Resortes,paper

Stress relaxation of nickel-based superalloy helical springs athigh temperatures

Simon P.A. Gill a,n, Gordon McColvin b, Andrew Strang a

a Department of Engineering, University of Leicester, University Road, Leicester LE1 7RH, UKb Alstom Power Ltd, Newbold Road, Rugby CV21 2NH, UK

a r t i c l e i n f o

Article history:Received 17 January 2014Received in revised form19 May 2014Accepted 21 June 2014Available online 28 June 2014

Keywords:Stress relaxationTorsional creepGeometrically necessary dislocationsHelical springs

a b s t r a c t

The creep resistance of materials in spring applications is generally acknowledged to be well below thatobserved in other applications. Helical springs formed from three candidate nickel-based superalloys,Nimonic 90, René 41 and Haynes 282, have been tested under compression in order to gain some insightinto this phenomenon. Stress relaxation tests conducted at 600–700 1C found that, under constantdisplacement, the degradation of the spring force is one to three orders of magnitude faster than wouldbe predicted from creep data from extruded samples under equivalent tensile loading. An analyticalmodel for torsional creep in helical springs is derived from a modified version of the Dyson creep model.The effects of various microstructural features on the deformation rate are considered. Effects such as thecoarsening of the precipitate-strengthening gamma-prime phase, tertiary creep due to dislocationmultiplication, damage evolution and hardening due to transfer of the stress to the particles from thematrix are concluded to make negligible contributions. It is predicted that the poor performance ofthe springs is due to the very high population of geometrically necessary dislocations that result fromthe bending and twisting of the wire into a helical coil. It is expected that these dislocations are resistantto conventional heat treatments, resulting in a persistent residual stress field and a large number ofdislocations to facilitate the creep process. In some cases, the stress relaxation is found to be so fast thatthe precipitate hardening of the alloy is too slow to prevent significant initial degradation of the spring.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Metallic wire, helical coil compression springs continue to be ofconsiderable practical interest in many high temperature plantapplications, but to date have failed to live up to expectationsbased upon axial creep properties [6]. Springs of this type are usedextensively in power plant surface clearance control systems. Theinspiration to delve further into the issues of springs for hightemperature service arises from a potential requirement for springmaterials to match the needs of ever increasing power planttemperatures. This will enable increased operating temperaturesand reduced CO2 emissions per megawatt.

A mechanical modelling approach has been adopted to high-light those materials and engineering characteristics that wouldbest lead to a wire wound helical coil compression spring that willoperate at higher temperatures more in keeping with the knownhigh temperature properties of the parent material. The modellingapproach has been matched with practical investigation of coil

springs in three common usage nickel-based superalloys wellknown for their high temperature creep strength Nimonic 90,René 41 and Haynes 282. The materials selected are available asfine wire products and all exhibit good fabricability in coil forming.

Nimonic 90 was developed as an age-hardenable creep resis-tant alloy for temperatures up to about 920 1C. It has good ductilityand is typically used in turbine blades, ring sections, hot workingtools, forgings and high temperature springs [15]. René 41 wasdesigned to be used in high stress applications within the 650–980 1C temperature range. René-41 is difficult to fabricate due toits higher gamma-prime fraction and it also work hardens rapidly,requiring frequent annealing stages. As a material with goodresistance to oxidation and corrosion, it is often used in applica-tions such as after-burner components, turbine casings andfasteners [17]; however, it is known to suffer from strain agecracking. Haynes 282 is a more recent alloy developed as asuccessor to Waspaloy and René-41. It combines good thermalstability, weldability, and fabricability not found in many com-mercial alloys, and has excellent creep strength equal to that ofRené-41, in the temperature range of 649–927 1C. Slow gamma-prime precipitation kinetics result in excellent ductility in the as-annealed state. It is often used in components in critical areas of

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Materials Science & Engineering A

http://dx.doi.org/10.1016/j.msea.2014.06.0800921-5093/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author.E-mail address: [email protected] (S.P.A. Gill).

Materials Science & Engineering A 613 (2014) 117–129

gas turbines, combustors, compressors and exhaust/nozzle appli-cations [17].

The experimental procedure for measuring the stress relaxa-tion in the springs is outlined in Section 2. A simple, analyticalmodel for this stress relaxation is developed in Section3. Section 4compares the predictions of the model with the experimentalresults and proposes explanations for the lower than expectedperformance of nickel-based superalloy springs and how it can beimproved.

2. Materials and experimental procedure

The primary objective of the study was to determine thecompressive stress relaxation behaviour of closely wound helicalsprings suitable for operation at temperatures in the range of 600–750 1C. Accordingly, three creep resistant nickel-based alloys,Nimonic 90, Rene 41 and Haynes Alloy 282, were selected aspotential candidates for applications within this temperaturerange. In each case the alloys were supplied to spring manufac-turers as solution treated 30% cold drawn to 2.5 mm diameterwire, i.e., in the ‘spring tempered condition,’ as is normal for wirecoil spring manufacture. The springs were then cold wound to therequired dimension on a mandrel and subsequently given aprecipitation hardening treatment appropriate to each alloy. TheNimonic 90 springs were manufactured by Alstom Power and theRené 41 and Haynes 282 springs were manufactured by EuropeanSprings and Pressings Ltd. Details of the nominal compositions ofthe alloys are shown in Table 1.

The solution treatment and post spring winding heat treat-ments for the three alloys are given in Table 2. Note that Haynes282 is the only alloy that undergoes a two-stage precipitationhardening heat treatment. The René 41 requires the highestprecipitation hardening conditions to achieve the optimum pre-cipitate distribution for high temperature strength and ductility, asit has the highest concentration of gamma-prime formingelements.

In the present studies the helical springs wound from 2.5 mmdiameter wire were 29 mm in height with outside and insidediameters of 19 mm and 14 mm, respectively, as shown in Fig. 1.Each spring consisted of five free coils and two end coils whichwere tapered and flattened. Following measurement of the overallfree length of each of the springs, they were fully compressiontested in an Instron Tensometer to determine their load/displace-ment characteristics, as shown in Fig. 2. This enabled the springconstant for each candidate material to be determined as well asthe load required for any predetermined compression of eachspring. No significant difference between the spring constants forthe three alloys was observed.

The compressive stress relaxation tests were conducted onhelical springs using the procedure described by Betteridge [22]. Inorder to determine the effects of long-term thermal exposure onthe compressive stress relaxation of the springs, samples of eachalloy were placed on a 12 mm diameter stainless steel bolt withwashers at each end and pre-compressed to lengths of 19 mm,21 mm and 23 mm, as shown in Fig. 1. Nuts were screwed down tolock the displacements and the loads corresponding to each

displacement condition recorded. The bolted assemblies for eachof the three alloys and three compressive displacements were thenplaced in furnaces set at 600 1C, 650 1C and 700 1C for exposuredurations of 1, 3, 10, 30, 100, 300 and 1000 h. Three springs wereused for every test and the remnant force reported for each test isthe average over the three results. In every case the standard

Table 1Nominal compositions in wt% of the three nickel-based superalloys in the experi-mental trial: Nimonic 90 (N90), René 41 (R41) and Haynes 282 (H282).

wt% Ni Cr Co Mo Ti Al Fe Mn Si C B

H282 57.7 19.57 10.23 8.5 2.1 1.43 0.37 0.02 0.05 0.061 0.006N90 53 19.5 18 N/A 2.4 1.4 3 1 1.5 0.13 N/AR41 52.5 19 11.06 9.67 3.11 1.56 2.93 0.07 0.05 0.08 0.006

Table 2Solution and final precipitation heat treatments for the superalloy wire and helicalsprings (AC¼air cooled).

