A» x2

‚x/‚»

x1

‚a

v

FIGURE 6.2 Uniform surface pressure.

tion (6.4b) gives,

ρ0

f

δWext (φ, δv) = g · δv dv = ρ0 g · δv dV (6.17)

J

v V

It is clear that none of the terms in this expression depend on the current ge-

f

ometry, and hence its linearization is super¬‚uous, that is, DδWext (φ, δv)[u] =

0.

6.5.2 SURFACE FORCES

Although a wide variety of traction forces exist, only the important case

of uniform normal pressure will be addressed. The techniques, however,

illustrated by this simple example are relevant to more complex situations

such as frictional contact.

Figure 6.2 shows a general body with an applied uniform pressure p

acting on a surface a having a pointwise normal n. The traction force

vector t is therefore pn and the corresponding virtual work component is,

p

δWext (φ, δv) = pn · δv da (6.18)

a

In this equation the magnitude of the area element and the orientation of the

normal are both displacement-dependent. Consequently, any change in ge-

ometry will result in a change in the equilibrium condition and the emergence

of a sti¬ness term. Although it may be tempting to attempt the linearization

8 LINEARIZED EQUILIBRIUM EQUATIONS

of Equation (6.18) by a pull back to the initial con¬guration in the usual

manner, a more direct approach is available by using an arbitrary parame-

terization of the surface as shown in Figure 6.2. (An understanding of this

approach is facilitated by imagining the surface area a to be a single isopara-

metric element.) In terms of this parameterization the normal and area ele-

ments can be obtained in terms of the tangent vectors ‚x/‚ξ and ‚x/‚· as,

‚x ‚x

‚ξ — ‚· ‚x ‚x

n= ; da = dξd· (6.19)

—

‚ξ ‚·

‚x ‚x

‚ξ — ‚·

which enables Integral (6.18) to be expressed in the parameter plane as,

‚x ‚x

p

δWext (φ, δv) = pδv · dξd· (6.20)

—

‚ξ ‚·

Aξ

Note that the only displacement-dependent items in this equation are the

vectors ‚x/‚ξ and ‚x/‚·, which linearize to ‚u/‚ξ and ‚u/‚· respectively.

Hence the use of the product rule and a cyclic manipulation of the triple

product gives,

‚x ‚u ‚x ‚u

p

DδWext (φ, δv)[u] = p dξd·

—δv ’

· · —δv

‚ξ ‚· ‚· ‚ξ

Aξ

(6.21)

It is clear that Equation (6.21) is unsymmetric in the sense that the

terms u and δv cannot be interchanged without altering the result of the

integral. Hence the discretization of this term would, in general, yield an

unsymmetric tangent matrix component. However, for the special but fre-

quently encountered case where the position of points along the boundary ‚a

is ¬xed or prescribed, a symmetric matrix will indeed emerge after assembly.

This is demonstrated by showing that the integration theorems discussed in

Section 2.4.2 enable Equation (6.21) to be rewritten as,

‚x

‚x ‚δv ‚δv

p

DδWext (φ, δv)[u] = p dξd·

—u ’

· · —u

‚ξ ‚· ‚· ‚ξ

Aξ

‚x ‚x

+ p(u—δv) · ν· ’ νξ dl (6.22)

‚ξ ‚·

‚Aξ

where ν = [νξ , ν· ]T is the vector in the parameter plane normal to ‚Aξ . For

the special case where the positions along ‚a are ¬xed or prescribed, both

the iterative displacement u and the virtual velocity δv are zero a priori

along ‚Aξ and the second integral in the above expression vanishes. (Addi-

tionally, if a symmetry plane bisects the region a, then it is possible to show

that the triple product in the second term of Equation (6.22) is also zero

9

6.6 VARIATIONAL METHODS AND INCOMPRESSIBILITY

along this plane.) Anticipating closed boundary conditions a symmetric ex-

p

pression for DδWext (φ, δv)[u] can be constructed by adding half Equations

(6.21) and (6.22) to give,

1 ‚x ‚u ‚δv

p

DδWext (φ, δv)[u] = p —δv + dξd·

· —u

2 ‚ξ ‚· ‚·

Aξ

1 ‚x ‚u ‚δv

p —δv + dξd·

’ · —u

2 ‚· ‚ξ ‚ξ

Aξ

(6.23)

Discretization of this equation will obviously lead to a symmetric component

of the tangent matrix.

