional forces....
of thin plates the equation of equilibrium is
, whereis called flexural or cylindrical rigidity of the plate.