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Derivation of novel creep-integrated fatigue equations
International Journal of Fatigue 128 (2019) 105184
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Derivation of novel creep-integrated fatigue equations
T
E.H. Wong
⁎
Sino-Singapore International Joint Research Institute, China Energy Research Institute at NTU, Singapore
ARTICLE INFO
ABSTRACT
Keywords: Creep Creep fatigue Environmental fatigue Solder Stainless steel
This manuscript presents a systematic derivation of novel creep-integrated fatigue equations that possess the following unique characteristics: (i) creep damage and creep-fatigue interaction damage are embedded into the fatigue equation and requires no deliberate computation; (ii) the embedded creep damage is formulated from the constitutive equation of creep; and (iii) the integrated equation is capable of modelling the full range of creep fatigue from pure creep to pure fatigue. These characteristics ensure that these creep-integrated fatigue equations are fundamentally sound, easy to use, and applicable over a wide range of creep fatigue conditions. The creep-integrated fatigue equations based on three constitutive equations of steady-state creep: Dorn equation, Larson-Miller equation, and Manson-Haferd equation are derived and successfully validated using two metals of vastly different creep and fatigue characteristics – the SS316 stainless steel and the Sn37Pb solder.
1. Introduction Most engineering structural elements are known to fail by the mechanism of fatigue. But pure fatigue is rare; most incidences of engineering fatigue are accompanied by environmental damaging factors such as creep, corrosion, and radiation; etc. Among these, creep assisted fatigue or simply creep fatigue has attracted the most interest as it is responsible for the structural failure of large number of critical engineering components; such as turbine blades and rotors in aircraft and steam engines, vessels and tubing in nuclear reactor and fossil power plants, and, no least, solder joints in electronic assemblies. The study of the creep fatigue of metals has been built on the fundamental understanding of the fatigue and the creep of metals. The main challenge has been in compounding the damages due to these two mechanisms to arrive at a reliable model for life prediction. A simple approach that is widely practised in the electronic assembly community is to treat creep as time-dependent plasticity [1–3]. Thus, the life prediction models for fatigue, for example the Coffin-Manson equation [4,5], is extended to creep fatigue: pl
= CN
pl
+
c
= CN ,
(1)
wherein εpl and εc are the cumulative plastic and creep strains respectively in a single cycle of loading; N is the “life” measured as the number of cycles to failure; and C and β are fitting constants. However, this simplistic act of lumping creep damage together with fatigue damage falls apart in the face of microstructural evidence [6,7]: creep
⁎
damage in metals, especially those with low creep ductility, proceeds in the form of grain boundary cavitation in the bulk of the metals. Fatigue damage on the other hand proceeds from transgranular microcracks on the surface of metals. The early practice in the power industry was to treat creep and fatigue as two independent damages and assuming the two damages can be added up scalarly. This may be expressed mathematically as
Df + Dc
(2)
1,
where Df and Dc are the damages due to fatigue and creep, respectively. This is known as the linear damage summation rule [8]. Microstructural evidence suggests that these two kinds of microstructural damage tend to interact in the later stage and aggravate each other. This is generally referred to as creep-fatigue interaction (CFI), which tends to be particularly intense for metals with low creep ductility [7]. The damage equation is more appropriately expressed as [9–12]
Df + Dc + Dcf
1,
(3)
where Dcf is the damage due to CFI, which is a function of Dc and Df. The extensive experimental effort required to define this function is rather challenging and near impossible for some industries; for example, it is impractical to undertake comprehensive CFI characterisation for the considerable number of solder alloys used in electronic assemblies. The aerospace industry practices the strain range partitioning approach of Manson et al. [13] who proposed that metals have different tolerances for damages due to fatigue, pure cyclic creep, and cyclic
Address: Energy Research Institute at NTU, Singapore. E-mail address: [email protected].
https://doi.org/10.1016/j.ijfatigue.2019.06.044 Received 25 April 2019; Received in revised form 6 June 2019; Accepted 30 June 2019 Available online 02 July 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
Nomenclature
tc; tref
cycle time; reference cycle time - the cycle time below which creep is dormant time at rupture under steady stress condition; cumulated tR; tcR cycle time at rupture under pure cyclic creep t∞ a constant in the steady-stress Manson-Haferd parameter it is the intercept of iso-σs lines on T-log(tR) plot tcR∞; tc∞ a constant in the cyclic-rupture Manson-Haferd parameter tcR∞ = ϕεRt∞, a constant in the cyclic-damage MansonHaferd parameter tc∞ = ϕcϕεRt∞ β, βo, βb fatigue exponent; fatigue exponent in a pure fatigue condition, in a creep fatigue condition cumulated creep strain in a single cycle; the plastic strain εc; εpl range or the cumulated plastic strain in a single cycle εp; εp,ref (the residual capacity of) plastic strain amplitude in a creep-fatigue condition; (the capacity of) plastic strain amplitude in a pure fatigue condition, corresponding to cyclic life N εc,eff the effective cyclic creep strain defined as c, eff = p, ref p ¯ secondary stage strain rate under steady-stress condition; II ; c the average creep strain rate in a cyclic condition ¯c = εc/ tc εR; εcR creep strain at rupture under steady stress condition; cumulated cyclic creep strain at rupture creep strain factors: ϕc = εc/εcR, ϕεR = εcR/εR ϕc, ϕεR σ; σ(t); σs cyclic stress amplitude; stress-time function; steady uniaxial tensile stress
B; BR, BcR, Bc a coefficient in the creep constitutive of Larson-Miller; a coefficient in the steady-stress Larson-Miller parameter BR = B/εR, in the cyclic-rupture Larson-Miller parameter BcR = B/εcR, in the cyclic-damage Larson-Miller parameter Bc = B/εc C; Co, Cb, fatigue coefficient; fatigue coefficient in a pure fatigue condition, in a creep fatigue condition Df; Dc; Dcf; Dc,eff damage due to fatigue; damage due to creep; damage due to creep-fatigue interaction; effective creep damage that includes damages due to creep and creep-fatigue interaction f cyclic frequency N; Nexp; Ncom cyclic life; experimental cyclic life; computed cyclic life Pk(σs); PcRk(σ), PcDk(σ) stress parameter for creep rupture under steady-stress σs; stress parameter for creep rupture, for creep damage, under cyclic stress amplitude σ; the subscript k may be “D”, “LM”, or “MH” and refers to the steady-stage creep constitutive equation of Dorn, LarsonMiller, and Manson-Haferd, respectively PcRk(εp), PcDk(εp) plastic-strain parameters transformed from the corresponding stress parameters using Ramberg–Osgood relation T; Tref temperature; reference temperature - the temperature below which creep is dormant creep fatigue; and the different tolerances may be expressed as
= Cp N
pl
p,
c
= Cc N
c,
cp
= Ccp N
cp .
constants, and tc is the cycle time. However, the fitting constants in the Engelmaier equation have frequently been reported to be dependent on the designs of the electronic assemblies and the cyclic temperatures [21]. The literal act of making the fatigue coefficients a function of temperature and/or cycle time does not integrate creep damage into the fatigue equation. Wong & Mai [22] suggested that a fatigue equation that has fully integrated creep damage must be capable of modelling the extreme conditions of creep fatigue; that is, the conditions of pure fatigue and pure creep. This condition is neither satisfied by the equation of Coffin nor Engelmaier. Wong & Mai [22] proposed the following form of integrated creep-fatigue equation:
(4)
The total damage is again defined by Eq. (3) except that Dcf is now assumed to be independent of Dc and Df. LeMaitre & Plumtree [14] and Chaboche [15,16] have advocated the continuum damage model, which gives, in the absence of mean stress, the rates of creep and fatigue damages as: dDf
=
dN dDc dt
=
r A (1 B (1
s (t ) D)
p
(1
D)
(1
D) q ,
D)
(5)
where σ(t) is the instantaneous stress; σ is the amplitude of a cyclic stress; A, B, p, q, r and s are temperature dependent material characteristics. While the total damage is given by Df + Dc, the interaction between the creep and the fatigue damages is embedded in Eq. (5). Coffin [17] proposed the “frequency-modified fatigue life” equation: pl
= C (T )(Nf k (T ) 1 )
p
= Co N
where
E ( T , tc )
=
o [1
+ b1 T
b2 ln(1 + b3/ tc )],
Tref )
c2 log(tc/ tref ), T > Tref , t > tref ,
Tref and tref are the reference temperature and the reference cycle time, respectively, below which creep is dormant. The function c(T, tc) incorporates creep damage and has a value that lies between 0 and 1; the extreme conditions c(T, tc) = 0 and 1 represent pure creep and pure fatigue, respectively. Liu et al. [23] proposed that c1 and c2 may be expressed in terms of the Manson-Haferd stress parameter, PMH(σs) [24]:
c1 ( ) =
c2 1 , c2 = , PMH ( s ) log(t / tref )
(11)
wherein σs is a time-steady uniaxial tensile stress applied to a specimen while characterizing the time to creep rupture, tR; t∞ is the time at which the iso-σs lines converge in the T-log(tR) space. A critical flaw in the above integrated creep-fatigue equations, Eqs. (6)–(11), is that the embedded creep damage functions are inconsistent with the constitutive equations of creep. We shall present in this manuscript a systematic approach to deriving novel creep-integrated fatigue equations that possess the following unique characteristics: (i) creep damage and creep-fatigue
(7)
E (T , t c ) ,
c 1 (T
(10)
where f is the cyclic frequency and C(T), β(T) and k(T) are functions of temperature. While it lacks the rigor of the continuum damage mechanics approach, its simplicity – there is no need to compute creep strain explicitly – has attracted much interest in the electronic assembly industry. Solomon [18] and Shi et al. [19] have modelled the creepfatigue life of eutectic SnPb solder using this technique. Ironically, both researchers have reported a dependence of the frequency exponent, k (T), on the cyclic frequency. If this is true, then Eq. (6) is fundamentally flawed. Analysing the thermomechanical fatigue of eutectic SnPb solder joints in electronic assemblies, Engelmaier [20] proposed a different form of integrated creep-fatigue equation, which may be expressed more concisely as p
(9)
o,
where c (T , tc ) = 1
(6)
(T ) ,
= Co c (T , tc ) N
(8)
εp is the amplitude of plastic strain; Co, βo, b1, b2, and b3 are fitting 2
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
interaction damage are embedded into the fatigue equation and requires no deliberate computation; (ii) the embedded creep damage is formulated from the constitutive equation of creep; and (iii) the equation is capable of modelling the full range of creep fatigue from pure creep to pure fatigue. We shall also generalise the method to model general environment aggravated fatigue.
therefore not a reliable equation for modelling creep-fatigue. It is worth highlighting that the Engelmaier equation, Eq. (7), falls under this class of equation. 2.2. Creep parameters 2.2.1. Steady-stress creep rupture parameters Assuming the dominance of the secondary stage of creep, the steady-stress creep strain rate is frequently described in the form:
2. Derivations 2.1. The forms of creep-integrated fatigue equation
II
= Co N
(12)
o,
II
where εp,ref is the plastic strain capacity – define here as the amplitude of the alternating plastic strain. The subscript “ref” emphasizes the condition of pure fatigue. Creep damage can be integrated into the fatigue equation by modifying the fatigue coefficients Co and βo. A generic creep-integrated fatigue equation may take the form: p
= Co c ( p, T , tc ) N
o b ( p, T , t c ) ,
= c ( p , T , tc ) N
o [b ( p, T , tc ) 1]
=
p
+
(13)
R
(14)
= Co c ( p, T , tc ) N
o;
= Co N
Let or
p / p, ref
=1
N
o [b ( p, T , tc ) 1] .
