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: 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 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. 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. {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of equilibrium equations for small rotations