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a ξ
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 ‚·


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

(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


‚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 ‚· ‚·


1 ‚x ‚u ‚δv
p —δv + dξd·
’ · —u
2 ‚· ‚ξ ‚ξ

(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 ‚·


‚ ‚x ‚ ‚x
= p · (u—δv) ’ · (u—δv) dξd·
‚· ‚ξ ‚ξ ‚·


‚x ‚δv ‚x ‚δv
p dξd·
’ ’
· u— · u—
‚ξ ‚· ‚· ‚ξ


‚x ‚δv ‚x ‚δv
= p dξd·
—u ’
· · —u
‚ξ ‚· ‚· ‚ξ


‚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.

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