Biomedical Engineering Reference
In-Depth Information
3J
′
2
Pure shear
Uniaxial compression
Uniaxial tension
Biaxial compression
Biaxial tension
Triaxial compression
Triaxial tension
J
1
Not defined
Figure 6.9
Schematic representation of basic tests in the
J
1
-
3
J
2
invariant space.
In the case that a material has the same uniaxial yield stress in tension and
compression, the influence of the third invariant is often disregarded since the
maximum error is in the indicated range of the outer and inner bounds, cf. Eqs
(6.61) and (6.62). In such a case, the mathematical description can be performed
in a
J
1
-
3
J
2
invariant space as indicated in Figure 6.9.
In the
J
1
-
3
J
2
invariant space, basic material tests can be identified as lines
through the origin as indicated in Figure 6.9. Table 6.9 summarizes the linear
equations for these basic experiments. As one can see, a uniaxial tensile test is
represented, for example, by the bisecting line in the first quadrant of the Cartesian
J
1
-
3
J
2
coordinate system. Performing a uniaxial tensile test would mean to
''walk'' from the origin along this straight line (by monotonically increasing the
load) until a material limit, for example, initial yield or failure, is reached. This
point (indicated in Figure 6.9 by
◦
) makes part of the yield or limit surface and the
Table 6.9
Definition of basic tests in the
J
1
-
3
J
2
invariant space.
3
J
2
1
Case
J
Comment
Uniaxial tension (
σ
)
σ
σ
Slope: 1
Uniaxial compression (
−
σ
)
−
σ
σ
Slope:
−
1
Biaxial tension (
σ
)
2
σ
σ
Slope: 0
.
5
Biaxial compression (
−
σ
)
−
2
σ
σ
Slope:
−
0
.
5
Triaxial tension (
σ
)
3
σ
0
Horizondal axis
Triaxial compression (
−
σ
)
−
3
σ
0
Horizondal axis
√
3
τ
Pure shear (
τ
)
0
Vertical axis
