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FIGURE 6.1 Linearized equilibrium.



Considering a trial solution φk , the above equation can be linearized in the
direction of an increment u in φk as,

δW (φk , δv) + DδW (φk , δv)[u] = 0 (6.2)

Consequently it is necessary to ¬nd the directional derivative of the virtual
work equation at φk in the direction of u. It is worth pausing ¬rst to ask
what this means! To begin, a virtual velocity δv(φ(X)) is associated with
every particle labeled X in the body, and it is not allowed to alter during
the incremental change u(x) (see Figure 6.1). At a trial solution position
φk , δW (φk , δv) will have some value, probably not equal to zero as required
for equilibrium. The directional derivative DδW (φk , δv)[u] is simply the
change in δW due to φk changing to φk + u. Since δv remains constant
during this change, the directional derivative must represent the change in
the internal forces due to u (assuming that external forces are constant).
This is precisely what is needed in the Newton“Raphson procedure to adjust
the con¬guration φk in order to bring the internal forces into equilibrium
with the external forces. Hence, the directional derivative of the virtual
work equation will be the source of the tangent matrix.
The linearization of the equilibrium equation will be considered in terms
3
6.3 LAGRANGIAN LINEARIZED INTERNAL VIRTUAL WORK



of the internal and external virtual work components as,
DδW (φ, δv)[u] = DδWint (φ, δv)[u] ’ DδWext (φ, δv)[u] (6.3)
where,

δWint (φ, δv) = σ : δd dv (6.4a)
v

δWext (φ, δv) = f · δv dv + t · δv da (6.4b)
v ‚v



6.3 LAGRANGIAN LINEARIZED INTERNAL VIRTUAL
WORK

Although the eventual discretization of the linearized equilibrium equations
will be formulated only for the Eulerian case, it is nevertheless convenient
to perform the linearization with respect to the material description of the
equilibrium equations, simply because the initial elemental volume dV is
constant during the linearization. This will then be transformed by a push
forward operation to the spatial con¬guration. Recall from Equation (4.43)
that the internal virtual work can be expressed in a Lagrangian form as,

δWint (φ, δv) = S : δ E dV (6.5)
V
Using the product rule for directional derivatives and the de¬nition of the
material elasticity tensor, the directional derivative is obtained as,

DδWint (φ, δv)[u] = D(δ E : S)[u] dV
V

™ ™
= δ E : DS[u] dV + S : Dδ E[u] dV
V V

™ ™
= δ E : C : DE[u] dV + S : Dδ E[u] dV (6.6)
V V

where DE[u] is given by Equation (3.71). The term Dδ E[u] in the second

integral emerges from the fact that δ E, as given by Equation (3.97), is a
function not only of δv but also of the con¬guration φ as,
1 ™T ‚δv
™ ™ ™
δ E = (δ F F + F T δ F ); δF = = 0 δv (6.7)
2 ‚X
The directional derivative of this equation can be easily found recalling from
Equation (3.70) that DF [u] = 0 u to give,
1
™ T T
Dδ E[u] = [( 0 δv) +( 0 δv] (6.8)
0u 0 u)
2
4 LINEARIZED EQUILIBRIUM EQUATIONS



Observe that because the virtual velocities are not a function of the con-
¬guration the term 0 δv remains constant. Substituting Equation (6.8)
into (6.6) and noting the symmetry of S gives the material or Lagrangian
linearized principle of virtual work as,

™ T
DδWint (φ, δv)[u] = δ E : C : DE[u] dV + S : [( 0 δv] dV
0 u)
V V
(6.9)

Given the relationship between the directional and time derivatives as ex-

plained in Section 3.10.3, note that δ E can be expressed as DE[δv], which
enables Equation (6.9) to be written in a more obviously symmetric form
as,

T
DδWint (φ, δv)[u] = DE[δv] : C : DE[u] dV + S : [( 0 δv] dV
0 u)
V V
(6.10)


