Theory of Fracture

In principle, the force necessary to cause a brittle break F = E jL can be calculated from the energy E required to separate chemical and physical bonds by an interbond-partner distance L. To break extended-chain poly-(ethylene) crystals perpendicular to the chain direction (i.e., breaking covalent bonds), a force of about 20 000 MPa is necessary, whereas to cause a break parallel to the chain direction (i.e., working only against dispersion forces), only 200 MPa is required. Experimentally, however, a maximum tensile strength of 20 MPa is observed (the so-called crystal paradox). Consequently, the break must occur at inhomogeneities, since these lead to an inhomogeneous distribution of the tensile stress onto disruption points and thus lead to stress concentrations. 

The break behavior of energy-elastic and entropy-elastic bodies is different. According to the break theory of Ingles for energy-elastic bodies, there is a relationship between the critical break stress (an)crit, the stress operating at the top of a crack ai i, the geometry of the crack, and the modulus of elasticity. In the simplest case of a crack of length L with a round tip of radius R, we have 

The Ingles theory offers a very good description of the break behavior of silicate glass, since silicate glasses are practically solely energy elastic and the 

Crazes occur perpendicular to the stress direction shortly before a destructive break. They may be up to 100 pm long and up to 10 pm wide. Crazes are not hairline cracks, that is, they are not totally void between the break surfaces, in contrast to what are known as white breaks. 

Because certain materials whiten with crack formation on flexing, these 

---

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. 

Fracture mechanic The fracture mechanics theory developed for metals is also adaptable for use with plastics. The basic concepts remain the same, but since metals and plastics are different they require different techniques to describe their fatigue-failure behaviors. Some of the comments made about crack and fracture influences on fatigue performance relate to the theory of fracture mechanics. The fracture mechanics theory method, along with readily... 

In many instances, 7 increases with temperature. The 7 of pure liquids always decreases when temperature T rises94. Thus the temperature coefficient of 7 is different from what would be expected from the true surface energy on the other hand, positive values oid y /dT agree with the new theory of fracture energy, because e0 of many materials is greater the higher the temperature. 

Their analysis of experimental data shows that tensile strength was the only parameter that varied as a function of particle size. Model simulation indicate that larger lumps were stronger than smaller lumps which is contradictory to Waters et al. [8], Teo and Waters [9], and Griffith [10] theory of fracture, which implies that larger particle are more likely to contain larger cracks and hence be more susceptible to breakage. 

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). 

This situation has been resolved, at least in part, by a recent generalized theory of fracture mechanics which gives. 

Developed chiefly in Russia, the kinetic theory of fracture at first appears to represent an entirely different account of fracture phenomena to that discussed in Section 1.2. In fact some Russian authors have claimed that the kinetic theory contradicts the Griffith theory of fracture. As we shall see, however, this is not the case. 

We shall see that the situation is far from simple and that the data on molecular processes cannot be used to make quantitative predictions about macroscopic deformation and failure properties. In particular it will beconw evident that the kinetic theory of fracture initiation is an oveRimpliflcation. 

Our discussion of the kinetic theory of fracture in Section 1 has already indicated the manner in which applied stress can bring about a net accumulation of nwlecular breakages in a jxrlymeric solid. Since the stress is continuous throughout a specimen loaded in tension, these breakages are distributed throughout the material and can be detected by ESR in terms of a volume concentration of free radicals. 

Applications of linear elastic fracture mechanics (primarily) to the brittle fracture of solid polymers is discussed by Professor Williams. For those not versed in the theory of fracture mechanics, this paper should serve as an excellent introduction to the subject. The basic theory is developed and several variants are then introduced to deal with weak time dependence in solid polymers. Previously unpublished calculations on failure times and craze growth are presented. Within the framework of brittle fracture mechanics and testing this paper provides for a systematic approach to the faOure of engineering plastics. 

In the case of lead azide, Andreev [42] and Bowden and Yoffe [43] suggest that lead azide detonates immediately after being ignited and that a burning regime is absent. The theory of fracture that was subsequently developed to explain the initiation of fast reaction [44,45], and the previous observations lead to the conclusion that the shock initiation mechanism of this primary explosive is not likely to exhibit the same characteristics as those exhibited by the secondary explosives. However, examination of the shock sensitivity of dextrinated and polyvinyl lead azide to pulse durations vaiying from 0.1 to 4.0 psec shows that the initiation characteristics are indeed similar to those observed for heterogeneous explosives. 

The quantitative theories of fracture which are currently in use are based on a fundamental principle of continuum thermodynamics, namely the first law or the energy balance which states that... 

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). 

Bazant formulated a statistical theory of fracture for quasibrittle materials [5, 23, 24]. He assumed that there exist several hierarchical orders which each can be described by parallel and serial linking of so-called representative volume elements (RVEs). For large specimens (and low probability of failures) the fracture statistics is equal to the Weibull statistics, i.e. if the specimens size is larger than 500 to 1000 times of the size of one RVE. In the actual case this is similar to the diameter of the critical flaw. For smaller specimens the volume effect disappears and the fracture... 

Kinetic Theory of Fracture. Catastrophic failure of a polymeric material is a complex process in which a sequence of partially understood events occurs at both the molecular and macroscopic levels. The stress-induced cleavage of the main-chain polymer bond is one event occurring on the molecular level which has been studied by both stress MS and electron spin resonance spectroscopy (ESR). 

The major thrust of the stress MS studies to date has focused on gathering evidence to support the kinetic theory of fracture proposed by Zhurkov. For the Zhurkov kinetic theory of fracture to be applicable to the material in question, not only must the mechanical and thermal degradation products be identical, but both processes must follow the same degradation mechanism. 

Figure 11. Important questions in the use of stress MS data as evidence for the kinetic theory of fracture...

Only if the answers to all these questions are aflBrmative can we use the Zhurkov kinetic theory of fracture to explain the degradation processes. The evidence gathered to date which is germane to this subject is summarized below. 

The generalized theory of fracture mechanics of Andrews (35) predicts that the cohesive fracture energy per unit surface area J is given by the energy required to break the bonds crossing the fracture plane, Jq, multiplied by a loss function, 6. 

R. Thomson, Lattice theory of fracture and crack creep, J. Appl. Phys. 44 (1973) 2051-2063. 

---