The **Kirchhoff–Love theory of plates** is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and was developed in 1888 by Love ^{ [1] } using assumptions proposed by Kirchhoff. The theory assumes that a mid-surface plane can be used to represent a three-dimensional plate in two-dimensional form.

- Assumed displacement field
- Quasistatic Kirchhoff-Love plates
- Strain-displacement relations
- Equilibrium equations
- Boundary conditions
- Constitutive relations
- Small strains and moderate rotations
- Isotropic quasistatic Kirchhoff-Love plates
- Pure bending
- Bending under transverse load
- Cylindrical bending
- Dynamics of Kirchhoff-Love plates
- Governing equations
- Isotropic plates
- References
- See also

The following kinematic assumptions that are made in this theory:^{ [2] }

- 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.

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

The vectors form a Cartesian basis 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 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

If are the angles of rotation of the normal 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 expansion of the displacement around the mid-surface.

The original theory developed by Love was valid for infinitesimal strains and rotations. The theory was extended by von Kármán to situations where moderate rotations could be expected.

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

where as .

Using the kinematic assumptions we have

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

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 stresses.

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

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.

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

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 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

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

where is Poisson's Ratio and is Young's Modulus. 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

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

Here we have assumed that the in-plane displacements do not vary with and . In index notation,

and in direct notation

which is known as the biharmonic equation. 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.

If a distributed transverse load is applied to the plate, the governing equation is . Following the procedure shown in the previous section we get^{ [3] }

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.

Derivation of equilibrium equations for transverse loading For 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

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^{ [3] }

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

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

- mode
*k*= 0,*p*= 1 - mode
*k*= 0,*p*= 2 - mode
*k*= 1,*p*= 2

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):

where is the bending stiffness of the plate. For a uniform plate of thickness ,

In direct notation

For free vibrations, the governing equation becomes

Derivation of dynamic governing equations for isotropic Kirchhoff-Love plates For an isotropic and homogeneous plate, the stress-strain relations are

where are the in-plane strains. The strain-displacement relations for Kirchhoff-Love plates are

Therefore, the resultant moments corresponding to these stresses are

The governing equation for an isotropic and homogeneous plate of uniform thickness in the absence of in-plane displacements is

Differentiation of the expressions for the moment resultants gives us

Plugging into the governing equations leads to

Since the order of differentiation is irrelevant we have . Hence

If the flexural stiffness of the plate is defined as

we have

For small deformations, we often neglect the spatial derivatives of the transverse acceleration of the plate and we are left with

Then, in direct tensor notation, the governing equation of the plate is

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In applied mechanics, **bending** characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.

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The **Föppl–von Kármán equations**, named after August Föppl and Theodore von Kármán, are a set of nonlinear partial differential equations describing the large deflections of thin flat plates. With applications 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:

In continuum mechanics, a **compatible** deformation **tensor field** in a body is that *unique* tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. **Compatibility** is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.

In continuum mechanics, ** plate theories** are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions. The typical thickness to width ratio of a plate structure is less than 0.1. A plate theory takes advantage of this disparity in length scale to reduce the full three-dimensional solid mechanics problem to a two-dimensional problem. The aim of plate theory is to calculate the deformation and stresses in a plate subjected to loads.

**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 plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load.

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. This suggests that a two-dimensional plate theory will give an excellent approximation to the actual three-dimensional motion of a plate-like object, and indeed that is found to be true.

The **Mindlin–Reissner theory** of plates is an extension of Kirchhoff–Love plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1951 by Raymond Mindlin. A similar, but not identical, theory had been proposed earlier by Eric Reissner in 1945. Both theories are intended for thick plates in which the normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface. The Mindlin–Reissner theory is used to calculate the deformations and stresses in a plate whose thickness is of the order of one tenth the planar dimensions while the Kirchhoff–Love theory is applicable to thinner plates.

In theoretical physics, **relativistic Lagrangian mechanics** is Lagrangian mechanics applied in the context of special relativity and general relativity.

In theoretical physics, the **dual graviton** is a hypothetical elementary particle that is a dual of the graviton under electric-magnetic duality, as an S-duality, predicted by some formulations of supergravity in eleven dimensions.

- ↑ A. E. H. Love,
*On the small free vibrations and deformations of elastic shells*, Philosophical trans. of the Royal Society (London), 1888, Vol. série A, N° 17 p. 491–549. - ↑ Reddy, J. N., 2007,
**Theory and analysis of elastic plates and shells**, CRC Press, Taylor and Francis. - 1 2 Timoshenko, S. and Woinowsky-Krieger, S., (1959),
**Theory of plates and shells**, McGraw-Hill New York.

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