(22)
II tR,
R
f ( s)
=
tR , e H / RT
(23)
H ( s) ln BR + ln tR = , R 1/ T
(24)
T
Tref
ln(tR/ t )
.
(25)
II
=
R
tR
= De (T
Tref ) r ( s ) ,
ln( R/ tR) = ln D + (T
T
Tref ;
Tref ) r ( s ).
(26a) (26b)
Rearranging gives
(17)
PMH ( s ) =
T Tref 1 = , r ( s) ln(tR D / R )
(27)
which suggests D = εR/t∞. For convenience, we shall refer to Eqs. (20), (21), and (26) as Dorn equation, Larson-Miller equation, and MansonHaferd equation respectively. 2.2.2. Cyclic-stress creep rupture parameters If we could perform a cyclic-stress experiment at a stress amplitude σ and a cyclic creep strain, εc, such that pure cyclic creep rupture of a test specimen occurs after n cycles, then, using the Dorn parameter as an illustration, the average creep strain rate experienced by the test specimen shall take the form of Eq. (20) as
while making b(εp,T,tc) = 1 at pure fatigue and b(εp,T,tc)=∞ at pure creep. The fractional creep damage is given by
1
(21b)
H ( s )/ RT ,
Eq. (25) suggests the mechanism of creep is dormant at temperature below the reference temperature, Tref. The corresponding steady-stress creep strain rate equation may be deduced as follows:
(18)
o b ( p, T , tc ),
=
PMH ( s ) =
c(εp,T,tc) is now the residual fractional fatigue capacity and 1 − c (εp,T,tc) the fractional creep damage. Mathematically, the conditions of pure fatigue and pure creep may also be modelled by letting c(εp,T,tc) in Eq. (13) equate to unity; that is, p
(21a)
0,
wherein BR = B/εR. The Dorn parameter, PD(σs), describes the gradients of the iso-σs lines in the tR vs. eH/RT domain and the Larson-Miller parameter, PLM(σs), describes the gradients of the iso-σs lines in the ln (tR) vs. 1/T domain. Based on the observation that the iso-σs plot of the creep rupture data on the T vs. ln(tR) domain appeared to converge at a single point, (Tref, t∞), Manson-Haferd [24] came to the arbitrary stress parameter:
herein Dc,eff captures the damages due to creep and creep-fatigue interaction. Referring to Eq. (14), the condition b(εp,T,tc) > 1 models increasing fractional damage of creep with an increasing number of cycles; while the condition b(εp,T,tc) < 1 models the reverse. Thus, the function b (εp,T,tc) is effectively a damage evolution function. Ignoring the evolution of material damage, which is standard practice for the strain-life model, the residual fractional fatigue capacity ought to be independent of the number of cycles. This implies b(εp,T,tc) in Eq. (14) equals to unity and Eq. (13) is reduced to p
T
= ln B
II
and PLM ( s ) =
(16)
1;
ln
PD ( s ) =
It should be highlighted that εc,eff is the effective cyclic creep strain. Thus, Eq. (15) does not equate to the direct addition of the two inelastic strains as suggested by Eq. (1). The damage equation is given by
Df + Dc , eff
(20)
0,
where tR is the time to creep rupture; and assuming εR to be independent of the applied stress and temperature, Eqs. (20) and (21) may be rearranged into:
(15)
c, eff .
T
where B is a material constant. Eqs. (20) and (21) suggest that the mechanism of creep is active at any temperature above zero Kelvin. Defining the eventual creep rupture strain, εR, as
gives the residual fractional fatigue capacity. The fractional creep damage is defined as 1-εp/εp,ref, which suggests p, ref
H ( s )/ RT ,
= Be
or
where εp is effectively the residual capacity of plastic strain amplitude in the presence of creep; c(εp,T,tc) and b(εp,T,tc) are creep functions that are responsible for the reduced capacity of plastic strain amplitude. The condition c(εp,T,tc) = b(εp,T,tc) = 1 models pure fatigue, while the condition c(εp,T,tc) = 0 models pure creep. The ratio p / p, ref
H / RT ,
wherein the activation energy, H, is a material constant and f(σs) may take the form of a power law [25], a hyperbolic sine [26], or an exponential function [27]. In contrast, Larson-Miller [28] have postulated that the energy barrier is dependent on the applied stress; that is,
In the absence of a mean stress, the life of a metal experiencing pure fatigue with significant plastic deformation may be expressed in the form of strain-life equation, p, ref
= f ( s) e
(19)
Noting that b(εp,T,tc) ≥ 1, Eq. (19) suggests that the fractional creep damage increases with increasing number of cycle. More importantly, Eq. (19) suggests that there is absolutely no creep damage at N = 1, which is against all known experimental observations. Eq. (18) is
¯c = n c = cR =g ( ) e ntc tcR 3
H / RT ,
T
0,
(28)
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
where εcR is the cumulated cyclic creep strain at rupture and tcR is the cumulated cycle time at rupture. Rearranging Eq. (28) gives the cyclic rupture Dorn parameter, PcRD(σ): cR
g( )
H / RT .
= PcRD ( ) = tcR e
2.2.4. Cyclic strain of triangular wave form The steady-stress parameters can be most conveniently expressed in the form of a power-law function:
where a and b are fitting constants; and the subscript k represents D, LM, and MF, respectively. For a cyclic stress of triangular wave form with an amplitude σ and a cycle time tc and nil mean stress, and assuming identical creep damage due to tensile and compressive stresses, the stress-time cycle is adequately represented by a quarter cycle described by σ(t) = 4σt/tc. Referring to Eq. (39) and substituting Pk(σ(t)) = a(4σt/tc)b into Eqs. (31)–(33), but over a quarter cycle, and then into Eqs. (34)–(36), the cyclic-stress damage parameters can be induced from their respective steady-stress rupture parameter:
By performing the cyclic creep rupture test at varied stress amplitude, temperature, and cycle time, and assuming the magnitude of εcR to be independent of these varied conditions, then PcRD(σ) could be extracted from the gradients of the iso-σ lines in the tcR-eH/RT domain. In practice, it is impossible to perform pure cyclic creep rupture test – there would be accompanying fatigue damage in any cyclic test. On the other hand, with reasonable assumptions the cyclic creep rupture parameter can be induced from the steady-stress creep rupture parameter. Referring to Eqs. (20) and (28), if we assume (i) identical creep damage due to tensile and compressive stresses, and (ii) linear cumulation of creep strain over varied magnitudes of stress, then we may express the function g(σ) in terms of f(σ) as tc
g ( )=
0
f (| (t )|) dt tc
PcDD ( ) = PcDLM ( ) =
PcDMH ( ) = (1
,
= K (T )
0
PcRLM ( ) =
PLM ( (t )) dt tc
tc tc dt 0 PMH ( (t ))
PcRMH ( ) =
;
PcDk ( p) = aK¯ b
H / RT
=
c
=
c PcRD (
),
c ( p, T , tc ) = 1
PcDMH ( ) =
T
Tref
ln(tc /tc )
=
T
Tref
ln(tcR/ tcR )
=
1 = PcRMH ( ), s( )
= [1
thus,
c ( p, T , tc )] Co N
= tc e H / RT ( T , t ) c = T [ln(BR /( LM MF (T , tc ) =
(35)
c
R
= Co [1
c ( p, T , tc )] N
o / R.
T
D (T , tc )
c
R )) + ln tc ]
= 0 for T ( T , t ) c = 0 for T LM
0 0
(T , tc ) = 0 for T
Tref
Tref
ln(tc / ( c R t ))
MF
(44)
satisfy condition (ii) as well as the pure-fatigue-modelling requirement of condition (i). It is worth highlighting that the coefficients BR and t∞ can be extracted from the steady-stress creep rupture test while the multiplying factor, ϕcϕεR, is given by Eq. (38). The pure-creep-modeling requirement of condition (i) requires that χ(εp)η(T,tc) = 1 at pure creep. Referring to Eqs. (34)–(36), it is evident that the function
(36)
(37)
o;
(43)
( p) (T , tc ).
D (T , tc )
where Bc = B/εc = BR/(ϕcϕεR), BcR = B/εcR = BR/ϕεR, tc∞ =ϕcϕεRt∞ and tcR∞=ϕεRt∞. The multiplying factors ϕcϕεR equates to εc/εR. Referring to Eqs. (15) and (12), the effective cyclic creep strain is given by c
(42)
Herein χ(εp)η(T,tc) = 1 − c(εp,T,tc) is the fractional creep damage. Condition (i) requires that χ(εp)η(T,tc) = 0, or simply η(T,tc) = 0, at pure fatigue. Referring to the constitutive equations for cyclic creep damage, Eqs. (34)–(36), it is evident that the functions:
where ϕc = εc/εcR. Referring to Eqs. (A2) and (A6) in Appendix A, the cyclic damage Larson-Miller parameter and the cyclic damage MansonHaferd parameter can be written in terms of their respective cyclic rupture parameter:
PcDLM ( ) = T (ln Bc + ln tc ) = T (ln BcR
nb ¯ p .
We shall ensure that the creep-integrated fatigue equation, Eq. (17), satisfies the two necessary conditions: (i) capable of modelling the full range of creep fatigue from pure creep to pure fatigue; and (ii) the integrated creep damage is constituted from the constitutive equation of creep. Let the creep function c(εp,T,tc) takes the form:
(34)
Q( ) + ln tcR) = = PcRLM ( ), R
(41)
2.3. Expressing creep function c(εp,T,tc) in terms of creep constitutive equations
(33)
g( )
= K¯ pn¯ ,
Thus, Eq. (40) is transformed into cyclic-strain creep damage parameters, PcDk(εp), wherein the subscript k represents D, LM, and MF, respectively.
(32)
.
n (T ) p
and n, respectively. Substituting Eq. (41) into Pk(σ) = aσb yields
2.2.3. Cyclic-stress creep damage parameters From the above section and again using the Dorn parameter as an illustration, the magnitude of creep damage in a single cycle – refer to as the cyclic damage Dorn parameter – can be expressed in terms of the cyclic rupture Dorn parameter:
PcDD ( ) = tc e
(40)
b) PMH ( ),
where K(T) and n(T) are temperature-dependent material constants; and K¯ and n¯ are fitting constants that are the temperature-average of K
(31)
where ϕεR = εcR/εR. Similarly, the cyclic rupture Larson-Miller parameter, PcRLM(σ), and the cyclic rupture Manson-Haferd parameter, PcRMH(σ), may be induced from the respective steady-stress parameters (Appendix A): tc
b) PD ( )
wherein Pk(σ) = aσ and σ is the amplitude of the cyclic triangular stress. The cyclic-stress damage parameters may be expressed in terms of cyclic plastic strain using the Ramberg–Osgood relation for cyclic stress-strain:
(30)
From Eq. (23), substituting f(σ(t)) in Eq. (30) with εR/PD(σ(t)) and then substituting into Eq. (29), the cyclic rupture Done parameter, PcRD(σ), can be induced from the steady-stress Dorn parameter, PD(σs), as
PcRD ( ) =
c R (1 PLM ( ) 1+b
b
.
