PSI - Issue 13

Fuzuli Ağrı Akçay / Procedia Structural Integrity 13 (2018) 1695 – 1701 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

1699

5

Additionally, characteristic length (relevant to brittle fracture) can be written in terms of Mode I plane stress fracture toughness, , and the tensile strength, , as well, as in the following: 2 ,0 2 Ic I Ic Ic K l E   = + (13b) Eq. (13b) is obtained based on the assumption that critical energy dissipation (during the formation of new fracture surfaces) associated to initially crack-free bodies is equal to that of pre-cracked bodies. This is a reasonable assumption, as critical energy release rate is considered as a material property (see e.g., Anderson, 2005). An interesting and useful feature of Eq. (13b) is that this equation relates a microscopic quantity, ,0 , to macroscopic material parameters: , , . Furthermore, and more importantly, Eq. (13b) provides necessary foundation to obtain Mode I fracture toughness of a material simply by uniaxial tensile test specimen. The characteristic length appearing in Eq. (13b) differs from the critical distance L , implemented in the theory of critical distances (TCD). The main reason for this difference is attributed the fact that TCD is developed for notched bodies, and half critical distance ( L /2) is defined as the distance from the notch root where condition for failure is defined (Taylor, 2007). In other words, half-critical distance, L /2, is established as the point where two linear elastic stress-distance curves of different notches (a sharp notch and a blunt notch, for example), at the onset of failure, intersect each other (Pelekis & Susmel, 2017). On the other hand, the characteristic length defined in this article represents a microscopic material feature that is relevant to brittle fracture of initially crack-free bodies. Although it may not be of practical interest, mathematical interest triggers the evaluation of Eq. (13) for the upper limit case when tensile strength and Young’s modulus are on the same order of magnitude. In fact, physically, this represents the case of which tensile strength is on the same order of magnitude of cohesive strength. Hence, in this case, Eq. (13a) reduces to

E



l



(14a)

2 2 I Ic 

I

,0

E



2

(14b)





,0 2 I I l

Ic

Substituting Γ I = 2 into Eq. (14b) yields

I E l

s 

Ic  

(15)

,0

Tensile strength given in Eq. (15) is consistent with the cohesive strength equation. Theoretical cohesive strength is given as (Anderson, 2005)

E

s 

cs  =

0 (16) where 0 is atomic spacing (at equilibrium). Therefore, the above analysis suggests that atomic spacing represents the characteristic length scale when tensile strength is on the same order of magnitude of cohesive strength, as would be expected. Surprisingly, although the fracture criterion is developed at the continuum scale, it provides insight on fracture process at particular small scales as well. On the other hand, in practical cases, Young’s modulus is approximately two to three times higher order of magnitude than the tensile strength (i.e., ≫ ). Hence, in this case, Eq. (13b) reduces to 2 ,0 Ic I Ic l K E   (17a) x