Alloy Wire solution treatmenttemperatures (1C)

Spring precipitation treatmenttemperatures (1C)

Haynes 282 1150 1C rapid cool 2 h 1010 1C ACþ8 h 788 1C ACNimonic 90 1150 1C rapid cool 4 h 750 1C ACRené 41 1080 1C rapid cool 16 h 760 1C AC

Fig. 1. The spring in the bolt assembly. The ends of the springs are flattened andtapered. The spring is subjected to a fixed compressive displacement δ using atightened nut and bolt.

0 2 4 6 8 10 12 140

50

100

150

200

250

300

350

sprin

g fo

rce,

P (N

)

displacement, δδ (mm)

Rene 41Haynes 282Nimonic 90

Fig. 2. Compressive load–displacement plots for the springs in the as-suppliedcondition.

S.P.A. Gill et al. / Materials Science & Engineering A 613 (2014) 117–129118

deviation was very small, indicating that the results were veryreproducible.

Following each period of exposure, batches of samples wereremoved from the furnaces and allowed to cool to room tempera-ture. The assemblies were then disassembled and the overalllength of each spring recorded. This was followed by compressiondisplacement tests on each of the exposed samples which allowedthe residual force in the spring and the spring constant to bedetermined, as shown in Fig. 3. The ratio of the load supplied bythe thermally exposed spring relative to that in the original as-supplied spring is used to indicate the degree of load relaxationthat occurred during the test. Data for each alloy for the threeinitial spring compression conditions can then be plotted asdiagrams of load relaxation against exposure time up to 1000 hfor each of the tests at 600 1C, 650 1C and 700 1C. This is presentedin Section 4.1 alongside the model predictions. The spring constantwas not found to be significantly affected by the stress relaxationprocess. Also, it was found that the distance between the free coilsalways stays a fixed proportion of the distance between the endsof the spring during the stress relaxation tests. This implies thatthe ends of the springs behave in a similar fashion to the rest ofthe spring. This is an important observation for the next section,where a model is developed for predicting the stress relaxationresponse in the free coils of the spring such that the end effects areneglected.

3. Modelling the stress relaxation of springs

This section develops and explores theoretical models [2] thatcan explain the observed experimental data. First the initialstresses in the deformed linear, elastic spring are established inSection 3.1. The creep model for the nickel-based superalloysunder study is introduced in Section 3.2. These two models arethen combined in Section 3.3 to develop a simplified, analyticalmodel for the stress relaxation in a cylindrical wire under torsion.Full finite element modelling of the stress relaxation in a helicalspring is undertaken in Section 3.4 for comparison. The effect offurther physical processes is considered in Sections 3.5 and 3.6.First the development of a back stress is included due to partialtransfer of the load from the matrix to the particulate phase.Tertiary creep in the context of dislocation multiplication is alsoconsidered in Section 3.5. Section 3.6 looks at the evolution of thegamma prime particles in these precipitate strengthened materi-als, namely, particle coarsening and further post-heat treatment

precipitation of gamma prime. Finally, a summary of the finalmodel is given in Section 3.7.

3.1. Stresses in the spring

Neglecting the small contribution from the pitch of the spring,the length of wire in a spring of coil radius R consisting of N coils isL¼ 2πRN, where R is defined to be the average of the minimum(inside) and maximum (outside) coil radii. If the spring is verticallycompressed a distance δ then the total torsional rotation of thewire due to this downward movement is δ=R. The rotation per unitlength (twist) is therefore

ϕ¼ δ2πR2N

: ð3:1Þ

Initially we neglect the contribution from the shear force actingon the wire, as well as the effect of the curvature (R) and pitch ðψ Þ.This essentially assumes that the response of the helical spring isequivalent to that of a straight, cylindrical wire of radius a, subjectto a pure torsion of ϕ per unit length. In this case, the only stress inthe wire is the torsional shear stress

γθsðrÞ ¼ rϕ ð3:2Þwhere r is the radial distance from the central longitudinal s-axisrunning along the wire length and θ is the angle between r andthe horizontal xy plane, as shown in Fig. 4. The torque per unitlength required to achieve this twisting is

Q0 ¼Z 2π

0

Z a

0r2τθsdr dθ¼ 2πGϕ

Z a

0r3dr¼ πGϕa4

2ð3:3Þ

where the shear stress τθs ¼ Gγθs and G is the shear modulus.Given that the torque applied to the spring by a vertical compres-sive force P0 is Q0 ¼ P0R, the linear elastic force–displacementrelation for the straight wire spring approximation is therefore

P0 ¼ kδ ð3:4Þwhere the spring (stiffness) constant

k¼ Ga4

4R3N: ð3:5Þ

For the given geometry of a¼1.25 mm and R¼8.25 mm, it isfound that the spring constant of the springs both before and aftertesting (there is little significant change) is 17.5 N/mm. Given aroom temperature shear modulus of G0 ¼ 82:5 GPa this equates toa spring with N¼5.12 effective coils. The true number of ‘complete’coils in the spring is not clearly defined, as the ends of the springwire are tapered and flattened.

0 2 4 6 8 10 12 140

50

100

150

200

250

300

350sp

ring

forc

e, P

(N)

displacement, δδδδ (mm)

as-received1000 hours at 700oC

Fig. 3. Comparison of the load–displacement response for an as-received Haynes282 spring and a similar spring following 1000 h exposure at 700 1C under a fixed10 mm compressive displacement.

Fig. 4. The spring geometry. (a) Side view showing the distance travelled along thelength of the wire, s, and the pitch angle ψ . The exploded view of the wire sectionshows the distance from the centre of the wire, r, at an angle θ to the xy-plane. Thebase of the spring is in the xy-plane and Z is the helical axis. (b) Plan view showingthe mean radius of the coil R. The dashed lines indicate a small wire segment.

