Page 387 - Mechanical Behavior of Materials

Section 8.7 Plastic Zone Size, and Plasticity Limitations on LEFM 389

on the in-plane dimensions of Eq. 8.39 are less stringent than Eq. 8.40, so that the limits on the use
of LEFM are automatically satisfied if plane strain is satisfied.

Example 8.6
For the situation of Ex. 8.1:

(a) For a = 10 mm, determine whether or not plane strain applies and whether or not LEFM
is valid. Also estimate the plastic zone size.
(b) Do the same for the estimated a c = 16.3 mm.

Solution (a) Plane strain applies if Eq. 8.40 is satisfied. Use t = 5 mm, b = 50 mm, and K
as calculated in Ex. 8.1(a), and also σ o = 415 MPa for 2014-T651 Al, to obtain

2 √ 2
K 17.7MPa m

t, a,(b − a), h ≥ 2.5 = 2.5 ?
σ o 415 MPa
5, 10, 40, large h ≥ 0.0045 m = 4.5mm?

Yes, the test is successful, so plane strain applies and LEFM is applicable (Ans.). The plastic
zone size is then estimated as the value for plane strain from Eq. 8.38:

√ 2
1 K 2 1 17.7MPa m
2r oε = = = 0.19 mm Ans.
3π σ o 3π 415 MPa
√
(b) For a c = 16.3 mm and K = K Ic = 24 MPa m, the plane strain test is similarly applied.
√ 2
24 MPa m

5, 16.3, 33.7, large h ≥ 2.5 = 8.4mm?
415 MPa
No, the test fails, and plane strain does not apply (Ans.). But LEFM may still be applicable if
Eq. 8.39 is satisfied. Thus, we ask,
2 √ 2

4 K 4 24 MPa m
a,(b − a), h ≥ = ?
π σ o π 415 MPa
16.3, 33.7, large h ≥ 4.3mm?
Yes, the test is successful, and LEFM is applicable (Ans.). The plastic zone size is then estimated
as the value for plane stress from Eq. 8.37:
2 √ 2
1 K 1 24 MPa m

2r oσ = = = 1.06 mm Ans.
π σ o π 415 MPa