Thinplate

Thin plate refers to a flat structural element whose thickness is small relative to its other dimensions, allowing bending and vibration to be analyzed with two-dimensional theories applied to a mid-surface. In engineering practice, thin plates are modeled using plate theories that reduce a three-dimensional elasticity problem to a problem defined on the mid-surface.

The classical approach is Kirchhoff–Love (or simply classical plate theory), which assumes that straight lines normal to the mid-surface remain straight and perpendicular after deformation, and that transverse shear deformations are negligible. For small deflections under a transverse load q(x,y), the governing equation is D ∇^4 w = q, where w(x,y) is the transverse deflection and ∇^4 is the biharmonic operator. The plate stiffness D is D = E h^3 / [12(1 − ν^2)] for an isotropic plate, with E the Young’s modulus, ν the Poisson ratio, and h the thickness. Boundary conditions depend on edge support and can be simply supported, clamped, or free.

For thicker plates, Mindlin–Reissner theory introduces transverse shear deformation corrections, improving accuracy when h is not negligible.

Thin plate splines, used in interpolation and smoothing, adopt a related variational principle: minimize a bending-energy functional ∫∫ [(w_xx)^2 + 2(w_xy)^2 + (w_yy)^2] dx dy subject to interpolating data constraints. The resulting function has the form w(x) = a0 + a1 x + a2 y + Σ_i b_i φ(||x − x_i||), where φ(r) = r^2 log r in two dimensions. This connects to the biharmonic operator and to surface fitting in graphics and geostatistics.

Applications of thin-plate models span aerospace, civil engineering, automotive design, and computer graphics, where accurate bending behavior and smooth surface fitting are required.