Page 70 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 50 Thursday, April 14, 2005 6:28 PM

50 Fracture Mechanics: Fundamentals and Applications

FIGURE 2.20 Stress concentration effects due to a through crack in finite and infinite width plates: (a)
infinite plate and (b) finite plate.

One technique to approximate the finite width boundary condition is to assume a periodic array
of collinear cracks in an infinite plate (Figure 2.21). The Mode I stress intensity factor for this
situation is given by
/
K I a = σπ 2 W tan  π a    12 (2.45)

 π a  2 W  
The stress intensity approaches the infinite plate value as a/W approaches zero; K is asymptotic
I
to a/W = 1.
More accurate solutions for a through crack in a finite plate have been obtained from finite-element
analysis; solutions of this type are usually fit to a polynomial expression. One such solution [12] is given by

  π a  / 12    a  2  a  4 
K I a = σπ  sec   1 − . 0 025 + . 0 06  (2.46)
  2 W     W   w  

Figure 2.22 compares the finite width corrections in Equation (2.45) and Equation (2.46). The
secant term (without the polynomial term) in Equation (2.46) is also plotted. Equation (2.45) agrees
with the finite-element solution to within 7% for a/W < 0.6. The secant correction is much closer
to the finite element solution; the error is less than 2% for a/W < 0.9. Thus, the polynomial term
in Equation (2.46) contributes little and can be neglected in most cases.
Table 2.4 lists stress intensity solutions for several common configurations. These K solutions
I
are plotted in Figure 2.23. Several handbooks devoted solely to stress intensity solutions have been
published [12–14].