R tc tc dt 0 PD ( (t ))
(39)
Pk ( s ) = a sb,
(29)
k
(38)
( p) =
1 PcDk ( p)
satisfies this condition. 4
(45)
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
3. Demonstration and validation
Dorn equation is significantly larger than the other two constitutive equations. It is also noted that the magnitude of the fatigue coefficient Co for the Larson-Miller equation is significantly larger than that of the Manson-Haferd equation. Intuitively, this was attributable to the fact that the magnitude of Co was projected to 0 K for the Larson-Miller equation and to 500 K for the Manson-Haferd equation. The intuition was verified by modifying the Larson-Miller equation to be
The creep-integrated strain-life equation, p = Co c ( p, T , tc ) N o , with the creep function c(εp,T,tc) derived from the constitutive equation of creep as described in Section 2.3 will be demonstrated using two metal alloys that are hugely different in creep and fatigue properties: stainless steel SS316 and eutectic tin-lead solder Sn37Pb. 3.1. Stainless steel SS316
II
pi )
(T , tc )j],
p, ref
ln(Nexp, ij )]2 /
ij.
(48)
c
R )ij
Nij o,
.
(51)
Table 1 SS316: Fitting constants for the steady-stress creep rupture parameters.
It should be noted that the function χ(εp) or η(T,tc) is also dependent on the magnitude of ϕcϕεR, which is given by Eq. (38). However, the circular dependence of the factor ϕcϕεR and the function c(εp,T,tc) renders some difficulty in the regression operation. To ease this difficulty, the factor ϕcϕεR was approximated as
(
p
c ( p , T , tc )
3.2.1. Evaluating the steady-stress creep rupture parameters The experimental steady-stress creep rupture data of Sn37Pb solder at six magnitudes of tensile stress and three temperatures were extracted from Figs. 2 and 3 of Shi et al. [31]. These were expressed in the forms of Dorn parameter, the Larson-Miller parameter, and the MansonHaferd parameter. The creep-dormant temperature was chosen to be Tref = 150 K for the last parameter. The power-law fitting constants, a and b, and the fitting constants H/R, BR, and t∞ for the stress parameters were evaluated and are tabulated in Table 3. Similar to the case of SS316, the magnitude of the regression difference for the Dorn parameter is much larger (more than one order) than the Larson-Miller and the Manson-Haferd parameters.
This was regressed against the experimental creep fatigue life, Nexp,ij, by minimizing the average of the difference-square of all data,
[ln(Ncom, ij )
=
3.2. Eutectic tin-lead solder Sn37Pb
(46)
(47)
1/ o .
(50)
Tref ,
The transformed pure fatigue data for the equations of Dorn, LarsonMiller, and the Manson-Haferd are shown in Fig. 2. The transformed data for the last two equations collapsed into power-law curves. The creep integrated fatigue equation, Eq. (17), in which the creep damage is modelled using the constitutive equations of Larson-Miller and Manson-Haferd in the forms described in Eqs. (43)–(45) is thus validated for stainless steel SS316. A scrutiny of Fig. 2(a) suggests that the Dorn equation has over-weighted the effect of time on creep damage.
wherein χ(εp) is given by Eq. (45) and η(T,tc) by Eq. (44). It is worth highlighting that χ(εp) is dependent on the power-law fitting constants, a and b, while η(T,tc) is dependent on the fitting constants H/R, BR, or t∞, which have been evaluated from the steady-stress creep rupture test as listed in Table 1. The corresponding creep-fatigue life, Ncom,ij, was computed using Eq. (17) as
Ncom, ij = ( pi/ Cij )
T
3.1.3. Transforming to pure fatigue equation, p, ref = Co N o With the creep function c(εp,T,tc) fully defined, the experimental creep fatigue data, εp-N, of Fig. 1 was transformed to pure fatigue data, εp,ref-N, using the transformation equation:
3.1.2. Evaluating the pure fatigue coefficients, Co and βo, and the ramberg–osgood coefficients, K¯ and n¯ The experimental creep fatigue data of SS316 at four temperatures and multiple strain rates are extracted from Fig. 266.6 of Kanazawa & Yoshida [30] and these are plotted in Fig. 1. It is noted that the residual plastic strain capacity of the metal decreases with increasing temperatures and decreasing strain rates. Being tested under cyclic triangular wave, the cycle time of the data may be estimated from the known magnitudes of plastic strain and strain rate as tc = 4εp/ . Let the creepfatigue coefficient, Cij, corresponding to a particular plastic strain amplitude, εpi, and a particular temperature-cycle time, (T, tc)j, be
(
H ( s )/ R (T Tref ) ,
and setting Tref = 500 K. Designated as Larson-Miller-Tref, the corresponding fitting constants for the experimental steady-stress creep rupture data and the experimental creep fatigue data have been tabulated in Tables 1 and 2, respectively. It is noted that the magnitude of the fatigue coefficient, Co, for the Tref-modified Larson-Miller equation has indeed been reduced to a magnitude close to that of the MansonHaferd equation.
3.1.1. Evaluating the steady-stress creep rupture parameters The experimental steady-stress creep rupture time of SS316 at multiple temperatures and stress magnitudes were extracted from Fig. 12 of Ref. [29]. These were expressed in the forms of Dorn parameter, the Larson-Miller parameter, and the Manson-Haferd parameter; the creep-dormant temperature was chosen to be Tref = 500 K for the latter. The power-law fitting constants, a and b, and the respective fitting constants H/R, BR, and t∞ for the stress parameters were evaluated through regression and these are tabulated in Table 1. The regression difference – defined as the average of the sum of the difference of the squares between the experimental and the computed data – was also reported in the table. It is noted that the regression difference for the Dorn parameter is orders of magnitude larger than the other two parameters.
Cij = Co c ( pi, Tj, tc, j ) = Co [1
= Be
Constitutive equations
Others b
Dorn
1.81 × 104
−1.94
Larson-Miller
3.64 × 104
−0.118
Manson-Haferd
−146
−0.252
Larson-Miller-Tref
2.18 × 104
−0.248
Note: σs in MPa. 5
P(σs) = aσs
b
a
(49)
where λ is a fitting constant. The magnitudes of the pure fatigue coefficients, Co and βo, the Ramberg–Osgood coefficients, K¯ and n¯ , the fitting constant λ, and the corresponding regression difference are tabulated in Table 2. It is worth noting that the Ramberg–Osgood coefficients in this exercise are dependent on the magnitude of the fitting constant λ; thus, these should be treated as fitting constants and not material constants. It is noted that the magnitude of the regression difference for the
Fitting constants
H/R (K) 1.50 × 104 BR (s) 362 t∞ (s) 2.69 × 1011 BR (s) 1
Regression difference
2.3 × 10−1 7.1 × 10−4 2.6 × 10−3 5.2 × 10−3
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
Fig. 1. SS316: Experimental creep fatigue data extracted from Ref. [30].
3.2.2. Evaluating the pure fatigue coefficients, Co and βo, and the Ramberg–Osgood coefficients, K¯ and n¯ The creep fatigue coefficients (Cb, βb)j at five temperatures and five cycle times were extracted from Figs. 6 and 12, respectively, of Shi et al. [19], who had performed creep fatigue experiment on Sn37Pb solder using triangular strain cycles. The coefficients are tabulated in Table 4. The coefficients (Cb, βb)j were used to generate pseudo experimental creep fatigue data, (εpi, Nexp)j, where i = 1 to 5. These are shown in Fig. 3. Following the procedures described for SS316 in Section 3.1.2, the pure fatigue coefficients, Co and βo, and the Ramberg–Osgood coefficients, K¯ and n¯ , for the constitutive equations of Dorn, LarsonMiller, and Manson-Haferd are tabulated in Table 5 - it is worth mentioning that the factor ϕcϕεR was approximated as a constant in this demonstration. Similar to the case of SS316, the Dorn equation returns the largest magnitude of the regression difference. 3.2.3. Transforming to pure fatigue equation, p, ref = Co N o The regenerated experimental creep fatigue data, εp-N, of Fig. 3 was transformed to pure fatigue data, εp,ref-N, using Eq. (51). Ignoring the Dorn equation, the transformed pure fatigue data for the Larson-Miller and the Manson-Haferd equations are shown in Fig. 4. The transformed data for these two equations collapsed into power-law curves. The creep integrated fatigue equation, Eq. (17), in which the creep damage is modelled using the constitutive equations of Larson-Miller and MansonHaferd in the forms described in Eqs. (43)–(45) is thus validated for Sn37Pb solder.
Fig. 2. SS316: Transformed pure fatigue data using the constitutive equations of (a) Dorn, (b) Larson-Miller and (c) Manson-Haferd.
3.2.4. Benchmarking The creep-integrated fatigue equations embedded with the LarsonMiller and the Manson-Haferd equations, respectively, are
Table 2 SS316: Pure fatigue coefficients and Ramberg–Osgood coefficients for creep-integrated fatigue equations. Constitutive equations
Dorn Larson-Miller Manson-Haferd Larson-Miller-Tref
Fatigue coefficients
Ramberg–Osgood coefficients
Fitting constant
Co
βo
K¯ (MPa)
n¯
λ
0.471 1.09 0.684 0.662
0.563 0.548 0.544 0.543
688 207 68 200
0.097 0.025 0.031 0.025
1 × 10−3 2.1 × 10−3 5.3 × 10−5 2.2 × 10−3
Note: (ϕcϕεR)ij = λN−βo . ij 6
Regression difference
3.1 × 10−1 5.9 × 10−2 3.2 × 10−2 5.9 × 10−2
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
Fig. 3. Sn37Pb: Regenerated experimental creep fatigue data. Table 3 Sn37Pb: Fitting constants for the steady-stress creep rupture parameters. Constitutive equations
Fitting constants P(σs) = aσsb
Others
a
b
Dorn
7.81
−4.08
Larson-Miller
1.29 × 104
−0.198
Manson-Haferd
−32.6
−0.389
H/R (K) 5.04 × 103 BR (s) 3.14 × 107 t∞ (s) 4.33 × 109
Regression difference
5.3 × 100 1.2 × 10−2 4.6 × 10−2
Note: σs in MPa. Fig. 4. Sn37Pb: Transformed pure fatigue data using the constitutive equations of (a) Larson-Miller and (b) Manson-Haferd.
