Linear Elastic Fracture Mechanics: Development and advancement in study of catastrophic failures & Fracture analysis.

## Related Theories

Fracture is one of the most serious problems that the society has been facing from a very long time. Fractures have started occurring ever since the inception of manmade structures in the history of the world. The problem now is even worse than what is used to be centuries back since more and more technologies and advancement in manmade structures are coming up with increased complexities. Much research has been carried out in the past, which has only generalized the preexisting technologies. However not much research has been done on the specific trends of elastic fractures.

This topic is extremely important to be further researched on. Different recommendations as well as feedbacks regarding minimizing catastrophic accidents in the future can be obtained. The different kinds of micro mechanisms that leads to the fracture failures can be understood with an in depth analysis of this topic (Kang et al. 2015). The roles of “process zone” as well as the two dimensions of the fracture phenomena, which are deterministic and the probabilistic dimensions, can be addressed.

Linear Elastic Fracture Mechanics or LEFM assumes the material to be isotropic as well as linear elastic in nature and based on this assumption, the field of stress by the side of the crack tip can be calculated using the elasticity theory. When the stresses in the crack tip area exceeds the toughness of the material fracture, the crack grows (Minhajuddin, Saha and Biligiri 2015). Most of the formulas are derived either for either plane stress or for plane strains, which are associated with three loading modes of loadings on a body that is cracked: opening, sliding, as well as tearing.

Formulas and assumptions used in LEFM:

The basic analysis of LEFM is the stress in the area of the cracked tip. It is the location function, the conditions of loading conditions as well as the specimen geometry, which can be shown using the formula as shown below in the diagram. It is assumed that there is no stress intensity factor in the idea scenario.

Figure 1: Basic analysis formula for LEFM (Wanhill, Molent and Barter 2013).

However, in practice the stress intensity factor or “K” is calculated based on the crack tip stress and it is compared against the fracture toughness as a function of the location, loading as well as the geometry. The location is represented as r and is the coordinate system.

Figure 2: Calculation of stress intensity factor “k” in LEFM (Weißgraeber et al. 2016).

## Advantages and Drawbacks

The fracture toughness can be calculated using a material specific stress intensity factor K_{c }(Nejati, Paluszny and Zimmerman 2015). It is assumed that the stress intensity factor or K is lesser than K_{c.}

_{}

Figure 3: Material specific experiment for calculating K_{c }(Munz and Fett 2013).

Figure 4: Correlation of Stress Intensity Factor Vs. Crack Tip Stresses (Periasamy and Tippur 2013).

As far as we are concerned with the determination of the failure load, the advantage of using Finite Fracture Mechanics with respect to the Cohesive Crack Model is evident, since a troublesome analysis of the softening taking place in the fracture process zone is not necessary (Vu-Bac et al. 2013). There are nearly neither restrictions with regards to constitutive behavior of adjacent materials or the cohesive zone itself nor if geometric non linearity’s (large strains, rotations) play a more or less important role.

The characteristics of growth for small fatigue cracks were investigated under an alloy steel with grain sizes, which are 15 μm and 91 μm. When crack length on the surface is shorter than three-grain diameters or 3d, rate of crack growth decreases. Aspect ratios also vary widely. Cracks that are longer than the 3d remain uninfluenced by microstructure, but do not reach the speed of growth as expected form LEFM. This is due to the crack closure difference seen in LEFM between small and large cracks, which is a drawback. Since crack closure calculation is a difficult task, 3d+ 150 μm can be considered as the critical crack length for any engineering task. Also in another experiment, the authenticity of the LEFM methodology was verified using the ratio between intensity factor of linear elastic stress and the plane strain fracture toughness (Anderson 2017). This procedure can be only considered for primary loading which is a major drawback (Askes and Susmel 2015). The load calculated by LEFM and elastic-plastic behavior varied a lot, which deemed LEFM invalid.

The theoretical background of this topic that involved mathematical calculations of stress as well as strain dated back to a century ago. The spectacular 24 W incident marked the beginning of the early research works. Brocks and K.-H. Schwalbe accidents has made Irwin publish his first seminal paper in the 20th century (Gross and Seelig 2017). Irwin proved the relationship between stress intensity and the energy approach and gave a dual stress field description, long before similar research within the fracture-mechanics community.

## Gradual Development in this Field

Rice’s T-stress was the first actual parameter, and the driving force of crack is strain-energy date of release for Griffith, which took another 10 years to be launched in Germany marking the foundation of DVM Working Group Fracture Mechanics in 1969.

Heinz Neuber was considered as the “father” of a specialized field called notch mechanics, who later research this topic in his seminal book 3^{rd} edition, which provided evidences for the catastrophic failures in fracture mechanics (Guo and Chen 2013). His research was primarily about the stress distribution violations in the notch vicinity or crack tip.

Later in the 1980s, the retired president from the Federal Institute of Materials Testing (BAM) in Berlin had stated fracture mechanics as a form of pseudo-science, since it was an introduction to surface energy, which was Griffith’s date of energy release, and a materialistic parameter with the strange dimension called MPa√m. These arguments showed similarities with Newtonian and relativistic mechanics.

