Issue 77

M. Rehaman et alii, Fracture and Structural Integrity, 77 (2026) 45-55; DOI: 10.3221/IGF-ESIS.77.04

A NALYTICAL ESTIMATES OF STRESS INTENSITY FACTORS

T

he analytical formulation for the asymmetric three-point bend specimen is not available in the literature. The formulations proposed by He and Hutchinson [23] for estimating K for an asymmetric four-point bend specimen are used to modify for the asymmetric 3PB specimen as under:

2 3 Qs hW

a       W





a

(7)

K

F

I

I

where, 3 Qs / hW 2 is the applied bending stress in the case of the 3PB specimen Similarly,     3/2 1/2 1/2 / 6 1 / II II Q a W a F K W hW a W        

(8)

In the case of the 3PB specimen, Q F  Q = Shear Force where, h is the thickness of the specimen, a is the crack length, and W is the width of the specimen. The Fi = (a/W) polynomial equations by Murakami [24]:

I     for 0 / 0.7 a W   a W   F II     for 0 / 1 a W   a W   F





















2

3

4

5

1.122 1.121 / 3.740 / a W a W   



19.05 / a W a W a W   22.5 /

3.873 /

(9)

















2

3

4

7.264 9.37 / 2.74 /   

1.87 / a W a W a W a W   1.04 /

(10)

Using the above equations, a C program has been developed to compute K I and K II for various applied loads.

R ESULTS AND D ISCUSSION

T

o estimate K, T, and to accomplish the contour and magnitude of PZ ahead of the crack-tip, various load steps were applied on an asymmetric TPB specimen of different a/W and β eq . For different load steps and β eq , the K and T for the asymmetric TPB specimen have been computed using the ABAQUS postprocessor. The analyses of K I and K II for each β eq will be handled using the effective stress intensity factor ( K eff ), given by 2 2 eff I II K K K   . In the analyses, the K eff will be normalized by yield stress and the square root of the thickness ( σ y h 1/2 ). The premeditated values of normalized K eff for a / W = 0.4-0.7 in asymmetric TPB specimens , typically for a 1kN applied load, are plotted against β eq in Fig. 3. The finite element analysis (FEA) results and theoretical results of normalized K eff vs. β eq show good agreement. The analytical values of K eff for various load steps, a / W , and β eq of the asymmetric TPB specimen have been calculated from Eqns. (7) and (8). The percentage error between the FEA and theoretical results is less than 3.5%. This error may be attributed to the varied s / L ratio and bend loading condition of the asymmetric TPB specimen. Fig. 3 demonstrates that the normalized K eff increases as β eq changes from 0 o (Mode II) to 90 o (Mode I), and for the same applied load, the normalized K eff is almost 20 times higher for β eq = 90 o than for β eq = 0 o . Also, Fig. 3 shows that normalized K eff depends on a / W and the applied load. The T-stress will be evaluated using the dimensionless biaxiality ratio (B) eff B T a K   and K eff [12, 25]. The variation of B vs. a/W is shown in Fig. 4 for different β eq . From Fig. 4, the computed B exactly matches Sham’s [26] results for the TPB

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