S.P.A. Gill et al. / Materials Science & Engineering A 613 (2014) 117–129 119

The simple model for a straight cylindrical wire suggests thatthe maximum shear strain due to torsion is γmax

θs ¼ aϕ. For thecurved wire, with average radius of curvature R, the variation inthe width of a curved arc segment, as indicated by the dashed linesin Fig. 4b, relative to the mean width is 1þ r=R

� �cos θ. A simple

model for the strain variation through the cross-section that takesinto account the curvature of the wire is therefore

γθsðr;θÞ ¼rϕ

1þðr=RÞ cos θ: ð3:6Þ

This first order correction to the model for wire curvatureshows that the shear strain at the innermost point of the springwire γmax

θs ¼ ðc=ðc�1ÞÞaϕ is now the maximum shear strain, wherec¼ R=a is known as the spring index and c=ðc�1Þ is known as thestress correction factor [8]. Higher order corrections to the stresscorrection factor have been proposed and investigated [7,8]. In thispaper we have c¼6.6 which suggests that to first order there is astress correction factor of 1.179. This compares favourably withresults from finite element calculations which find 1.213 [8]. If thepitch of the spring, ψ ¼ sin �1ðh=LÞ, is included then the radius ofcurvature of the coils increases slightly from R to R= cos ψ . Herewe have h¼29 mm so the pitch correction factor 1= cos ψ is1.006. The effect of this adjustment is negligible. Therefore, to firstorder, we expect the stresses in a zero-pitch (toroidal) wire undertorsion to be very similar to those in the helical spring. Thisconcurs with the results of finite element calculations for smallpitches [8]. To first order, the torque induced in the wire is nowwritten as

Q0 ¼ GZ 2π

0

Z a

0r2γθs 1þ r

R

� �cos θ

� �dr dθ¼ πGϕa4

2ð3:7Þ

This is identical to (3.3) demonstrating that although the stressdistribution in a straight wire is different from that in a helicalspring [11], the net effect on the linear elastic response of thespring is a second order effect in c.

As well as the torsional shear stress, τθs, there is also a shearstress τzs ¼ P0=Az to balance the applied shear force P0, whereAz ¼ πa2= cos ψ is the resolved cross-sectional area of the wire inthe z-direction. As we have seen above, the pitch effect isnegligible such that the effective shear stress at a point is

τ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτ2θsþτ2zs

q. We have ðτmax

zs =τmaxθs Þ2 ¼ 1=ð4c2Þ ¼ 0:005 and there-

fore it is found that 99.5% of the load carried by a spring withc¼6.6 is due to torsion of the wire and hence in a simple analyticalmodel the contribution from the shear force can be neglected.Consequently we simply write γ ¼ γθs and τ¼ τθs for the subse-quent duration of the paper. This is why (3.4) and (3.5) have beenwidely and effectively used in the analysis of springs, althoughthey strictly apply to a straight wire under pure torsion.

For comparison with standard uniaxial tensile creep tests weconsider the effective strain ϵe ¼ γ=

ffiffiffi3

pand the effective (von

Mises) stress σe ¼ffiffiffi3

pτ in the spring [3]. The applied end displace-

ments used here are δ¼ 6; 8 and 10 mm. These equate to max-imum effective strains of ϵmax

e ¼ 0:34%, 0.45% and 0.57% for astraight wire and ϵmax

e ¼ 0:41%, 0.54% and 0.68% for a curvedspring. To determine the maximum effective stresses it is impor-tant to take the effect of temperature on the modulus into account.We use a standard approximation that the modulus halves as themelting temperature is approached, GðTÞ ¼ G0ð1�ðT�300=2ðTm�300ÞÞÞ; where Tm¼1643 K. This is a reasonable fit to theexperimental data over the temperature range of interest [15].Therefore σmax

e ¼ 382 MPa, 511 MPa and 638 MPa at 600 1C for astraight wire and σmax

e ¼ 449 MPa, 602 MPa and 752 MPa for acurved spring. Although the applied load on the springs is small,P0¼105–175 N, the stresses are very large. For reference, the 0.1%proof stress for Nimonic 90 at 600 1C is approximately 700 MPa [5]

and the tensile strength is 1040 MPa [4]. This suggests that someinitial plastic yielding should occur in a very small area around theinnermost section of the spring wire at 600 1C at the highestapplied displacement of 10 mm although there is no evidence forthis. This is thought to be due to residual stresses in the wire (seeSection 4.3). The inelastic response of the spring materials is nowinvestigated in the context of uniaxial tensile creep.

3.2. The Dyson creep model

The physics-based creep law of Dyson [5] is adopted to modelthe time-dependent relaxation of the stresses in the springs. Thishas been widely applied to high gamma-prime fraction nickel-based superalloys such as CMSX-4 [10], as well as medium gamma-prime fraction superalloys such as Nimonic 90 [5]. One of theadvantages of this model is that it is derived from physicalprinciples, so the material constants are related to measurablematerial properties. The physical model assumes dislocation climbas the dominant mechanism. Harrison et al. [4] show in a physics-based deformation map that dislocation climb is expected to be animportant creep mechanism for effective stresses in the range of80–640 MPa for Nimonic 90. At high stresses particle shearingbecomes more important and dominates above 640 MPa. At lowerstresses diffusion creep is more active and below 80 MPa it isexpected to be the dominant mechanism. Atkinson and Gill [9] haveextended the Dyson creep model to include particle shearing (highvolume fraction superalloys) and strengthening of the matrix (lowvolume fraction superalloys), although incorporation of either effectis not expected to be necessary for the medium volume fractionsuperalloys considered here. The uniaxial creep strain rate in thiscase is derived to be the following function of the applied stress σ

dϵc

dt¼ A sinhðασÞ: ð3:8Þ

where the two material parameters

A¼ ρð1þηÞð1� f ÞDs=M: ð3:9Þand

α¼ b2ηrpMkBT

ð1�HÞ ð3:10Þ

are defined in terms of the dislocation density ρ, the gamma-prime

volume fraction f , the diffusion coefficient, Ds ¼ 10�4expð�ð3:2�105=RGTÞÞ, Burger's vector b¼0.25 nm, the particle spacing-to-radius ratio, η¼ 1:6ðð

ffiffiffiffiffiffiπ=

p4f Þ�1Þ and the particle radius, rp, where

kB is Boltzmann’s constant, RG is the gas constant, T is thetemperature in Kelvin,M¼3 is Taylor’s constant for a polycrystallinematerial and H is a hardening parameter which models thedevelopment of a back stress and the transition from primary tosecondary creep. Note that Dyson proposes a slightly different pre-factor A¼ ρηf ð1� f ÞDs=M. Eq. (3.9) is a modified expression pro-posed by Atkinson and Gill [9] which takes into account glide inaddition to climb and was found to more closely fit the creep datapresented in this paper. The accumulation of damage is notrepresented in (3.8). This is reasonable as the total plastic strains(o1%) here are too small for this to be a necessary consideration.Dyson [5] proposes the following expression for the equilibriumvolume fraction as a function of temperature

f ðTÞ ¼ c0�cEðTÞ0:23�cEðTÞ

ð3:11Þ

where c0 is the combined mol% concentration of gamma-primeforming elements (Al and Ti for the alloys considered here) andcE ¼ 17expð�ð7250=TÞÞ is their combined equilibrium matrix con-centration. In general the volume fraction increases with c0 anddecreases with temperature.

