assembled as,

R(x) = T(x) ’ F(x) = 0 (7.22)

These equations represent the ¬nite element discretization of the pointwise

di¬erential equilibrium Equation (4.16).

7.3.2 DERIVATION IN MATRIX NOTATION

The discretized equilibrium equations will now be recast in the more famil-

iar matrix-vector notation. To achieve this requires a reinterpretation of

the symmetric stress tensor as a vector comprising six independent compo-

nents as,

σ = [σ11 , σ22 , σ33 , σ12 , σ13 , σ23 ]T (7.23)

Similarly, the symmetric rate of deformation tensor can be re-established in

a corresponding manner as,

d = [d11 , d22 , d33 , 2d12 , 2d13 , 2d23 ]T (7.24)

where the o¬-diagonal terms have been doubled to ensure that the product

dT σ gives the correct internal energy as,

dT σ dv

σ : d dv = (7.25)

v v

The rate of deformation vector d can be expressed in terms of the usual

10 DISCRETIZATION AND SOLUTION

B matrix and the nodal velocities as,

® ‚Na

0 0

‚x1

‚Na

0 0

‚x2

‚Na

0 0

n

‚x3

d= Ba v a ; Ba = ‚Na (7.26)

‚Na

0

‚x2 ‚x1

a=1

‚Na ‚Na

0

° ‚x3 ‚x1 »

‚Na ‚Na

0 ‚x3 ‚x2

Introducing Equation (7.26) into Equation (7.25) for the internal energy

enables the discretized virtual work Equation (7.13) to be rewritten as,

(Ba δv a )T σ dv ’

δW (φ, Na δv a ) = f · (Na δv a ) dv

(e) (e)

v v

’ t · (Na δv a ) da (7.27)

(e)

‚v

Following the derivation given in the previous section leads to an alternative

(e)

expression for the element equivalent nodal forces T a for node a as,

T (e) = BT σ dv (7.28)

a a

v (e)

Observe that because of the presence of zeros in the matrix Ba , Expres-

sion (7.15a) is computationally more e¬cient than Equation (7.28).

7.4 DISCRETIZATION OF THE LINEARIZED

EQUILIBRIUM EQUATIONS

Equation (7.22) represents a set of nonlinear equilibrium equations with

the current nodal positions as unknowns. The solution of these equations

is achieved using a Newton“Raphson iterative procedure that involves the

discretization of the linearized equilibrium equations given in Section 6.2.

For notational convenience the virtual work Equation (7.12) is split into

internal and external work components as,

δW (φ, δv) = δWint (φ, δv) ’ δWext (φ, δv) (7.29)

which can be linearized in the direction u to give,

DδW (φ, δv)[u] = DδWint (φ, δv)[u] ’ DδWext (φ, δv)[u] (7.30)

11

7.4 LINEARIZED EQUILIBRIUM EQUATIONS

where the linearization of the internal virtual work can be further subdivided

into constitutive and initial stress components as,

DδWint (φ, δv)[u] = DδWc (φ, δv)[u] + DδWσ (φ, δv)[u]

σ : [( u)T

= δd : c : µ dv + δv] dv (7.31)

v v

Before continuing with the discretization of the linearized equilibrium Equa-

tion (7.30), it is worth reiterating the general discussion of Section 6.2 to

inquire in more detail why this is likely to yield a tangent sti¬ness matrix.

(e) (e)

Recall that Equation (7.15), that is, δW (e) (φ, Na δv a ) = δv a · (T a ’ F a ),

(e)

essentially expresses the contribution of the nodal equivalent forces T a

(e) (e)

and F a to the overall equilibrium of node a. Observing that F a may be

position-dependent, linearization of Equation (7.15) in the direction Nb ub ,

with Na δv a remaining constant, expresses the change in the nodal equiv-

(e) (e)

alent forces T a and F a , at node a, due to a change ub in the current

position of node b as,

DδW (e) (φ, Na δv a )[Nb ub ] = D(δv a · (T (e) ’ F (e) ))[Nb ub ]

a a

= δv a · D(T (e) ’ F (e) )[Nb ub ]

a a

(e)

= δv a · K ab ub (7.32)

The relationship between changes in forces at node a due to changes in

the current position of node b is furnished by the tangent sti¬ness matrix

(e)

K ab , which is clearly seen to derive from the linearization of the virtual

work equation. In physical terms the tangent sti¬ness provides the Newton“

Raphson procedure with the operator that adjusts current nodal positions so

that the deformation-dependent equivalent nodal forces tend toward being

in equilibrium with the external equivalent nodal forces.

