Chapter 1 Introduction
1.7 Methods for the determination of stress intensity factors
The SIF plays a vital role in the application of the principles of linear elastic fracture mechanics to practical problems in design and analysis. Determination of SIF for real life components with a crack is therefore very important for prediction of the crack condition. The SIFs can be determined through analytical, numerical and experimental methods.
1.7.1 Analytical methods
The analytical methods are mostly suitable for idealized geometries, loading and boundary conditions. There are number of techniques used in the solution of the SIFs of two-dimensional crack problems. Amongst, very notable methods are conformal mapping, Laurent series expansion, boundary collocation method, integral
transform method and weight function method. Theoretical methods are essential for two reasons. First, they provide the correct form of singularities and asymptotic solutions that are useful in analyzing and interpreting the experimental observations and in improving the accuracy of numerical solutions. Secondly, they provide accurate solutions for relatively simple geometries and for certain idealized material behavior and could be used as benchmark problems for validation of numerical methods and experimental methods.
1.7.2 Numerical methods
Numerical methods for determination of SIF are extremely useful in dealing with the complex configurations usually found in a great many practical problems.
Finite element and boundary element methods are widely used for numerical estimations of SIFs. The SIFs can be estimated either by the displacement based methods or energy based methods. The use of numerical methods, particularly finite element methods, has vastly broadened the range of problems that can be solved by computational approaches. A major advantage is that engineers can easily calculate SIFs at their desks, using personal computers and large number of commercially available finite element codes.
1.7.3 Experimental methods
Experimental methods provide new alternatives and opportunities for solving fracture mechanics problems. Many factors make the experimental determination of the SIFs indispensable. Analytical methods for determination of the SIFs are usually based on simplifying assumptions which imply certain detachment from reality and those theories can be verified only by experimentation to convince whether such idealization has not resulted in an undue distortion of the essential features of the problem. The aim of all experimental methods is to extract the SIFs from the measurable data. Several experimental methods have been developed for the measurement of SIFs in the past and some of the widely used experimental methods are caustics, moiré interferometry, photoelasticity and strain gage techniques.
1.7.3.1 The method of caustics
The method of caustics or shadow spot method relies on the deflection of light rays due to stress field gradient. Caustics are three dimensional surfaces in space enclosing a dark region and along which a high intensity of light occurs [13,14].
When a uniform beam of light is incident on a reflective surface containing geometric nonlinearities, the beam is reflected in such a way that its intensity varies spatially to form the caustic. This occurs when light is reflected from the region around the crack tip in a polished specimen subjected to load. The caustic can be seen by placing a screen in the reflected light path. A thin ring of high intensity light the caustic, surrounding a dark spot will be observed. The size and shape of the ring, or caustic can be related to the magnitude of SIFs. The SIF is related to diameter of the caustic, specimen thickness, the distance between the reference plane and the specimen and also Young’s modulus and Poisson’s ratio of the specimen.
The advantage of caustics method is that it can be applied to both transparent and opaque materials. It has also been used with variety of materials such as isotropic, anisotropic and composite material for the determination of SIFs under static and dynamic loading condition. This method is also widely employed for the determination of mixed mode SIFs. However, the data produced were not as reliable as other methods such as moiré interferometry, photoelasticity and strain gage method [15].
1.7.3.2 Moire interferometry
The two basic principles that govern the formation of moiré interferometry fringes are the interference of light and the diffraction of light. It provides contour maps of the in-plane displacement field, from which small strains can be determined [13]. This technique is based on the interference of two regular gratings and widely employed in quasi-static and dynamic loading problems. Out of two gratings, first grating acts as reference which is undeformed. The second grating which is called active grating is affixed to the surface of the specimen and is deformed by the strains experienced in the specimen. Simultaneous viewing of the two gratings produces the
displacement. Then, SIF is calculated by substitution of the order of fringe and the pitch of the reference grating in appropriate equation. This method is not consistent as compared to other optical method such as photoelasticity and is also more expensive [15].
1.7.3.3 Photoelasticity
Photoelasticity method is by far the most widely used whole-field technique for studying cracked bodies. It is an optical method of experimental stress analysis, which yields a whole field representation of principal stress difference [13]. The difference in the principal stresses is related to the fringe order, material fringe constant and the length of the light path. In this method the SIF can be determined by measuring the fringe order and position parameters on a fringe loop. This method has also been applied for determination of mixed mode SIFs. Many investigators used this technique to study crack tip stress field for both static and dynamic conditions.
In comparisons of optical analyses of crack tip stress field, the photoelastic results were the most consistent and in a fully equipped laboratory, photoelasticity would be the first choice to provide accurate, repeatable data [15]. Although the idea of using the photoelastic method of stress analysis to the solution of crack problems seems to be attractive, many difficulties are encountered during measurements due to high concentration of the isochromatic fringes near the crack tip which alter the real meaning of the isochromatic pattern of the corresponding problem of the cracked plate.
1.7.3.4 Electrical resistance strain gage
Of all of the techniques of experimental mechanics, by far the most commonly used techniques are those based on electrical resistance strain gage. Moreover, among the experimental techniques for the determination of SIFs strain gage based techniques are relatively simple, easy to use and inexpensive [13]. Electrical resistance strain gages are the most commonly used type of strain sensors and hereafter these gages are named as only strain gages for simple representation. The
change in resistance of a conductor with its change in length forms the basis for strain measurement using strain gages.
Strain gages are bonded to the surface of the body and forms a part of it. When the body deforms, the gage also subjected to the same deformation. As a result, resistance of the gage material changes due to changes in its length due to deformation. This resistance change due to deformation is measured in terms of voltage change using a Wheatstone bridge circuit. The output voltage of the Wheatstone bridge circuit can be calibrated to give the axial strain along the strain