Fracture mechanics Griffith theory

Analysis of Failure Failure of "Flawless" Materials Fracture Mechanics Griffith Theory Stress Intensity Factors Fracture Energy Viscoelastic Effects Examples Fatigue Conclusion... 

A quantitative explanation of the effect requires an advance in fracture mechanics. Griffiths theory explains fracture in a perfectly elastic material as dependent on crack depth. This has been extended to cover the situation where there is a small zone of plastic deformation ahead of the crack. The problem is more difficult when the pla.stic deformation is large compared to the crack size, and, as far as I know, there has been no treatment of the situation when plastic deformation covers the whole thickness of the specimen over an appreciable length. Any analysis would also require an understanding of the transition in material from crystalline yielding to looking and chain breakage and the form of the local stress-strain curve beyond that which is measured. 

The importance of inherent flaws as sites of weakness for the nucleation of internal fracture seems almost intuitive. There is no need to dwell on theories of the strength of solids to recognize that material tensile strengths are orders of magnitude below theoretical limits. The Griffith theory of fracture in brittle material (Griflfith, 1920) is now a well-accepted part of linear-elastic fracture mechanics, and these concepts are readily extended to other material response laws. 

According to Eqs. (13.145) and (13.148) the fracture stress in plane strain is a factor 1 /(1-v2) 1 /0.84 1.2 higher than in plane stress. Experimentally, however, the difference is much bigger. The reason for this discrepancy is that Griffith s equations were developed in linear fracture mechanics, which is based on the results of linear elasticity theory where the strains are supposed to be infinitesimal and proportional to the stress. 

The idea that the strength of bulk solids is controlled by flaws was advanced by Griffith in 1921 and has led to the development of a mudi more sophisticated continuum approach to fracture, known as fracture mechanics. Fracture mechanics is concerned always with the conditions for the propagation of an existing crack, and it is important to bear this in mind when comparing different theories of fracture. Griffith s ideas are well known and do not need to be elaborated here. There are some aspects of his theory which are relevant to the present discussion, however. Griffith s equation for the fracture stress of an elastic material is (for plane stress). 

Thus, one needs a theory of fracture that is based on the stability of the largest (or dominant) flaw or crack in the material. Such formalism was first introduced by A. A. Griffith in 1920 [1] and forms the basis of what is now known as linear (or linear elastic) fracture mechanics (LEFM). 

Stress Intensity Factors. A seemingly different approach to fracture on the basis of stress intensity factors, (, 1 6, 17) is frequently encountered in the fracture mechanics literature. It is, therefore, appropriate to briefly discuss this approach and its relation to the Griffith theory. 

To see how the fracture energy may be used in the initiation of chemical reactions, the concepts of fracture mechanics are introduced, including the strain rate and temperature dependence of the ductile-brittle behavior. The starting point is the Griffith theory which in its simplest form applies to perfectly brittle materials and states that for a crack to form, the elastic strain energy available must be at least sufficient to provide the energy of the new surfaces formed [74]. 

There are two principal theories, or models, that attempt to describe what happens during brittle fracture, the Griffith fracture theory and the Irwin model. Both assume that fracture takes place through the presence of preexisting cracks or flaws in the polymer and are concerned with what happens near such a crack when a load is applied. Each leads to the definition of a fracture-toughness parameter and the two parameters are closely related to each other. The Griffith theory is concerned with the elastically stored energy near the crack, whereas the Irwin model is concerned with the distribution of stresses near the crack. Both theories apply strictly only for materials that are perfectly elastic for small strains and are therefore said to describe linear fracture mechanics. 

Recall that the whole theory of Griffith has been developed for elastic bodies—what applies to metals within a certain range of imposed stresses. Thus, Eqs. (24.17)-(24.23) form the essence of linear elastic fracture mechanics (LEFM). In Eq. (24.24) a plastic termr p has been added to the elastic term F metals exhibit also plasticity, hence the improvement displayed in Eq. (24.24). If we make a further step and assume that F includes all nonelastic contributions, we shall have an equation usable also for viscoelastic materials. We, therefore, have to use Eq. (24.24) instead of (24.17) while in Eqs. (24.18)-(24.23) we need to put F + F instead... 

One general comment is that defects are not as strong a controlling feature of breakage in these extensible textile fibres as in many other materials. Rupture forces cannot be calculated from modulus and crack depth as in Griffiths brittle fracture, or even from the later theories of fracture mechanics. As described below, Moseley (1963) showed that severe damage could be imposed on nylon and polyester fibres with no effect on strength at room temperature. 

Fracture mechanics for brittle materials is basically derived from the theory presented by Griffith (1920). Initially it was proposed for an elliptical crack in an infinite elastic and homogeneous medium subjected to distant tensile forces and the conditions for crack propagation were formulated for that case. Later, the brittle fracture mechanics were developed for various situations in real structural elements, with non-negligible plastic deformations and with several complications necessary to account for heterogeneity of materials, time effects, etc. In that approach the crack s appearance and propagation is considered as a basic effect of loading and as phenomena directly related to final failure. 

The development of the fracture mechanics approach to metals lead to Kaplan s (1961) proposal to apply fracture mechanics to concretes, using also relations applied in other fields of materials science. The Griffith s approach to cracks in perfectly brittle and homogeneous materials was extensively followed in numerous studies of cracking processes in cementitious materials, (cf. Section 10.1.1). Then the question arose whether the cracks themselves, and which of them, were governed by the Griffith s approach. Later, however, it became clear that the situation in these materials is much more complex and requires a more diversified approach. The answer for the above question is therefore also more complex the Griffith s approach is applicable to those materials, but with several restrictions and complementary assumptions. Various related questions are the subject of further developments of new theories and proposals (cf. Section 10.1). 

The well-known Griffith s theory constitutes the principle of fracture mechanics. In fracture mechanics, the fracture of a brittle material is caused by the increase of tension-stress at the edges of micro-cracks (disfigurements) in the material. The breaking strength of a catalyst follows the Griffith equation. 

Ivanova, V. S. (1982). From the Griffith Theory to Fracture Fractal Mechanics. Fiziko-Khimicheckaya Mekhanika Materialov, 1993,29(3) 432 p. 

Cracking occurs when the stress in the network exceeds its strength. Since the classic work of Griffith [70], it has been understood that fracture of brittle materials depends on the presence of flaws that amplify the stress applied to the body. That is, if a uniform stress is applied to a body containing a crack with a length of c, the stress at the tip of the crack is proportional to ct,Vc, and failure occurs when that stress exceeds the strength of the material. The theory of linear elastic fracture mechanics (LEFM), which is discussed in several excellent textbooks [71-73], indicates that catastrophic crack propagation occurs when... 

Modem fracture mechanics was pioneered by A.A. Griffith who showed that brittle materials could fail catastrophically by cracks that become self-propagating even at stresses much lower than their tensile strength. Griffith s theory was expanded to include ductile materials and has led to the definition of a material property called fracture toughness that allows the prediction of critical crack lengths. 

Methods for mechanical testing of materials are briefly introduced along with various strengthening mechanisms. The number and siu-face area of the slip systems in metals and in ceramics are shown to be responsible for the ductility (or the lack of it) and for ductile-to-brittle transitions. Griffith s theory of brittle fracture is used to introduce fracture mechanics and to develop the concept of fracture toughness. The viscoelastic behavior of polymers is briefly discussed. 

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