S.P.A. Gill et al. / Materials Science & Engineering A 613 (2014) 117–129120

Mustata et al. [12] express (3.8) for a multiaxial stress state σij

as

dϵcijdt

¼ 3sij2σe

A sinhðασeÞ: ð3:12Þ

where the deviatoric stress, sij ¼ σij�1=3ðσkkδijÞ, determines thecreep direction and the von Mises effective stress, σe ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3sijsij=2

p,

determines the creep rate.The evolution of the back stress during primary creep,

σb ¼Hσe, is given by [5]

dσb

dt¼ Efð1� f Þ 1�σb

σn

b

!dϵcedt

: ð3:13Þ

where the effective creep strain ϵce ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2ϵcijϵcijÞ=3

qand the max-

imum back stress σn

b ¼Hnσe with Hn ¼ 2f =ð1þ2f Þ. Tertiary creep isincorporated into the model via dislocation multiplication suchthat

dρdt

¼ ρ0K0dϵcedt

ð3:14Þ

where ρ0 is the initial dislocation density and K0 ¼ 300 [5] is amultiplication constant.

The calibration for this model is shown for Nimonic 90 in Fig. 5against data for an extruded section subsequently cold rolled [15].The fit is extremely good at predicting the applied stressesrequired to generate creep strains of 0.1% and 0.2% at given times,and is reasonable for 0.5%, although the stress required is slightlyoverestimated, i.e., the stress-dependence is underestimated.Similar calibrations were conducted for René 41 [17] and Haynes282 [17] from available creep data. The material input parametersare given in Table 3. As discussed in Section 1, the manufacturersquoted creep performance of Haynes 282 and René 41 extrudedsections in the chosen temperature range are comparable andmarginally better than that of Nimonic 90. The creep modeloutlined here for uniaxial tensile loading is now developed forthe case of torsional stress relaxation in the spring wire.

3.3. An analytical solution for stress relaxation in a straight,cylindrical wire under pure torsion due to steady state creep

For a purely torsional stress state (3.2), (3.12) can be used toexpress the local shear creep strain γc in terms of the local shearstress τ as

dγc

dt¼

ffiffiffi3

pA sinhð

ffiffiffi3

pατÞ: ð3:15Þ

This is similar in form to the uniaxial case (3.8) withM replacedby M=

ffiffiffi3

p. The instantaneous shear stress arises from the elastic

shear strain, τ¼ Gγe, where the total shear strain γ ¼ γeþγc is thesum of the elastic shear strain and the creep shear strain [1]. From(3.6) we write γc ¼ grϕ, where gðr;θÞ ¼ ð1þðr=RÞ cos θÞ�1. Hence

dγc

dt¼

ffiffiffi3

pA sinhð

ffiffiffi3

pαGðgrϕ�γcÞÞ: ð3:16Þ

To make analytical progress, we now make some simplifyingassumptions about the constitutive law. These will be validatedagainst the full model in Sections 3.4 and 3.5. Firstly, steady stateprimary creep is assumed by neglecting the back stress (H¼0) anddislocation multiplication ðρ¼ ρ0Þ. As the initial creep rate istypically the fastest (for plastic strainso1%) this assumption isexpected to slightly overestimate the creep rate. As the stressrelaxation of the spring is controlled for the majority ofthe process by the highly stressed regions near the outside ofthe wire (large r) we make the additional assumption thatsinhðxÞ � 1=2ðex�1Þ for x⪢1, where the one is included to givethe correct asymptotics as x-0. Therefore (3.16) is approximatedas

dγc

dt¼ 12A0½expðβðgrϕ�γcÞÞ�1�: ð3:17Þ

where A0 ¼ffiffiffi3

pρoð1þηÞð1� f ÞDs=M and β¼

ffiffiffi3

pGb2ηrp=MkBT are

constants. This can be integrated over time such that

γcðr; tÞ ¼ grϕþ1βln 1�exp �1

2A0βt

� �ð1�expð�βgrϕÞÞ

ð3:18Þ

The torque applied to the spring to maintain the torsion ϕ isderived from (3.7)

Q ðtÞ ¼ PðtÞR¼ GZ 2π

0

Z a

0r2ðgrϕ�γcÞ 1þ r

R

� �cos θ

� �dr dθ ð3:19Þ

and therefore the relative force in the spring (compared to itsinitial value) is

PðtÞP0

¼ � 2βπϕa4

Z 2π

0

Z a

0

r2

gln ½1�e�ð1=2ÞA0βtð1�e�βg r ϕÞ�dr dθ:

ð3:20Þ

Numerical calculations show that the inclusion of g has anegligible effect (some terms cancel due to symmetry and othersare small) such that it is reasonable to assume g¼1. The integralscan then be evaluated analytically to give

PðtÞP0

¼ � 43h

lnð1�yÞþ ∑2

n ¼ 0

6

n!h3�n S 4�n;ye�h

y�1

� �� 6

h3S 4;

yy�1

� � ð3:21Þ

where h¼ βaϕ is a dimensionless group, y¼ expð�ð1=2ÞA0βtÞ isthe measure of time and Sðm; zÞ ¼∑1

p ¼ 1 zp=pm is the polylogarith-

mic function. Even further simplification can provide more insightinto what controls the creep process. If we assume that e�βrϕ⪡1

10 100 1000 10000 1000000

100

200

300

400

500

600

7000.1%0.2%0.5%

appl

ied

stre

ss (M

Pa)

time(hours)

Fig. 5. The calibration of the uniaxial creep model for Nimonic 90 at 600 1C (uppercurves) and 700 1C (lower curves) using creep stress data for accumulated plasticstrains of 0.1%, 0.2% and 0.5% for an extruded section subsequently cold rolled [15].

Table 3Material parameters for the variant Dyson creep model (3.8) calibrated fromuniaxial tensile creep tests on extruded sections.

Alloy Particleradius,rp (nm)

Dislocationdensity,ρ0

Concentration,c0 (mol%)

Nimonic90

10 0.3�1010 m�2 5.7

René 41 10 1010 m�2 7.0Haynes 282 7.5 1010 m�2 5.7

S.P.A. Gill et al. / Materials Science & Engineering A 613 (2014) 117–129 121

then (3.20) is easily evaluated as

PðtÞP0

� � 43h

lnð1�yÞ ð3:22Þ

which is equivalent to ignoring the last two terms in (3.21). Notethat this expression does not have the correct asymptotics as t-0,and hence this expression is not expected to be valid at earlytimes. In addition, at later times e�βrϕ⪡1 is not valid, so it isexpected that (3.22) only applies for roughly 0:3o ðP=PoÞo0:7. Inthis regime it is expected that A0βt⪡1 so a further simplification is

PðtÞP0

� � 43h

lnðtÞþ ln12A0β

� � : ð3:23Þ

This is a useful expression as it implies that a plot of P=P0

against ln tð Þ will yield a linear fit (in the region of validity) wherethe creep exponent β can be determined from the slope and thecreep pre-factor Ao can be determined from the intercept.