Table 4 Sn37Pb: Creep-fatigue coefficients extracted from Ref. [19]. Creep-fatigue coefficients
233 K
298 K
348 K
398 K
423 K
Cb βb
2.76 0.775
2.22 0.755
1.70 0.745
1.43 0.75
0.97 0.715
Creep-fatigue coefficients
Cycle time, tc (s) at T = 298 K
Cb βb
the Ramberg–Osgood coefficients, the regression have yielded the pure fatigue coefficients, Co and βo, and the fitting constants b1, b2, and b3; and these are tabulated in Table 5. The Engelmaier equation has returned a regression difference that is much larger in magnitude than the other two equations. The magnitude of the evaluated fatigue coefficient Co is also unrealistically small. The fallacy of the Engelmaier equation is even more evident when the creep fatigue data are transformed to the pure fatigue space, which is shown in Fig. 5. It is noted that the equation has under-collapsed the data at small number of cycles while over-collapsing the data at large number of cycles. The reason for this has been explained in Section 2.1: this type of equations model “increasing fractional damage of creep with increasing number of cycles”. The superiority of the creep-integrated fatigue equations embedded with the Larson-Miller and the Manson-Haferd equations are thus demonstrated.
Temperature (K) at tc = 1 sec
100 s
101 s
102 s
103 s
104 s
2.22 0.755
1.92 0.735
1.67 0.715
1.23 0.715
0.60 0.670
benchmarked against the Engelmaier equation, Eqs. (7) and (8), which is popularly used in the microelectronic packaging industry for modelling the creep fatigue of solder joints. Following the procedures described in Section 3.1.2 for evaluating the pure fatigue coefficients and
Table 5 Sn37Pb: Pure fatigue coefficients and Ramberg–Osgood coefficients for creep-integrated fatigue equations. Constitutive equations
Dorn Larson-Miller Manson-Haferd Engelmaier (not based on constitutive equation)
Fatigue coefficients
Ramberg–Osgood coefficients
Fitting constant
Co
βo
K¯ (MPa)
n¯
λ
1.66 6.58 3.75 1.49
0.736 0.806 0.773 0.719
14.9 24.6 6.56 b1 = 9.26 × 10−4 b2 = 1.77 × 10−2 b3 = 7.46 × 107
0.020 0.223 0.128
1.0 0.72 5.4 × 10−3
Note: ϕcϕεR = λ. 7
Regression difference
8.3 × 10−2 2.4 × 10−2 8.8 × 10−3 8.4 × 10−2
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
wherein ϕ = xc/xF and xc is the cumulated environmental damage in a cycle of loading. For the environmental factors ξ ,ψ,…. that are noncyclical in nature the cyclic damage parameter PcD(σ, ξ ,ψ,….) can be derived from the steady-state parameter P(σs, ξ ,ψ,….) using Eqs. (31)–(36). We may further generalized the rate equations for environmental damage to
x =
xF x = h ( s, , , ...) k (T ) or x = F = Ae tF tF
h ( s , , ,...) k (T ) .
(58)
The steady-state parameter is given by the gradients of the iso-(σs, ξ ,ψ, ….) lines in the tF vs. k(T)−1 or the ln tF vs. k(T) space; that is,
xF tF = or P ( s, , , ...) h ( s, , , ...) k (T ) 1 ln A/ xF + ln tF = h ( s, , , ...) = . k (T )
P ( s, , , ...) =
Fig. 5. Sn37Pb: Transformed pure fatigue data using the Engelmaier equation.
The damage functions are given by
4. Generalizing the formulation
( , , , ...) =
The technique for formulating the creep-integrated fatigue equation may be generalised to model environment-assisted fatigue if (i) under the condition of steady environment the environmental damage occurs at a steady rate, x , that can be expressed in the form of the Dorn or the Larson-Miller or the Manson-Haferd equation:
x x = F = h ( s, , , ...) e tF = Ae h (
H / RT
x or x = F = Ae tF
h ( s, , ,...)/ RT
and
(52)
wherein xF is the magnitude of the environmental damage at failure and tF is the time at failure and (ii) xF is independent of the magnitudes of the temperature T, the applied stress σs, and the environmental factors ξ ,ψ,….. If there exist simple time-temperature transformation relations, then the plots of iso-(σs, ξ ,ψ,….) lines in the tF vs. eH/RT or the ln tF vs. 1/ (RT) or the T vs. ln tF space shall appear as straight lines, whose gradients are given by
T Tref ln A/ xF + ln tF 1 or = . 1/(RT ) h ( s, , , ...) ln(tF / t )
o,
(53)
(54)
wherein σ is the cyclic stress amplitude; and the environmental damage function is given by
c ( , , , ...,T , tc ) = 1 where
and
( , , , ...) (T , tc ),
( , , , ...) =
(T , tc ) = tc e
H / RT
1 PcD ( , , , ...)
or
T Tref log(A/ xc ) + log tc or , 1/(RT ) ln(tc /( t ))
log(A/ xc ) + log tc . k (T )
(61)
A systematic derivation of creep-integrated fatigue equations has been presented. These creep-integrated fatigue equations possess three unique characteristics: (i) the integrated creep damage is constituted from the constitutive equations of creep; the fitting constants in the creep damage function can be evaluated with ease by performing steady-stress creep rupture test; (ii) creep damage and creep-fatigue interaction damage are embedded naturally into the equation and requires no separate computation; and (iii) the equations are capable of modelling the full range of creep fatigue from pure creep to pure fatigue. These characteristics ensure that these novel equations are fundamentally sound, easy to use, and applicable over a wide range of creep fatigue conditions. The creep-integrated fatigue equations based on three constitutive equations of creep: Dorn equation, Larson-Miller equation, and Manson-Haferd equation have been derived; and these have been applied to the experimental creep fatigue data of two metal alloys that have vastly different creep and fatigue characteristics: the SS316 stainless steel and the Sn37Pb solder. The creep-integrated fatigue equations based on the constitutive equation of Larson-Miller and Manson-Haferd have been found to have modelled very well the creep fatigue of these two metal alloys. The methodology of deriving the creep-integrated fatigue equation can be generalised to model environment assisted fatigue.
These functions of gradients may be represented by a steady-state parameter, P(σs, ξ ,ψ,….). The environment-integrated fatigue equation is given by
= Co c ( , , , ...,T , tc ) N
(T , tc ) = tc k (T ) or
(60)
5. Conclusions
xF t = HF/ RT or h ( s, , , ...) h ( s, , , ...) e =
1 PcD ( , , , ...)
We have learnt from the experience of creep-integrated fatigue equations that the three creep constitutive equations have returned varied performance. Thus, the proposed environment-integrated fatigue equations ought to be validated against experimental data.
x or x = F tF
s, , ,...)(T Tref ),
(59)
(55) (56) (57)
Appendix A. Theoretical derivation of cyclic creep rupture parameters A.1. Cyclic rupture Larson-Miller parameter The average creep strain rate experienced by a test specimen undergoing pure cyclic creep rupture is assumed to take the form of Eq. (21); that is,
¯c =
cR
tcR
= Be
Q ( )/ RT ,
T
0.
(A1)
Rearranging gives the cyclic rupture Larson-Miller parameter: 8
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
PcRLM ( ) =
Q( ) ln BcR + ln tcR = . R 1/T
(A2)
Assuming identical creep damage due to tensile and compressive stresses, and linear cumulation of creep strain over varied magnitudes of stress one arrives at tc 0
Q( )=
H (| (t )|) dt tc
.
(A3)
From Eq. (24), substituting H(σ(t)) in Eq. (A3) with RPLM(σ(t)) and then substituting into Eq. (A2) yields tc
PcRLM ( ) =
0
PLM (| (t )|) dt tc
.
(A4)
A.2. Cyclic rupture Manson-Haferd parameter The average creep strain rate experienced by a test specimen undergoing pure cyclic creep rupture is assumed to take the form of Eq. (26); that is,
¯c =
cR
tcR
= De (T
Tref ) s ( ) ,
T
Tref .
(A5)
Rearranging gives the cyclic rupture Manson-Haferd parameter:
T Tref 1 = , s( ) ln(tcR /tcR )
PcRMH ( ) =
(A6)
Following the same assumptions presented in the previous section gives
s ( )=
tc 0
r (| (t )|) dt tc
.
(A7)
From Eq. (27), substituting r(σ(t)) in Eq. (A7) with −1/PMH(σ(t)) and then substituting into Eq. (A6) yields
PcRMH ( ) =
tc tc dt 0 PMH ( (t ))
.
(A8)
Eng Des 1987;105:19–33. [17] Coffin LF Jr. Fatigue at high temperature. ASTM STP 520 Fatigue at elevated temperature; 1973. p. 5–34. [18] Solomon H. Fatigue of 60/40 Solder. IEEE Trans Comp Hybrids Manuf Technol 1986;9(4):423–32. [19] Shi XQ, Pang HLJ, Zhou W, Wang ZP. Low cycle fatigue analysis of temperature and frequency effects in eutectic solder alloy. Int J Fatigue 2000;22:217–28. [20] Engelmaier W. Fatigue life of leadless chip carrier solder joint during power cycling. IEEE Trans Comp Hybrids Manuf Technol 1983;6:232–7. [21] Chauhan P, Osterman M, Ricky SWR, Pecht M. Critical review of the Engelmaier model for solder joint creep fatigue reliability. IEEE Trans Comp Packag Technol 2009;32(3):693–700. [22] Wong EH, Mai Y-W. A unified equation for creep fatigue. Int J Fatigue 2014;68:186–94. [23] Liu D, Pons DJ, Wong EH. Creep-integrated fatigue equation for metals. Int J Fatigue 2017;98:167–75. [24] Manson SS, Haferd AM. A linear time-temperature relation for extrapolation of creep and stress-rupture data NACA TN 2890 National Advisory Committee for Aeronautics; 1953. [25] Norton FH. The creep of steel at high temperatures. London: McGraw-Hill; 1929. [26] McVetty PG. Creep of metals at elevated temperatures: the hyperbolic sine relation between stress and creep rate. Trans ASME 1943;65:761–9. [27] Dorn JE. Some fundamental experiments on high temperature creep. J Mech Phys Solids 1954;8:85–116. [28] Larson F, Miller J. A time-temperature relationship for rupture and creep stresses. Trans ASME 1952;174(5):765–75. [29] High temperature characteristics of stainless steels. Available online: https://www. nickelinstitute.org/~/Media/Files/TechnicalLiterature/High_ TemperatureCharacteristicsofStainlessSteel_9004_.pdf. [30] Kanazawa K, Yoshida S. Effect of temperature and strain rate on the high temperature low-cycle fatigue behavior of austenitic stainless steels. Creep and fatigue in elevated temperature applications. International conference sponsored by the Institution of Mechanical Engineers, American Society of Mechanical Engineers, American Society for Testing Materials, Philadelphia, 23–27 September 1973 and Sheffield, 1–5 April 1974 1974;vol. 1. [31] Shi XQ, Wang ZP, Yang QJ, Pang HLJ. Creep behaviour and deformation mechanism map of SnPb eutectic solder alloy. J Eng Mater Techno 2003;125:81–8.
