Some Basics in Fracture Mechanics

Ultimately, fracture can only occur if all atomic bonds in an area are pulled apart and break. The stress necessary to break a bond (the theoretical strength) is between E/S and E/20, where E is the elastic modulus of the material [51]. Typical tensile stresses applied to highly loaded components are in the order of /1000 or even smaller, and yet fracture of components still occurs. To explain this discrepancy, it is necessary that certain strong local stress concentrations exist these are termed Jlaivs. The action of flaws can be discussed by using the simple example of an elliptical hole in a uniaxial tensile-loaded plate. At the tip of its major semi-axis (which is perpendicularly oriented to the stress direction), the stress concentration is [52]  

It is clear that the stress concentration may become very high, if the ellipse is extremely elongated that is, if c3 b. For that case, the equation can be approximated 

An elliptical hole must be extremely elongated to create the stress concentration necessary to reach the theoretical strength. To give a simple example, for an applied stress of o = E/1000 the ratio of major to minor semi-axis must be c/h w 50 (c/q 2500) to create the stress concentration necessary to reach the theoretical strength at a notch tip /10. 

consider the stress concentration produced by cracks. The stress field around a crack tip can be determined by linear elastic fracture mechanics (common solutions for this are available in standard text books [51, 53]). Here, a material is treated as a linear elastic continuum, and a crack is assumed to be a mathematical section through it (having a crack tip radius of zero). Under plane strain conditions, the components of the local stress field on a volume element, o,j, in a region near the crack tip is space dependent and can be expressed as (polar coordinates r and 0 origin at the crack tip) [51, 53]  

The tensor is a function of the angle 0 only, and values can be found in various textbooks [51, 53]. The stress field scales with the stress intensity factor, K, given by 

---