4.5 Tangent Stiffness Matrix at Zero Deformation

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The tangent stiffness matrix at zero deformation can be obtained by substituting F = I in

the expression for _KLMN. The expressions in the previous section take the form

I1 = 3 = I 1 (4.233)

I2 = 3 = I 2 (4.234)

I3 = 1 (4.235)

λ1 = λ2 = λ3 = λ1 = λ2 = λ3 = 1. (4.236)

The derivatives of the invariants take the value

∂I1

∂CKL

= GKL (4.237)

∂I2

∂CKL

= 2GKL (4.238)

∂I3

∂CKL

= GKL (4.239)

∂2I1

∂CKL∂CMN

= 0 (4.240)

∂2I2

∂CKL∂CMN

= GKLGMN − 12

(GKMGLN + GKNGLM) (4.241)

∂2I3

∂CKL∂CMN

= ∂2I2

∂CKL∂CMN

(4.242)

∂I 1

∂CKL

= 0 (4.243)

∂I 2

∂CKL

= 0 (4.244)

∂Jel

∂CKL

= 12

GKL (4.245)

∂2I 1

∂CKL∂CMN

= −13

GKLGMN + 12

(GKMGLN + GKNGLM) (4.246)

∂2I 2

∂CKL∂CMN

= ∂2I 1

∂CKL∂CMN

(4.247)

∂2J el

∂CKL∂CMN

= 14

GKLGMN − 14

(GKMGLN + GKNGLM). (4.248)

210 HYPERELASTIC MATERIALS

Finally, the derivatives of the principal stretches satisfy for F = I :

3

_

j=1

λj,C = 12

G_. (4.249)

3

_

j=1

λj,CC = −14

I

G_ (4.250)

3

_

j=1

λj,C ⊗ λj,C = 14

I

G_ (4.251)

3

_

j=1

λj,C = 0 (4.252)

3

_

j=1

λj,CC = − 1

12G_ ⊗ G_ + 14

I

G_ (4.253)

3

_

j=1

λj,C ⊗ λj,C = − 1

12G_ ⊗ G_ + 14

I

G_ . (4.254)

Comparison with Equation (1.436) reveals that, in the initial configuration, an equivalent λ

and μ can be defined as the coefficient of the terms GKLGMN and GKMGLN + GKNGLM

respectively.

4.5.1 Polynomial form

Substitution of the above expressions into Equation (4.123) yields

_KLMN =       −43

(B10 + B01) + 2

D1

 

GKLGMN + 2(B10 + B01)(GKMGLN + GKNGLM).

(4.255)

Hence,

λeq = −43

(B10 + B01) + 2

D1

(4.256)

μeq = 2(B10 + B01). (4.257)

Frequently, an equivalent bulk modulus Keq is defined, satisfying (cf Equation (1.454))

Keq = λeq + 23

μeq. (4.258)

Hence,

Keq = 2

D1

. (4.259)

HYPERELASTIC MATERIALS 211

4.5.2 Arruda–Boyce form

Equation (4.129) yields

_KLMN =       −4μ

3

_12

+ 3

10λ2

m

+ 297

1050λ4

m

+ 2052

7000λ6

m

+ 210 195

673 750λ8

m

_ − 2

D

 

GKLGMN

+ 2μ

_12

+ 3

10λ2

m

+ 297

1050λ4

m

+ 2052

7000λ6

m

+ 210 195

673 750λ8

m

_ ·

(GKMGLN + GKNGLM) (4.260)

Hence,

μeq = 2μ

_12

+ 3

10λ2

m

+ 297

1050λ4

m

+ 2052

7000λ6

m

+ 210 195

673 750λ8

m

_ (4.261)

Keq = 2

D

. (4.262)

4.5.3 Ogden form

Substitution into Equation (4.218) leads to

_ =

N

_

i=1

8μi

αi

           

αi

_− 1

12

G_ ⊗ G_ + 1

4

I

G_

_ + 2

D1

G_ ⊗ G_ (4.263)

from which

μeq =

N

_

i=1

μi (4.264)

Keq = 2

D1

. (4.265)

4.5.4 Elastomeric foam behavior

Equation (4.232) yields

_ = 8

N

_

i=1

μi

αi

           

(αi − 1)

1

4

I

G_ − 1

4

I

G_ + (αiβi + 1)

4

G_ ⊗ G_ − 1

4

G_ ⊗ G_ + 1

2

I

G_

 

=

N

_

i=1

μi &2I

G_ + 2βiG_ ⊗ G_' . (4.266)

This leads to the following expressions for the equivalent constants:

μeq =

N

_

i=1

μi (4.267)

212 HYPERELASTIC MATERIALS

λeq = 2

N

_

i=1

βiμi (4.268)

Keq =

N

_

i=1

2μi _βi + 13

_ . (4.269)

4.5.5 Closure

For the polynomial model, the Arruda–Boyce model and the Ogden model, the equivalent

bulk modulus is related to the coefficient D1 (polynomial model, Ogden model) or D

(Arruda–Boyce model). Incompressible behavior corresponds to D1 = D = 0. To avoid

the resulting singularities in the material law, the CalculiXcode (CalculiX 2003) replaces

this behavior by a nearly incompressible behavior corresponding to an equivalent Poisson

coefficient μeq = 0.475 at zero deformation. One finds (Equation (1.455)),

Keq =

2μeq(1 + νeq)

3(1 − 2νeq)

. (4.270)

Accordingly, for a polynomial material,

D1 =

3(1 − 2νeq)

μeq(1 + νeq)

= 0.1017

μeq

. (4.271)

If N >1 in the polynomial model, the following numerical relationship (disregarding the

dimensions) is proposed:

Di =      3(1 − 2νeq)

μeq(1 + νeq)

i

= _0.1017

μeq

_i

. (4.272)

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