1. Show that the linearized internal virtual work can also be expressed as:

‚2Ψ

‚P

DδW (φ, δv)[u] = ( 0 δv) :A:( 0 u) dV ; A= =

‚F ‚F ‚F

V

where P is the ¬rst Piola“Kirchho¬ tensor.

2. Show that for the case of uniform pressure over an enclosed ¬xed bound-

ary, the external virtual work can be derived from an associated potential

21

6.6 VARIATIONAL METHODS AND INCOMPRESSIBILITY

as δWext (φ, δv) = DΠp (φ)[δv] where,

p

ext

1

Πp (φ) = px · n da

ext

3 a

Explain the physical signi¬cance of this integral.

3. Prove that for two-dimensional applications, Equation (6.23) becomes,

1 ‚u ‚δv

p

DδWext (φ, δv)[u] = pk · —δv + d·

—u

2 ‚· ‚·

L·

where k is a unit vector normal to the two-dimensional plane and · is a

parameter along the line L· where the pressure p is applied.

4. Prove that by using a di¬erent cyclic permutation than that used to

derive Equation (6.22), the following alternative form of Equation (6.23)

can be found for the case of an enclosed ¬xed boundary with uniform

surface pressure,

‚δv ‚u ‚δv ‚u

p

DδWext (φ, δv)[u] = px · dξd·

’

— —

‚· ‚ξ ‚ξ ‚·

Aξ

5. Prove that by assuming a constant pressure interpolation over the integra-

tion volume in Equation (6.41), a constant pressure technique equivalent

to the mean dilatation method is obtained.

6. A six-¬eld Hu-Washizu type of variational principle with independent

volumetric and deviatoric variables is given as,

¯ ˆ ¯ ¯

ΠHW (φ, J, F , p, P , γ) = Ψ (C) dV + U (J) dV + p(J ’ J) dV

V V V

+ P :( ’ F ) dV + γ P : F dV

0φ

V V

where C = F T F , J = det( 0 φ), and P denotes the deviatoric com-

ponent of the ¬rst Piola“Kirchho¬ stress tensor. Find the stationary

conditions with respect to each variable. Explain the need to introduce

the Lagrange multiplier γ. Derive the formulation that results from as-

suming that all the ¬elds except for the motion are constant over the

integration volume.

CHAPTER SEVEN

DISCRETIZATION AND SOLUTION

7.1 INTRODUCTION

The equilibrium equations and their corresponding linearizations have been

established in terms of a material or a spatial description. Either of these

descriptions can be used to derive the discretized equilibrium equations and

their corresponding tangent matrix. Irrespective of which con¬guration is

used, the resulting quantities will be identical. It is, however, generally

simpler to establish the discretized quantities in the spatial con¬guration.

Establishing the discretized equilibrium equations is relatively standard,

with the only additional complication being the calculation of the stresses,

which obviously depend upon nonlinear kinematic terms that are a function

of the deformation gradient. Deriving the coe¬cients of the tangent ma-

trix is somewhat more involved, requiring separate evaluation of constitu-

tive, initial stress, and external force components. The latter deformation-

dependant external force component is restricted to the case of enclosed

normal pressure. In order to deal with near incompressibility the mean

dilatation method is employed.

Having discretized the governing equations, the Newton“Raphson so-

lution technique is reintroduced together with line search and arc length

method enhancements.

7.2 DISCRETIZED KINEMATICS

The discretization is established in the initial con¬guration using isopara-

metric elements to interpolate the initial geometry in terms of the parti-

1

2 DISCRETIZATION AND SOLUTION

X 3, x 3 ·

ub »

δva

a b

(e)

`

a

b

(e)

time = t

X 1, x 1

X 2, x 2

time = 0

FIGURE 7.1 Discretization.

