EQUATIONS OF ELASTIC HYPERSURFACES

3. CALCULUS OF DIFFERENTIAL OPERATORS ON HYPERSURFACES 85
In other words, (3.80) is the Euler-Lagrange equation associated with the integral functional
(3.78).
Similarly, minimizers of the energy functional
∫
E [U] :=
S
[‖dU‖ 2 + ‖δU‖ 2 ]
dS, U ∈ Λ l TS , (3.81)
are null-solutions of the Hodge-Laplacian (cf. later (3.124)), while minimizers of the energy
functional
∫
E [U] := ‖∇U‖ 2 dS, U ∈ TS , (3.82)
S
are null-solutions of the Bochner-Laplacian ((cf. later (3.125)).
Our aim is to adopt a similar point of view in the case of anisotropic and isotropic (Lamé)
system of elasticity on S . The departure point is to consider the total free (elastic) energy
∫
E [U] := E(x, D S U(x)) dS, D S U := [ Dj S U k , U ∈ TS (3.83)
]n×n
S
(cf. (3.68)), ignoring at the moment the displacement boundary conditions (Koiter’s model)).
As before, equilibria states correspond to minimizers of the above variational integral (see
[NH1, § 5.2]). First we should identify the correct form of the stored energy density
E(x, D S U(x)). We shall restrict attention to the case of linear elasticity. In this scenario,
E = (S S , Def S ) depends bi-linearly on the stress tensor S S = [ S jk] and the deformation
(strain)
n×n
tensor
Def S = [ D jk] n×n , Djk U := 1 2[
D
S
j U k + D S k U j
]
, j, k = 1, . . . , n (3.84)
which, according to Hooke’s law, satisfy S S = TDef S , for some linear, fourth-order tensor
T. If the medium is also homogeneous (i.e. the density and elastic parameters are positionindependent),
it follows that E depends quadratically on D S U, i.e.
for some linear operator
E(x, D S U(x)) = 〈TD S U(x), D S U(x)〉 (3.85)
T : M n,n (R) −→ M n,n (R), (3.86)
where M n,n (R) stands for the vector space of all n × n matrices with real entries. Hereafter,
we organize M n,n (R) as a real Hilbert space with respect to the inner product
〈A, B〉 := Tr(AB ⊤ ) = ∑ i,j
a ij b ij , ∀ A = [a ij ] i,j
, B = [b ij ] i,j
∈ M n,n (R), (3.87)
where B ⊤ denotes transposed matrix, and Tr is the usual trace operator for square matrices.
A linear operator (3.86 is a tensor of order 4, i.e., T = [ c ijkl
]ijkl , and
[ ]
∑
TA = c ijkl a kl , for A = [a kl ] kl
∈ M n,n (R) . (3.88)
k,l
ij