3.8 Incompressibility Constraint

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Many materials such as rubber or organic tissue are either incompressible or can be

viewed as such. In Chapter 1, it was shown that this condition is equivalent to J = 1.

Denoting the undeformed position of X by the rectangular coordinates (X, Y,Z), the

deformed position by (x, y, z) and the displacements by (u, v,w), this condition is equivalent

to

J =

______

x,X x,Y x,Z

y,X y,Y y,Z

z,X z,Y z,Z

______

= 1 (3.142)

GEOMETRIC NONLINEAR EFFECTS 175

or, using the local coordinates γ (ξ, η, ζ),

J =

______

x,ξ x,η x,ζ

y,ξ y,η y,ζ

z,ξ z,η z,ζ

______

・

______

ξ,X ξ,Y ξ,Z

η,X η,Y η,Z

ζ,X ζ,Y ζ,Z

______

= 1. (3.143)

This is a function of the displacement components of all nodes belonging to the element

at stake. Indeed (cf Equation (2.9) and (2.10)),

x =

N

 

i=1

ϕi(ξ, η, ζ)xi =

N

 

i=1

ϕi(ξ, η, ζ )(Xi + ui) (3.144)

y =

N

 

i=1

ϕi(ξ, η, ζ )(Yi + vi ) (3.145)

z =

N

 

i=1

ϕi(ξ, η, ζ )(Zi + wi ). (3.146)

Notice that Equations (3.144) to (3.146) only apply if the formulation is isoparametric,

that is, the undeformed position and the displacements are interpolated in the same way.

Accordingly, one finds

x,ξ =

N

 

i=1

∂ϕi

∂ξ

(ξ, η, ζ )(Xi + ui ) (3.147)

and similarly for the other terms. If we write Equation (3.143) as

f (u1, v1,w1, u2, v2,w2, . . . , uN, vN,wN) = 0 (3.148)

the linearization yields

f (u01

, v0

1,w0

1, u02

, v0

2,w0

2, ・ ・ ・ , u0

N, v0

N,w0

N)

+

i

_ ∂f

∂ui

____

0

(ui − u0

i ) + ∂f

∂vi

____0

(vi − v0

i ) + ∂f

∂wi

____

0

(wi − w0

i )

_ ≈ 0. (3.149)

Substitution of Equations (3.147) into Equation (3.143) reveals that f is a linear function

of ui if all vi and wi are kept constant, that is, vi = v0

i and wi = w0

i , ∀i. Accordingly,

∂f

∂ui

=

______

∂ϕi

∂ξ

∂ϕi

∂η

∂ϕi

∂ζ

y,ξ y,η y,ζ

z,ξ z,η z,ζ

______

・

______

ξ,X ξ,Y ξ,Z

η,X η,Y η,Z

ζ,X ζ,Y ζ,Z

______

. (3.150)

Equation (3.149) can be applied at any internal point of the element and leads to one

equation in all the degrees of freedom belonging to the element (e.g. 60 degrees of freedom

for the 20-node brick element). If it is applied to the points on the border, the degrees of

freedom of the adjoining elements must be considered too. In that case, it sounds feasible

176 GEOMETRIC NONLINEAR EFFECTS

to require that the mean of the Jacobian determined for each of the adjoining elements

separately, must be 1.

The question remains, at what points should Equation (3.142) be applied to yield valid

results. Application to too many points leads to volumetric locking of the element. Taking

hybrid elements as reference, where the pressure is usually interpolated with a lower degree

than the displacements, it is proposed to apply the incompressibility condition to the corner

nodes for quadratic elements, and to the center of the element for linear type elements.

4

Hyperelastic Materials

In this chapter, hyperelastic materials will be discussed. They are defined as materials for

which a free energy function

_(C, θ,X) (4.1)

exists such that Equations (1.393) and (1.394) apply. The function _ is sometimes called the

stored-energy function (Ciarlet 1993), (Simo and Hughes 1997). Because of the functional

dependence in Equation (4.1), hyperelastic materials have no memory (Figure 4.1). After

unloading, they return without time delay to their starting position. The determination of

the second Piola–Kirchhoff stress is straightforward through Equation (1.393):

S = 2

∂_

∂C

. (4.2)

The question naturally arises whether the function _ can be freely chosen or whether

physical considerations impose any restrictions. This is treated in the first section. Then,

a few popular isotropic models are discussed and applied to simulate a shear test and

the inflation of a balloon. Finally, the theory is extended to anisotropic materials such as

fiber-reinforced tissues. For further reading, the reader is particularly referred to (Holzapfel

2000) and (Bonet and Wood 1997).

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