Linear elastic fracture mechanic

Brittle solids fracture because the applied stress is amplified by minute cracks—of order 1/im in size— whicb occur naturally, as a result of fabrication, solidification, fatigue damage, etc. These cradts are frequent termed Griffith cracks, after the originator of the theory we are about to describe. 

The presence of the crack modifies the elastic stress distribution in its vicinity. 

From the fracture viewpoint, the stress distribution along the indicated line Oxj is particularly significant. In the Xj-direction, the stress 0-22 reaches a maximum value r at the ellipse surface falls as Xj increases, and ultimate obtains the value at a distance from the crack (as expected). In the x,-direction, (Th is zero at the ellipse surface and rises to a value of the order of a before falling again to zero with 

9 An elliptical, through-thickness crack in an elastic sheet subject to a stress a in the X2-direction causes a stress distribution along Ox, as shown is amplified from to at both tips of the crack. 

10 A narrow, through-thickness crack tn a tNn, wide sheet subject to a stress or. 

When the crack propagates, the crack surface can be formed by either shear or cleavage fracture, or a mixture of both, leading to fracture surfaces as discussed in section 3.5. If fracture occurs by crack propagation at stresses below the yield strength, the global plastic deformation of the component is usually small because plastic deformation is localised at the crack tip. 

As the name suggests, linear-elastic material behaviour is the precondition to allow applying the theory of linear-elastic fracture mechanics (lefm), discussed in this section. Strictly speaking, this precondition is fulfilled only in brittle materials like ceramics. In good approximation, it can also be used in ductile materials if the region of plastic deformation is restricted to the vicinity of the crack tip. Therefore, it can in many cases also be used to analyse metals. 

We start by considering the stress field near the crack tip and the energy release during crack propagation. Next, we will discuss how to design components against failure by crack propagation and how to determine relevant material parameters. 

In the late forties, Irwin suggested the energy balance of Eq. (8.7) could be considered in a slightly different way. In Eq. (8.9), the mechanical energy terms, involving and f/, were coupled because they involve terms that act to promote crack extension, whereas the surface term was treated separately because it represents the resistance of the material to fracture. One can, therefore, define a parameter 

It is useful at this point to consider the energetics involved when a crack increases its length by a small amount. Consider the loading geometry shown in 

These equations can be shown to be equivalent by setting u= F. Equations (8.15) and (8.16) also allow ( to be determined experimentally from compliance measurements. This is accomplished by measuring the compliance of bodies with various crack sizes which allows dA/dc to be evaluated. It is important to 

At the same time as the strain energy release rate concept was being developed, it became clear that an alternative approach could be used to describe fracture. This alternative was based on the idea that a variety of elastic problems involving cracks could be solved. All stress fields in the vicinity of a crack can be derived from three modes of loading, which are illustrated in Fig. 8.11. In the linear elastic solutions for all three modes, the stresses a.j and displacements , in the vicinity of a crack tip take the form 

One might expect that K or the magnitude of the crack tip stresses would be related to the energy-based parameter G. Irwin (1958) was able to show that for plane stress 

---

In moie ductile materials the assumptions of linear elastic fracture mechanics (LEFM) are not vahd because the material yields more at the crack tip, so that... 

Fracture Mechanics. Linear elastic fracture mechanics (qv) (LEFM) can be appHed only to the propagation and fracture stages of fatigue failure. LEFM is based on a definition of the stress close to a crack tip in terms of a stress intensification factor K, for which the simplest general relationship is... 

Substantial work on the appHcation of fracture mechanics techniques to plastics has occurred siace the 1970s (215—222). This is based on earlier work on inorganic glasses, which showed that failure stress is proportional to the square root of the energy required to create the new surfaces as a crack grows and iaversely with the square root of the crack size (223). For the use of linear elastic fracture mechanics ia plastics, certaia assumptioas must be met (224) (/) the material is linearly elastic (2) the flaws within the material are sharp and (J) plane strain conditions apply ia the crack froat regioa. 

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. 

Linear Elastic Fracture Mechanics Behavior of Graphite... 

Linear elastic fracture mechanics (LEFM) is based on a mathematical description of the near crack tip stress field developed by Irwin [23]. Consider a crack in an infinite plate with crack length 2a and a remotely applied tensile stress acting perpendicular to the crack plane (mode I). Irwin expressed the near crack tip stress field as a series solution ... 

Although Griffith put forward the original concept of linear elastic fracture mechanics (LEFM), it was Irwin who developed the technique for engineering materials. He examined the equations that had been developed for the stresses in the vicinity of an elliptical crack in a large plate as illustrated in Fig. 2.66. The equations for the elastic stress distribution at the crack tip are as follows. 

Composite materials have many distinctive characteristics reiative to isotropic materials that render application of linear elastic fracture mechanics difficult. The anisotropy and heterogeneity, both from the standpoint of the fibers versus the matrix, and from the standpoint of multiple laminae of different orientations, are the principal problems. The extension to homogeneous anisotropic materials should be straightfor-wrard because none of the basic principles used in fracture mechanics is then changed. Thus, the approximation of composite materials by homogeneous anisotropic materials is often made. Then, stress-intensity factors for anisotropic materials are calculated by use of complex variable mapping techniques. 

The above results are derived from linear elastic fracture mechanics and are strictly valid for ideally brittle materials with the limit of the process zone size going to zero. In order to apply this simple framework of results, Irwin (1957) proposed that the process zone, r be treated as an effective increase in crack length, Sc. With this modification, the fracture toughness becomes... 

Linear combination of atomic orbitals (LCAO) method, 16 736 Linear condensation, in silanol polycondensation, 22 557-558 Linear congruential generator (LCG), 26 1002-1003 Linear copolymers, 7 610t Linear density, 19 742 of fibers, 11 166, 182 Linear dielectrics, 11 91 Linear elastic fracture mechanics (LEFM), 1 509-510 16 184 20 350 Linear ethoxylates, 23 537 Linear ethylene copolymers, 20 179-180 Linear-flow reactor (LFR) polymerization process, 23 394, 395, 396 Linear free energy relationship (LFER) methods, 16 753, 754 Linear higher a-olefins, 20 429 Linear internal olefins (LIOs), 17 724 Linear ion traps, 15 662 Linear kinetics, 9 612 Linear low density polyethylene (LLDPE), 10 596 17 724-725 20 179-211 24 267, 268. See also LLDPE entries a-olefin content in, 20 185-186 analytical and test methods for,... 

ISO 15850 2002 Plastics - Determination of tension-tension fatigue crack propagation -Linear elastic fracture mechanics (LEFM) approach... 

The term fracture toughness or toughness with a symbol, R or Gc, used throughout this chapter refers to the work dissipated in creating new fracture surfaces of a unit nominal cross-sectional area, or the critical potential energy release rate, of a composite specimen with a unit kJ/m. Fracture toughness is also often measured in terms of the critical stress intensity factor, with a unit MPay/m, based on linear elastic fracture mechanics (LEFM) principle. The various micro-failure mechanisms that make up the total specific work of fracture or fracture toughness are discussed in this section. 

In the case of linear-elastic-fracture-mechanics, and nearly all epoxy polymers obey the requirements for LEFM to be employed, a simple relationship exists between KIc and G,c... 

In graphic presentation of Kk results, the error bars given for the control are typical of all those data points which do not have their own error bars. In cases where error exceeded 10%, individual error bars are provided and labelled with the corresponding symbol. Such large deviations are thought to result from the violation of the homogeneity criterion of linear elastic fracture mechanics at 15% of certain oligomers. (See, for example, Fig. 7). 

---