by judiciously choosing the interpolating functions used for the volumetric

¯ ¯

variables J, p, and their variations, δ J and δp. In particular, the simplest

possible procedure involves using constant interpolations for these variables

over a given volume, typically a ¬nite element. The resulting method is

known as the mean dilatation technique.

EXAMPLE 6.3: Hu-Washizu variational principle

Equation (6.45) is a Hu-Washizu type of variational equation in the

sense that it incorporates three independent variables, namely, the mo-

tion, a volumetric strain ¬eld, and its corresponding volumetric stress.

It is, however, a very particular case as only the volumetric stress and

strain components are present in the equation. A more general Hu-

Washizu variational principle involving the motion, a complete stress

¬eld such as the ¬rst Piola“Kirchho¬ tensor P , and its associated strain

F is given as,

ΠHW (φ, F , P ) = Ψ(F ) dv + P :( ’ F ) dV ’ Πext (φ)

0φ

V V

(continued)

17

6.6 VARIATIONAL METHODS AND INCOMPRESSIBILITY

EXAMPLE 6.3 (cont.)

where now F is an independent variable as yet unrelated to the defor-

mation gradient of the motion 0 φ. The stationary condition of this

functional with respect to a variation δv in motion φ gives the principle

of virtual work as,

DΠ(φ, F , P )[δv] = P: 0 δv dV ’ δWext (φ)[δv] = 0

V

whereas the stationary conditions with respect to F and P give a

constitutive equation and a relationship between the strain and the

motion as,

‚Ψ

DΠ(φ, F , P )[δF ] = : δF dV = 0

’P

‚F

V

DΠ(φ, F , P )[δP ] = ( ’ F ) : δP dV = 0

0φ

V

These expressions are the weak forms equivalent to the hyperelastic re-

lationship P = ‚Ψ/‚F and the strain equation F = 0 φ respectively.

6.6.5 MEAN DILATATION PROCEDURE

For convenience the following discussion is based on an arbitrary volume V ,

which after discretization will inevitably become the volume of each element.

¯ ¯

Assuming that p, J and δp, δ J are constant over the integration volume,

Equations (6.48“9) yield,

1 v

¯

J= JdV = (6.50a)

V V

V

dU

p= (6.50b)

¯

dJ ¯

J=v/V

¯

Observe that Equation (6.50a) shows that the surrogate Jacobian J is the

integral equivalent of the pointwise Jacobian J = dv/dV . A typical ex-

pression for the volumetric strain energy has already been introduced in

¯

Section 5.5.3, Equation (5.57). In terms of J this equation now becomes,

1¯

¯

U (J) = κ(J ’ 1)2 (6.51)

2

18 LINEARIZED EQUILIBRIUM EQUATIONS

from which the mean pressure emerges as,

v’V

¯

p(J) = κ (6.52)

V

At this juncture it is convenient to acknowledge that the mean dilatation

method will be used in the FE formulation presented in the next chapter.

This enables us to incorporate a priori the mean pressure derived above

using the Hu-Washizu functional directly into the principle of virtual work

to give,

δW (φ, δv) = σ : δd dv ’ δWext (φ, δv) = 0, σ = σ + pI;

v

dU

p= (6.53)

¯

dJ ¯

J=v/V

which can also be expressed in the initial con¬guration as,

S = S + pJC ’1

δW (φ, δv) = S : DE[δv] dV ’ δWext (φ, δv) = 0,

V

(6.54)

Given that the pressure is constant over the volume, Equation (6.53) can

also be written as,

δW (φ, δv) = σ : δd dv + p ·δv dv ’ δWext (φ, δv)

v v

= σ : δd dv + pv(div δv) ’ δWext (φ, δv) = 0 (6.55)

v

where the notation div implies the average divergence over the volume v,

for instance,

1

div δv = ·δv dv (6.56)

v v

EXAMPLE 6.4: Mean deformation gradient method

Using the Hu-Washizu variational principle introduced in Example 6.3,

it is possible to derive a technique whereby the complete deformation

gradient F , rather than just its volumetric component, is taken as

constant over the integration volume. Finite element discretizations

based on this type of technique are sometimes used to avoid shear

(continued)

19

6.6 VARIATIONAL METHODS AND INCOMPRESSIBILITY

EXAMPLE 6.4 (cont.)

locking as well as volumetric locking. By assuming that P , F , and

their variations are constant in Example 6.3, the following equations

are obtained for F and P ,

1

F= 0 φ dV = 0φ

V V

‚Ψ

P=

‚F F= φ

0

where 0 represents the average gradient over the integration volume V .

With the help of these equations the principle of virtual work becomes,

δW (φ, δv) = V P : 0 δv ’ δWext (φ, δv) = 0

Note that this can be written in terms of the corresponding second

Piola“Kircho¬ tensor S = ( 0 φ)’1 P as,

δW (φ)[δv] = V S : DE[δv] ’ δWext (φ, δv) = 0;

T T

2DE[δv] = ( 0 δv +( 0 δv)

0 φ) 0φ

T

or in terms of the Cauchy stress tensor vσ = V P ( as,

0 φ)

’1

δW (φ)[δv] = vσ : δv ’ δWext (φ, δv) = 0; δv = ( 0 δv)( 0 φ)

As usual it is now necessary to linearize the modi¬ed virtual work equa-

tion in preparation for a Newton“Raphson iteration and the development of

a tangent matrix. Again this linearization is ¬rst obtained using the initial

con¬guration Equation (6.54) for which the integral limits remain constant.

Disregarding the external force component, this gives,

‚S ™

DδW int (φ, δv)[u] = DE[δv] : 2 : DE[u] dV + S : Dδ E[u] dV

‚C

V V

‚(JC ’1 )

‚S

= DE[δv] : 2 +p : DE[u] dV

‚C ‚C

V

™

(DE[δv] : JC ’1 )Dp[u] dV +

+ S : Dδ E[u] dV

V V

ˆ

= DE[δv] : (C + C p ) : DE[u] dV

V

T

DE[δv] : JC ’1 dV

+ S : [( 0 δv] + Dp[u]

0 u)

V V

(6.57)

20 LINEARIZED EQUILIBRIUM EQUATIONS

Observe that, via Equation (6.50b), the pressure is now an explicit function

of the current volume and thus of φ and hence is subject to linearization

in the direction of u. With the help of Equation (3.129) and the usual

push forward operations (6.11), this equation is rewritten in the current

con¬guration as,

σ : [( u)T

ˆ

DδW int (φ, δv)[u] = δd : (c +c p ) : µ dv + δv] dv

v v

+ v(div δv)Dp[u] (6.58)

The linearization of the pressure term follows from Equation (6.50b) as,

d2 U

Dp[u] = ¯2 D(v/V )[u]

dJ ¯

J=v/V

1 d2 U

= DJ[u] dV

¯

V dJ 2 ¯ V

J=v/V

1 d2 U

= J · u dV

¯

V dJ 2 ¯ V

J=v/V

v d2 U

= κv div u;

¯ κ=

¯ (6.59)

¯

V dJ 2 ¯

J=v/V

from which, ¬nally,

ˆ

DδW int (φ, δv)[u] = δd : (C + cp ) : µ dv

v

σ : [( u)T

+ δv] dv + κv(div δv)(div u)6.60)

¯ (

v

where, for instance for the volumetric potential shown in Equation (6.51),

κ is

¯

vκ

κ=

¯ (6.61)

V

The discretization of the mean dilatation technique will be considered

in the next chapter.