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Definition of Crack Tip Opening Displacement
There are two common definitions of the crack tip opening displacement (CTOD):
1. The opening displacement of the original crack tip.
2. The displacement at the intersection of a 90° vertex with the crack flanks.
These two definitions are equivalent if the crack blunts in a semicircle.
CTOD in Specimen
The crack tip opening displacement (CTOD) of a crack at the edge of a threepoint bending specimen is shown below:
where CTOD_{m} is the measured crack tip opening displacement, usually near the edge of the specimen for ease of access, CTOD is the real crack tip opening displacement, a is the length of the crack, and b is the width of the rest of the specimen. Please note that the figure is for illustration purpose only and not to scale. From simple geometry of two similar triangles:
where is a dimensionless rotational factor used to locate the center of the hinge.
For simplicity, let's assume that the center of the hinge locates at the center of b, i.e., ~ 1/2. The CTOD then becomes
The above hinge model may not be accurate when the displacement is mostly elastic. A more accurate approach is to separate the CTOD into an elastic part and a plastic part:
where is the small scale yielding stress and m is a dimensionless constant that depends on the material properties and the stress states.
Relationship between J and CTOD
Consider a linear elastic body containing a crack, the J integral and the crack tip opening displacement (CTOD) have the following relationship
where and m are defined in the previous section. For plane stress and nonhardening materials, m = 1. Hence, for a through crack in an infinite plate subjected to a remote tensile stress (Mode I), the crack tip opening displacement is
Shih, C. F., 1981, took a step further and showed that a unique relationship exists between J and CTOD beyond the validity limits of LEFM. He introduced the 90° intercept definition of CTOD, as illustrated below.
The displacement field is
The CTOD is evaluated from u_{x} and u_{y} at r = r^{*} and :
Since
The CTOD becomes
Definition of J Integral
Consider a nonlinear elastic body containing a crack,
the J integral is defined as
where is the strain energy density, is the traction vector, is an arbitrary contour around the tip of the crack, n is the unit vector normal to ; , , and u are the stress, strain, and displacement field, respectively.
Rice, J. R., 1968, showed that the J integral is a pathindependent line integral and it represents the strain energy release rate of nonlinear elastic materials
where is the potential energy, the strain energy U stored in the body minus the work W done by external forces and A is the crack area.
The dimension of J is
J vs. and K
For linear elastic materials, the J integral J is in fact the strain energy release rate, , and both are related to the stress intensity factor K in the following fashion:
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