Excursion into Fracture Mechanics

Before the theoretical concepts for the description and understanding of stress crack formation will be discussed in detail, some fracture-mechanics terms and relationships should be reviewed and explained to help understand these theories (Knott 1973). 

The calculations of stress and strain fields can be simplified by taking into account special symmetries of the test specimen and configuration of the forces. Let us consider e.g. a very thin plate of large dimension. The stress and strain field only depends on the two coordinates within the plane of the plate. Let x- and y-axes be in the plane of the plate and the z-axis 

The state of a thin plate loaded on the narrow sides is an example of a so-called plane state of stress one of the components of the stress is all over identical to zero. Similar simplifications are possible in the case of the so-called plane state of strain. In such a state, one of the components of the displacement vector, that describes the spatial displacement of the points of the body at elastic deformation, is all over identical to zero and other components again depend only on x and y. A large but very thick plate can be taken as an example. In the plane state of strain, in addition to and O, component does also exist. This component stems from the condition that no displacement should take place in z direction. 

For plane states of stress and strain the methods of function theory can be used that enable the analytical calculation of the stresses and deformation fields for various fracture modes and various forms of notch and crack tips (Westergaard 1939). 

For an opening mode fracture, stresses and in the plane of the crack (0=0) can be given by  

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