These expressions are fitted to the results for a single stressrelaxation test on a Nimonic 90 spring at 600 1C in Fig. 6. The forcevs. time plot in Fig. 6a shows that the model using the uniaxialcreep data ðρ0 ¼ 0:3� 1010 m�2Þ is clearly very poor. However, agood fit is obtained in this case (in the absence of the hardeningfactor H) for ρ0 ¼ 1012. This is simply equivalent to rescaling thetime axis, but indicates that the stress relaxation in the spring isroughly 300 times faster than would be expected in an extrudedbar in tension. For the remaining duration of this paper, only forcevs. log 10ðtÞ plots will be presented, such as that shown in Fig. 6b.This more clearly shows the short term and long term asymptoticsof the behaviour. It also demonstrates the predicted linear regime

in the middle of the profile for 0:3o ðP=PoÞo0:7 and shows thelimits of the simple expressions (3.22) and (3.23), which areidentical for ðP=P0Þ40:2. The predictions from this simple analy-tical model are now compared with the exact solution obtainedfrom finite element simulations.

3.4. Comparison between stress relaxation due to steady state creepin a straight wire, a zero-pitch spring and a helical spring by finiteelement analysis

In the derivation of the analytical model (3.21) it has beenassumed that the spring wire is subjected to a purely torsionalstress state (3.2), with the conclusion that the first order correction(3.6) has no net elastic effect, and that higher order corrections aresmall. In addition, the shear stress τrs has also been ignored assmall. In this section, the general form of the creep law (3.12) isemployed in finite element calculations with the same steady statecreep rate assumptions of Section 3.3. Firstly pure torsional creepin a straight cylindrical wire was simulated. The stress relaxationof the wire was found to be identical to the predictions of theanalytical model (3.21) as required. The effect of the wire curva-ture R was evaluated by simulating a 2% length of a single helicalcoil (a 7.21) section, as shown in Fig. 7. Both end faces weredisplaced in the z-direction (the direction of compression) by adistance w¼ ðδ=2Þð0:02=5:15Þ at the centre point of each face, oneupward and the other downward. This induces the initial torsionalrotation per unit length ϕ and the corresponding torque Q0 andthe shear force P0. The end faces must also be rigidly rotated by anamount w=l around the radial tangent to each end face to avoidinducing bending in the segment, where l¼ 0:02n2πR is the lengthof the segment. The faces were allowed to move radially outward.The effect of varying the pitch angle ψ was evaluated. Thecalculations were conducted in COMSOL Multiphysics v4.3 usingthe non-linear materials module with the Garofalo (hyperbolicsine) creep model. The stress relaxation response of a straight wire(3.21), a zero-pitch (toroidal) wire and the actual helical springgeometry was determined [20]. They are not illustrated here, as allthe geometries gave almost identical, visually indistinguishableresults. Given the negligible discrepancy between the differentapproaches, it is therefore proposed that for the rest of the paperthe simple geometry of a straight wire subject to pure torsion isamply sufficient to represent the stress relaxation of a helicalspring. In the next section the creep hardening/softening of thematerial is examined in this context.

3.5. Effect of back stress and tertiary creep on stress relaxation

In this section, the time-dependence of the material creepparameters AðtÞ and αðtÞ is considered. Including dislocationmultiplication using (3.14) is typically very important for nickel-based superalloys which demonstrate only a small amount ofprimary creep (up to strains of 0.2–0.3%), rapidly switching to adistinctive tertiary creep regime with an exponential strain ratewithout a significant secondary creep regime. Eq. (3.14) is readilyintegrated with respect to time to give

ρðtÞ ¼ ρ0 1þ 1ffiffiffi3

p K0γcðtÞ� �

: ð3:24Þ

Under the most extreme circumstances of a maximum shearstrain of 0.68%, this suggests a maximum dislocation density of2:2ρ0. Whilst this is not insignificant, it is only in a small region ofthe wire cross-section, and hence in general tertiary creep is notexpected to be of great importance in this application. This isdemonstrated in Fig. 8 where the difference between the modelwith back stress and the model with back stress and tertiary creepis small.

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g fo

rce,

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0

time(hours)

Fig. 6. Comparison of the analytical stress relaxation models (3.21)–(3.23) againstexperimental data for Nimonic 90 at 600 1C with δ¼ 9 mm for (a) time t and (b)log 10ðtÞ on the horizontal axis.

S.P.A. Gill et al. / Materials Science & Engineering A 613 (2014) 117–129122

A back stress is generated due to transfer of the load from thematrix to the particles as creep of the matrix progresses. If theapplied load τ¼ τ0 is constant then (3.13) predicts that

HðtÞ ¼Hn 1�exp �ð1þbÞGγc

τ0

� �� �ð3:25Þ

where b¼ ð1þνÞð1þ2f Þ=ð3ð1� f ÞÞ�1 is a constant. This describesa monotonic progression from zero to the maximum value of Hn

(which is always between 0 and 1). However, for the particularcase of stress relaxation considered here, the applied load is not aconstant and hence (3.13) can be expressed as

dHdtτþH

dτdt

¼ Ef3ð1� f Þ 1� H

Hn

� �dγc

dt: ð3:26Þ

Given τ¼ Gðgrϕ�γcÞ this becomes

Ef3ð1� f Þ 1� H

Hn

� �þHG

�1dHdt

¼ 1Gðgrϕ�γcÞ

dγc

dtð3:27Þ

which can be integrated over time to give

HðtÞ ¼ ab

1� 1�γcðtÞgrϕ

� �b !

ð3:28Þ

where E¼ 2ð1þνÞG and a¼ 2ð1þνÞf =3ð1� f Þ. The two contribu-tions to b arise from the rate of increase in the back stress and therate of decrease in the applied load (due to stress relaxation). Thebehaviour of (3.28) is interesting as b can be either negative orpositive. To investigate the long time predictions of the model welet H0 ¼ limt-1HðtÞ. For a large volume fraction ðf 40:3Þ one hasb40 for which HðtÞ monotonically increases from zero toH0 ¼ minðða=bÞ;1Þ. For materials in the temperature range con-sidered in this paper one has a smaller volume fraction ðf o0:3Þ forwhich bo0. In this case the back stress continues to develop untilthe maximum value H0 ¼ 1 is reached, at which point the creepprocess prematurely stops. The total creep strain at this point isγc ¼ ð1�eÞgrϕ, where e¼ ð1�ðb=aÞÞ1=b. The final residual force inthe spring once creep has stopped is therefore eP0. The value of eonly varies slightly in the context of the materials and testconditions considered here. It has a minimum value of 5% for thelow gamma-prime alloy Nimonic 90 at 700 1C and a maximumvalue of 7% for the higher gamma-prime alloy René 41 at 600 1C.The associated evolution of the hardness parameter (3.28) is

illustrated in Fig. 8. Note that this is purely a stress relaxationphenomenon.