References [1] Akay HU, Paydar NH, Bilgic A. Fatigue life predictions for thermally loaded solder joints using a volume-weighted averaging technique. Trans ASME J Electron Packag 1997;119:228–35. [2] Liang J, Golhardt N, Lee PS, Heinrich S, Schroeder S. A integrated fatigue life prediction methodology for optimum design and reliability assemssment of solder integ-connections. Int Intersoc Electron Photo Packag 1997:1583–92. [3] Darveaux R. Effects of simulation methodology on solder joint crack growth correlation and faituge life prediction. Trans ASME J Electron Packag 2002;124:147. [4] Coffin Jr. LF. A study of cyclic thermal stresses in a ductile metal. Trans ASME 1954;76. [5] Manson SS. Behavior of materials under conditions of thermal stress Technical Report NACA-TR-1170 National Advisory Committee for Aeronautics; 1954 [6] Penny RK, Marriott DL. Design for creep. 2nd ed. NY: McGraw Hill; 1971. p. 148–50. [7] Holdsworth SR. Creep-fatigue failure diagnosis. Materials 2015;8:7757–69. [8] Taira S. Lifetime of structures subjected to varying load and temperature. Berlin: Springer-Verlag; 1962. p. 96–124. [9] TRD 301. Annex I—Design: Calculation for cyclic loading due to pulsating internal pressure or combined changes of internal pressure and temperature, Technical rules for steam boilers. Essen, Germany: Vereinigung der Technischen Überwachungsvereine; 1978. [10] R5 Panel, Revision 3; Assessment procedure for the high temperature response of structures, Gloucester, UK, British Energy, 2003. [11] ASME boiler and pressure vessel code, Section III rules for the construction of nuclear facility components, Div. 1 - Subsection NH, Class 1 components in elevated temperature service, ASME, New York, 2007. [12] RCC-MR. Design and construction rules for mechanical components of nuclear installations. Paris: AFCEN; 2012. [13] Manson SS, Halford GR, Hirschberg MH. Creep-fatigue analysis by strain range partitioning. Proc First symposium on design for elevated temperature environment, San Francisco. 1971. p. 12–28. [14] LeMaitre J, Plumtree A. Application of damage concepts to predict creep-fatigue failure. Trans ASME J Eng Mater Technol 1979;101(3):284–92. [15] Chaboche JL. Une loi différentielle d’endommagement de fatigue avec cumulation non linéaire. Revue Francaise de Mécanique 1974:50–1. [16] Chaboche JL. Continnum damage mechanics: present state and future trends. Nucl
9
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Derivation of novel creep-integrated fatigue equations
T
E.H. Wong
⁎
Sino-Singapore International Joint Research Institute, China Energy Research Institute at NTU, Singapore
ARTICLE INFO
ABSTRACT
Keywords: Creep Creep fatigue Environmental fatigue Solder Stainless steel
This manuscript presents a systematic derivation of novel creep-integrated fatigue equations that possess the following unique characteristics: (i) creep damage and creep-fatigue interaction damage are embedded into the fatigue equation and requires no deliberate computation; (ii) the embedded creep damage is formulated from the constitutive equation of creep; and (iii) the integrated equation is capable of modelling the full range of creep fatigue from pure creep to pure fatigue. These characteristics ensure that these creep-integrated fatigue equations are fundamentally sound, easy to use, and applicable over a wide range of creep fatigue conditions. The creep-integrated fatigue equations based on three constitutive equations of steady-state creep: Dorn equation, Larson-Miller equation, and Manson-Haferd equation are derived and successfully validated using two metals of vastly different creep and fatigue characteristics – the SS316 stainless steel and the Sn37Pb solder.
1. Introduction Most engineering structural elements are known to fail by the mechanism of fatigue. But pure fatigue is rare; most incidences of engineering fatigue are accompanied by environmental damaging factors such as creep, corrosion, and radiation; etc. Among these, creep assisted fatigue or simply creep fatigue has attracted the most interest as it is responsible for the structural failure of large number of critical engineering components; such as turbine blades and rotors in aircraft and steam engines, vessels and tubing in nuclear reactor and fossil power plants, and, no least, solder joints in electronic assemblies. The study of the creep fatigue of metals has been built on the fundamental understanding of the fatigue and the creep of metals. The main challenge has been in compounding the damages due to these two mechanisms to arrive at a reliable model for life prediction. A simple approach that is widely practised in the electronic assembly community is to treat creep as time-dependent plasticity [1–3]. Thus, the life prediction models for fatigue, for example the Coffin-Manson equation [4,5], is extended to creep fatigue: pl
= CN
pl
+
c
= CN ,
(1)
wherein εpl and εc are the cumulative plastic and creep strains respectively in a single cycle of loading; N is the “life” measured as the number of cycles to failure; and C and β are fitting constants. However, this simplistic act of lumping creep damage together with fatigue damage falls apart in the face of microstructural evidence [6,7]: creep
⁎
damage in metals, especially those with low creep ductility, proceeds in the form of grain boundary cavitation in the bulk of the metals. Fatigue damage on the other hand proceeds from transgranular microcracks on the surface of metals. The early practice in the power industry was to treat creep and fatigue as two independent damages and assuming the two damages can be added up scalarly. This may be expressed mathematically as
Df + Dc
(2)
1,
where Df and Dc are the damages due to fatigue and creep, respectively. This is known as the linear damage summation rule [8]. Microstructural evidence suggests that these two kinds of microstructural damage tend to interact in the later stage and aggravate each other. This is generally referred to as creep-fatigue interaction (CFI), which tends to be particularly intense for metals with low creep ductility [7]. The damage equation is more appropriately expressed as [9–12]
Df + Dc + Dcf
1,
(3)
where Dcf is the damage due to CFI, which is a function of Dc and Df. The extensive experimental effort required to define this function is rather challenging and near impossible for some industries; for example, it is impractical to undertake comprehensive CFI characterisation for the considerable number of solder alloys used in electronic assemblies. The aerospace industry practices the strain range partitioning approach of Manson et al. [13] who proposed that metals have different tolerances for damages due to fatigue, pure cyclic creep, and cyclic
Address: Energy Research Institute at NTU, Singapore. E-mail address: [email protected].
https://doi.org/10.1016/j.ijfatigue.2019.06.044 Received 25 April 2019; Received in revised form 6 June 2019; Accepted 30 June 2019 Available online 02 July 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
Nomenclature
tc; tref
cycle time; reference cycle time - the cycle time below which creep is dormant time at rupture under steady stress condition; cumulated tR; tcR cycle time at rupture under pure cyclic creep t∞ a constant in the steady-stress Manson-Haferd parameter it is the intercept of iso-σs lines on T-log(tR) plot tcR∞; tc∞ a constant in the cyclic-rupture Manson-Haferd parameter tcR∞ = ϕεRt∞, a constant in the cyclic-damage MansonHaferd parameter tc∞ = ϕcϕεRt∞ β, βo, βb fatigue exponent; fatigue exponent in a pure fatigue condition, in a creep fatigue condition cumulated creep strain in a single cycle; the plastic strain εc; εpl range or the cumulated plastic strain in a single cycle εp; εp,ref (the residual capacity of) plastic strain amplitude in a creep-fatigue condition; (the capacity of) plastic strain amplitude in a pure fatigue condition, corresponding to cyclic life N εc,eff the effective cyclic creep strain defined as c, eff = p, ref p ¯ secondary stage strain rate under steady-stress condition; II ; c the average creep strain rate in a cyclic condition ¯c = εc/ tc εR; εcR creep strain at rupture under steady stress condition; cumulated cyclic creep strain at rupture creep strain factors: ϕc = εc/εcR, ϕεR = εcR/εR ϕc, ϕεR σ; σ(t); σs cyclic stress amplitude; stress-time function; steady uniaxial tensile stress
B; BR, BcR, Bc a coefficient in the creep constitutive of Larson-Miller; a coefficient in the steady-stress Larson-Miller parameter BR = B/εR, in the cyclic-rupture Larson-Miller parameter BcR = B/εcR, in the cyclic-damage Larson-Miller parameter Bc = B/εc C; Co, Cb, fatigue coefficient; fatigue coefficient in a pure fatigue condition, in a creep fatigue condition Df; Dc; Dcf; Dc,eff damage due to fatigue; damage due to creep; damage due to creep-fatigue interaction; effective creep damage that includes damages due to creep and creep-fatigue interaction f cyclic frequency N; Nexp; Ncom cyclic life; experimental cyclic life; computed cyclic life Pk(σs); PcRk(σ), PcDk(σ) stress parameter for creep rupture under steady-stress σs; stress parameter for creep rupture, for creep damage, under cyclic stress amplitude σ; the subscript k may be “D”, “LM”, or “MH” and refers to the steady-stage creep constitutive equation of Dorn, LarsonMiller, and Manson-Haferd, respectively PcRk(εp), PcDk(εp) plastic-strain parameters transformed from the corresponding stress parameters using Ramberg–Osgood relation T; Tref temperature; reference temperature - the temperature below which creep is dormant creep fatigue; and the different tolerances may be expressed as
= Cp N
pl
p,
c
= Cc N
c,
cp
= Ccp N
cp .
