Polyharmonic boundary value problems

1.1 Classical problems from elasticity 5
1.1.2 The Kirchhoff-Love model for a thin plate
As for the beam we assume that the plate, the vertical projection of which is the
planar region Ω ⊂ R2 , is free to move horizontally at the boundary. Then a simple
model for the elastic energy is
12 J(u) = (∆u)
Ω
2 + (1 − σ) u 2
xy − uxxuyy − f u dxdy, (1.5)
where f is the external vertical load. Again u is the deflection of the plate in vertical
direction and, as above for the beam, first order derivatives are left out which
indicates that the plate is free to move horizontally.
This modern variational formulation appears already in [173], while a discussion
for a boundary value problem for a thin elastic plate in a somehow old fashioned
notation is made already by Kirchhoff [249]. See also the two papers of Birman
[57, 58], the books by Mikhlin [303, §30], Destuynder-Salaun [141], Ciarlet [102],
or the article [103] for the clamped case.
In (1.5) σ is the Poisson ratio, which is defined by σ = λ
2(λ+µ) with the so-called
Lamé constants λ,µ that depend on the material. For physical reasons it holds that
µ > 0 and usually λ ≥ 0 so that 0 ≤ σ < 1 2 . Moreover, it always holds true that
σ > −1 although some exotic materials have a negative Poisson ratio, see [265].
For metals the value σ lies around 0.3 (see [280, p. 105]). One should observe that
for σ > −1, the quadratic part of the functional (1.5) is always positive.
For small deformations the terms in (1.5) are taken as approximations being
purely quadratic with respect to the second derivatives of u of respectively twice
the squared mean curvature and the Gaussian curvature supplied with the factor
σ − 1. For those small deformations one finds
1
2 (∆u) 2 + (1 − σ) u 2
1
xy − uxxuyy ≈ 2 (κ1 + κ2) 2 − (1 − σ)κ1κ2
= 1 2 κ2 1 + σκ1κ2 + 1 2 κ2 2 ,
where κ1, κ2 are the principal curvatures of the graph of u. Variational integrals
avoiding such approximations and involving the original expressions for the mean
and the Gaussian curvature are considered in Section 1.8 and lead as a special case
to the Willmore functional.
Which are the appropriate boundary conditions? For the clamped and hinged
boundary condition the natural settings, that is the Hilbert spaces for these two situations,
are respectively H = H2 0 (Ω) and H = H2 ∩ H1 0 (Ω). Minimising the energy
functional leads to the weak Euler-Lagrange equation 〈dJ(u),v〉 = 0, that is
Ω
(∆u∆v + (1 − σ)(2uxyvxy − uxxvyy − uyyvxx) − f v) dxdy = 0 (1.6)
for all v ∈ H. Let us assume both that minimisers u lie in H 4 (Ω) and that the exterior
normal ν = (ν1,ν2) and the corresponding tangential τ = (τ1,τ2) = (−ν2,ν1) are
well-defined. Then an integration by parts of (1.6) leads to