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 »

»