σ
πθ θ θ
y
= I
+
σ
πθ θ θ
x
=
For more information, see Stress-Intensity Factor Calculation (p. 363).
11.1.2.4. T-Stress
The asymptotic expansion of the stress field in the vicinity of the crack tip, expressed in the local polar
coordinate system described in Figure 11.2: Schematic of a Crack Tip (p. 342), is represented as:
ij
ij
ij
ij
( ) ( ) ( ) 1 i 1j 33 3 i 3 j1
( ) ^2
where the first singular terms of this eigen-expansion (the terms involving ) are the stress-intensity
factors, and the first non-singular term (T ) is the elastic T-stress.
T-stress is the stress acting parallel to the crack faces. It is tightly linked to the level of crack-tip stress
triaxiality, thus its sign and magnitude can substantially change the size and shape of the crack-tip
plastic zone [ 14 ]. Negative T-stress values decrease the level of crack-tip triaxiality (leading to larger
plastic zones), while positive values increase the level of triaxiality (leading to smaller plastic zones). A
higher crack-tip triaxiality promotes fracture because the input of external work is dissipated less by
the global plastic deformation and is thus available to augment local material degradation and damage
[ 15 ].
T-stress also plays an important role in the stability of straight crack paths submitted to Mode I loading
conditions. For a small amount of crack growth, cracks with T < 0 have been shown to be stable,
whereas cracks with T > 0 tend to deviate from their initial propagation plane [ 16 ].
For more information, see T-Stress Calculation (p. 370).
11.1.2.5. Material Force
Used primarily to analyze material defects such as dislocations, voids, interfaces and cracks, material
force (also known as configurational force) can be understood by considering the presence of an inclusion
in an elastic solid (matrix material), as shown in this figure:
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