Constitutive Modeling of the Mechanical Behavior of Trabecular Bone – Continuum Mechanical Approaches - Biomechanics of Hard Tissues - page 159

for the definition of the three scalar principal stress invariants I 1 , I 2 and I 3 .
The three roots

λ i of Eq. (6.7) are the principal stresses: σ = max( λ 1 , λ 2 , λ 3 ), I

σ III ). It may be noted here that the

principal stress state is the state that has no shear components, that is, only the
stress components are the principal stresses σ i ( i = I, II, III) on the main diagonal
of the stress tensor. Another interpretation of the principal invariants is given by
• I 1 =trace 4) of σ ij :
I 1

= min( λ 1 , λ 2 , λ 3 ), and σ = ( I 1 − σ − σ III II I

= σ + σ + σ (6.8) xx yy zz

• I 2 = sum of the two-row main subdeterminants of σ ij :

+ +

σ σ σ σ σ σ xx xy yy yz xx xz

I 2 = (6.9)

σ σ σ σ σ σ

xy yy yz zz xz zz

• I 3 = determinant of σ ij :

σ σ σ xx xy xz

I 3 = σ σ σ (6.10)

xy yy yz

σ σ σ xz yz zz

In addition to these principal invariants, there is also often another set of invariants
used. This set is included in the principal invariants I i and called basic invariants J i :

=

J 1 I 1 (6.11)

1
2 I 1 −

J 2 = I 2 (6.12)

1

3 I 1 −

J 3 = I 1 I 2 + I 3 (6.13)

The definition of both principal and basic invariants is summarized and compared
in Table 6.1.
ItcanbeseenfromTable6.1thatthesphericaltensor

o
ij is completely
characterized by its first invariant, because the second and third invariants are
its powers. The stress deviator tensor s ij is completely characterized by its second
and third invariants. Therefore, the physical contents of the stress state σ ij can be
completely described either by the three invariants or, if we use the decomposition
in its spherical and deviatoric parts, by the first invariant of the spherical tensor and
the second and third invariants of the stress deviator tensor. It should be noted here
that this statement is valid for both, that is, the principal and basic invariants. In
the following, we will only use the basic invariants, and thus, the physical content
of stress state will be described by the following set of invariants:

σ

J 1 , J 2 , J 3 σ → (6.14)

ij

Table 6.2 summarizes formulae based on the general stress components σ ij for the

practical calculation of the basic stress invariants.
Finally, it should be mentioned here that it is also possible to derive the same
sets of invariants from the strain tensor ε ij .

4) The trace of a tensor is the sum of the
diagonal terms.