The plate equation are the assumptions and mathematical formulations that reduce a 3D elasticity problem to a 2D plate bending problem, governed by a fourth-order PDE linking deflection and applied loads, with boundary conditions and material properties defining the solution. The classical Kirchhoff-Love theory remains the cornerstone, with various refinements addressing its limitations for thicker plates or complex loading .

The plate equation governs the bending and deformation behavior of plates in solid mechanics, are rooted in classical plate theory and its mathematical formulation. Here are the main points:

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Physical and Geometrical Assumptions

• The plate is assumed to be elastic, homogeneous, and isotropic.
• The plate is initially flat, and deflections are small compared to the plate thickness.
• The slope of the deflected surface is small, allowing linearization of strain-displacement relations.
• Stresses vary linearly through the thickness, with zero transverse normal stress ( $\sigma_{zz} =0$ ).
• The plate thickness is small relative to its other dimensions, justifying a two-dimensional approximation of a three-dimensional solid.

Governing Equations

• The plate bending is described by a fourth-order partial differential equation relating the plate deflection $w$ to the applied transverse load $q$.

• The classical Kirchhoff-Love plate theory provides the fundamental governing equation:

  $$ \nabla^2 \nabla^2 w=\frac{q}{D} $$

  where $D$ is the flexural rigidity of the plate, defined as

  $$ D=\frac{E H^3}{12\left(1-\nu^2\right)} $$

  with $E$ being Young's modulus, $\nu$ Poisson's ratio, and $H$ the plate thickness.

• The moment-curvature relations connect bending moments $M_{\alpha \beta}$ to the curvatures (second derivatives of deflection), analogous to beam bending theory but extended to two dimensions.

Boundary Conditions

• Plates can have various boundary conditions such as simply supported, clamped, or free edges.
• For example, simply supported edges have zero deflection and bending moment conditions, while clamped edges have zero deflection and zero slope (rotation).

Strain-Displacement and Stress Relations

• The strain components in the plate include membrane strains and bending strains, with bending strains proportional to the distance from the mid-plane.
• The Kirchhoff-Love theory assumes normals to the mid-surface remain straight and normal after deformation (no transverse shear deformation).
• Stress components are related to moments and curvatures through linear elastic constitutive relations.