# Give a detailed proof of Eqs. (2.1.20) and (2.1.21)

2.2 Derive the equivalent of equation (2.1.3) for the energy release rate in circular cracks of growing radius a in an axisymmetrie stress field. (Answer: 2жа0 Sa = 6WH.)

2.3 Show that the generalized displacement associated with the resultant force of a uniform traction distribu­tion is the average displacement in the direction of the tractions over the area of their application. (Hint: write U = aa, where a is a variable stress and et – is a fixed unit vector.)

2.4 To simulate rock fracture in the laboratory, a very large panel of thickness h with a relatively small crack of length 2a is tested by injecting a fluid into the crack. From Inglis (1913) results it is known that under pressure p the straight crack adopts an elliptical shape, with minor axis c = 2pa/E’, where the effective modulus E’ is equal to the Young modulus E for generalized plane stress and equal to E/( 1 – is2) for plane strain, with is = Poisson’s ratio, (a) Find the complementary energy as a function of p and a. (b) Find the energy release rate for this case (note that a is the half crack length, not the crack length).

2.5 To test the fracture behavior of rock, a large 50-mm-thick slab will be tested in a laboratory by injecting a fluid into a central crack of initial length 2ao = 100mm. Let p be the fluid pressure and V the volume expansion of the crack. In the assumption of full linear elastic behavior with Gf = 20 N/m, find and plot the p-V and Q-a curves the panel experiences when it is subjected to a controlled-volume injection until the crack grows up to 1000 mm, after which it is unloaded to zero pressure. Use effective elastic modulus E’ = 60 MPa in the expression for Q obtained in exercise 2.5.

2.6 For the panel of exercise 2.6, find and plot the p-V and Q-a curves for a test in which the crack extends from 100 mm to 1000 mm under volume expansion control and then the panel is unloaded. Assume that resistance to crack growth varies with crack extension Да in t&e form

TC= 267/^1-^] for 0 < Да < A (2.1.54)

Ц — Gj for Да > A (2.1.55)

where Gf = 100 N/rrrand A = 276 mm. Find the peak pressure and the maximum increase in volume. Use an effective elastic modulus E’ — 60 GPa.

2.7 Find the J integral for an infinite strip, of thickness b and width 2h, with a symmetric semi-infinite crack subjected to imposed zero displacements on its lower face and constant vertical displacement и on its upper face (Fig. 2.1.12; Rice 1968a). Assume linear elasticity and plane strain with known elastic modulus E and Poisson’s ratio is.

M ttt? tttHtf t MM M

2.8

A double cantilever beam specimen with arm depths h = 10 mm, thickness b ~ 10 mm, and initial crack length ao = 50 mm, is made of a material with a fracture energy Gj — 180 J/m2 and an elastic modulus E = 250 GPa. The specimen is tested at a controlled displacement rate so that the load goes through the maximum and then decreases, at still increasing displacement, down to 25% of the peak load. When this point is reached, the specimen is completely unloaded. Assuming that LEFM and the beam theory apply, find the P(u) and G(a) curves. Give the equations of the different arcs and the coordinates of the characteristic points.