constants, and tc is the cycle time. However, the fitting constants in the Engelmaier equation have frequently been reported to be dependent on the designs of the electronic assemblies and the cyclic temperatures [21]. The literal act of making the fatigue coefficients a function of temperature and/or cycle time does not integrate creep damage into the fatigue equation. Wong & Mai [22] suggested that a fatigue equation that has fully integrated creep damage must be capable of modelling the extreme conditions of creep fatigue; that is, the conditions of pure fatigue and pure creep. This condition is neither satisfied by the equation of Coffin nor Engelmaier. Wong & Mai [22] proposed the following form of integrated creep-fatigue equation:
(4)
The total damage is again defined by Eq. (3) except that Dcf is now assumed to be independent of Dc and Df. LeMaitre & Plumtree [14] and Chaboche [15,16] have advocated the continuum damage model, which gives, in the absence of mean stress, the rates of creep and fatigue damages as: dDf
=
dN dDc dt
=
r A (1 B (1
s (t ) D)
p
(1
D)
(1
D) q ,
D)
(5)
where σ(t) is the instantaneous stress; σ is the amplitude of a cyclic stress; A, B, p, q, r and s are temperature dependent material characteristics. While the total damage is given by Df + Dc, the interaction between the creep and the fatigue damages is embedded in Eq. (5). Coffin [17] proposed the “frequency-modified fatigue life” equation: pl
= C (T )(Nf k (T ) 1 )
p
= Co N
where
E ( T , tc )
=
o [1
+ b1 T
b2 ln(1 + b3/ tc )],
Tref )
c2 log(tc/ tref ), T > Tref , t > tref ,
Tref and tref are the reference temperature and the reference cycle time, respectively, below which creep is dormant. The function c(T, tc) incorporates creep damage and has a value that lies between 0 and 1; the extreme conditions c(T, tc) = 0 and 1 represent pure creep and pure fatigue, respectively. Liu et al. [23] proposed that c1 and c2 may be expressed in terms of the Manson-Haferd stress parameter, PMH(σs) [24]:
c1 ( ) =
c2 1 , c2 = , PMH ( s ) log(t / tref )
(11)
wherein σs is a time-steady uniaxial tensile stress applied to a specimen while characterizing the time to creep rupture, tR; t∞ is the time at which the iso-σs lines converge in the T-log(tR) space. A critical flaw in the above integrated creep-fatigue equations, Eqs. (6)–(11), is that the embedded creep damage functions are inconsistent with the constitutive equations of creep. We shall present in this manuscript a systematic approach to deriving novel creep-integrated fatigue equations that possess the following unique characteristics: (i) creep damage and creep-fatigue
(7)
E (T , t c ) ,
c 1 (T
(10)
where f is the cyclic frequency and C(T), β(T) and k(T) are functions of temperature. While it lacks the rigor of the continuum damage mechanics approach, its simplicity – there is no need to compute creep strain explicitly – has attracted much interest in the electronic assembly industry. Solomon [18] and Shi et al. [19] have modelled the creepfatigue life of eutectic SnPb solder using this technique. Ironically, both researchers have reported a dependence of the frequency exponent, k (T), on the cyclic frequency. If this is true, then Eq. (6) is fundamentally flawed. Analysing the thermomechanical fatigue of eutectic SnPb solder joints in electronic assemblies, Engelmaier [20] proposed a different form of integrated creep-fatigue equation, which may be expressed more concisely as p
(9)
o,
where c (T , tc ) = 1
(6)
(T ) ,
= Co c (T , tc ) N
(8)
εp is the amplitude of plastic strain; Co, βo, b1, b2, and b3 are fitting 2
International Journal of Fatigue 128 (2019) 105184
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interaction damage are embedded into the fatigue equation and requires no deliberate computation; (ii) the embedded creep damage is formulated from the constitutive equation of creep; and (iii) the equation is capable of modelling the full range of creep fatigue from pure creep to pure fatigue. We shall also generalise the method to model general environment aggravated fatigue.
therefore not a reliable equation for modelling creep-fatigue. It is worth highlighting that the Engelmaier equation, Eq. (7), falls under this class of equation. 2.2. Creep parameters 2.2.1. Steady-stress creep rupture parameters Assuming the dominance of the secondary stage of creep, the steady-stress creep strain rate is frequently described in the form:
2. Derivations 2.1. The forms of creep-integrated fatigue equation
II
= Co N
(12)
o,
II
where εp,ref is the plastic strain capacity – define here as the amplitude of the alternating plastic strain. The subscript “ref” emphasizes the condition of pure fatigue. Creep damage can be integrated into the fatigue equation by modifying the fatigue coefficients Co and βo. A generic creep-integrated fatigue equation may take the form: p
= Co c ( p, T , tc ) N
o b ( p, T , t c ) ,
= c ( p , T , tc ) N
o [b ( p, T , tc ) 1]
=
p
+
(13)
R
(14)
= Co c ( p, T , tc ) N
o;
= Co N
Let or
p / p, ref
=1
N
o [b ( p, T , tc ) 1] .
(22)
II tR,
R
f ( s)
=
tR , e H / RT
(23)
H ( s) ln BR + ln tR = , R 1/ T
(24)
T
Tref
ln(tR/ t )
.
(25)
II
=
R
tR
= De (T
Tref ) r ( s ) ,
ln( R/ tR) = ln D + (T
T
Tref ;
Tref ) r ( s ).
(26a) (26b)
Rearranging gives
(17)
PMH ( s ) =
T Tref 1 = , r ( s) ln(tR D / R )
(27)
which suggests D = εR/t∞. For convenience, we shall refer to Eqs. (20), (21), and (26) as Dorn equation, Larson-Miller equation, and MansonHaferd equation respectively. 2.2.2. Cyclic-stress creep rupture parameters If we could perform a cyclic-stress experiment at a stress amplitude σ and a cyclic creep strain, εc, such that pure cyclic creep rupture of a test specimen occurs after n cycles, then, using the Dorn parameter as an illustration, the average creep strain rate experienced by the test specimen shall take the form of Eq. (20) as
while making b(εp,T,tc) = 1 at pure fatigue and b(εp,T,tc)=∞ at pure creep. The fractional creep damage is given by
1
(21b)
H ( s )/ RT ,
Eq. (25) suggests the mechanism of creep is dormant at temperature below the reference temperature, Tref. The corresponding steady-stress creep strain rate equation may be deduced as follows:
(18)
o b ( p, T , tc ),
=
PMH ( s ) =
c(εp,T,tc) is now the residual fractional fatigue capacity and 1 − c (εp,T,tc) the fractional creep damage. Mathematically, the conditions of pure fatigue and pure creep may also be modelled by letting c(εp,T,tc) in Eq. (13) equate to unity; that is, p
(21a)
0,
wherein BR = B/εR. The Dorn parameter, PD(σs), describes the gradients of the iso-σs lines in the tR vs. eH/RT domain and the Larson-Miller parameter, PLM(σs), describes the gradients of the iso-σs lines in the ln (tR) vs. 1/T domain. Based on the observation that the iso-σs plot of the creep rupture data on the T vs. ln(tR) domain appeared to converge at a single point, (Tref, t∞), Manson-Haferd [24] came to the arbitrary stress parameter:
herein Dc,eff captures the damages due to creep and creep-fatigue interaction. Referring to Eq. (14), the condition b(εp,T,tc) > 1 models increasing fractional damage of creep with an increasing number of cycles; while the condition b(εp,T,tc) < 1 models the reverse. Thus, the function b (εp,T,tc) is effectively a damage evolution function. Ignoring the evolution of material damage, which is standard practice for the strain-life model, the residual fractional fatigue capacity ought to be independent of the number of cycles. This implies b(εp,T,tc) in Eq. (14) equals to unity and Eq. (13) is reduced to p
T
= ln B
II
and PLM ( s ) =
(16)
1;
ln
PD ( s ) =
It should be highlighted that εc,eff is the effective cyclic creep strain. Thus, Eq. (15) does not equate to the direct addition of the two inelastic strains as suggested by Eq. (1). The damage equation is given by
Df + Dc , eff
(20)
0,
where tR is the time to creep rupture; and assuming εR to be independent of the applied stress and temperature, Eqs. (20) and (21) may be rearranged into:
(15)
c, eff .
T
where B is a material constant. Eqs. (20) and (21) suggest that the mechanism of creep is active at any temperature above zero Kelvin. Defining the eventual creep rupture strain, εR, as
gives the residual fractional fatigue capacity. The fractional creep damage is defined as 1-εp/εp,ref, which suggests p, ref
H ( s )/ RT ,
= Be
or
where εp is effectively the residual capacity of plastic strain amplitude in the presence of creep; c(εp,T,tc) and b(εp,T,tc) are creep functions that are responsible for the reduced capacity of plastic strain amplitude. The condition c(εp,T,tc) = b(εp,T,tc) = 1 models pure fatigue, while the condition c(εp,T,tc) = 0 models pure creep. The ratio p / p, ref
H / RT ,
wherein the activation energy, H, is a material constant and f(σs) may take the form of a power law [25], a hyperbolic sine [26], or an exponential function [27]. In contrast, Larson-Miller [28] have postulated that the energy barrier is dependent on the applied stress; that is,
In the absence of a mean stress, the life of a metal experiencing pure fatigue with significant plastic deformation may be expressed in the form of strain-life equation, p, ref
= f ( s) e
(19)
Noting that b(εp,T,tc) ≥ 1, Eq. (19) suggests that the fractional creep damage increases with increasing number of cycle. More importantly, Eq. (19) suggests that there is absolutely no creep damage at N = 1, which is against all known experimental observations. Eq. (18) is
¯c = n c = cR =g ( ) e ntc tcR 3
H / RT ,
T
0,
(28)
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
where εcR is the cumulated cyclic creep strain at rupture and tcR is the cumulated cycle time at rupture. Rearranging Eq. (28) gives the cyclic rupture Dorn parameter, PcRD(σ): cR
g( )
H / RT .
= PcRD ( ) = tcR e
2.2.4. Cyclic strain of triangular wave form The steady-stress parameters can be most conveniently expressed in the form of a power-law function:
where a and b are fitting constants; and the subscript k represents D, LM, and MF, respectively. For a cyclic stress of triangular wave form with an amplitude σ and a cycle time tc and nil mean stress, and assuming identical creep damage due to tensile and compressive stresses, the stress-time cycle is adequately represented by a quarter cycle described by σ(t) = 4σt/tc. Referring to Eq. (39) and substituting Pk(σ(t)) = a(4σt/tc)b into Eqs. (31)–(33), but over a quarter cycle, and then into Eqs. (34)–(36), the cyclic-stress damage parameters can be induced from their respective steady-stress rupture parameter:
By performing the cyclic creep rupture test at varied stress amplitude, temperature, and cycle time, and assuming the magnitude of εcR to be independent of these varied conditions, then PcRD(σ) could be extracted from the gradients of the iso-σ lines in the tcR-eH/RT domain. In practice, it is impossible to perform pure cyclic creep rupture test – there would be accompanying fatigue damage in any cyclic test. On the other hand, with reasonable assumptions the cyclic creep rupture parameter can be induced from the steady-stress creep rupture parameter. Referring to Eqs. (20) and (28), if we assume (i) identical creep damage due to tensile and compressive stresses, and (ii) linear cumulation of creep strain over varied magnitudes of stress, then we may express the function g(σ) in terms of f(σ) as tc
g ( )=
0
f (| (t )|) dt tc
PcDD ( ) = PcDLM ( ) =
PcDMH ( ) = (1
,
= K (T )
0
PcRLM ( ) =
PLM ( (t )) dt tc
tc tc dt 0 PMH ( (t ))
PcRMH ( ) =
;
PcDk ( p) = aK¯ b
H / RT
=
c
=
c PcRD (
),
c ( p, T , tc ) = 1
PcDMH ( ) =
T
Tref
ln(tc /tc )
=
T
Tref
ln(tcR/ tcR )
=
1 = PcRMH ( ), s( )
= [1
thus,
c ( p, T , tc )] Co N
= tc e H / RT ( T , t ) c = T [ln(BR /( LM MF (T , tc ) =
(35)
c
R
= Co [1
c ( p, T , tc )] N
o / R.
T
D (T , tc )
c
R )) + ln tc ]
= 0 for T ( T , t ) c = 0 for T LM
0 0
(T , tc ) = 0 for T
Tref
Tref
ln(tc / ( c R t ))
MF
(44)
satisfy condition (ii) as well as the pure-fatigue-modelling requirement of condition (i). It is worth highlighting that the coefficients BR and t∞ can be extracted from the steady-stress creep rupture test while the multiplying factor, ϕcϕεR, is given by Eq. (38). The pure-creep-modeling requirement of condition (i) requires that χ(εp)η(T,tc) = 1 at pure creep. Referring to Eqs. (34)–(36), it is evident that the function
(36)
(37)
o;
(43)
( p) (T , tc ).
