This module solves for the radius of the region of plastic strain near a crack tip.

Materials develop plastic strains as the yield strength is exceeded in the region near a crack tip. The amount of plastic deformation is restricted by the surrounding material which remains elastic. The size of the plastic zone is dependent on the stress condition of the body.

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 the free surface, Sz = 0 throughout the section and a biaxial state of stress results. This is termed the plane stress condition.

 In a thick body, the material is constrained in the z direction due to the thickness of the cross section and strain in the z direction equals zero. This results in the plane strain condition. Due to Poisson's effect, a stress, Sz, 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.

Linear elastic stress analysis of sharp cracks predicts infinite stresses at the crack tip. However, in real materials, stresses at the crack tip are finite because the crack tip radius must be finite. Inelastic material deformation, such as plasticity in metals and crazing in polymers, leads to further relaxation of crack tip stresses. The elastic stress analysis becomes increasingly inaccurate as the inelastic region at the crack tip becomes larger. The basic assumption of linear elastic fracture mechanics (LEFM) remains valid if the plastic zone size remains small in relation to the overall dimension of the crack (crack size) and the cracked body (thickness). The size of the crack tip yielding zone, termed the plastic zone, can be estimated. Simple corrections to LEFM are available when moderate crack tip yielding occurs (see the stress intensity factor module or the references). When the stress or displacement field is desired at a point within the plastic zone, an effective stress intensity factor, K_eff, that accounts for moderate crack tip yielding must be used (used in ETB Crack Stress module). For more extensive yielding, one must apply alternative crack tip parameters that take nonlinear material behavior into account (elastic - plastic fracture mechanics), such as the Crack Tip Opening Displacement (CTOD). This module can calculate the CTOD for plane stress and plane strain loadings.

The plastic zone size can be estimated by two methods: the Irwin approach, where the elastic stress analysis is used to estimate the elastic-plastic boundary and the strip yield model, also known as the Dugdale model. Both of these approaches are available in this module and both lead to simple corrections for crack tip yielding. The term plastic zone usually applies to metals, but is used here to describe inelastic crack tip behavior in a more general sense.

Irwin Approach:

On the crack plane (q = 0) the normal stress, S_y, in a linear elastic material is given by:

As a first approximation, we can assume that the boundary between elastic and plastic behavior occurs when the maximum stress given by the above equation satisfies a yield criterion. For plane stress loading, yielding occurs when S_y = S_yield, the uniaxial yield strength of the material. Setting the above equation equal to the yield strength and solving for r results in the first order estimate of the plastic zone size (r_y). In plane strain, yielding is suppressed by the triaxial state of stress, and the approximate plastic zone correction is smaller by a factor of three.

If we neglect strain hardening, the stress distribution for r < r_y can be represented by a horizontal line at S_y = S_yield. Thus, the stress singularity is truncated by yielding at the crack tip. The simple first order analysis is not strictly correct because it is based on an elastic crack tip solution. When yielding occurs, stresses must redistribute in order to satisfy equilibrium. The figure in the module shows this. The cross-hatched region represents forces that would be present in an elastic material, but cannot be carried in the elastic-plastic material because the stress cannot exceed yield. The plastic zone must increase in size in order to accommodate these forces. A simple force balance shown below leads to a second order estimate of the plastic zone size, r_p.

All the plastic zone sizes calculated in this module are estimates for the plastic zone size under monotonic loading. However, 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 placed into compression by the surrounding elastic body. The change in stress at the crack tip due to the reversed loading is twice the value of the yield strength. The cyclic plastic zone size is more characteristic of a plane strain state even in thin plates. Therefore, LEFM concepts can be 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 size and the size of the cracked body usually remains valid.