Eq. (3.16) is solved numerically with AðtÞ defined by (3.9) and(3.24) and αðtÞ defined by (3.10) and (3.28). The relative effects ofincluding the different physical processes in the model areillustrated in Fig. 8. It can be seen that inclusion of the back stressslows the predicted stress relaxation rate, typically by a factor ofabout two. As predicted from (3.28), the hardness parameter Hconverges to unity for the case of stress relaxation with bo0. Thisresults in a cessation of the relaxation, with a finite residual forceremaining in the spring. As described previously, consideration ofthe tertiary creep process due to dislocation multiplication (3.24)slightly accelerates the stress relaxation process toward its end butis generally of little significance. These two effects are clearly notsignificant enough to explain the very large discrepancy betweenthe stress relaxation predictions for the experimental spring data(fit) and those from the extruded sample data (uniaxial) in Fig. 6.The effects of microstructural features such as the gamma-primevolume fraction and particle size are now investigated in the nextsection.

Fig. 7. Finite element calculations for the stress relaxation at 600 1C at δ¼ 10 mm for a 7.21 section of the helical spring at (a) before relaxation and (b) after 1000 s. Theslightly asymmetrical stress state is due to the curvature of the spring wire, with the larger stresses on the inside of the wire.

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1.0re

lativ

e sp

ring

forc

e, P

/P0

time(hours)

basic model (3.20)with back stressback stress+tertiaryhardness, H

Fig. 8. The contribution of back stress and tertiary creep to stress relaxation of thespring. The evolution of the hardness parameter H is also shown. As H approachesunity the relaxation of the spring force stops at 5% of the initial value.

S.P.A. Gill et al. / Materials Science & Engineering A 613 (2014) 117–129 123

3.6. Predicted influence of coarsening and further precipitation onstress relaxation

The sinh term in the creep law (3.10) is very sensitive to thevalue of α . Therefore it is of some interest to determine thesensitivity of the stress relaxation of a spring to various para-meters that affect this term. Firstly, Dyson [13] proposes thatparticle coarsening occurs due to classic Ostwald ripening accord-ing to the following expression:

r3pðtÞ ¼ r3pð0Þ 1þ3� 106exp �3� 105

RT

!t

" #ð3:29Þ

At 600 1C this predicts that an increase in radius of 10% willtake over 3000 years and even at 750 1C it is expected to take over7 years, so it seems reasonable to neglect this.

The other contribution from the precipitate-hardening particlesis their volume fraction. Previously it has been assumed that this isa constant, f ðTÞ, determined from the operating temperature Tusing (3.11). However, the volume fraction at the start of the test isas in the as-received state. This depends on the previous heattreatment of the sample. Here it is assumed that the volumefraction of gamma-prime particles in the as-received state is theequilibrium value at the final heat treatment temperature, THT ,given in Table 2. The change in the volume fraction over time canbe described by the Johnson–Avrami–Mehl equation [14]. Assum-ing that the growth is nucleation-controlled and that new pre-cipitates rapidly grow to the same size as the existing precipitates,rp, then the growth exponent is 1. In this case the volume fractionof particles evolves as

dfdt

¼ μðf ðTÞ� f Þ ð3:30Þ

subject to f ð0Þ ¼ f ðTHT Þ, where the constant μ is related to thenucleation rate (and hence is a function of temperature). Precipi-tation of gamma-prime in the temperature range under considera-tion is usually quite rapid, typically occurring over the first 1–10 h.

It is commonly assumed in alloys that they will precipitationharden during service, but in this case the creep is so rapid thatsignificant deformation can take place before precipitation hard-ening occurs. The influence of the initial gamma-prime volumefraction on the early stages of stress relaxation in a Nimonic 90spring at 600 1C is illustrated in Fig. 9 for μ¼ 0.1 per hour. Thelower curve in Fig. 9a shows the change in f for THT ¼ 900 1C, fromits initial low value of 11% (during which significant stressrelaxation occurs) up to its final value of 24% (at which point thestress relaxation slows). It is clear that a large difference betweenthe age-hardening temperature and the in-service temperature ofthe alloy can have a huge effect on the early stages of the stressrelaxation. It is expected that this effect will have the mostinfluence on the behaviour of the René 41 springs, as they havethe highest final age-hardening temperature of 760 1C, as shown inTable 2. Fig. 9b shows how this can also affect the evolution of theback stress in the springs.

3.7. Modelling summary

The stresses on the outside of the spring wire have been shownto be very large, within the vicinity of the yield strength of thematerial in the stipulated temperature range. A basic stressrelaxation model for a cylindrical wire under pure torsion basedon a variant of the Dyson creep model has been derived in (3.21).This has been shown to predict a linear regime (3.23) in a plot ofrelative spring force PðtÞ=Pð0Þ against log 10ðtÞ. In Fig. 7, it has beenshown that this simple model is a good approximation for thestress relaxation of a helical spring, although the creep rate mustbe adjusted to be significantly larger than that observed in uniaxial

extruded creep specimens. Solving the full Eq. (3.16) numericallyallows for the development of a back stress (3.28) and thedistinctive tertiary creep of nickel-based superalloys (3.27). Fig. 8demonstrates that inclusion of the back stress can result in adecrease in the creep rate, although well within the range ofuncertainty for the creep model. It does, however, result in thespring force relaxing to a finite non-zero value of roughly 5–7% ofthe initial value. As expected, tertiary creep is of little consequenceas the strains here are well below 1%. As creep is rapid, particlecoarsening (3.29) is negligible. One significant factor that is notincluded in the basic model (3.21) is the effect of further pre-cipitation in the first 1–10 h of service if the material is functioningwell below the heat treatment temperature, as illustrated inFig. 9a. The predictions of the full model (3.16) are now comparedwith the experimental results in the following section.

4. Discussion

The predictions of the model developed in Section 3 arecompared with experimental data in Section 4.1. The sensitivityof the test specimens to stress is found to be much less than onewould expect from the stress relaxation model. This is explored inSection 4.2. The consequence of this is to look at other factors thatcould induce this type of response. Section 4.3 considers the effectof residual stresses in the springs to offer an explanation for thisphenomenon.

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ness

par

amet

er, H

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Fig. 9. The effect of further gamma-prime precipitation on the stress relaxation of aNimonic 90 spring at 600 1C (a) stress relaxation and gamma-prime volumefraction and (b) harness parameter H.

S.P.A. Gill et al. / Materials Science & Engineering A 613 (2014) 117–129124

4.1. Comparison with experimental results

The predicted stress relaxation responses for the three materi-als introduced in Section 2, Nimonic 90, Haynes 282 and René 41,are shown for different temperatures as the dashed lines inFigs. 10–12, respectively, for the case of δ¼ 10 mm. In every case,the measured relative spring force decays much faster than theuniaxial creep test data predicts. In the extreme case of Nimonic90 at 700 1C in Fig. 10c, the stress relaxation in the springs is threeorders of magnitude faster than expected in a uniaxial tensile

creep test specimen. This suggests that one or more of the materialparameters in the spring wire are substantially different fromthose in the uniaxial test specimens.

