Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 290

§ 2. Hyperelastic Materials

By Cauchy's theorem, the equilibrium state of an elastic body is characterized by

− div T(x) = f(x), x ∈ , ( 2 . 1 )

and the boundary conditions

φ(x) = φ 0 (x), x ∈ 0 ,

( 2 . 2 )

T(x) · n = g(x), x ∈ 1 .

Here f is the applied body force and g is the surface traction on the part 1 of
the boundary. 0 denotes the part of the boundary on which the displacement is
given.
We regard these equations as a boundary-value problem for the deformation
φ , and write

div T(x, ∇ φ(x)) = f(x), − x ∈ ,

T(x, ∇ φ(x))n = g(x),

( 2 . 3 )

x ∈ 1 ,

φ(x) = φ(x 0 ), x ∈ 0 .

For simplicity, we neglect the dependence of the forces f and g on φ , i.e., we
consider them to be dead loads; cf. Ciarlet [1988, §2.7].
To be more precise, is the domain occupied by the deformed body, and is
also unknown. For simplicity, we identify with the reference configuration, and
restrict ourselves to an approximation which makes sense for small deformations.

2.1 Definition. An elastic material is called hyperelastic if there exists an energy
functional W :

3

×M

+ −→ R

such that

∂ W
∂F (x, F )

T(x,F) =

3
+

for x ∈ , F ∈ M .

There is a variational formulation corresponding to the boundary-value prob-
lem (2.3) for hyperelastic materials, provided that the vector fields f and g can be
written as gradient fields: f =

grad F and g = grad G . In this case the solutions

of (2.3) are stationary points of the total energy

[ W(x, I(ψ) = ∇ ψ(x)) − F (ψ(x)) ] dx +

1 G

(ψ(x))dx. ( 2 . 4 )