»2

(2,3) 1 2

(0,3)

(10,3)

3 »1

time = 0

»2

1 2

X1 , x1

(0,0) (4,0)

»1

(continued)

5

7.2 DISCRETIZED KINEMATICS

EXAMPLE 7.1 (cont.)

The initial X and current x nodal coordinates are,

X1,1 = 0; X2,1 = 4; X3,1 = 0;

X1,2 = 0; X2,2 = 0; X3,2 = 3;

x1,1 = 2; x2,1 = 10; x3,1 = 10

x1,2 = 3; x2,2 = 3; x3,2 = 9

The shape functions and related derivatives are,

N1 = 1 ’ ξ1 ’ ξ2

‚N1 ‚N2 ‚N3

’1 1 0

; = ; = ; =

N2 = ξ1

’1 0 1

‚ξ ‚ξ ‚ξ

N3 = ξ2

Equations (7.1) and (7.6b) yield the initial position derivatives with

respect to the nondimensional coordinates as,

’T

X1 = 4ξ1 ‚X ‚X 130

40

; = ; =

03 12 0 4

‚ξ ‚ξ

X2 = 3ξ2

from which the derivatives of the shape functions with respect to the

material coordinate system are found as,

‚N1 130 13

’1

= =’ ;

12 0 4 ’1 12 4

‚X

‚N2 13 ‚N3 10

= ; =

12 0 12 4

‚X ‚X

A similar set of manipulations using Equations (7.2) and (7.11) yields

the derivatives of the shape functions with respect to the spatial coor-

dinate system as,

‚N1 1 13

30 ’1

= =’ ;

24 ’4 4 ’1 24 0

‚x

‚N2 1 ‚N3 10

3

= ; =

24 ’4 24 4

‚x ‚x

6 DISCRETIZATION AND SOLUTION

EXAMPLE 7.2: Discretized kinematics

Following Example 7.1 the scene is set for the calculation of the defor-

mation gradient F using Equation (7.7) to give,

‚N1 ‚N2 ‚N3 168

FiJ = x1,i + x2,i + x3,i ; i, J = 1, 2; F=

306

‚XJ ‚XJ ‚XJ

Assuming plane strain deformation, the right and left Cauchy“Green

tensors can be obtained from Equation (7.9) as,

® ®

36 48 0 100 48 0

1 1

C = F T F = ° 48 100 0 » ; b = F F T = ° 48 36 0 »

9 9

0 09 0 09

Finally, the Jacobian J is found as,

«®

680

1

J = det F = det ° 0 6 0 » = 4

3

003

7.3 DISCRETIZED EQUILIBRIUM EQUATIONS

7.3.1 GENERAL DERIVATION

In order to obtain the discretized spatial equilibrium equations, recall the

spatial virtual work Equation (4.27) given as the total virtual work done by

the residual force r as,

δW (φ, δv) = σ : δd dv ’ f · δv dv ’ t · δv da (7.12)

v v ‚v

At this stage, it is easier to consider the contribution to δW (φ, δv) caused

by a single virtual nodal velocity δv a occurring at a typical node a of ele-

ment (e). Introducing the interpolation for δv and δd given by Equations

(7.3) and (7.10) gives,

δW (e) (φ, Na δv a ) = σ : (δv a — Na )dv

(e)

v

’ f · (Na δv a ) dv ’ t · (Na δv a ) da (7.13)

(e) (e)

v ‚v

where the symmetry of σ has been used to concatenate the internal energy

term. Observing that the virtual nodal velocities are independent of the

7

7.3 DISCRETIZED EQUILIBRIUM EQUATIONS

integration and Equation (2.51b); that is, σ : (u — v) = u · σv for any

vectors u, v, enables the summation to be rearranged to give,

δW (e) (φ, Na δv a ) = δv a · σ Na dv ’ Na f dv ’ Na t da

(e) (e) (e)

v v ‚v

(7.14)

The virtual work per element (e) per node a can, alternatively, be expressed

(e) (e)

in terms of the internal and external equivalent nodal forces T a and F a

as,

δW (e) (φ, Na δv a ) = δv a · (T (e) ’ F (e) ) (7.15a)

a a

where,

3

‚Na

(e)

T (e) = σ Na dv; Ta,i = σij dv

a

‚xj

v (e) v (e)

j=1

(e)

Fa = Na f dv + Na t da

(e) (e)

v ‚v

In this equation the Cauchy stress σ is found from the appropriate consti-

tutive equation given in Chapter 5, which will involve the calculation of the

left Cauchy“Green tensor b = F F T ; for example, see Equations (5.29) or

(5.51) and Example 7.3.

EXAMPLE 7.3: Equivalent nodal forces T a

Building on the previous example, the calculation of the equivalent

nodal forces is now demonstrated. A compressible neo-Hookean mate-

rial will be consider for which µ = 3 and » = 2, which, using Equa-

tion (5.29), yields the Cauchy stresses (rounded for convenience) for

this plane strain case as,

® ®

σ11 σ12 0 84 0

µ »

σ = ° σ21 σ22 0 » = (b ’ I) + (ln J)I ≈ ° 4 3 0»

J J

0 0 σ33 0 0 0.8

8 DISCRETIZATION AND SOLUTION

EXAMPLE 7.3 (cont.)

From Equation (7.15b) the equivalent nodal internal forces are,

‚Na ‚Na

Ta,i = σi1 + σi2 dv;

‚x1 ‚x2

(e)

v

a = 1, 2, 3 T1,1 = ’24t T2,1 = 8t T3,1 = 16t

; ; ;

i = 1, 2 T1,2 = ’12t T2,2 = 0 T3,2 = 12t

where t is the element thickness. Clearly these forces are in equilibrium.

The contribution to δW (φ, Na δv a ) from all elements e (1 to ma ) con-

taining node a (e a) is,

ma

δW (e) (φ, Na δv a ) = δv a · (T a ’ F a )

δW (φ, Na δv a ) = (7.16a)

e=1

ea

where the assembled equivalent nodal forces are,

ma ma

(e)

F (e)

Ta = Ta ; Fa = (7.16b,c)

a

e=1 e=1

ea ea

Finally the contribution to δW (φ, δv) from all nodes N in the ¬nite element

mesh is,

N N

δW (φ, δv) = δW (φ, Na δv a ) = δv a · (T a ’ F a ) = 0 (7.17)

a=1 a=1

Because the virtual work equation must be satis¬ed for any arbitrary virtual

nodal velocities, the discretized equilibrium equations, in terms of the nodal

residual force Ra , emerge as,

Ra = T a ’ F a = 0 (7.18)

Consequently the equivalent internal nodal forces are in equilibrium with

the equivalent external forces at each node a = 1, 2, . . . , N .

For convenience, all the nodal equivalent forces are assembled into single

arrays to de¬ne the complete internal and external forces T and F respec-

tively, as well as the complete residual R, as,

T1 F1 R1

® ® ®

T F R

2 2 2

T = . ; F = . ; R= . (7.19)

°.» °.» °.»

. . .

Tn Fn Rn

9

7.3 DISCRETIZED EQUILIBRIUM EQUATIONS

These de¬nitions enable the discretized virtual work Equation (7.17) to be

rewritten as,

δW (φ, δv) = δvT R = δvT (T ’ F) = 0 (7.20)

where the complete virtual velocity vector δvT = [δv T , δv T , . . . , δv T ].

1 2 n

Finally, recalling that the internal equivalent forces are nonlinear func-

tions of the current nodal positions xa and de¬ning a complete vector of

unknowns x as the array containing all nodal position as,

®

x1

x

2

x= . (7.21)

°.» .

xn