Issue 66

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 66 (2023) 233-260; DOI: 10.3221/IGF-ESIS.66.15

From the above formulae, it is seen that K I apart from α , σ ∞ and β , depends also on the material (i.e., on ν and E), while K II is material independent. In this context, the variation of K I against ν is plotted in Fig.19a. To plot K I ( ν ), E was kept fixed equal to 3.2 GPa (PMMA), while it was assumed that σ ∞ = –1 MPa, τ =0, and β =45 o (at which K I assumes its maxi mum value). In Fig.19b, K I is plotted against E for the as above σ ∞ , τ and β . In this case, ν was kept fixed equal to 0.3.

18

52 54 56 58 60 62 K I ꞏ 10 -6 [MPa ꞏ m 1/2 ]

6 10 K I ꞏ 10 -6 [MPa ꞏ m 1/2 ] 14

2

0.1

0.2

0.3

0.4

10

20

30

40

50

(a)

(b)

Poisson's ratio ν

Young's modulus E [GPa]

Figure 19: The variation of K I at the crack tip, for σ ∞ = –1 MPa, τ =0, and β =45

o : (a) against ν , for a fixed E=3.2 GPa, and (b) against

E, for a fixed ν =0.3. It should be noticed, however, that K I assumes extremely small values compared to K II since tan λ is a very small quantity. Actually, for β =0 ο , K I vanishes, in accordance with the zero crack opening displacement (see the second of Eqns.(9) or of Eqns.(43), for k=0 and β =0 ο ). For β =90 ο , K I vanishes as well, because λ vanishes. For all other values, i.e., for 0 ο < β <90 ο (in case the cracked plate is under uniaxial compression, i.e., k=0), K I assumes (very small) finite positive values, so that, upon combined with the negative K II -values it suffices full contact of the crack lips. Assuming again that E=3.2 GPa, ν =0.36, and σ ∞ = –1 MPa, K I and K II are plotted, for various values of τ , against β in Fig.20a. K I values are only visible in the magnified view of Fig.20b. As mentioned previously, K I attains its maximum values at β =45 o . In Fig.20a, the unnatural case of negative K I (corresponding to the unnatural overlapping) is also shown for overview reasons.

0.0 0.1

6

0 

I Naturally acceptable K

2 KI ꞏ 10 5 [MPa ꞏ m 1/2 ] 4

0.25

-0.4 -0.3 -0.2 -0.1 SIFs [MPa ꞏ m 1/2 ]

II K

0.50

0  I Unacceptable negative K 0.25

0.75

0.50

0.75

0

0 153045607590

0 153045607590

(a)

(b)

Crack inclination angle β [deg]

Crack inclination angle β [deg]

Figure 20: (a) The naturally unacceptable negative K I -value for a uniaxially compressed cracked plate, and the variation of the naturally acceptable K I - and K II -values due to the ‘general problem’, for various τ -values, against the crack inclination angle β . (b) A magnified view of the naturally acceptable K I -values. The findings of the present analysis regarding SIFs are in qualitative agreement with existing approaches addressing over lapping [29-31], but here, in particular for uniaxial compression of the cracked plate, K II is always accompanied with nor mal contact stresses on the crack. D ISCUSSION AND CONCLUSIONS ome classical concepts of LEFM were revisited in this study, in the light of a novel approach for quantifying the contact stresses developed on the lips of a ‘mathematical’ crack in an elastic plate, in case the crack lips are forced to mutual contact against each other. The inconsistency of this configuration with the classical solution of plane LEFM, prohibits determination of these contact stresses (recall the boundary condition of stress free crack lips), leading to unnat ural overlapping of the crack lips, and, moreover to unnatural negative values for the mode-I SIF. S

255