Kirchhoff–Love plate theory
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 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: Assumed displacement field. Let the position vector of a point in the undeformed plate  be formula_1. Then The vectors formula_3 form a Cartesian basis with origin on the mid-surface of the plate, formula_4 and formula_5 are the Cartesian coordinates on the mid-surface of the undeformed plate, and formula_6 is the coordinate for the thickness direction. Let the displacement of a point in the plate be formula_7. Then This displacement can be decomposed into a vector sum of the mid-surface displacement formula_9 and an out-of-plane displacement formula_10 in the formula_6 direction. We can write the in-plane displacement of the mid-surface as Note that the index formula_13 takes the  values 1 and 2 but not 3. Then the Kirchhoff hypothesis implies that If formula_14 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 formula_16 as the first order Taylor series 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 formula_18 as formula_13. 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 formula_20 pointing towards positive formula_6 direction, these equations are where the thickness of the plate is formula_23. In index notation, where formula_24 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 formula_26 is an effective shear force. Constitutive relations. The stress-strain relations for a linear elastic Kirchhoff plate are given by Since formula_28 and formula_29 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 10formula_38 to 15formula_39, 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 formula_44 is Poisson's Ratio and formula_45 is Young's Modulus. The moments corresponding to these stresses are In expanded form, where formula_48 for plates of thickness formula_49. 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 formula_51, 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 do not vary with formula_4 and formula_5. In index notation, and in direct notation which is known as the biharmonic equation. The bending moments are given by Bending under transverse load. If a distributed transverse load formula_20 pointing along positive formula_6 direction is applied to the plate, the governing equation  is formula_60. 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. 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 formula_63. 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 formula_67, and 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. 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): where formula_71 is the bending stiffness of the plate. For a uniform plate of thickness formula_23, In direct notation For free vibrations, the governing equation becomes