This module calculates the stresses and displacements at a point near a crack tip. The closed-form solutions used in this module assume isotropic linear elastic material behavior, and are only valid at points near the crack tip (r < 0.1 * crack length).

The module defines a polar coordinate system with the origin at the crack tip, as shown in Figure 1. The displacements near the crack tip vary as , where r is the distance from the crack tip. The stresses and strains are singular at the crack tip, varying as . Since the stress is asymptotic to r = 0, it is called a stress singularity.

Figure 1: Crack coordinate system.

The three basic types of loading that a crack can experience are shown in Figure 2.

• Mode I loading is normal to the crack plane, and tends to open the crack. The crack surfaces tend to separate symmetrically with respect to the crack plane.

• Mode II corresponds to in-plane shear loading and tends to slide one crack face with respect to the other (shearing mode). The stress is parallel to the crack growth direction.

• Mode III corresponds to out-of-plane shear, or tearing.

Figure 2: The three basic modes of fracture.

A cracked body can be loaded in any of these modes, or a combination of two or three modes. Each mode of loading produces the singularity at the crack tip. In a mixed mode problem (more than one mode is present), due to the principle of linear superposition, the individual contributions to the stress component are additive. This module can find the stress and displacement fields for Mode I, Mode II, Mode III, and mixed mode loadings. Note that Mode III loading is the same in both plane stress and plane strain loadings.

This module uses the Williams and Westergaard Stress Functions. These equations can be converted to polar stresses (Srr, Stt, Srt, Srz, Stz) and displacements (ur, ut, uz) if required. For these equations please refer to the references listed at the end of the help file.

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 described above becomes increasingly inaccurate as the inelastic region at the crack tip becomes larger.

The size of the crack tip yielding zone, termed the plastic zone, can be estimated using the Plastic Zone Size module.  Simple corrections to linear fracture mechanics 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. For more extensive yielding, one must apply alternative crack tip parameters that take nonlinear material behavior into account, such as the Crack Tip Opening Displacement (CTOD).  CTOD can be calculated from the Plastic Zone Size module.

Figure 3: The module input form.

The crack 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 radius and theta input parameters are the polar coordinates of the point corresponding to the calculated stress and displacement field.

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, and the radius in inches, then the stresses would be calculated in psi and the displacements would be calculated in inches.

• The Stress Intensity Factors for the selected loading modes
• The polar coordinates (r, q) of the point in the stress and displacement field
• The material properties including the Modulus of Elasticity, the Poisson's ratio, the Shear Modulus, and a Field Constant, C, that depends on if the loading is plane strain (thick plate) or plane stress (thin plate)
• The stress field and the contribution made by each mode.
• The displacement field and the contribution made by each mode.

Figure 4. Module tabulated results.