STRESS CONCENTRATION

Author: Anita Noble

Stress Concentration

The fracture of a material is dependent upon the forces that exist between the atoms. Because of the forces that exist between the atoms, there is a theoretical strength that is typically estimated to be one-tenth of the elastic modulus of the material. However, the experimentally measured fracture strengths of materials are found to be 10 to 1000 times below this theoretical value. The discrepancy is explained to exist because of the presence of small flaws or cracks found either on the surface or within the material. These flaws cause the stress surrounding the flaw to be amplified where the magnification is dependent upon the orientation and geometry of the flaw. Looking at fig. 1, one can see a stress profile across a cross section containing an internal, elliptically-shaped crack. One can see that the stress is at a maximum at the crack tip and decreased to the nominal applied stress with increasing distance away from the crack. The stress is concentrated around the crack tip or flaw developing the concept of stress concentration. Stress raisers are defined as the flaws having the ability to amplify an applied stress in the locale.

Fig. 1: (a) The geometry of surface and internal cracks. (b) Schematic stress profile along the line X-X' in (a), demonstrating stress amplification at crack tip positions.

Determination of the Maximum Stress at the Crack Tip

If the crack is assumed to have an elliptical shape and is oriented with its long axis perpendicular to the applied stress, the maximum stress, sm can be approximated at the crack tip by Equation 1.

Eqn. 1: Determination of the maximum stress surrounding a crack tip.

The magnitude of the nominal applied tensile stress is so; the radius of the curvature of the crack tip is r; and a represents the length of a surface crack, or half the length of an internal crack.

Determination of Stress Concentration Factor

The ratio of the maximum stress and the nominal applied tensile stress is denoted as the stress concentration factor, Kt, where Kt can be calculated by Equation 2. The stress concentration factor is a simple measure of the degree to which an external stress is amplified at the tip of a small crack.

Eqn. 2: Determination of the stress concentration factor.

Stress Concentration Considerations

It is important to remember that stress amplification not only occurs on a microscopic level (e.g. small flaws or cracks,) but can also occur on the macroscopic level in the case of sharp corners, holes, fillets, and notches. Fig. 2 depicts the theoretical stress concentration factor curves for several simple and common material geometries.

Fig. 2: Stress concentration factor plots for three different macroscopic flaw situations.

Stress raisers are typically more destructive in brittle materials. Ductile materials have the ability to plastically deform in the region surrounding the stress raisers which in turn evenly distributes the stress load around the flaw. The maximum stress concentration factor results in a value less than that found for the theoretical value. Since brittle materials cannot plastically deform, the stress raisers will create the theoretical stress concentration situation.

Reference:

Callister, William D. Materials Science and Engineering: An Introduction - 3rd Edition. John Wiley & Sons, Inc.: New York, 1994.