D (T , tc )
where Bc = B/εc = BR/(ϕcϕεR), BcR = B/εcR = BR/ϕεR, tc∞ =ϕcϕεRt∞ and tcR∞=ϕεRt∞. The multiplying factors ϕcϕεR equates to εc/εR. Referring to Eqs. (15) and (12), the effective cyclic creep strain is given by c
(42)
Herein χ(εp)η(T,tc) = 1 − c(εp,T,tc) is the fractional creep damage. Condition (i) requires that χ(εp)η(T,tc) = 0, or simply η(T,tc) = 0, at pure fatigue. Referring to the constitutive equations for cyclic creep damage, Eqs. (34)–(36), it is evident that the functions:
where ϕc = εc/εcR. Referring to Eqs. (A2) and (A6) in Appendix A, the cyclic damage Larson-Miller parameter and the cyclic damage MansonHaferd parameter can be written in terms of their respective cyclic rupture parameter:
PcDLM ( ) = T (ln Bc + ln tc ) = T (ln BcR
nb ¯ p .
We shall ensure that the creep-integrated fatigue equation, Eq. (17), satisfies the two necessary conditions: (i) capable of modelling the full range of creep fatigue from pure creep to pure fatigue; and (ii) the integrated creep damage is constituted from the constitutive equation of creep. Let the creep function c(εp,T,tc) takes the form:
(34)
Q( ) + ln tcR) = = PcRLM ( ), R
(41)
2.3. Expressing creep function c(εp,T,tc) in terms of creep constitutive equations
(33)
g( )
= K¯ pn¯ ,
Thus, Eq. (40) is transformed into cyclic-strain creep damage parameters, PcDk(εp), wherein the subscript k represents D, LM, and MF, respectively.
(32)
.
n (T ) p
and n, respectively. Substituting Eq. (41) into Pk(σ) = aσb yields
2.2.3. Cyclic-stress creep damage parameters From the above section and again using the Dorn parameter as an illustration, the magnitude of creep damage in a single cycle – refer to as the cyclic damage Dorn parameter – can be expressed in terms of the cyclic rupture Dorn parameter:
PcDD ( ) = tc e
(40)
b) PMH ( ),
where K(T) and n(T) are temperature-dependent material constants; and K¯ and n¯ are fitting constants that are the temperature-average of K
(31)
where ϕεR = εcR/εR. Similarly, the cyclic rupture Larson-Miller parameter, PcRLM(σ), and the cyclic rupture Manson-Haferd parameter, PcRMH(σ), may be induced from the respective steady-stress parameters (Appendix A): tc
b) PD ( )
wherein Pk(σ) = aσ and σ is the amplitude of the cyclic triangular stress. The cyclic-stress damage parameters may be expressed in terms of cyclic plastic strain using the Ramberg–Osgood relation for cyclic stress-strain:
(30)
From Eq. (23), substituting f(σ(t)) in Eq. (30) with εR/PD(σ(t)) and then substituting into Eq. (29), the cyclic rupture Done parameter, PcRD(σ), can be induced from the steady-stress Dorn parameter, PD(σs), as
PcRD ( ) =
c R (1 PLM ( ) 1+b
b
.
R tc tc dt 0 PD ( (t ))
(39)
Pk ( s ) = a sb,
(29)
k
(38)
( p) =
1 PcDk ( p)
satisfies this condition. 4
(45)
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
3. Demonstration and validation
Dorn equation is significantly larger than the other two constitutive equations. It is also noted that the magnitude of the fatigue coefficient Co for the Larson-Miller equation is significantly larger than that of the Manson-Haferd equation. Intuitively, this was attributable to the fact that the magnitude of Co was projected to 0 K for the Larson-Miller equation and to 500 K for the Manson-Haferd equation. The intuition was verified by modifying the Larson-Miller equation to be
The creep-integrated strain-life equation, p = Co c ( p, T , tc ) N o , with the creep function c(εp,T,tc) derived from the constitutive equation of creep as described in Section 2.3 will be demonstrated using two metal alloys that are hugely different in creep and fatigue properties: stainless steel SS316 and eutectic tin-lead solder Sn37Pb. 3.1. Stainless steel SS316
II
pi )
(T , tc )j],
p, ref
ln(Nexp, ij )]2 /
ij.
(48)
c
R )ij
Nij o,
.
(51)
Table 1 SS316: Fitting constants for the steady-stress creep rupture parameters.
It should be noted that the function χ(εp) or η(T,tc) is also dependent on the magnitude of ϕcϕεR, which is given by Eq. (38). However, the circular dependence of the factor ϕcϕεR and the function c(εp,T,tc) renders some difficulty in the regression operation. To ease this difficulty, the factor ϕcϕεR was approximated as
(
p
c ( p , T , tc )
3.2.1. Evaluating the steady-stress creep rupture parameters The experimental steady-stress creep rupture data of Sn37Pb solder at six magnitudes of tensile stress and three temperatures were extracted from Figs. 2 and 3 of Shi et al. [31]. These were expressed in the forms of Dorn parameter, the Larson-Miller parameter, and the MansonHaferd parameter. The creep-dormant temperature was chosen to be Tref = 150 K for the last parameter. The power-law fitting constants, a and b, and the fitting constants H/R, BR, and t∞ for the stress parameters were evaluated and are tabulated in Table 3. Similar to the case of SS316, the magnitude of the regression difference for the Dorn parameter is much larger (more than one order) than the Larson-Miller and the Manson-Haferd parameters.
This was regressed against the experimental creep fatigue life, Nexp,ij, by minimizing the average of the difference-square of all data,
[ln(Ncom, ij )
=
3.2. Eutectic tin-lead solder Sn37Pb
(46)
(47)
1/ o .
(50)
Tref ,
The transformed pure fatigue data for the equations of Dorn, LarsonMiller, and the Manson-Haferd are shown in Fig. 2. The transformed data for the last two equations collapsed into power-law curves. The creep integrated fatigue equation, Eq. (17), in which the creep damage is modelled using the constitutive equations of Larson-Miller and Manson-Haferd in the forms described in Eqs. (43)–(45) is thus validated for stainless steel SS316. A scrutiny of Fig. 2(a) suggests that the Dorn equation has over-weighted the effect of time on creep damage.
wherein χ(εp) is given by Eq. (45) and η(T,tc) by Eq. (44). It is worth highlighting that χ(εp) is dependent on the power-law fitting constants, a and b, while η(T,tc) is dependent on the fitting constants H/R, BR, or t∞, which have been evaluated from the steady-stress creep rupture test as listed in Table 1. The corresponding creep-fatigue life, Ncom,ij, was computed using Eq. (17) as
Ncom, ij = ( pi/ Cij )
T
3.1.3. Transforming to pure fatigue equation, p, ref = Co N o With the creep function c(εp,T,tc) fully defined, the experimental creep fatigue data, εp-N, of Fig. 1 was transformed to pure fatigue data, εp,ref-N, using the transformation equation:
3.1.2. Evaluating the pure fatigue coefficients, Co and βo, and the ramberg–osgood coefficients, K¯ and n¯ The experimental creep fatigue data of SS316 at four temperatures and multiple strain rates are extracted from Fig. 266.6 of Kanazawa & Yoshida [30] and these are plotted in Fig. 1. It is noted that the residual plastic strain capacity of the metal decreases with increasing temperatures and decreasing strain rates. Being tested under cyclic triangular wave, the cycle time of the data may be estimated from the known magnitudes of plastic strain and strain rate as tc = 4εp/ . Let the creepfatigue coefficient, Cij, corresponding to a particular plastic strain amplitude, εpi, and a particular temperature-cycle time, (T, tc)j, be
(
H ( s )/ R (T Tref ) ,
and setting Tref = 500 K. Designated as Larson-Miller-Tref, the corresponding fitting constants for the experimental steady-stress creep rupture data and the experimental creep fatigue data have been tabulated in Tables 1 and 2, respectively. It is noted that the magnitude of the fatigue coefficient, Co, for the Tref-modified Larson-Miller equation has indeed been reduced to a magnitude close to that of the MansonHaferd equation.
3.1.1. Evaluating the steady-stress creep rupture parameters The experimental steady-stress creep rupture time of SS316 at multiple temperatures and stress magnitudes were extracted from Fig. 12 of Ref. [29]. These were expressed in the forms of Dorn parameter, the Larson-Miller parameter, and the Manson-Haferd parameter; the creep-dormant temperature was chosen to be Tref = 500 K for the latter. The power-law fitting constants, a and b, and the respective fitting constants H/R, BR, and t∞ for the stress parameters were evaluated through regression and these are tabulated in Table 1. The regression difference – defined as the average of the sum of the difference of the squares between the experimental and the computed data – was also reported in the table. It is noted that the regression difference for the Dorn parameter is orders of magnitude larger than the other two parameters.
Cij = Co c ( pi, Tj, tc, j ) = Co [1
= Be
Constitutive equations
Others b
Dorn
1.81 × 104
−1.94
Larson-Miller
3.64 × 104
−0.118
Manson-Haferd
−146
−0.252
Larson-Miller-Tref
2.18 × 104
−0.248
Note: σs in MPa. 5
P(σs) = aσs
b
a
(49)
where λ is a fitting constant. The magnitudes of the pure fatigue coefficients, Co and βo, the Ramberg–Osgood coefficients, K¯ and n¯ , the fitting constant λ, and the corresponding regression difference are tabulated in Table 2. It is worth noting that the Ramberg–Osgood coefficients in this exercise are dependent on the magnitude of the fitting constant λ; thus, these should be treated as fitting constants and not material constants. It is noted that the magnitude of the regression difference for the
Fitting constants
H/R (K) 1.50 × 104 BR (s) 362 t∞ (s) 2.69 × 1011 BR (s) 1
Regression difference
2.3 × 10−1 7.1 × 10−4 2.6 × 10−3 5.2 × 10−3
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
Fig. 1. SS316: Experimental creep fatigue data extracted from Ref. [30].
3.2.2. Evaluating the pure fatigue coefficients, Co and βo, and the Ramberg–Osgood coefficients, K¯ and n¯ The creep fatigue coefficients (Cb, βb)j at five temperatures and five cycle times were extracted from Figs. 6 and 12, respectively, of Shi et al. [19], who had performed creep fatigue experiment on Sn37Pb solder using triangular strain cycles. The coefficients are tabulated in Table 4. The coefficients (Cb, βb)j were used to generate pseudo experimental creep fatigue data, (εpi, Nexp)j, where i = 1 to 5. These are shown in Fig. 3. Following the procedures described for SS316 in Section 3.1.2, the pure fatigue coefficients, Co and βo, and the Ramberg–Osgood coefficients, K¯ and n¯ , for the constitutive equations of Dorn, LarsonMiller, and Manson-Haferd are tabulated in Table 5 - it is worth mentioning that the factor ϕcϕεR was approximated as a constant in this demonstration. Similar to the case of SS316, the Dorn equation returns the largest magnitude of the regression difference. 3.2.3. Transforming to pure fatigue equation, p, ref = Co N o The regenerated experimental creep fatigue data, εp-N, of Fig. 3 was transformed to pure fatigue data, εp,ref-N, using Eq. (51). Ignoring the Dorn equation, the transformed pure fatigue data for the Larson-Miller and the Manson-Haferd equations are shown in Fig. 4. The transformed data for these two equations collapsed into power-law curves. The creep integrated fatigue equation, Eq. (17), in which the creep damage is modelled using the constitutive equations of Larson-Miller and MansonHaferd in the forms described in Eqs. (43)–(45) is thus validated for Sn37Pb solder.