7.4.1 CONSTITUTIVE COMPONENT “ INDICIAL FORM

The constitutive contribution to the linearized virtual work Equation (7.31)

for element (e) linking nodes a and b is,

DδWc(e) (φ, Na δv a )[Nb ub ]

1 1

= (δv a — Na + Na — δv a ) : c : (ub — Nb + Nb — ub ) dv

v (e) 2 2

(7.33)

In order to make progress it is necessary to temporarily resort to indicial

12 DISCRETIZATION AND SOLUTION

notation, which enables the above equation to be rewritten as,

DδWc(e) (φ, Na δv a )[Nb ub ]

3

1 ‚Na ‚Na 1 ‚Nb ‚Nb

= δva,i + δva,j ub,k + ub,l dv

c ijkl

v (e) 2 ‚xj ‚xi 2 ‚xl ‚xk

i,j,k,l=1

3

‚Na 1 ‚Nb

= δva,i (c ijkl +c ijlk +c jikl +c jilk ) dv ub,k

‚xj 4 ‚xl

v (e)

i,j,k,l=1

(e)

= δv a · K c,ab ub (7.34)

where the constitutive component of the tangent matrix relating node a to

node b in element (e) is,

3

‚Na sym ‚Nb

[K c,ab ]ij = dv; i, j = 1, 2, 3 (7.35)

c

‚xk ikjl ‚xl

v (e) k,l=1

in which the symmetrized constitutive tensor is,

1

sym

c ikjl = (c ikjl +c iklj +c kijl +c kilj ) (7.36)

4

EXAMPLE 7.4: Constitutive component of tangent matrix

[K c, ab ]

The previous example is revisited in order to illustrate the calcula-

tion of the tangent matrix component connecting nodes two to three.

Omitting zero derivative terms the summation given by Equation (7.35)

yields,

1 1 1 1

sym sym

[K c,23 ]11 = ’ (24t)

c 1112 c 1212

8 6 6 6

1 1 1 1

sym sym

[K c,23 ]12 = ’ (24t)

c 1122 c 1222

8 6 6 6

1 1 1 1

sym sym

[K c,23 ]21 = ’ (24t)

c 2112 c 2212

8 6 6 6

1 1 1 1

sym sym

[K c,23 ]22 = ’ (24t)

c 2122 c 2222

8 6 6 6

(continued)

13

7.4 LINEARIZED EQUILIBRIUM EQUATIONS

EXAMPLE 7.4 (cont.)

where t is the thickness of the element. Substituting for c sym from

ijkl

Equations (7.36) and (5.37“8) yields the sti¬ness coe¬cients as,

2 1 1

[K c,23 ]11 = ’ » t; [K c,23 ]12 = µ t; [K c,23 ]21 = µ t;

3 2 2

2

[K c,23 ]22 = ’ (» + 2µ )t

3

where » = »/J and µ = (µ ’ » ln J)/J.

7.4.2 CONSTITUTIVE COMPONENT “ MATRIX FORM

The constitutive contribution to the linearized virtual work Equation (7.31)

for element (e) can alternatively be expressed in matrix notation by de¬ning

the small strain vector µ in a similar manner to Equation (7.26) for d as,

n

µ = [µ11 , µ22 , µ33 , 2µ12 , 2µ13 , 2µ23 ]T ; µ= (7.37a,b)

B a ua

a=1

The constitutive component of the linearized internal virtual work “ see

Equation (7.31) “ can now be rewritten in matrix-vector notation as,

δdT Dµ dv

DδWc (φ, δv)[u] = δd : C : µ dv = (7.38)

V v