The parameters that most depend on the particular processingconditions are the radius of the gamma-prime particles, rp, and theinitial dislocation density, ρ0. As rp appears inside the sinh term, asmall change in its value can have a very significant effect. Byincreasing α, it can dramatically increase the stress-dependenceand temperature-dependence of the stress relaxation rate. This iscommensurate with the observations for Nimonic 90, where therapid stress relaxation at 700 1C is not picked up by the modelwith rp¼10 nm. It is therefore tempting to increase the radius. Theother consequence of increasing rp, however, is that it furtheraccentuates the difference between the δ¼ 6; 8 and 10 mmcurves. Overall the model substantially overestimates the variationin the spring force relaxation curves for the different applieddisplacements. This strong δ-dependence is to be expected for thecreep of most metals, as they generally have a highly non-linearstress-dependence. However, all the experimental measurementsshowing reduced δ-dependence, particularly Nimonic 90 at 700 1Cand René 41 at 650 1C and 700 1C 1C. For this reason, the value ofthe gamma-prime radius rp is assumed to remain unchanged, withthe initial dislocation density ρ0 as the fitting parameter.

The different parameters used are compared in Table 4. Thisshows that the spring relaxation model is a factor of 5 (Haynes282), 20 (René 41) and 67 (Nimonic 90) times faster than expectedfrom the uniaxial tensile creep test data. A potential explanation

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Nimonic 90 at 650oC

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δδδδ=6mmδδδδ=8mmδδδδ=10mm

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tive

sprin

g fo

rce,

P/P

0

time(hours)

Fig. 10. Comparison between the stress relaxation model and the experimentaldata for Nimonic 90 at (a) 600 1C, (b) 650 1C and (c) 700 1C. The solid lines are thepredictions for δ¼ 6 mm (top), 8 mm (middle) and 10 mm (lower). The dashed lineis the prediction for δ¼ 10 mm using the model data fitted to the uniaxial creepmodel shown in Fig. 5. In (c) the dotted line is the prediction for ρ0 ¼ 1012 m�2.

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Haynes 282 at 650oC

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Haynes 282 at 700oC

Fig. 11. Comparison between the stress relaxation model and the experimentaldata for Haynes 282 at (a) 650 1C and (b) 700 1C. The solid lines are the predictionsfor δ¼ 6 mm (top), 8 mm (middle) and 10 mm (lower). The dashed line is theprediction using the model data fitted to the uniaxial creep model for δ¼ 10 mm.

S.P.A. Gill et al. / Materials Science & Engineering A 613 (2014) 117–129 125

for this substantial difference is the different processing conditionsof the spring versus the extruded uniaxial test sample. Onceextruded into spring wire, the alloys are then wound into a helicalcoil of radius R. This bending induces a strain gradient of 1=R.Torsional investigations of strain gradient plasticity theory [19,25–28] suggest that there must be an associated density of geome-trically necessary dislocations of 1=Rb¼ 50� 1010 m�2 to accom-modate this deformation, where b¼0.25 nm is Burger’s vector.There is also a strain gradient of ϕc ¼ h=LR due to the twisting ofthe wire upon coiling, where h and L are the height and wirelength of the resulting spring. This has an associated dislocationdensity of ϕc=b¼ 5� 1010 m�2. The predicted total dislocationdensity due to the formation of the spring is therefore55�1010 m�2. This is clearly much higher than the values

anticipated from the calibration of the uniaxial creep model inTable 4.A proportion of these dislocations will annihilate upon the post-forming heat treatment, although as these are geometricallynecessary dislocations, required to accommodate the plastic strain,they are typically all of the same orientation. This makes themmore resistant to annealing, as whilst statistically stored disloca-tions exist in populations of equally distributed orientations,geometrically necessary dislocations are all of a similar orientationand hence unable to annihilate each other by combination.

As indicated in Table 2, the creep responses and microstruc-tures of the three alloys are only expected to be marginallydifferent prior to the coiling process, or in fact after the coilingprocess. The main difference between them is expected to be afterthe subsequent post-coiling heat treatment. It is telling that thealloy that provides the best performance, closest to its uniaxialcreep test benchmark, is Haynes 282, which has a two-stageprecipitation heat treatment. The two stage process providesgreater control of the precipitating species, particularly carbidesand the gamma-prime phase. The next closest alloy is René 41which has a heat treatment nearly 200 1C higher than the poorestperforming alloy, Nimonic 90. The resistance to creep of René 41 isinitially very poor, with a large reduction in the spring force in thefirst 10 h. The model fits the data closely, and demonstrates thatthis early drop can be explained by the low gamma-prime fractionthat exists in the alloy at the commencement of the test due to thelarge difference between the heat treatment temperature (f¼18%at THT¼899 1C) and the test temperatures (f¼27% at T¼650 1C andf¼29% at T¼700 1C). The subsequent hardening of the alloy isconsistent with further precipitation of gamma-prime at the loweroperating temperatures. So a large difference in the dislocationdensity between the extruded section (used for uniaxial creepdata) and the coiled wire (used in springs) would appear to be avalid explanation for the reduction in creep-resistance of thespring materials. Confirmation of this will be the subject of furthermicrostructural investigation. One outstanding unexplained fea-ture, however, is the lack of variation in the experimental data forthe different loading cases of δ¼ 6; 8 and 10 mm. The modelshows a large separation between the three (upper, middle andlower) curves for these loading cases, which is not reflected in theexperimental data. Reasons for this discrepancy are investigated inthe next section.

4.2. Investigation of the apparent low sensitivity of the springs tostress

The unexpectedly low stress-dependence in the experimentaldata is investigated through the predictions of a simple power lawcreep model

dγcdt

¼ Brϕ�γcγ0

� �n

ð4:1Þ

where B and γ0 are temperature-dependent material parameters,and the stress exponent, n41, can be varied to see the influenceof the magnitude of the non-linearity on the stress relaxationresponse. Integrating (4.1) with respect to time and inserting theresult in (3.20) defines the relative spring force evolution forpower law creep to be

PðtÞP0

¼ 2F 11m

;4m

; 1þ 4

m

; �m

aϕγ0

� �m Bγ0

� �t

� �ð4:2Þ

where 2F 1ð½i; j�; ½k�; xÞ is the hypergeometric function, and m¼ n�1is assumed to be a positive integer. For the linear case (n¼1) thisreduces to PðtÞ=Po ¼ expð�Bγ�1

0 tÞ.The effect of the stress exponent on the shape of the spring

force relaxation curve is shown in Fig. 13a. The value of the

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δδδδ=6mmδδδδ=8mmδδδδ=10mm

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tive

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g fo

rce,

P/P

0

time(hours)

Fig. 12. Comparison between the stress relaxation model and the experimentaldata for René 41 at (a) 650 1C and (b) 700 1C. The solid lines are the predictions forδ¼ 6 mm (top), 8 mm (middle) and 10 mm (lower). The dashed line is theprediction using the model data fitted to the uniaxial creep model for δ¼ 10 mm.

