Constitutive Modeling of the Mechanical Behavior of Trabecular Bone – Continuum Mechanical Approaches - Biomechanics of Hard Tissues - page 174

Table 6.8

Values of the stress Lode angle for basic tests.

Case Component

θ according to Eq. (6.58) θ as given in Figure 6.6

0 ◦ 0 ◦

Uniaxial tension σ I = σ or

0 ◦ 120 ◦ ( − 60 ◦ )

σ II = σ or

0 ◦ 240 ◦ (60 ◦ ) σ III = σ

60 ◦ 0 ◦ (180 ◦ ) Uniaxial compression σ I =− σ or

60 ◦ 60 ◦ (120 ◦ )

σ II =− σ −

or

60 ◦ 60 ◦ σ III =− σ

60 ◦ 60 ◦ Biaxial tension σ I = σ II = σ or

60 ◦ − 60 ◦ σ I = σ III = σ or

60 ◦ 180 ◦ σ II = σ III = σ

0 ◦ 240 ◦

Biaxial compression σ I = σ II =− σ or

0 ◦ 120 ◦

σ I = σ III =− σ or

0 ◦ 0 ◦

σ II = σ III =− σ

Triaxial tension σ I = σ II = σ III = σ − −

Triaxial compression σ I = σ II = σ III =− σ − −

30 ◦ − 30 ◦ Pur shear σ I = τ ; σ II =− τ

)
σ I = τ ; σ III =− τ

( σ xy = τ

30 ◦ 30 ◦

)
σ II = τ ; σ III =− τ

( σ xz = τ

30 ◦ 90 ◦

σ yz = τ

( )

shape along the hydrostatic axis, then all evaluated points can be drawn in a single
octahedral plane. However, if one expects a dependency, then only stress states with
the same hydrostatic stress are allowed to be represented in the same octahedral
plane where J 1 = const. holds. As a result, for example, uniaxial tensile ( J 1 = σ I )
and pure shear tests ( J 1 = 0) cannot be represented in the same octahedral

plane. In order to draw the shape of the yield surface for a pressure-sensitive
material, the following differing multiaxial stress states with J 1 = 0, for example,

σ 0), are a possibility to

obtain values in the same deviatoric plane. It should be noted here again that the
shape changes only its size along the hydrostatic axes but remains similar in the
mathematical sense. Typical Lode angles for basic experiments are summarized in
Table 6.8, where the evaluation of Eq. (6.58) and the graphical representation in
the octahedral plane are given.
For an isotropic material, the labels I, II, III attached to the principal coordi-
nate axes are arbitrary. It follows that the yield condition must have threefold
symmetry and it is only required to investigate the sector

=− σ II ( σ = 0), or σ =− 2 σ =− 2 σ III ( σ > 0 ∨ σ < I III I II I I

0 ◦ to 60 ◦ ,cf.
Figure 6.7. The other sectors follow directly from the symmetry. If the uniaxial
yield stress is in addition the same in tension and compression, a sixfold sym-
metry is obtained and only the sector

θ =

0 ◦ to 30 ◦ needs to be investigated, cf. θ =

Figure 6.7.