Figure 5: Models developed by a Dugdale and Barenblatt to avoid stress singularity (Wang 2015).

LEFM Characterization of an Anisotropic Shale:

The existence of the natural shale bedding planes makes fracture characterization extremely complicated. To understand it better, recently four groups bending tests in three-point on the Longmaxi shale in the southeast Chongqing were conducted from different inclination angles. It was found that the cracks ere induced when the bedding plane inclination angle was high (Gupta et al. 2013). The results also showed that that the crack stress of the anisotropic filed shale is dependent on stress intensity factor as well as elastic constants and inclination angle of the plane.

Figure 6: Visible bedding planes of Longmaxi shale (Guo et al. 2014).

The outcome of the research showed typical graph of measured load versus loading-point displacement for different inclination angles (Kammer et al. 2015). Initially, the curves were relatively gentle due to the initial loading stages, whereas the later stages showed a linear growth.

Figure 7: three point bending clamp of shale (Guo et al. 2014).

Conclusion:

Therefore, it can be concluded that despite its drawbacks, Linear elastic fracture mechanics eliminates the need of complex analysis of fracture process zone softening. Although the results derived from LEFM is not the desired outcome of any given experiment, still it has multiple applications in the industries. The early researches of the 20^{th} century also showed the relationship between stress and energy as proved by Irwin. It can also be concluded form the above paragraphs that in case of faster development of larger plastic deformation zones compared to the rate of crack growth, this technique must be effectively used. Considering possible future work, being able to use crack tip stresses can be used to greatly change the way in which the cracked material behavior can be modelled for instance in transportation and energy industries.

References:

Anderson, T.L., 2017. Fracture mechanics: fundamentals and applications. CRC press.

Askes, H. and Susmel, L., 2015. Understanding cracked materials: is linear elastic fracture mechanics obsolete?. Fatigue & Fracture of Engineering Materials & Structures, 38(2), pp.154-160.

Gross, D. and Seelig, T., 2017. Fracture mechanics: with an introduction to micromechanics. Springer.

Guo, T. and Chen, Y.W., 2013. Fatigue reliability analysis of steel bridge details based on field-monitored data and linear elastic fracture mechanics. Structure and Infrastructure Engineering, 9(5), pp.496-505.

Guo, X., Li, Y., Liu, R. and Wang, Q., 2014. Characteristics and controlling factors of micropore structures of the Longmaxi Shale in the Jiaoshiba area, Sichuan Basin. Natural Gas Industry B, 1(2), pp.165-171.

Gupta, V., Duarte, C.A., Babuška, I. and Banerjee, U., 2013. A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics. Computer methods in applied mechanics and engineering, 266, pp.23-39.

Kammer, D.S., Radiguet, M., Ampuero, J.P. and Molinari, J.F., 2015. Linear elastic fracture mechanics predicts the propagation distance of frictional slip. Tribology letters, 57(3), p.23.

Kang, Z., Bui, T.Q., Saitoh, T. and Hirose, S., 2015. An extended consecutive-interpolation quadrilateral element (XCQ4) applied to linear elastic fracture mechanics. Acta Mechanica, 226(12), pp.3991-4015.

Minhajuddin, M., Saha, G. and Biligiri, K.P., 2015. Crack propagation parametric assessment of modified asphalt mixtures using linear elastic fracture mechanics approach. Journal of Testing and Evaluation, 44(1), pp.471-483.

Munz, D. and Fett, T., 2013. Ceramics: mechanical properties, failure behaviour, materials selection (Vol. 36). Springer Science & Business Media.

Nejati, M., Paluszny, A. and Zimmerman, R.W., 2015. On the use of quarter-point tetrahedral finite elements in linear elastic fracture mechanics. Engineering Fracture Mechanics, 144, pp.194-221.

Periasamy, C. and Tippur, H.V., 2013. Measurement of crack-tip and punch-tip transient deformations and stress intensity factors using digital gradient sensing technique. Engineering Fracture Mechanics, 98, pp.185-199.

Vu-Bac, N., Nguyen-Xuan, H., Chen, L., Lee, C.K., Zi, G., Zhuang, X., Liu, G.R. and Rabczuk, T., 2013. A phantom-node method with edge-based strain smoothing for linear elastic fracture mechanics. Journal of Applied Mathematics, 2013.

Wang, H., 2015. Numerical modeling of non-planar hydraulic fracture propagation in brittle and ductile rocks using XFEM with cohesive zone method. Journal of Petroleum Science and Engineering, 135, pp.127-140.

Wanhill, R.J.H., Molent, L. and Barter, S.A., 2013. Fracture mechanics in aircraft failure analysis: Uses and limitations. Engineering Failure Analysis, 35, pp.33-45.

Weißgraeber, P., Felger, J., Geipel, D. and Becker, W., 2016. Cracks at elliptical holes: stress intensity factor and finite fracture mechanics solution. European Journal of Mechanics-A/Solids, 55, pp.192-198.

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