EQUATIONS OF ELASTIC HYPERSURFACES

172 SHELLS
If S S := [ S kj denotes the stress tensor, Hook’s law relates the stress and the
]n×n
deformation tensors through the linear relation
S l,m =
3∑
c j,k,l,m D j,k u for j, l = 1, . . . , n (5.137)
j,k=1
and the stiffness coefficients posses the following symmetry
c j,k,l,m = c k,j,l,m = c l,m,j,k ∀j, k, l, m = 1, . . . , n . (5.138)
These symmetry property reduce the number of distinct coefficients from 81 to 21 and yield
the symmetry of the stress tensor S l,m = S m,l , S ⊤ = S.
We suppose that the body is homogeneous, which implies that the stiffness coefficients
c j,k,l,m are all constant.
The medium satisfies the equilibrium equations
Def ∗ τ(t) − Φ(t) ≡ 0 , ∀t ∈ S . (5.139)
after application of volume forces Φ(t) = (Φ 1 (t), . . . , Φ n (t)) ⊤ . By inserting the representation
of the deformation tensor into Hook’s law (5.137) and the latter into the equilibrium
equations (5.139), we get the second order partial differential equation for the displacement
vector u
A S (t, D)u(t) = Φ(t) in S , (5.140)
where A S (t, D) is n × n matrix differential operator of order 2:
∥ ∥∥∥∥ n∑
A S (t, D)u = Def ∗ τ = Def ∗ c j,k,l,m D j,k u∥
j,k=1
∥
n×n
∥ ∥∥∥∥ n∑
= Def ∗ c j,k,l,m [D k u j + ν k (∂ j ν) · u]
∥
j,k=1
n×n
{ n∑
[
]
n∑
= c j,k,l,m Dm ∗ + ν m c j,k,q,m (∂ l ν q ) [D k u j + ν k (∂ j ν) · u]
j,k,m=1
q=1
} n
l=1
. (5.141)
Note that we have applied the symmetry properties of the deformation tensor (5.136) and
also the equality
n∑
c j,k,l,m D j,k u = 1 n∑
c j,k,l,m [D k u j + D j u k + u · ∇ C (ν j ν k )]
2
j,k=1
[
]
n∑
n∑
= c j,k,l,m D k u j + ν k (∂ j ν q )u q
j,k=1
=
j,k=1
q=1
n∑
c j,k,l,m [D k u j + ν k (∂ j ν) · u] , (5.142)
j,k=1