Issue 2

Al. Carpinteri et al., Frattura ed Integrità Strutturale, 2 (2007) 10-16

when increasing the parameter R [28]. Therefore, assum- ing again an incomplete self-similarity in 5 Π , we have:

their influence on fatigue crack growth can be taken into account as a degradation of the material properties. Fi- nally, the last category includes geometric parameters re- lated to the material microstructure, such as the internal characteristic length, h , and to the tested geometry, such as the characteristic structural size, D , and the initial crack length, a 0 . Considering a state with no explicit time dependence, it is possible to apply the Buckingham’s Π Theorem [19] to reduce by n the number of parameters involved in the problem (see e.g. [8, 20–26] for some relevant applica- tions of this method in Solid Mechanics). As a result, we have:

a N

d

=

d

2

β

1 K K R K K β β σ σ − − ⎛ ⎞ ⎛ ⎞Δ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 1 IC y IC 2 2 (1 ) (1 )

(6)

(

)

β

2 − Φ Π Π Π = 2 2 3 4 , ,

=

(

)

(

)

2 R K − Δ Φ Π Π Π 1 β 2 2 3 4 , ,

.

=

IC

y

Comparing Eq. (6) with the expression of the Paris’ law, we find that our proposed formulation encompasses Eq. (1) as a limit case when:

2

IC K a K D h a R K K K K σ σ σ σ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ Δ ⎜ ⎟ ⎟ y 2 2 2 0 2 2 2 IC IC IC IC , , , ;1 y y y

d

( m C K β = = 1 ,

N

d

)

(7)

(

)

m σ − −

2 R β − Φ Π Π Π 2 2 3 4 , ,

2

2

(1 )

.

IC y

As a consequence, from Eq. (7) it is possible to notice that the parameter C is dependent on two material pa- rameters, such as the fracture toughness, IC K , and the yield stress, y σ , as well as on the loading ratio, R , and on the nondimensional parameters 2 Π , 3 Π , and 4 Π . More- over, Eq. (7) demonstrates, from the theoretical stand- point, the existence of a relationship between the parame- ters C and m . 3 CORRELATION DERIVED ACCORDING TO THE CRACK GROWTH INSTABILITY CONDITION In this Section we derive a correlation between the Paris’ law parameters similar to that in Eq. (7) on the basis of the condition of crack growth instability. In fact, as firstly pointed out by Forman et al. [4], the crack propagation rate, d / d a N , is not only a function of the stress-intensity factor range, K Δ , but also on the condition of instability of the crack growth when the maximum stress-intensity factor approaches its critical value for the material. Focusing our attention on this dependence, Forman et al. [4] observed that the crack propagation rate must tend to infinity when max IC K K → , i.e. (8) This rapid increase in the crack propagation rate is then responsible for the fast deviation from the linear part of the Region II in the fatigue plot (see e.g. Fig. 1). Consid- ering the transition point labeled CR in Fig. 1 between Region II and Region III, the following relationship be- tween the crack growth rate and the stress-intensity factor range can be derived according to the Paris’ law: (9) max IC d lim d K K a N → = ∞ d m a v C K N ⎛ ⎞ = = Δ ⎜ ⎟ ⎝ ⎠ ( ) CR CR d

= Φ

− =

(4)

( ⎜ ⎝

⎠

)

1 2 3 4 5 = Φ Π Π Π Π Π , , , ,

.

At this point, we want to see if the number of the quanti- ties involved in the relationship (4) can be reduced fur- ther from five. Considering the nondimensional parame- ter IC / K K Δ , it has to be noticed that this is usually small in the Region II of fatigue crack growth. However, since it is well-known that the fatigue crack growth phenome- non is strongly dependent on this variable (see e.g. the Paris’ law in Eq. (1)), a complete self-similarity in this parameter cannot be accepted. Hence, assuming an in- complete self-similarity in 1 Π , we have:

2

1 β

( IC σ ⎛ ⎞ ⎛ ⎞ Δ = Φ Π Π Π Π ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 1 2 3 4 5 y IC , , , K a K K

d

)

,

(5)

N

d

and, consequently, the nondimen-

where the exponent β 1

sional parameter 1 Φ , cannot be determined from consid- erations of dimensional analysis alone. Moreover, the ex- ponent β 1 may depend on the nondimensional parameters i Π . It has to be noticed that 2 Π takes into account the ef- fect of the specimen size and it corresponds to the square of the nondimensional number Z defined in [8], and to the inverse of the square of the brittleness number s in- troduced in [20, 21, 27]. Moreover, the parameter 4 Π is responsible for the dependence of the fatigue phenome- non on the initial crack length, as recently pointed out in [10]. Repeating this reasoning for the parameter (1 ) R − , which is a small number comprised between zero and unity, a complete self-similarity in 5 Π would imply that fatigue crack growth is independent of the loading ratio. How- ever, this behavior is in contrast with some experimental results indicating an increase in the response d / d a N

CR

12