# Relationship Between Kj and Q

Since the asymptotic near-tip stress field is unique, and since the rate of energy flow into the crack tip must depend only on this field, there must exist a unique relationship between the energy release rate Q, and the stress intensity factor Kj. There are various way&to derive it. The simplest is to calculate the work of stress during the opening of a short slit ahead of the crack. We consider Mode I and imagine the crack tip to be advanced by an infinitesimal distance Sa in the direction of axis sq (Fig. 2.2.3a, b). Let A and В be the initial and final state. We use the procedure illustrated in Fig. 2.2.4, where the initial stress state A has been preserved by introducing a line slit of length Sa ahead of the preexisting crack, but keeping it closed by means of external surface tractions equal to the stresses existing in the actual state A (Fig. 2.2.4a). The final state В is then reached by reducing these stresses proportionally down to zero. In doing so, the intermediate states are such as the one depicted in Fig. 2.2.4b, in which the surface traction (closing tractions) are reduced to where r is a scalar load parameter varying from 1 in the initial state A to 0 in the final state В (Fig. 2.2.4d). The crack openings in the intermediate states must vary linearly with r because the structure is elastic, and so they must be proportional to (1 — r) and, therefore, equal to (1 — r)wB (Fig. 2.2.4d). Since the remote boundaries are assumed clamped, the only energy supply in going from the initial to the final state is the work done by the closing surface tractions. The elemental work per unit area at a given location on the crack surface when the crack opens dw under tractions <722 is —<t22dw, the minus sign coming from the different orientation of <t22 (closing) and dw (opening). Using this for an elemental intermediate step in which the load factor r varies by dr, the work (external energy supply) per unit surface of the slit turns out to be

df Tt) = -(r<721)d[(1 – r) «,*] ~ – т<72л2(-с/т wls) (2.2.17)   where b is the thickness of the body. Integration yields the total work per unit surface done by the surface tractions in passing from state A to B:

Therefore, the total external work supply — thus also the elastic energy variation at clamped boundaries — is obtained by integration of the previous equation with respect to r:

1 fSa

Ub ~ Ma = Wa-b — ■ b / crf2 wB dr (2.2.19)

2 Jo   Since Sa is vanishingly small, one may now use the near-tip field expressions (2.2.4) and (2.2.9),

in which the integration has been performed by means of the substitution r — Sa sin2 t. Noting that under fixed boundary conditions QbSa = — SU, and that —> Kf = Kj for Sa —> 0, we obtain the

celebrated Irwin’s result:  (2.2.22)

This shows that Griffith’s and Irwin’s approaches are equivalent, and allows us to discuss fracture criteria.

2.2.1 Local Fracture Criterion for Mode I: K]c.

Mode I is quite simple. Since the stress state of the material surrounding a very small fracture process zone —the crack tip— is uniquely determined by Kj, the crack will propagate when this stress intensity
factor readies a certain critical value K/c, called fracture toughness. Kjc for the given material may be determined performing a fracture test and determining the Ki value that provoked failure. Because the energy fracture criterion must also hold, and indeed does according to the fundamental relationship (2.2.22), Kic is related. to the fracture energy Gj by.

 Kic = %/Fg7 (2.2.23) With this definition, the local fracture criterion for pure mode I may be stated in criterion: indexfracture criterionlin local approach analogy to the energy if K] < Kic then: No crack growth (stable) (2.2.24) if Kj = Kic then: Quasi-static growth possible (2.2.25) if К і > Kic then: Dynamic growth (unstable) (2.2.26)

For loadings that are not pure mode I, the problem becomes more difficult because, in general, an initially straight crack kinks upon fracture and the criteria must give not only the loading combination that produces the fracture, but also the kink direction. This is still an open problem today, and the interested reader may find a summary of the most widely used criteria in Chapter 4. For most of the discussions in this book, LEFM mode 1 fracture is all that is needed.

Exercises

2.9 Estimate the strength of a large plate under uniaxial tensile stress if it contains through cracks of up to 10 mm. The plate is made of a brittle steel with К и — 60 MPa^/m.

2.10 Assuming plane stress and applicability of beam theory, find the expression for the stress intensity factor (mode 1) of a double cantilever beam specimen of thickness i>, arm length a, and arm depth h subjected to two opposite bending moments M (see Fig. 2.1.3).

2.11 Determine the stress intensity factor of the center-cracked panel subjected to internal pressure, described in exercise 2.5 (a) Use Irwin’s relationship, (b) use the near-lip expansion for the crack opening,

2.12 To test the fracture behavior of rock, a large 50-mm-thick panel of this material will be tested in a laboratory by injecting a fluid into a central crack of initial length 2cm = 100mm. Let p be the fluid pressure and V the volume expansion of the crack. Assuming linear elastic behavior with К/с = 35 кРа^Дп, find and plot the p-V and К-a curves the panel would experience if it were subjected to a controlled volume injection until the crack grew up to 1000 mm, and was then unloaded to zero pressure. Use Inglis result given in exercise

2.5 and an effective elastic modulus E’ — 60 MPa in the determination of the volume increase.

2.13 Find the stress intensity factor for the infinite strip of exercise 2.8.

2.14 Check that the coefficients of the near-tip power expansions for 022 and w for the center cracked panel subjected to remote equiaxial stress satisfy the relationship (2.2.16).