Constitutive Modeling of the Mechanical Behavior of Trabecular Bone – Continuum Mechanical Approaches - Biomechanics of Hard Tissues - page 160

Table 6.1 Definition of principal and basic invariants.

Stress tensor

ij

σ ij σ s ij

Principal invariants

I 1 , I 2 , I 3 I 1 , I 2 , I 3 I 1 , I 2 , I 3

I 1 = σ ii I 1 =

I 1 = σ ii 0

1
2 ( σ ii σ jj − σ ij σ ji )

I 2 =

1

3 ( σ ii ) 2

I 2 =−

1
2 s ij s ji

I 2 =

3 2 σ ii σ jj σ kk + σ ij σ jk σ ki

1 1

27 (

1
3 s ij s jk s ki

I 3 = σ ii ) 3 I 3 = I 3 =

2 σ ij σ ji σ kk

3

−

I 1 I 2 , I 3 ⇒ I 1 , I 2 , I 3 ⇒ ⇒

Basic invariants

J 1 , J 2 , J 3 J 1 , J 2 , J 3 J 1 , J 2 , J 3

J 1 = σ ii J 1 = 0 J 1 = σ ii

1
2 σ ij σ ji

J 2 =

1
6 (

J 2 =

1
2 s ij s ji

J 2 = σ ii ) 2

1
3 σ ij σ jk σ ki

J 3 =

1

9 ( σ ii ) 3

J 3 =

1
3 s ij s jk s ki

J 3 =

J 1 J 2 , J 3 ⇒ J 1 , J 2 , J 3 ⇒ ⇒

Basic invariants in terms of σ ij . Table 6.2

Invariants General stress values

Stress tensor

J 1 σ xx + σ yy + σ zz

2

zz

1

xx + σ yy + σ xy + σ xz + σ yz

J 2 σ + σ

3 σ

1 xx + σ yy + σ zz + 3 σ xy σ xx +

J 3

xy σ yy + 3 σ xz σ xx + 3 σ xz σ zz +

+ 3 σ

yz σ zz + 6 σ xy σ xz σ yz
Spherical tensor

yz σ yy + 3 σ

+ 3 σ

J 1 σ xx + σ yy + σ zz

6 σ xx + σ yy + σ zz 2

J 2

1

9 σ xx + σ yy + σ zz 3
Stress deviator tensor

J 3

1

J 1

0

6 (

J 2

1

σ xx − σ yy ) 2 σ yy − σ zz ) 2

+ (

σ zz − σ xx ) 2 + σ

xy + σ yz + σ zx

+ (

J 3 s xx s yy s zz + 2 σ xy σ yz σ zx

yz − s yy σ zx − s zz σ xy

− s xx σ

1
3 (2 σ xx − σ yy − σ zz )

with s xx =

1
3 (

s yy = − σ xx + 2 σ yy − σ zz )

1
3 ( − σ xx − σ yy + 2 σ zz )

s zz =