Strip Yield Model (Dugdale Model):

The strip yield model, was first proposed by Dugdale and Barenblatt. They assumed a long, slender plastic zone at the crack tip in a nonhardening material in plane stress. The model was first applied to a through crack in an infinite plate. The strip yield plastic zone is modeled by assuming a crack length of 2a + 2r_p, with a closure stress equal to S_yield applied at each crack tip. This model approximates elastic-plastic behavior by superimposing two elastic solutions: a through crack under remote tension and a through crack with closure stresses at the tips. Thus, the strip yield model is a classical application of the principle of superposition. The plastic zone size is determined by closing the zone crack by S_yield. The model assumes the following:

• No resistance to extension
• The size of the plastic deformation is artificially confined to the rigid plastic strip

The plastic zone shape predicted by the strip yield model is not a good approximation of the actual plastic zones in metals. However, many polymers produce crack tip craze zones that are predicted accurately by the strip yield model. Thus, although Dugdale originally proposed the strip yield model to account for yielding in thin steel sheets, the model is better suited for polymers.

For the plastic zone size solutions, a polar coordinate system (r, q) is defined with the origin at the crack tip. NOTE: r refers to the plastic zone size and q equals zero at the crack plane. The preceding estimates of the plastic zone size consider only the crack plane. It is possible to estimate the extent of plasticity at all angles by applying an appropriate yield criterion to the singular stress field equations (see the Singular Stress and Displacement Field module).

The yield criterion used in this module is the von Mises criterion, where yielding occurs when the effective stress, S_e, is equal to S_yield. S_e is a function of the principal stresses. The plastic zone shape can now be determined by calculating the plastic zone size at all theta values. There is a significant difference in the size and shape of the plastic zones for plane stress and plane strain. The plane strain condition suppresses yielding, resulting in a smaller plastic zone size for a given stress intensity value.

For Mode III the plastic zone size is independent of the angle theta (same at all angles), thus the plastic zone shape is circular.

Crack Tip Opening Displacement (CTOD):

LEFM is valid only as long as nonlinear material deformation is confined to a small region surrounding the crack tip. In many materials it is virtually impossible to characterize the fracture behavior with LEFM, and an alternative fracture mechanics model is required. Elastic-plastic fracture mechanics applies to materials that exhibit time-independent, nonlinear behavior (plastic deformation). One elastic-plastic parameter is the CTOD, which can be used as a fracture criterion. Critical values of the CTOD give nearly size-independent measures of fracture toughness, even for relatively large amounts of crack tip plasticity.

While examining fractured test specimens, Wells noticed that the crack faces had moved apart prior to fracture; plastic deformation blunted an initially sharp crack tip. The degree of crack tip blunting increased in proportion to the fracture toughness of the material. This observation led Wells to propose the opening at the crack tip as a measure of fracture toughness. Today, this parameter is known as the CTOD. Wells performed an approximate analysis that related CTOD to the stress intensity factor in the limit of small scale yielding. Irwin showed that crack tip plasticity makes the crack behave as if it were slightly longer. Thus, we can estimate CTOD by solving for the displacement at the physical crack tip, assuming an effective crack length of a + r_y.

Figure 1: The module input form.

The plastic zone size problem is defined by first selecting the proper type of solution from the Solution Type pull-down menu. The required inputs fields for the analysis will become white in color so they can be edited. All the text fields that are gray in color are not required for that particular solution type.

The modulus of elasticity and the Poisson's ratio are material properties that must be specified for all solution types. If either the modulus of elasticity or Poisson's ratio are less than or equal to zero, a message will be displayed in a separate window. A link to the ETB Materials Database module is provided for automatic input of material properties. The module will verify that proper material properties were entered.

The user must be consistent with the units to get correct answers. For example, if the stress intensity factors were entered in psi(in.)^0.5, the modulus of elasticity in psi, then the plastic zone size would be calculated in inches.

• -Stress Intensity Factors for the Selected Mode
•  The angle (theta) at which the plastic zone size was calculated
• The material properties including the Modulus of Elasticity, the Poisson's ratio, the Shear Modulus, Yield Strength, and a Field Constant, C, that depends on if the loading is plane strain (thick plate) or plane stress (thin plate)
• The plastic zone size

Figure 2. Module tabulated results.