EQUATIONS OF ELASTIC HYPERSURFACES

52 SHELLS
In view of (2.221) it is rather natural to introduce the deformation tensor as the symmetrized
covariant derivative (cf., e.g., [Ta2, V. I, Ch. 5, § 12]).
(Def S U)(V, W )= 1 {
}
〈∂ V U, W 〉 + 〈∂ W U, V 〉
2
= 1 {
}
〈∂V S U, W 〉 + 〈∂W S U, V 〉 , ∀ V, W ∈ TS . (2.223)
2
It represents a tensor of type (0, 2) (the covariant derivative ∂U
S
(0, 2)-tensor), whereas the antisymmetric part
dU(V, W ) = 〈dU, V ∧ W 〉 = 1 2
itself is a non-symmetrized
{
}
〈∂V S U, W 〉 − 〈∂W S U, V 〉 , ∀ V, W ∈ TS , (2.224)
is the exterior differential (see § 2.6).
Thus, Def S U = [ ˜Djk U ] (n−1)×(n−1)
(0, 2). Similarly to (2.222) we obtain:
is a symmetric contravariant tensor field of type
˜D jk U = (Def S U) jk = 1 2 (U j;k + U k;j ) , ∀ j, k = 1, . . . , n − 1 (2.225)
∑n−1
Here, as usual, Θ : Ω → S is a surface, U is a tangential vector field U = U j g j ∈ TS ,
g j = ∂ j Θ and
U k;j := ∂ j U k − ∑ Γ m kjU m , U k = ∑
m
m
j=1
g km U m (2.226)
is the covariant derivative (cf. (2.187), (2.211) and (2.213)).
The adjoint of Def S is Def ∗ S defined in local coordinates by
(Def ∗ S Z) j = 1 2[ (∂
S
k
) ∗Z jk + ( ∂ S k
) ∗Z kj]
(2.227)
for each tensor field Z = [ Z jk] of type (0, 2), where ( )
∂k
S ∗
denotes the dual to the covariant
derivative ∂k S U = U ;k. In fact, assuming S is a closed surface, we get
∫
∫
〈Def S U, Z〉 dS = Tr [ (Def S U)Z ⊤] dS = 1 ∑
∫
[ ]
∂
S
S
S 2
k U j + ∂j S U k Z jk dS
j,k
S
= 1 ∑
∫
[( )
U j ∂
S ∗Z jk
2
k + ( ) ∗Z
∂ ] ∫
j
S kj
dS = 〈U, Def ∗ S Z〉 dS
j,k
S
holds for any U ∈ TS and any tensor field Z = [ Z jk] of type (0, 2) and Def ∗ S defined in
(2.227).
If S has a non-empty boundary Γ = ∂S ≠ ∅ and ν Γ = ( ν 1 Γ , . . . , νn Γ) ⊤
∈ TS is the
outward unit normal to Γ ↩→ S then, proceeding as above, we get the following integration
S