Definition of Crack Tip Opening Displacement

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 three-point 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 path-independent 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: