PSI - Issue 10

V.N. Kytopoulos et al. / Procedia Structural Integrity 10 (2018) 272–279

275

V.N. Kytopoulos et al. / Structural Integrity Procedia 00 (2018) 000 – 000

4

W

d

d

d

s

s

s

Fig 2. Sketch of the optimum hole arrangement of initial damage

and between hole and edge of the specimen is given. By this distance negligible strain field interactions between holes can be assumed (Krajcinovic et al. (1987); Lemaitre (1996)). Thereafter, if n is the number of holes then one can show that (2 n +1) d ≤ W . Furthermore, by putting A 0 =Wt =initial cross-sectional area of a specimen with thickness, t , and A d =( ndt )=damaged area, one can obtain the ratio:

d A n A (2n 1)  

0

which for n →∞ gives 1/2. This means that the initial damaged area cannot be larger than 50% of the total initial cross-sectional area of the specimen. Due to the complexity of the problem and for the present purposes some convenient and simplifying approximations and assumptions were needed to be taken into considerations. These were based on the following formulae:   f f f D D ( D ) lnD lnD       1 1 0 0 1 (5) f D e (7) where x=σ fd /σ f 0 and σ fd is the fracture stress of the initially-damaged material and σ f 0 is the fracture stress of the dam age-free (virgin) material. Eqs.(6) and (7) fulfill one of the basic theoretical damage conditions: At x → 1, i.e. fractured specimen, it holds D f → 1 and D f → 0 (Krajcinovic et al. (1987); Lemaitre (1996); Lemaitre and Desmorat (2005)). Eq.(5) presents the basic theoretical relation (Eq.(1)) between the initial damage D 0 and the fracture damage D f of the material, whereas Eqs.(6) and (7) correspond to the proposed simplifying approaches. On the basis that 0.5< D f <1 (Krajcinovic et al. (1987); Lemaitre (1996); Lemaitre and Desmorat (2005)) and using Eq.(5) for some given values of fracture damage D f , the corresponding theoretical values of the initial damage D 0 can be calculated. This was done at first for two fixed, most probable theoretical values of k=0 and k=0.5. Then, by means of the theoretical values of the ( D 0 , D f ) pair and using Eq.(6), constants A and B can be determined by a best fit procedure. The best fit procedure provided the corresponding correlation coefficient equal to about 0.985, which may reflect the relative correctness of Eq.(6). Combining Eqs. (6) and (7) one can obtain the following equation:    1 B f D A( D )   0 1 (6) ( x )c

B( x )c D A( e )     1 0 1

(8)

Introducing the experimental values of the ( D 0 , x ) pair into Eq.(8) the new constants A,B,C are calculated. In turn, by means of Eq.(7) the “true” fracture damage D f is evaluated. Bearing in mind the presented theoretical considerations, it follows that the calculation of all the previously mentioned additional four parameters were primarily based on the evaluation of “master sets” ( D 0 , D f ), obtained according to the above described procedure of operational approach.