EXAMPLE 6.2: Proof of Equation 6.22

Repeated use of cyclic permutations of the triple product on Equation

(6.21) and the integration theorem give,

‚u ‚x ‚u ‚x

p

DδWext (φ, δv)[u] = p · δv— · δv— dξd·

’

‚· ‚ξ ‚ξ ‚·

Aξ

‚ ‚x ‚ ‚x

= p · (u—δv) ’ · (u—δv) dξd·

‚· ‚ξ ‚ξ ‚·

Aξ

‚x ‚δv ‚x ‚δv

p dξd·

’ ’

· u— · u—

‚ξ ‚· ‚· ‚ξ

Aξ

‚x ‚δv ‚x ‚δv

= p dξd·

—u ’

· · —u

‚ξ ‚· ‚· ‚ξ

Aξ

‚x ‚x

+ p(u—δv) · ν· ’ νξ dl

‚ξ ‚·

‚Aξ

6.6 VARIATIONAL METHODS AND INCOMPRESSIBILITY

It is well known in small strain linear elasticity that the equilibrium equation

can be derived by ¬nding the stationary position of a total energy potential

with respect to displacements. This applies equally to ¬nite deformation

situations and has the additional advantage that such a treatment provides

a uni¬ed framework within which such topics as incompressibility, contact

boundary conditions and ¬nite element technology can be formulated. In

particular in the context of incompressibility a variational approach con-

10 LINEARIZED EQUILIBRIUM EQUATIONS

veniently facilitates the introduction of Lagrangian multipliers or penalty

methods of constraint, where the resulting multi¬eld variational principles

incorporate variables such as the internal pressure. The use of an inde-

pendent discretization for these additional variables resolves the well-known

locking problem associated with incompressible ¬nite element formulations.

6.6.1 TOTAL POTENTIAL ENERGY AND EQUILIBRIUM

A total potential energy functional whose directional derivative yields the

principle of virtual work is,

Π(φ) = Ψ(C) dV ’ f 0 · φ dV ’ t0 · φ dA (6.24)

V V ‚V

To proceed we assume that the body and traction forces are not functions of

the motion. This is usually the case for body forces f 0 , but it is unlikely that

traction forces t0 will conform to this requirement in a ¬nite deformation

context. (Obviously both these terms are independent of deformation in the

small displacement case.) Under these assumptions the stationary position

of the above functional is obtained by equating to zero its derivative in an

arbitrary direction δv to give,

‚Ψ

DΠ(φ)[δv] = : DC[δv] dV ’ f 0 · δv dV ’ t0 · δv dA

‚C

V V ‚V

= S : DE[δv] dV ’ f 0 · δv dV ’ t0 · δv dA = 0

V V ‚V

(6.25)

where Equation (5.7) for S has been used. Observe that this equation is

identical to the principle of virtual work, that is,

DΠ(φ)[δv] = δW (φ, δv) (6.26)

and consequently the equilibrium con¬guration φ renders stationary the to-

tal potential energy. The stationary condition of Equation (6.24) is also

known as a variational statement of equilibrium. Furthermore, the lin-

earized equilibrium Equation (6.10) or (6.15) can be interpreted as the sec-

ond derivative of the total potential energy as,

DδW (φ, δv)[u] = D2 Π(φ)[δv, u] (6.27)

11

6.6 VARIATIONAL METHODS AND INCOMPRESSIBILITY

6.6.2 LAGRANGE MULTIPLIER APPROACH TO

INCOMPRESSIBILITY

We have seen in Chapter 5 that for incompressible materials the consti-

tutive equations are only a function of the distortional component of the

deformation. In addition the incompressibility constraint J = 1 has to be

enforced explicitly. This is traditionally achieved by augmenting the varia-

tional Functional (6.24) with a Lagrange multiplier term to give,

ˆ

ΠL (φ, p) = Π(φ) + p(J ’ 1) dV (6.28)

V

where p has been used to denote the Lagrange multiplier in anticipation of

the fact that it will be identi¬ed as the internal pressure and the notation

ˆ

Π implies that the strain energy Ψ is now a function of the distortional

ˆ

component C of the right Cauchy“Green tensor (see Section 5.5.1), that is,

ˆ ˆ ˆ ˆ

Π(φ) = Ψ (C) dV ’ f 0 · φ dV ’ t0 · φ dA; Ψ (C) = Ψ(C)

V V ‚V

(6.29)

The stationary condition of the above functional with respect to φ and

p will be considered separately. Firstly the directional derivative of ΠL with

respect to p in the arbitrary direction δp is,

DΠL (φ, p)[δp] = δp(J ’ 1) dV = 0 (6.30)

V

Hence if this condition is satis¬ed for all functions δp the incompressibility

constraint J = 1 is ensured.