The crack tip opening displacement (CTOD) of a crack at the edge of a three-point bending specimen is shown below:
where CTODm 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.