Kirchhoff–Love plate theory - AbsoluteAstronomy.com

The Kirchhoff–Love theory of plates is a two-dimensional mathematical model

Mathematical model

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

that is used to determine the stresses and deformation

Deformation (mechanics)

Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body...

s in thin plate

Plate

Plate may refer to:* Plate , a broad, mainly flat vessel on which food is served* Silver or the plate, dishware and cutlery made of sterling, Britannia or Sheffield plate silver* Plating, the deposition of metallic layers...

s subjected to force

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

s and moment

Moment

- Science, engineering, and mathematics :* Moment , used in probability theory and statistics* Moment , several related concepts, including:** Angular momentum or moment of momentum, the rotational analog of momentum...

s. This theory is an extension of Euler-Bernoulli beam theory and was developed in 1888 by Love

Augustus Edward Hough Love

Augustus Edward Hough Love FRS , often known as A. E. H. Love, was a mathematician famous for his work on the mathematical theory of elasticity...

using assumptions proposed by Kirchhoff

Gustav Kirchhoff

Gustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects...

. The theory assumes that a mid-surface plane can be used to represent a three-dimensional plate in two-dimensional form.

The following kinematic assumptions that are made in this theory:

straight lines normal to the mid-surface remain straight after deformation
straight lines normal to the mid-surface remain normal to the mid-surface after deformation
the thickness of the plate does not change during a deformation.

Assumed displacement field

Let the position vector of a point in the undeformed plate be

. Then

The vectors

form a Cartesian

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

basis

Basis (linear algebra)

In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

with origin on the mid-surface of the plate,

and

are the Cartesian coordinates on the mid-surface of the undeformed plate, and

is the coordinate for the thickness direction.

Let the displacement

Displacement (vector)

A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...

of a point in the plate be

. Then

This displacement can be decomposed into a vector sum of the mid-surface displacement and an out-of-plane displacement

in the

direction. We can write the in-plane displacement of the mid-surface as

Note that the index

takes the values 1 and 2 but not 3.

Then the Kirchhoff hypothesis implies that

are the angles of rotation of the normal

Normal

Normal may refer to:* Normality , conformance to an average* Norm , social norms, expected patterns of behavior studied within the context of sociology* Normal distribution , the Gaussian continuous probability distribution...

to the mid-surface, then in the Kirchhoff-Love theory

Note that we can think of the expression for

as the first order Taylor series

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

expansion of the displacement around the mid-surface.

Quasistatic Kirchhoff-Love plates

The original theory developed by Love was valid for infinitesimal strains and rotations. The theory was extended by 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...

to situations where moderate rotations could be expected.

Strain-displacement relations

For the situation where the strains in the plate are infinitesimal and the rotations of the mid-surface normals are less than 10

the strains-displacement relations are

Using the kinematic assumptions we have

Therefore the only non-zero strains are in the in-plane directions.

Equilibrium equations

The equilibrium equations for the plate can be derived from the principle of virtual work. For a thin plate under a quasistatic transverse load

these equations are

where the thickness of the plate is

. In index notation,

where

are the stress

Stress (physics)

In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

es.

Derivation of equilibrium equations for small rotations

For the situation where the strains and rotations of the plate are small the virtual internal energy is given by

where the thickness of the plate is

and the stress resultants and stress moment resultants are defined as

Integration by parts leads to

The symmetry of the stress tensor implies that

. Hence,

Another integration by parts gives

For the case where there are no prescribed external forces, the principle of virtual work implies that

. The equilibrium equations for the plate are then given by

If the plate is loaded by an external distributed load

that is normal to the mid-surface and directed in the positive

direction, the external virtual work due to the load is

The principle of virtual work then leads to the equilibrium equations

Boundary conditions

The boundary conditions that are needed to solve the equilibrium equations of plate theory can be obtained from the boundary terms in the principle of virtual work. In the absence of external forces on the boundary, the boundary conditions are

Note that the quantity

is an effective shear force.

Constitutive relations

The stress-strain relations for a linear elastic Kirchhoff plate are given by

Since

and

do not appear in the equilibrium equations it is implicitly assumed that these quantities do not have any effect on the momentum balance and are neglected. The remaining stress-strain relations, in matrix form, can be written as

Then,

and

The extensional stiffnesses are the quantities

The bending stiffnesses (also called flexural rigidity) are the quantities

The Kirchhoff-Love constitutive assumptions lead to zero shear forces. As a result, the equilibrium equations for the plate have to be used to determine the shear forces in thin Kirchhoff-Love plates. For isotropic plates, these equations lead to

Alternatively, these shear forces can be expressed as

where

Small strains and moderate rotations

If the rotations of the normals to the mid-surface are in the range of 10

to 15

, the strain-displacement relations can be approximated as

Then the kinematic assumptions of Kirchhoff-Love theory lead to the classical plate theory with von Kármán

Theodore von Karman

strains

This theory is nonlinear because of the quadratic terms in the strain-displacement relations.

If the strain-displacement relations take the von Karman form, the equilibrium equations can be expressed as

Isotropic quasistatic Kirchhoff-Love plates

For an isotropic and homogeneous plate, the stress-strain relations are

The moments corresponding to these stresses are

In expanded form,

where

for plates of thickness

. Using the stress-strain relations for the plates, we can show that the stresses and moments are related by

At the top of the plate where

, the stresses are

Pure bending

For an isotropic and homogeneous plate under pure bending, the governing equations reduce to

Here we have assumed that the in-plane displacements are do not vary with

and

. In index notation,

and in direct notation

The bending moments are given by

Derivation of equilibrium equations for pure bending
For an isotropic, homogeneous plate under pure bending the governing equations are and the stress-strain relations are Then, and Differentiation gives and Plugging into the governing equations leads to Since the order of differentiation is irrelevant we have , , and . Hence In direct tensor notation, the governing equation of the plate is where we have assumed that the displacements are constant.

Bending under transverse load

If a distributed transverse load

is applied to the plate, the governing equation is

. Following the procedure shown in the previous section we get

In rectangular Cartesian coordinates, the governing equation is

and in cylindrical coordinates it takes the form

Solutions of this equation for various geometries and boundary conditions can be found in the article on bending of plates

Bending of plates

Bending of plates or plate bending refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the...

Derivation of equilibrium equations for transverse loading

or a transversely loaded plate without axial deformations, the governing equation has the form

where

is a distributed transverse load (per unit area). Substitution of the expressions for the derivatives of

into the governing equation gives

Noting that the bending stiffness is the quantity

we can write the governing equation in the form

In cylindrical coordinates

For symmetrically loaded circular plates,

, and we have

Cylindrical bending

Under certain loading conditions a flat plate can be bent into the shape of the surface of a cylinder. This type of bending is called cylindrical bending and represents the special situation where

. In that case

and

and the governing equations become

Dynamics of Kirchhoff-Love plates

The dynamic theory of thin plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes.

Governing equations

The governing equations for the dynamics of a Kirchhoff-Love plate are

where, for a plate with density

and

Derivation of equations governing the dynamics of Kirchhoff-Love plates

The total kinetic energy of the plate is given by

Therefore the variation in kinetic energy is

We use the following notation in the rest of this section.

Then

For a Kirchhof-Love plate

Hence,

Define, for constant

through the thickness of the plate,

Then

Integrating by parts,

The variations

and

are zero at

and

.
Hence, after switching the sequence of integration, we have

Integration by parts over the mid-surface gives

Again, since the variations are zero at the beginning and the end of the time interval under consideration, we have

For the dynamic case, the variation in the internal energy is given by

Integration by parts and invoking zero variation at the boundary of the mid-surface gives

If there is an external distributed force

acting normal to the surface of the plate, the virtual external work done is

From the principle of virtual work

. Hence the governing balance equations for the plate are

Solutions of these equations for some special cases can be found in the article on vibrations of plates

Vibration of plates

The vibration of plates is a special case of the more general problem of mechanical vibrations. The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two...

. The figures below show some vibrational modes of a circular plate.

Isotropic plates

The governing equations simplify considerably for isotropic and homogeneous plates for which the in-plane deformations can be neglected. In that case we are left with one equation of the following form (in rectangular Cartesian coordinates):