# Setting up Solutions from Closed-Form Expressions

3.1.1 Closed-Form Solutions from Handbooks

A large number of solutions for stress intensity factors have been collected in handbooks (Sih 1973; Rooke and Cartwright 1976; Tada, Paris and Irwin 1985; Murakami 1987). The energy release rates are not included in these handbooks because the expressions for Kj arc simpler, and the expressions for Q are very easily obtained from those for К/ using Irwin’s equation. Л few of the collected solutions are exact. Most of them are empirical fits of approximate but accurate numerical results. In a few cases no analytical expressions are given, but a graphical representation of the results is provided. In most cases of complex analytical expressions, a graphical representation is provided as well as the closed-form expression. Different fits, with different ranges of applicability and different accuracies, may be available for a given case, a point that must never be overlooked.

In this book, we write the expressions for К/ in the form (2.3.11) because of our interest in the size effect. This is to be taken into account when comparisons are made with handbooks. in which the prototype expression for a stress intensity factor is taken to be that for a center cracked infinite panel, so most handbooks use the form Ki — a^s/miF(a), where F(a) is a dimensionless function of the relative crack length. Comparing this with (2.3.11), it turns out that the relationship between k(a) and F(a) is k(a) = sfnaF(ct).

Example 3.1.1 For a single-edge cracked beam subjected to three-point bending (Fig. 3.1.1a), the expression for К і depends mildly on the shear force magnitude near the central cross-section, i. e., on the span-to-depth ratio. Fig. 3.1.1b shows a plot of k (a) for the limiting case of pure bending (formally equivalent to S/D —* oo) and for the case S/D — 4 (a standard ASTM testing geometry). Analytical approximate expressions for these two cases were produced by Tada, Paris and Irwin (1973), for pure bending, and by Srawley (1976), for S/D = 4. Recently, Pastor et al. (1995) produced expressions accurate within 0.5% for any a/D. The latter expressions have the advantage over the former that their structure is identical (additionally, they correct a 4% error that crept in the Srawley formula in the limit of short cracks). With the definition of any shown in Fig. 3.1.1, the shape factor takes the form (3.1.1)

where pr(cx) is a fourth degree polynomial in a. The expression of the polynomials for S/D = 4 and oo (pure bending) are

p4(a) = 1.900 – a [-0.089 + 0.603(l-a) – 0.441(1 – a)2 + 1.223(1 – a)3] (3.1.2)

Poo(a) = 1.989 – a( 1-а) [0.448 -0.458(1 – a) + 1.226(1 – a)2] (3.1.3)

Note that for very short cracks (a —> 0) the shape factors k(a) behave as со^/тта, where Co is a constant close to 1.12. For very deep cracks (a —» 1), k(a) oc cj(l — a)"3/2, where C] is a constant close to 2/3. This is the general trend for specimens in which the resultant force over the crack plane is zero.

D

Example 3.1.2 The stress intensity factor for the center cracked panel of Fig. 2.1.1 was obtained numerically by Isida (1973) with very high accuracy for II 3> D. These results may be approximated within 0.1% by the Fedderscn-Tada expression (Fedderscn 1966; Tada, Paris and Irwin (1973))

К і = aVDk(a). k( a) (1 – 0.1a2 + 0.96a4) (3.1.4)

V cos7ra

In this case, the behavior for short cracks coincides with that for an infinite panel, k(n) —* JnCt for long cracks (a —> 0.5), k(a) —> c2( 1 – 2a)-1/2 where c2 is a constant very close to 1. D

Example 3.1.3 The stress intensity factors for cracks with concentrated loads applied on the crack faces display a completely different type of dependence on the crack length. The simplest case is that of a center-cracked infinite panel loaded with two equal and opposite forces at the centers of the crack faces (Fig. 3.1.2, with а/II and a/D – C 1). The stress intensity factor is then written as (3.1.5) which shows that for a given load P the stress intensity factor decreases as the crack length increases. It is obvious, however, that this decrease cannot be indefinite for a real (finite) plate. Indeed, based on numerical results by Newman (1971), Tada, Paris and Irwin (1973) proposed the following modified formula for a finite panel of width D and height H = 2D:

with error less than 0.3% for any aj D. Note that for a relatively small crack length (a/D —> 0), this expression coincides with that for the infinite panel. On the other hand, for large cracks (a/D –> 0.5), k(a) —> c2(l — 2a)-1/2, which coincides with the previous example if one sets ay — P/bD. D