Issue 2

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

Figure 1. Scheme of the typical fatigue crack propagation curve

Variable

Definition

Symbol

Dimensions

FL –2 FL –3/2

q q q

Tensile yield stress of the material Material fracture toughness Frequency of the loading cycle

σ y

1

K IC

2

T –1

ω

3

FL –3/2

Δ K = K max

- K min

s

Stress-intensity range

1

D

s s s

Characteristic structural size Characteristic internal length

L L L

2

h

3

a 0

Initial crack length

4

max K K R = min

r

Loading ratio

–

1

Table 1. Main variables governing the fatigue crack growth phenomenon.

can be represented in terms of a product of powers of the dimensions of the remaining quantities. Parameters i s are such that their dimensions can be expressed as products of powers of the dimensions of the parameters i q . Fi- nally, parameters i r are nondimensional quantities. As regards the phenomenon of fatigue crack growth, it is possible to consider the following functional dependence:

dependent approaches, a relation between the Paris’ law parameters C and m is proposed. As a result, it is shown that only one macroscopic parameter is needed for the characterization of damage during fatigue crack growth. 2 CORRELATION DERIVED ACCORDING TO SELF-SIMILARITY CONCEPTS According to dimensional analysis, the physical phe- nomenon under observation can be regarded as a black box connecting the external variables (called input or governing parameters) with the mechanical response (output parameters). In case of fatigue crack growth in Region II, we assume that the mechanical response of the system is fully represented by the crack growth rate, 0 =d / d q a N , which is the parameter to be determined. This output parameter is a function of a number of variables: ( ) 0 1 2 1 2 1 2 , , , ; , , , ; , , , , n m k q F q q q s s s r r r = K K K (2) where i q are quantities with independent physical dimen- sions, i.e. none of these quantities has a dimension that

( y a F K K D h a R N σ ϖ = Δ − IC 0 , , ; , , , ;1 , )

d

(3)

d

where the governing variables are summarized in Tab. 1, along with their physical dimensions expressed in the Length-Force-Time class (LFT). From this list it is possi- ble to distinguish between three main categories of pa- rameters. The first category regards the material parame- ters, such as the yield stress, y σ , and the fracture toughness, IC K . The second category comprises the vari- ables governing the testing conditions, such as the stress- intensity factor range, K Δ , the loading ratio, R , and the frequency of the loading cycle, ω . Concerning environ- mental conditions and chemical phenomena, they are not considered as primary variables in this formulation and

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