Fig. 2. SS316: Transformed pure fatigue data using the constitutive equations of (a) Dorn, (b) Larson-Miller and (c) Manson-Haferd.
3.2.4. Benchmarking The creep-integrated fatigue equations embedded with the LarsonMiller and the Manson-Haferd equations, respectively, are
Table 2 SS316: Pure fatigue coefficients and Ramberg–Osgood coefficients for creep-integrated fatigue equations. Constitutive equations
Dorn Larson-Miller Manson-Haferd Larson-Miller-Tref
Fatigue coefficients
Ramberg–Osgood coefficients
Fitting constant
Co
βo
K¯ (MPa)
n¯
λ
0.471 1.09 0.684 0.662
0.563 0.548 0.544 0.543
688 207 68 200
0.097 0.025 0.031 0.025
1 × 10−3 2.1 × 10−3 5.3 × 10−5 2.2 × 10−3
Note: (ϕcϕεR)ij = λN−βo . ij 6
Regression difference
3.1 × 10−1 5.9 × 10−2 3.2 × 10−2 5.9 × 10−2
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
Fig. 3. Sn37Pb: Regenerated experimental creep fatigue data. Table 3 Sn37Pb: Fitting constants for the steady-stress creep rupture parameters. Constitutive equations
Fitting constants P(σs) = aσsb
Others
a
b
Dorn
7.81
−4.08
Larson-Miller
1.29 × 104
−0.198
Manson-Haferd
−32.6
−0.389
H/R (K) 5.04 × 103 BR (s) 3.14 × 107 t∞ (s) 4.33 × 109
Regression difference
5.3 × 100 1.2 × 10−2 4.6 × 10−2
Note: σs in MPa. Fig. 4. Sn37Pb: Transformed pure fatigue data using the constitutive equations of (a) Larson-Miller and (b) Manson-Haferd.
Table 4 Sn37Pb: Creep-fatigue coefficients extracted from Ref. [19]. Creep-fatigue coefficients
233 K
298 K
348 K
398 K
423 K
Cb βb
2.76 0.775
2.22 0.755
1.70 0.745
1.43 0.75
0.97 0.715
Creep-fatigue coefficients
Cycle time, tc (s) at T = 298 K
Cb βb
the Ramberg–Osgood coefficients, the regression have yielded the pure fatigue coefficients, Co and βo, and the fitting constants b1, b2, and b3; and these are tabulated in Table 5. The Engelmaier equation has returned a regression difference that is much larger in magnitude than the other two equations. The magnitude of the evaluated fatigue coefficient Co is also unrealistically small. The fallacy of the Engelmaier equation is even more evident when the creep fatigue data are transformed to the pure fatigue space, which is shown in Fig. 5. It is noted that the equation has under-collapsed the data at small number of cycles while over-collapsing the data at large number of cycles. The reason for this has been explained in Section 2.1: this type of equations model “increasing fractional damage of creep with increasing number of cycles”. The superiority of the creep-integrated fatigue equations embedded with the Larson-Miller and the Manson-Haferd equations are thus demonstrated.
Temperature (K) at tc = 1 sec
100 s
101 s
102 s
103 s
104 s
2.22 0.755
1.92 0.735
1.67 0.715
1.23 0.715
0.60 0.670
benchmarked against the Engelmaier equation, Eqs. (7) and (8), which is popularly used in the microelectronic packaging industry for modelling the creep fatigue of solder joints. Following the procedures described in Section 3.1.2 for evaluating the pure fatigue coefficients and
Table 5 Sn37Pb: Pure fatigue coefficients and Ramberg–Osgood coefficients for creep-integrated fatigue equations. Constitutive equations
Dorn Larson-Miller Manson-Haferd Engelmaier (not based on constitutive equation)
Fatigue coefficients
Ramberg–Osgood coefficients
Fitting constant
Co
βo
K¯ (MPa)
n¯
λ
1.66 6.58 3.75 1.49
0.736 0.806 0.773 0.719
14.9 24.6 6.56 b1 = 9.26 × 10−4 b2 = 1.77 × 10−2 b3 = 7.46 × 107
0.020 0.223 0.128
1.0 0.72 5.4 × 10−3
Note: ϕcϕεR = λ. 7
Regression difference
8.3 × 10−2 2.4 × 10−2 8.8 × 10−3 8.4 × 10−2
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
wherein ϕ = xc/xF and xc is the cumulated environmental damage in a cycle of loading. For the environmental factors ξ ,ψ,…. that are noncyclical in nature the cyclic damage parameter PcD(σ, ξ ,ψ,….) can be derived from the steady-state parameter P(σs, ξ ,ψ,….) using Eqs. (31)–(36). We may further generalized the rate equations for environmental damage to
x =
xF x = h ( s, , , ...) k (T ) or x = F = Ae tF tF
h ( s , , ,...) k (T ) .
(58)
The steady-state parameter is given by the gradients of the iso-(σs, ξ ,ψ, ….) lines in the tF vs. k(T)−1 or the ln tF vs. k(T) space; that is,
xF tF = or P ( s, , , ...) h ( s, , , ...) k (T ) 1 ln A/ xF + ln tF = h ( s, , , ...) = . k (T )
P ( s, , , ...) =
Fig. 5. Sn37Pb: Transformed pure fatigue data using the Engelmaier equation.
The damage functions are given by
4. Generalizing the formulation
( , , , ...) =
The technique for formulating the creep-integrated fatigue equation may be generalised to model environment-assisted fatigue if (i) under the condition of steady environment the environmental damage occurs at a steady rate, x , that can be expressed in the form of the Dorn or the Larson-Miller or the Manson-Haferd equation:
x x = F = h ( s, , , ...) e tF = Ae h (
H / RT
x or x = F = Ae tF
h ( s, , ,...)/ RT
and
(52)
wherein xF is the magnitude of the environmental damage at failure and tF is the time at failure and (ii) xF is independent of the magnitudes of the temperature T, the applied stress σs, and the environmental factors ξ ,ψ,….. If there exist simple time-temperature transformation relations, then the plots of iso-(σs, ξ ,ψ,….) lines in the tF vs. eH/RT or the ln tF vs. 1/ (RT) or the T vs. ln tF space shall appear as straight lines, whose gradients are given by
T Tref ln A/ xF + ln tF 1 or = . 1/(RT ) h ( s, , , ...) ln(tF / t )
o,
(53)
(54)
wherein σ is the cyclic stress amplitude; and the environmental damage function is given by
c ( , , , ...,T , tc ) = 1 where
and
( , , , ...) (T , tc ),
( , , , ...) =
(T , tc ) = tc e
H / RT
1 PcD ( , , , ...)
or
T Tref log(A/ xc ) + log tc or , 1/(RT ) ln(tc /( t ))
log(A/ xc ) + log tc . k (T )
(61)
A systematic derivation of creep-integrated fatigue equations has been presented. These creep-integrated fatigue equations possess three unique characteristics: (i) the integrated creep damage is constituted from the constitutive equations of creep; the fitting constants in the creep damage function can be evaluated with ease by performing steady-stress creep rupture test; (ii) creep damage and creep-fatigue interaction damage are embedded naturally into the equation and requires no separate computation; and (iii) the equations are capable of modelling the full range of creep fatigue from pure creep to pure fatigue. These characteristics ensure that these novel equations are fundamentally sound, easy to use, and applicable over a wide range of creep fatigue conditions. The creep-integrated fatigue equations based on three constitutive equations of creep: Dorn equation, Larson-Miller equation, and Manson-Haferd equation have been derived; and these have been applied to the experimental creep fatigue data of two metal alloys that have vastly different creep and fatigue characteristics: the SS316 stainless steel and the Sn37Pb solder. The creep-integrated fatigue equations based on the constitutive equation of Larson-Miller and Manson-Haferd have been found to have modelled very well the creep fatigue of these two metal alloys. The methodology of deriving the creep-integrated fatigue equation can be generalised to model environment assisted fatigue.
These functions of gradients may be represented by a steady-state parameter, P(σs, ξ ,ψ,….). The environment-integrated fatigue equation is given by
= Co c ( , , , ...,T , tc ) N
(T , tc ) = tc k (T ) or
(60)
5. Conclusions
xF t = HF/ RT or h ( s, , , ...) h ( s, , , ...) e =
1 PcD ( , , , ...)
We have learnt from the experience of creep-integrated fatigue equations that the three creep constitutive equations have returned varied performance. Thus, the proposed environment-integrated fatigue equations ought to be validated against experimental data.
x or x = F tF
s, , ,...)(T Tref ),
(59)
(55) (56) (57)
Appendix A. Theoretical derivation of cyclic creep rupture parameters A.1. Cyclic rupture Larson-Miller parameter The average creep strain rate experienced by a test specimen undergoing pure cyclic creep rupture is assumed to take the form of Eq. (21); that is,
¯c =
cR
tcR
= Be
Q ( )/ RT ,
T
0.
(A1)
Rearranging gives the cyclic rupture Larson-Miller parameter: 8
International Journal of Fatigue 128 (2019) 105184
E.H. Wong
PcRLM ( ) =
Q( ) ln BcR + ln tcR = . R 1/T
(A2)
Assuming identical creep damage due to tensile and compressive stresses, and linear cumulation of creep strain over varied magnitudes of stress one arrives at tc 0
Q( )=
H (| (t )|) dt tc
.
(A3)
From Eq. (24), substituting H(σ(t)) in Eq. (A3) with RPLM(σ(t)) and then substituting into Eq. (A2) yields tc
PcRLM ( ) =
0
PLM (| (t )|) dt tc
.
(A4)
A.2. Cyclic rupture Manson-Haferd parameter The average creep strain rate experienced by a test specimen undergoing pure cyclic creep rupture is assumed to take the form of Eq. (26); that is,
¯c =
cR
tcR
= De (T
Tref ) s ( ) ,
T
Tref .
(A5)
Rearranging gives the cyclic rupture Manson-Haferd parameter:
T Tref 1 = , s( ) ln(tcR /tcR )
PcRMH ( ) =
(A6)
Following the same assumptions presented in the previous section gives
s ( )=
tc 0
r (| (t )|) dt tc
.
(A7)
From Eq. (27), substituting r(σ(t)) in Eq. (A7) with −1/PMH(σ(t)) and then substituting into Eq. (A6) yields
PcRMH ( ) =
tc tc dt 0 PMH ( (t ))
.
(A8)
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