Issue 47

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 47 (2019) 348-366; DOI: 10.3221/IGF-ESIS.47.26

Parameter ρ ref

quantifies the relative contribution of hydrostatic to deviatoric stress components in a multiaxial stress. Two

=1 for tension or bending (only normal stress), ρ ref

limiting cases exist for uniaxial loading: a stress ratio ρ ref

=0 for torsion

(only shear stress). A purely hydrostatic state of stress would have ρ ref =0 or 1), the reference S-N line coincides with the line of the corresponding uniaxial loading (tension or torsion). In any other case in which the loading is multiaxial (i.e. for any other value 0≤ ρ ref ≤1), the reference S-N curve would lie between those for tension and torsion, its position being established by ρ ref .  . On either two limiting cases (i.e. ρ ref

J a

σ,τ(log)

(log)

k σ

tension

σ A

torsion

k τ

J A,τ

k τ

τ A

ρ =0 (torsion)

k ref

J A,ref

ρ = ρ ref

k σ

/  3

σ A

J A, 

tension (scaled)

ρ =1 (tension)

N (log)

N (log)

Figure 3 : Relationship between S-N lines in Wöhler diagram (left) and Modified Wöhler Diagram (right). The reference S-N line for a general multiaxial stress ( ρ = ρ ref ) is also shown. Symbol J a stands for a2, J .

, where C ref

= N A

·(J A,ref

) kref is a

In a log-log diagram, the reference S-N line is expressed by the equation

k

ref CN J  

ref

a

strength constant; J A,ref

is the amplitude strength (at N A

cycles) and k ref

the inverse slope. They are linearly interpolated as

[7]:





   J

J  

J

J





 A A ref ,

A,ref

A,

,

(12)





 k k

 

 

k k







ref

ref

from the amplitude strengths J A,  , J A,τ

and inverse slopes k σ

, k τ of the tension and torsion S-N curves.

Step 4 – Fatigue damage calculated for each stress projection Each stress projection Ω p,i ( t ), obtained in Step 2, is a uniaxial random stress. Its damage in time unit (damage/s), say d ( Ω p,i ( t )), can be estimated in the frequency-domain by uniaxial spectral methods. No restriction is imposed on which method to use from those available in the literature [15,16], although “wide-band methods” are recommended if the PSD of each projection Ω p,i ( t ) is not narrow-band. For example, fairly accurate estimations are given by the “Tovo-Benasciutti (TB) method” [15,16]:         1 2 i0 1 0 ref k , ref ,i TB,i p,i TB k Γ C d ref    (13)

 

  

2

where Γ ( - ) is the gamma function and η TB,i depends on a proper weighting coefficient b app

is a correction factor that accounts for the spectral bandwidth of S p,i (for their expressions, see [15,16]). In the limiting case of narrow-band ( f ) and it

PSD, η TB,i →1. Eqn. (13) has been proved to provide estimations close to those from Dirlik’s method, which the reader may be more familiar to [17]:                i k i i ref k ref k i i k ,i p,i ip DK D RD k k QD d ref ref ref ref ,3 ,2 ,1 0 , 1 2 1   (14)

  

 

  

  

C

2

ref

354