Fracture Toughness

Plastic Zone Size

Materials develop plastic strains as the yield stress is exceeded in the region near the crack tip (see Fig. 1). The amount of plastic deformation is restricted by the surrounding material, which remains elastic. The size of this plastic zone is dependent on the stress conditions of the body.

Figure 1. Yielding near crack tip.

Plane stress and plane strain conditions. In a thin body, the stress through the thickness (s_z) cannot vary appreciably due to the thin section. Because there can be no stresses normal to a free surface, s_z = 0 throughout the section and a biaxial state of stress results. This is termed a plane stress condition (see Fig. 2).

Figure 2. Plane stress and plane strain conditions

In a thick body, the material is constrained in the z direction due to the thickness of the cross section and e_z = 0, resulting in a plane strain condition. Due to Poisson`s effect, a stress, s_z, is developed in the z direction. Maximum constraint conditions exist in the plane strain condition, and consequently the plastic zone size is smaller than that developed under plane stress conditions.

Monotonic plastic zone size. The plastic zone sizes under monotonic loading have been estimated to be

	plane stress	(1)
	plane strain

where r is defined as shown in Fig. 3.

Figure 3. Monotonic plastic zone size

Cyclic plastic zone size. The reversed or cyclic plastic zone size is four times smaller than the comparable monotonic value. As the nominal tensile load is reduced, the plastic region near the crack tip is put into compression by the surrounding elastic body. As shown in Fig. 4, the change in stress at the crack tip due to the reversed loading is twice the value of the yield stress.

Equation 2 become

	plane stress	(2)
	plane strain

The cyclic plastic zone size is smaller than the monotonic and more characteristic of a plane strain state even in thin plates. Thus LEFM concepts can often be used in the analysis of fatigue crack growth problems even in materials that exhibit considerable amounts of ductility. The basic assumption that the plastic zone size is small in relationship to the crack and the cracked body usually remains valid.

Figure 4. Reversed plastic zone size

Fracture Toughness

As the stress intensity factor reaches a critical value (K_c), unstable fracture occurs. This critical value of the stress intensity factor is known as the fracture toughness of the material. The fracture toughness can be considered the limiting value of stress intensity just as the yield stress might be considered the limiting value of applied stress.

The fracture toughness varies with specimen thickness until limiting conditions (maximum constraint) are reached. Recall that maximum constraint conditions occur in the plane strain state.

The plane strain fracture toughness, K_Ic is independent on specimen geometry and metallurgical factors. ASTM Designation E-399, Standard Method of Test for Plane Strain Fracture Toughness of Metallic Materials, sets forth accepted procedures for determining this value.

It is often difficult to perform a valid test for K_Ic. For example, a valid test using a thin plate of high toughness material often cannot be performed. Rather the value, K_c at the given conditions is obtained.

The fracture toughness depends on both temperature and the specimen thickness. The following example shows the importance of the fracture toughness in designing against unstable fracture.