cles X a de¬ning the initial position of the element nodes as,

n

X= Na (ξ1 , ξ2 , ξ3 )X a (7.1)

a=1

where Na (ξ1 , ξ2 , ξ3 ) are the standard shape functions and n denotes the

number of nodes. It should be emphasized that during the motion, nodes

and elements are permanently attached to the material particles with which

they were initially associated. Consequently, the subsequent motion is fully

described in terms of the current position xa (t) of the nodal particles as (see

Figure 7.1),

n

x= Na xa (t) (7.2)

a=1

Di¬erentiating Equation (7.2) with respect to time gives the real or

virtual velocity interpolation as,

n n

v= Na v a ; δv = Na δv a (7.3)

a=1 a=1

Similarly, restricting the motion brought about by an arbitrary increment u

to be consistent with Equation (7.2) implies that the displacement u is also

3

7.2 DISCRETIZED KINEMATICS

interpolated as,

n

u= Na ua (7.4)

a=1

The fundamental deformation gradient tensor F is interpolated over an

element by di¬erentiating Equation (7.2) with respect to the initial coordi-

nates to give, after using Equation (2.135a),

n

F= 0 Na (7.5)

xa —

a=1

where 0 Na = ‚Na /‚X can be related to ξ Na = ‚Na /‚ξ in the standard

manner by using the chain rule and Equation (7.1) to give,

n

’T

‚Na ‚X ‚Na ‚X

= ; = ξ Na (7.6a,b)

Xa —

‚X ‚ξ ‚ξ ‚ξ

a=1

Equations (7.5) and (7.6b) are su¬ciently fundamental to justify expansion

in detail in order to facilitate their eventual programming. To this e¬ect,

these equations are written in an explicit matrix form as,

®

F11 F12 F13 n

‚Na

F = ° F21 F22 F23 »; FiJ = xa,i (7.7)

‚XJ

a=1

F31 F32 F33

and,

®

‚X1 /‚ξ1 ‚X1 /‚ξ2 ‚X1 /‚ξ3 n

‚X ‚XI ‚Na

= ° ‚X2 /‚ξ1 ‚X2 /‚ξ3 »; = Xa,I

‚X2 /‚ξ2

‚ξ ‚ξ± ‚ξ±

a=1

‚X3 /‚ξ1 ‚X3 /‚ξ2 ‚X3 /‚ξ3

From Equation (7.5) further strain magnitudes such as the right and left

Cauchy“Green tensors C and b can be obtained as,

3

T

C=F F = (xa · xb ) 0 Na 0 Nb ; CIJ = FkI FkJ (7.9a,b)

—

a,b k=1

3

T

b = FF = ( 0 Na 0 Nb ) xa — xb ; bij = FiK FjK (7.9c,d)

·

K=1

a,b

The discretization of the real (or virtual rate of deformation) tensor and the

linear strain tensor can be obtained by introducing Equation (7.3) into the

de¬nition of d given in Equation (3.101) and Equation (7.4) into Equation

4 DISCRETIZATION AND SOLUTION

(6.11a) respectively to give,

n

1

d= (v a — Na + Na — v a ) (7.10a)

2

a=1

n

1

δd = (δv a — Na + Na — δv a ) (7.10b)

2

a=1

n

1

µ= (ua — Na + Na — ua ) (7.10c)

2

a=1

where, as in Equation (7.6), Na = ‚Na /‚x can be obtained from the

derivatives of the shape functions with respect to the isoparametric coordi-

nates as,

n n

’T

‚Na ‚x ‚Na ‚x ‚xi ‚Na

= ; = ξ Na ; = xa,i

xa —

‚x ‚ξ ‚ξ ‚ξ ‚ξ± ‚ξ±

a=1 a=1

(7.11a,b)

Although Equations (7.10a“c) will eventually be expressed in a standard

matrix form, if necessary the component tensor products can be expanded

in a manner entirely analogous to Equations (7.6“7).

EXAMPLE 7.1: Discretization

This simple example illustrates the discretization and subsequent calcu-

lation of key shape function derivatives. Because the initial and current

geometries comprise right-angled triangles, these are easily checked.

(10,9)

3

X2, x 2