6.4 EULERIAN LINEARIZED INTERNAL VIRTUAL WORK

Equation (6.10) can perfectly well be used and may indeed be more appro-
priate in some cases for the development of the tangent sti¬ness matrix.
Nevertheless, much simpli¬cation can be gained by employing the equiva-
lent spatial alternative to give the same tangent matrix. To this end, the
materially based terms in Equation (6.10) must be expressed in terms of
spatially based quantities. These relationships are manifest in the following
pull back and push forward operations,

DE[u] = φ’1 [µ] = F T µF ; u + ( u)T
2µ = (6.11a)


DE[δv] = φ’1 [δd] = F T δd F ; δv + ( δv)T
2δd = (6.11b)


Jσ = φ— [S] = F SF T (6.11c)
3
Jc = φ— [C]; Jc = FiI FjJ FkK FlL CIJKL (6.11d)
ijkl
I,J,K,L=1
JdV = dv (6.11e)

With the help of these transformations it can be shown that the ¬rst inte-
grand in Equation (6.10) can be re-expressed in a spatial framework as,

DE[δv] : C : DE[u]dV = δd : c : µdv (6.12)

Additionally, the gradient with respect to the initial particle coordinates
appearing in the second integral in Equation (6.10) can be related to the
5
6.4 EULERIAN LINEARIZED INTERNAL VIRTUAL WORK



spatial gradient using the chain rule “ see Equation (3.69) “ to give,



= ( u)F (6.13a)
0u

0 δv = ( δv)F (6.13b)



Substituting these expressions into the second term of Equation (6.10) and
using Equation (6.11c) for the Cauchy and second Piola“Kirchho¬ stresses
reveals that the second integrand can be rewritten as,



T
= σ : [( u)T ( δv)]dv
S : [( 0 u) ( 0 δv)]dV (6.14)



Finally, Equation (6.10) can be rewritten using Equations (6.12) and (6.14)
to give the spatial or Eulerian linearized equilibrium equations as,




σ : [( u)T
DδWint (φ, δv)[u] = δd : c : µ dv + δv]dv (6.15)
v v




This equation will be the basis for the Eulerian or spatial evaluation of the
tangent matrix. Observe that the functional relationship between δd and
δv is identical to µ and u. This together with the symmetry of c and σ
implies that the terms u and δv can be interchanged in this equation without
altering the result. Consequently, the linearized virtual work equation is
symmetric in δv and u, that is,



DδWint (φ, δv)[u] = DδWint (φ, u)[δv] (6.16)



This symmetry will, upon discretization, yield a symmetric tangent sti¬ness
matrix.
6 LINEARIZED EQUILIBRIUM EQUATIONS




EXAMPLE 6.1: Proof of Equation (6.12)
In order to prove Equation (6.12), rewrite ¬rst Expressions (6.11a“b)
in indicial notation as,
DEIJ [δv] = FiI δdij FjJ ; DEKL [u] = FkK µkl FlL
i,j k,l

with the help of these expressions and Equation (6.11d), the left-hand
side of Equation (6.12) can be manipulated to give,
DE[δv] : C : DE[u] dV = DEIJ [δv]CIJKL DEKL [u] dV
I,J,K,L


= FiI δdij FjJ CIJKL
i,j
I,J,K,L


FkK µkl FlL J ’1 dv

k,l


FiI FjJ FkK FlL CIJKL J ’1 µkl dv
= δdij
I,J,K,L
i,j,k,l

= δdijc ijkl µkl dv
i,j,k,l

= δd : c : µ dv




6.5 LINEARIZED EXTERNAL VIRTUAL WORK

The external virtual work has contributions from body forces f and surface
tractions t. These two cases will now be considered separately.


6.5.1 BODY FORCES
The most common example of a body force is self-weight or gravity loading,
in which case f = ρg, where ρ is the current density and g is the acceler-
ation due to gravity. By a simple pull back of the body force component
in Equation (6.4b) it is easy to show that in this simple case the loading
is not deformation-dependent and therefore the corresponding directional
derivative vanishes. Recall for this purpose Equation (3.59) as ρ = ρ0 /J,
which when substituted in the ¬rst term of the external virtual work Equa-
7
6.5 LINEARIZED EXTERNAL VIRTUAL WORK



x3

·
p
ν ‚x/‚·
·
x(»,·)
n
dl

‚A »
»

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