Crack-extension force

Gc is a material property which is referred to as the toughness, critical strain energy release rate or crack extension force. It is effectively the energy required to increase the crack length by unit length in a piece of material of unit width. It has units of J/m. ... 

To obviate this inadequacy, Griffith s equation had to be modified to include the energy expended in plastic deformation in the fracture process. Accordingly, Irwin defined a parameter, G, the strain-energy-release rate or crack extension force, which he showed to be related to the applied stress and crack length by the equation ... 

If the material is not perfectly brittle, i.e., there are energy dissipating mechanisms in addition to the creation of new surfaces, then we introduce a term G, which is an energy (its units are J/m ) representing crack extension by all the available processes. You will find Gc referred to as total work of fracture, crack extension force, and strain energy release rate. 

R is the crack resistance of the material and is called the crack resistance force. The material will fracture when Gc = R, i.e., the crack extension force is equal to the crack resistance force. If the fracture is entirely brittle, the energy is only required to create new surfaces, then R = 2y. (Note The surface energy term used here differs from that in Eq. 18.11 by a factor of two because G, is associated with a single crack tip.)... 

In fact purely elastic solids do not exist. Solids like elastomers are viscoelastic they lose energy when subjected to a cycle of deformation, and particularly at a crack tip where stresses and strain rates are high. G - w is the crack extension force applied to the crack tip under this force the crack takes a limiting speed v, instead of continuously accelerate as for elastic solids, and one can write (6)... 

Most fracture experiments are conducted under normal loading conditions (mode I) and the fracture properties are described in terms of the strain energy release rate, or crack extension force, Gj. If a steadily increasing load, P, is applied to a joint, a critical value. Pc will be reached at which crack extension will just begin. At this condition, Gi is referred to as the fracture energy, or fracture toughness, G. Various test specimen... 

G, [kNm" ] is the elastic stress energy release rate, meaning that it describes the rate at which the energy for crack propagation is supplied. Gj as equal to dV/dc is also called the crack extension force and is related to elastic properties of the material and element and to the applied load system. 

If G 7 w, then the area of contact will change spontaneously so as to decrease the thermodynamic potential. If G < w, then Eqs. (3) and (4) show that A must increase, and the crack recedes. Conversely, if G > w, then the area of contact must decrease to give dg < 0 or dF < 0, and the crack extends. Quantity GdA is the mechanical energy released when the crack extends by dA. The breaking of interfacial bonds requires an amount of energy wdA, and the excess (G — w)dA is changed in kinetic energy if there is no dissipative factor G — w is the crack extension force, which is zero at equilibrium. 

Starting from a stable equilibrium with the two bodies compressed (P > 0, 8 > 0) let us decrease the cross-head displacement A one generally encounters a progressive reduction in the area of contact, i.e., a controlled rupture with (dG/dA) > 0 up to a point where (dGfdA) = 0 the equilibrium becomes unstable and the crack extends spontaneously toward the rupture under this given cross-head displacement. It extends with acceleration, for the crack extension force G-w increases as A decreases. 

As discussed in Section 2.3, a crack propagating under a constant crack extension force G — w undergoes a drag proportional to w and is a function of its velocity v one can... 

FIGURE 7. Reduced crack extension force vs. crack velocity for glass-polyurethane systems. (From Reference 10.)... 

As presented here, the argument applies to cracks in brittle materials. In fact, up to now, in this chapter, we have considered only such cracks. However, the Griffith formalism, with suitable reinterpretation, may be applied to materials which are not completely brittle. Orowan (1950) and Irwin (1948) argued that the plastic flow, even in the fracture of fairly brittle materials, absorbs far more energy than the surface creation process. To take account of this, one simply replaces 2 T by a term characterizing the plastic flow, taking it to be a material constant as a first approximation (but see Lardner (1974), p. 205). This constant is Irwin s crack extension force. 

Irwin considered the crack extension force for explaining the failure of materials. This force is given by ... 

As observed by various investigators [12-15, 180] an approximately linear relation holds between the logarithms of crack velocity and crack extension force ... 

Here G is basically an energy, or the crack extension force, and G does not depend on the loading type. We may be able to consider the fixed-grips case as one of the general situations. Then, Equation (12) becomes... 

The system which was chosen for this study was one in which the shrinkage of the adhesive was very small. Even so it was found that the effect of the shrinkage stresses was to reduce the crack extension force by over 40%. It may readily be seen that in many commercial adhesive systems the reduction in strength below that which is theoretically possible by the operation of shrinkage stresses may be very much greater. It is suggested that shrinkage stresses may often be the major cause of weakness in an adhesive joint. 

When composite patching is provided to the cracked surface of a steel member, the composite patching provides deformation constraint to the cracked mouth. Rose (1982) carried out an analytical study estimating the reduction of the crack extension force when a cracked plate is repaired by reinforcing patches bonded on its faces. Rose pointed out that there are two main effects of the bonded reinforcing... 

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