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 (sz) cannot vary appreciably due
to the thin section. Because there can be no stresses normal to a free surface,
sz = 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 ez = 0, resulting in
a plane strain condition. Due to Poisson`s effect, a stress,
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
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
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
As the stress intensity factor reaches a critical value (Kc), 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, KIc 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 KIc. For example, a
valid test using a thin plate of high toughness material often cannot be performed.
Rather the value, Kc 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.