The Kirchhoff–Love theory of plates is a two-dimensional
mathematical model
A mathematical model is an abstract and concrete, abstract description of a concrete system using mathematics, mathematical concepts and language of mathematics, language. The process of developing a mathematical model is termed ''mathematical m ...
that is used to determine the
stresses and
deformations in thin
plates subjected to
force
In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
s and
moments. This theory is an extension of
Euler-Bernoulli beam theory and was developed in 1888 by
Love
Love is a feeling of strong attraction and emotional attachment (psychology), attachment to a person, animal, or thing. It is expressed in many forms, encompassing a range of strong and positive emotional and mental states, from the most su ...
[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.] 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.
The following kinematic assumptions that are made in this theory:
[Reddy, J. N., 2007, Theory and analysis of elastic plates and shells, CRC Press, Taylor and Francis.]
* 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
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...
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
Displacement may refer to:
Physical sciences
Mathematics and physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
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
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
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 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
strain-displacement relations are
:
where
as
.
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
pointing towards positive
direction, these equations are
:
where the thickness of the plate is
. In index notation,
where
are the
stresses.
:
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 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
:
where
is
Poisson's Ratio
In materials science and solid mechanics, Poisson's ratio (symbol: ( nu)) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value ...
and
is
Young's Modulus
Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn ...
. The moments corresponding to these stresses are
:
In expanded form,
:
where