A statistical model of fatigue failure incorporating effects of specimen size and load amplitude on fatigue life

ABSTRACT

Among many contributing factors, the load range, number of load cycles and specimen geometry (including configuration and size) are three major variables for fatigue failure. Most existing statistical fatigue models deal with only one or two of these three variables. According to the statistical distribution of microcracks with respect to their size and spatial location, a weakest-link probabilistic model for fatigue failure is established to incorporate the combined effect of load range, number of load cycles and specimen size. The model reveals a compound parameter of load range and number of load cycles reminiscent of the empirical formulae of fatigue stress-life curve and its correlation with another compound parameter of cumulative failure probability and specimen size. Four sets of published fatigue test data are adopted to validate the model.

KEYWORDS:

• Fatigue
• probabilistic model
• microcrack distribution
• size effect

Acknowledgements

Wei-Sheng Lei is indebted to Professors Winfried Dahl and Wolfgang Bleck for hosting an earlier research stay at the Institute of Ferrous Metallurgy (IEHK), RWTH Acchen Technical University under the Alexander von Humboldt Research Fellowship.

Disclosure statement

No potential conflict of interest was reported by the authors.

Nomenclature

a	=	microcrack size
ai	=	initial microcrack size
$a_{max}$	=	maximum microcrack size
A, L, V	=	specimen surface area, length and volume in sequence
$A_0,L_0,V_0$	=	reference area, length and volume in sequence
$E$	=	elastic modulus
$F\left({\sigma }_e\ge s\right)$	=	fracture probability of an existing microcrack
$K_I$	=	mode-I stress intensity factor
$\mathrm{\Delta }K$	=	range of stress intensity factor
Kmax, Kmin	=	maximum and minimum stress intensity factors
$K_c$	=	critical stress intensity factor for local failure
$K_{th,L}$	=	local threshold of stress intensity factor
N	=	number of loading cycles
$n^{\mathrm{\prime }}$	=	cyclic strain-hardening exponent
P	=	cumulative probability
$p\left(\sigma ,a,V_0\right)$	=	fracture probability of volume element V0 with a microcrack
$R$	=	stress ratio
S	=	generalised cyclic load
$\overline{S}$	=	average value of $S$ on all the possible material planes
s	=	microscopic fracture strength
${\sigma }_{ys}$	=	yield stress
${\sigma }_1,{\sigma }_2,{\sigma }_3$	=	principal stresses
${\sigma }_{ij}\left(t\right)$	=	instant stress component at time t
${\sigma }_e$, $\mathrm{\Delta }{\sigma }_e$	=	effective stress and its range
${\sigma }_m$	=	mean stress
${\sigma }_n$	=	normal stress component
Δσ, $\mathrm{\Delta }{\sigma }_{nom}$	=	local and nominal stress ranges
σmax, σmin	=	maximum and minimum stresses
${\sigma }_a$, ${\sigma }_{a,nom}$	=	local and nominal tensile stress amplitudes
${\tau }_a,\tau {}_{a,nom}$	=	local and nominal shear stress amplitudes
${\dot{\epsilon }}_{ij}\left(t\right)$	=	instant strain rate at time t
εmax, εmin	=	maximum and minimum strains
$\mathrm{\Delta }\epsilon$, $\mathrm{\Delta }{\epsilon }_{nom}$	=	local and nominal strain ranges
${\epsilon }_a,{\epsilon }_{a,nom}$	=	local and nominal strain amplitudes
${\epsilon }_{p,a}$, $\mathrm{\Delta }{\epsilon }_p$	=	plastic strain amplitude and plastic strain range
$\mathrm{\Delta }\gamma$	=	shear strain range
${\epsilon }_f^{\mathrm{\prime }}$	=	fatigue ductility coefficient
$\mathrm{\Delta }{W}_t$	=	total strain energy range
$W_g$	=	strain work density per loading cycle with the period T
$B_{\alpha \beta }$	=	beta function
$H_b^{\ast }$	=	stress heterogeneity factor
$c_i$, $m_i$	=	model parameters in Paris’ law for microcracks ($i=I,II,III$)
$c_0,d$, m, $\chi$, $k_{g0}$, $\eta {}_1$, ${\eta }_2$, λ1	=	material constant
B, C, $N_{th},$ $S_{th}$, ${\sigma }_{th}$, $x_{th}$, Vth, $W_{th}$	=	thresholds
$a_0$, $N_0$, $N_u,$ $S_0,S_u$, $,\delta$, ${\sigma }_0{,}_0,W_u$	=	scale parameters
A1, $A_2,$ B, $Const,$ $c_1,c_2,D,{\beta }_1$, $k$,$q,W,\lambda {}_2$, ${\lambda }_3$, $\lambda {}_3$	=	constants
b, $c,\alpha ,\beta$	=	shape factors
Y	=	dimensionless parameter dependent on crack geometry
$y=f\left(x_1\right|x_2,x_3)$	=	$y$ as a function of variable $x_1$ with given values of variables $x_2\mathrm{and}{x}_3$
$\mathrm{\Omega }$, $\psi$	=	angles
HCF	=	high cycle fatigue
LCF	=	low cycle fatigue
VHCF	=	very high cycle fatigue
PDF	=	probability density function
f(a)	=	PDF of microcrack size (a)
g(s)	=	PDF of microscopic fracture strength (s)

Additional information

Funding

The financial support of the National Natural Science funding (No. 11872364) and CAS Pioneer Hundred Talents Program is acknowledged.