Moving Cracks

3.4.1 Steady Dynamic Crack Growth

In this section, the asymptotic solution for the problem of a crack propagating with constant velocity in an elastic body under steady state conditions is given. Consider a crack propagating in the planeX₂= 0and that the crack tip under study is moving in the direction ofe1with constant speedv. At timet= 0the crack tip passesX₁= 0.

Suppose that any fieldf(X₁, X₂, t)over the body is time invariant in a reference frame moving with the crack tip, that is

f(X₁, X₂, t) =f(x₁, x₂), (3.52)

where x₁ = X₁−vt andx₂ = X₂. Then a steady state solution for the prob-lem is sought and the number of independent variables in the probprob-lem can be reduced from three (X₁, X₂, t) to two (x₁, x₂). With this in mind, the problem can be rewritten in terms of complex potentials, see [Radok, 1956]. Express-ing these potentials in power series and after the imposition of the traction free condition, the asymptotic solution for the problem can be obtained. This proce-dure was used in [Nishioka and Atluri, 1983] to find the displacement and stress fields for all three fracture modes in a unified fashion, see also the works of [Cotterell, 1964, Rice, 1968a, Sih, 1970]. As a result, the displacement field for in-plane loading can be written as

⎡

θd= arctanαdx₂

The results presented here differ from those in [Nishioka and Atluri, 1983] in a factor (n+ 1)/2, in order to be consistent with the static displacement field (2.4), (2.5), satisfying thatlimv→0fij(r, θ, v) = fij(r, θ), wherefij(r, θ)are the functions for the static case defined in (2.5). As in the static case, the solution contains the rigid-body translations forn= 0.

In the case of the stresses, the asymptotic solution can be written as

⎡

(3.62) Again, as in the static case, the coefficientsKI1andKII1, forn= 1, will be denoted simply byKIandKII, which are, in this case, the dynamic stress intensity factors defined byKI = limr→0√

2πrσ₂₂(r, θ = 0)andKII = limr→0√

2πrσ₁₂(r, θ = 0), respectively. At this point, it is worth noting that since the solution is steady state, the SIFs are independent of time and the position of the crack tip. Forn= 2, the only nonzero component isg₁₁₂, leading toσⁿ₁₁⁼²= 4KI2BI(α_d²−α²_s)/√

2π.

3.4.2 Nonsteady Crack Growth

The problem of nonsteady crack growth is the most complicated of all problems considered so far. Despite this, it can be proved that for nonsteady crack growth, the spatial dependence of the singular term in the stress field is the same as in the case of steady crack growth and depends only on the instantaneous value of the crack propagation speed v. The only difference is then, that the stress inten-sity factors are now time dependent, see [Freund and Clifton, 1974, Nilsson, 1974, Achenbach and Bažant, 1975]. While the same result applies for the constant term, this is no longer true for higher order terms; the stress field up to orderO(√

r)is given in [Rosakis et al., 1991], while the solution up to orderO(r)can be found in [Nishioka, 1995]. The first term of the displacement and stress fields is given by

u₁= KI(t)

In the expression ofσ₁₂, the coefficient(1+α²_s)²multiplyingr⁻¹s ^/2cos(θs/2)differs from that given in [Nishioka and Atluri, 1983] (39c), but agrees with the expression given in a later work of NISHIOKA[Nishioka, 1995]. The difference is therefore attributed to a typographical error in the previous work of NISHIOKA.

3.4.3 Dynamic Stress Intensity Factors for Moving Cracks Consider an unbounded body and a semi-infinite crack propagating in the plane X₂= 0. The body is stress free and at rest for the timet≤0and the crack tip moves in thee1direction according toX₁^tip = l(t)fort > 0, with the only restriction that the crack tip speedv(t) = ˙l(t)lies in the interval0≤v(t)< CR. The crack surfaces are subjected to time dependent loading. Under these very general conditions of nonuniformly crack propagation, a remarkable result can be proved. The stress in-tensity factors are given by a universal function of the instantaneous crack tip speed vtimes the corresponding stress intensity factor for the given surface load applied to a stationary crack of lengthl(t). Mathematically

KI(t, l, v) =kI(v)KI(t, l,0), KII(t, l, v) =kII(v)KII(t, l,0),

(3.65) whereKI(t, l,0)andKII(t, l,0)are the stress intensity factors for a stationary crack of lengthl, andkI(v)andkII(v)are universal functions defined by

kI(v) =S₋(1/v) 1−v/CR

1−v/Cd

, kII(v) =S₋(1/v) 1−v/CR

1−v/Cs

, (3.66)

where the functionS₋(·)has been defined in (3.45). Freund [Freund, 1990] noted thatS₋(1/v)is close to unity for the range of velocities0 < v < CR, which can be used for practical applications. For the anti-plane case,kIII(v) =

1−v/Cs. In the light of these results, solutions for the SIFs for stationary cracks can be used to calculate SIFs for nonsteady crack propagation.

The described result was proved by FREUNDfor the particular case of time indepen-dent crack surface loading with general spatial variation and spatially uniform time dependent load, see [Freund, 1990] and the references therein. Later, KOSTROV

[Kostrov, 1975] proved the validity of these results for all three fracture modes in the case of general crack surface loading, see also [Burridge, 1976], and analysed the case of finite cracks. See [Melville, 1981] and references therein for an application of the results of KOSTROVto finite cracks. In this respect, it must be pointed out that the result mentioned above remains valid until the arrival at the crack tip under analysis of other waves, such as those generated or reflected by other cracks, reflected from the boundary, generated by other loads, etc.

At this point, four asymptotic solutions have been considered: stationary cracks un-der static loads, stationary cracks unun-der dynamic loads, steady state dynamic crack growth and nonsteady crack growth. From the point of view of the numerical experi-ments, it is always desirable to compare the numerical with the exact solution of the

problem. For stationary cracks and steady state crack growth, exact solutions can be constructed using the fact that the truncation of the asymptotic solution, at any level, satisfies the respective equilibrium equation. Then this truncated expansion can be used as exact solution by simply imposing its values at the boundary, as boundary conditions. For the case of dynamically loaded stationary cracks, this is no longer true, but considering the coefficients of the first fourth terms of the expansion as a polynomial int, the solution of DENGcan be used to construct an exact solution by truncating the general solution properly, this will be explored further in Chapter 5.

Finally, for nonsteady crack growth, the truncated series does not satisfy the equilib-rium equation, and given the complicated structure of the high order terms, an exact solution is difficult to find. Despite this, the result described in this section can be used to check the accuracy of the solution at least at the level of the SIFs.