Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 325

With the model (5.1) and the derivatives (5.3), the integration in (3.1) over the z
variable is easily evaluated: 22

t

t 3
12 a(θ, θ)

µt
2

2 dx 1 dx 2

(u) : = (θ, w) = +

ω |∇

w − θ | − fwdx 1 dx 2

( 5 . 4 )

ω

with

2 µ ε(θ) : ε(ψ) +

2 µ div θ div ψ dx 1 dx 2 .

λ
2

2 µ
λ +

a(θ, ψ) : = ( 5 . 5 )

ω

The symmetric gradient

∂θ i
∂x j +

, ,j

1
2

∂θ j
∂x i

ε ij (θ) : = = 1 , 2 ( 5 . 6 )

and the divergence are now based on functions of two variables. The first term in
(5.4) contains the bending part of the energy, and the second term contains the
shear term . Clearly the latter vanishes in the Kirchhoff model.
The solution of the variational problem does not change if the energy func-
tional is multiplied by a constant. Without altering the notation, we multiply by
t − 3 , replace t 2 f by f in the load, and normalize µ , leading to the (dimensionless)
expression

t − 2
2

1
2 a(θ, θ) +

2 dx −

(u) =

|∇ w − θ |

fwdx. ( 5 . 7 )

To stay with the usual notation, we have written instead of ω .
Now in the framework of the Kirchhoff model, (5.2) and (5.6) imply

ε ij (θ) = ∂ ij w,

and thus (5.5) gives the bilinear form

µ
i,j

∂ ij w∂ ij v + λ wv dx, a( ∇ w, ∇ v) = ( 5 . 8 )

with a suitable constant λ . This is a variational problem of fourth order of the
same structure as the variational formulation of the biharmonic equation.

22

Since D 33 u = 0, the model (5.1) is consistent with hypothesis H4 and with (4.2),

only if div θ = 0. The two hypotheses lead to slightly different factors in (5.5). Therefore

in computations of plates some so-called shear-correction factors may be found, but this
has no effect on our analysis.