Product: ABAQUS/Standard
The stress intensity factors 
, 
, and 
play an important role in linear elastic fracture mechanics. They characterize the influence of load or deformation on the magnitude of the crack-tip stress and strain fields and measure the propensity for crack propagation or the crack driving forces. Furthermore, the stress intensity can be related to the energy release rate (the J-integral) for a linear elastic material through
and 
is called the pre-logarithmic energy factor matrix (Shih and Asaro, 1988; Barnett and Asaro, 1972; Gao, Abbudi, and Barnett, 1991; Suo, 1990). For homogeneous, isotropic materials is diagonal and the above equation simplifies to
for plane stress and 
for plane strain, axisymmetry, and three dimensions. For an interfacial crack between two dissimilar isotropic materials with Young's moduli 
and 
, Poisson's ratios 
and 
, and shear moduli 
and 
,
for plane strain, axisymmetry, and three dimensions; and 
for plane stress. Unlike their analogues in a homogeneous material, and are no longer the pure Mode I and Mode II stress intensity factors for an interfacial crack. They are simply the real and imaginary parts of a complex stress intensity factor, whose physical meaning can be understood from the interface traction expressions: 
are polar coordinates centered at the crack tip. The bimaterial constant 
is defined as 
In this section we describe an interaction integral method (Shih and Asaro, 1988) to extract the individual stress intensity factors for a crack under mixed-mode loading. The method is applicable to cracks in isotropic and anisotropic linear materials.
In general, the J-integral for a given problem can be written as
correspond to 
when indicating the components of B. We define the J-integral for an auxiliary, pure Mode I, crack-tip field with stress intensity factor 
as 
Since the terms not involving or 
in 
and J are equal, the interaction integral can be defined as
If the calculations are repeated for Mode 
and Mode 
, a linear system of equations results:
are assigned unit values, the solution of the above equations leads to
The calculation of this integral is discussed next.Based on the definition of the J-integral, the interaction integrals 
can be expressed as
given as
represents three auxiliary pure Mode I, Mode II, and Mode III crack-tip fields for 
, respectively. 
is a contour that lies in the normal plane at position s along the crack front, beginning on the bottom crack surface and ending on the top surface (see Figure 2.16.2–1). The limit 
indicates that shrinks onto the crack tip.Figure 2.16.2–1 Definition of local orthogonal Cartesian coordinates at the point s on the crack front; the crack is in the 
–
plane.
Following the domain integral procedure used in ABAQUS/Standard for calculating the J-integral, we define an interaction integral for a virtual crack advance 
:
is a surface element on a vanishingly small tubular surface enclosing the crack tip (i.e., 
); 
is the outward normal to ; and 
is the local direction of virtual crack propagation. The integral 
can be calculated by the same domain integral method as that used for calculating the J-integral.To obtain at each node set P along the crack front line, 
is discretized with the same interpolation functions as those used in the finite elements along the crack front:
at the node set P and all other 
are zero. The result is substituted into the expression for 
. Finally, the interaction integral value at each node set P along the crack front can be calculated as