Table 4Comparison of the values of the initial dislocation density ρ0 used to fit the creepdata from uniaxial tensile tests in Fig. 5 and the values used to fit the spring forcerelaxation tests in Fig. 10–12.

Alloy Uniaxial creeptest fit

Stress relaxationspring fit

Factorincrease

Nimonic 90 0.3�1010 m�2 20�1010 m�2 67René 41 1010 m�2 20�1010 m�2 20Haynes 282 1010 m�2 5�1010 m�2 5

S.P.A. Gill et al. / Materials Science & Engineering A 613 (2014) 117–129126

relevant material constant Bγ�n0 is chosen so that the curves

coincide at P=P0 ¼ 0:7 at a similar time. The exponent that bestfits the material data shown is n¼8. This has a relatively shallowrate of decay, with the steepness of the curves increasing as theexponent decreases. With a low stress exponent, the high stresseson the outside of the wire are only marginally more importantthan the lower stresses near the centre, and hence, once initiated,the creep process continues quite rapidly. In the case of the higherstress exponents, the high stresses on the surface of the wiredominate the creep process initially, but are rapidly reduced. Theconsequence of this is a fairly rapid relaxation of the spring forceearly on, followed by a slower relaxation as the resulting lowerelastic stresses in the wire provide a significantly lower drivingforce for the creep process. The conclusion from Fig. 13a thereforeis that the observed spring relaxation force data is consistent witha highly stress-sensitive creep model, such as a high exponent(n¼8–10) power law creep law or the sinh-dependence of theDyson creep model (3.16). The linearity of the uniaxial creep testdata in Fig. 5 clearly demonstrates an exponential Dyson-typedependence, and in fact would, if anything, appear to under-estimate the stress-sensitivity for the higher creep strain data at0.5%. This conclusion is in contrast to the picture presented in

Fig. 13b however. This illustrates how the relaxation profiles areexpected to vary with the applied displacement δ for a test at aparticular temperature. It is clear that a strong dependence on δ isobserved for a high stress-sensitivity (n¼8) as expected, and thatthe strength of this dependence reduces with decreasing n untilthe profile becomes essentially stress-independent for the linearcase (n¼1) where there is just a single master curve for all δ.Comparison with the experimental data shows that the observedvariation between the δ¼ 6; 8 and 10 mm curves is more com-patible with a less stress-sensitive model (n¼2–4), with theextreme case of the data for René 41 in Fig. 12 demonstrating avariation more compatible with n¼1–2, although the curves are ofthe wrong shape.

We have established that a high stress-dependence is expectedfrom the uniaxial creep data and is consistent with the shapes ofthe profiles, if not the variance. Is there then a difference betweenthe creep process in uniaxial tension and the creep process intorsion? This seems unlikely as rigorous experiments on Nimonic80A in a similar stress and temperature range showed no sig-nificant difference between the tensile and torsional creep tests forthe range of strains that apply here of less than 1% [18]. Thisfinding is also supported by torsional creep tests on othermaterials [23,24].

Another explanation is the difference between creep and stressrelaxation, although again, in experimental tests there is nosubstantial observed difference between the two processes, espe-cially in terms of the stress exponent n [21]. A change in the creepmechanism is another factor to be considered. At very highstresses, particle shearing and even possibly initial yielding couldbe expected to dominate above 600 MPa or so [4]. However, thereis no evidence for the action of these processes as these wouldfurther differentiate between the low stress and high stress tests,particularly at the start, and this is not the case. Only a very smallregion on the surface of the δ¼10 mm test sample spring is likelyto experience such high stresses. At lower stresses, typically lessthan 100 MPa [4], diffusion creep is expected to be the operatingprocess. This mechanism is associated with a low stress exponentof n¼1–2. This would be expected to induce a slightly enhancedcreep rate in low stress regions, leading to a more rapid relaxationnear the end of the springs creep life with a smaller stress-variance. This would manifest as a transition from the highexponent profile (n¼8) to the lower exponent profile (n¼1–5)at the latter stages of relaxation, but it would not affect the profilewhen higher stresses dominate in the early stages. The remainingexplanation is the possibility of significant residual stresses exist-ing in the spring. This final consideration is explored in the nextsection.

4.3. Residual stresses

If the springs already have an internal stress state then theeffects of a superimposed stress state due to loading can have areduced (or possibly increased) effect. As discussed in Section 4.1,it is normally expected that any residual stress will be removed bythe heat treatment, although this is typically applied to caseswhere the residual stress arises from statistically stored disloca-tions which are more readily annihilated than geometricallynecessary dislocations as they have an equal population of positiveand negative Burger's vectors. Therefore the existence of a residualstress state is compatible with the explanation that a remnantpopulation of dislocations strongly enhances the stress relaxationrate of coiled wire. It is therefore also expected that the residualstresses in Haynes 282 will be lower than in the other two alloys,as this experiences a two-stage precipitation heat treatment. It istelling therefore that the relaxation profiles for Haynes 282 inFig. 11a and b demonstrate a larger δ-dependence than the other

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n=1n=3n=5 n=8

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n=1n=3n=5 n=8

Fig. 13. The dependence of (a) the slope and (b) the variance, of the relative springforce evolution on the stress exponent n, shown against the experimental data forNimonic 90 at 650 1C for (a) δ¼ 8 mm and (b) δ¼ 6 (upper), 8 (middle) and 10 mm(lower).

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two alloys at similar temperatures, in agreement with the theore-tical predictions.

Residual stresses will arise due to the bending and twisting ofthe wire into a helix, and both can interact with the relaxation ofthe torsional stresses through the effective stress according to(3.12). As the exact post-heat treatment residual stress state is notknown, a simple model is proposed here. Firstly, only the residualstresses due to twisting are considered as these are superimposedon the existing torsional shear stress state. It is also assumed thatthe forming operation will cause the torsional shear stresses toexceed yield across the entire wire cross-section, such that theshear strain γðrÞ ¼ γY is constant, where τY ¼ GγY is the shear yieldstress. Upon elastic unloading there is elastic recovery via arotation per unit length of ϕR. The net torque in the cross-section,Q ¼ 2πG

R a0 ðγY �rϕRÞr dr, must be zero. Therefore ϕR ¼ 4γY=3a and

the residual torsional stress state is represented by the strain fieldγRðrÞ ¼ γY ð1�ð4r=3aÞÞ. This is positive in the centre of the wire andnegative near the surface of the wire. As this has no net torque, ithas no effect on the force–displacement relationship for thespring. This stress field is also compatible with the observationthat the springs do not yield under high compressive loads whenthe stresses at the surface are expected to be of sufficientmagnitude for this to occur. The residual stresses therefore onlyappear in the creep rate, such that (3.17) becomes

dγc

dt¼ 12A0½expðβðγRðrÞþrϕ�γcÞÞ�1�: ð4:3Þ

The analytical solution for the basic creep model (3.20) istherefore

PðtÞP0

¼ � 4βϕa4

Z a

0r2 ln ½1�e�ð1=2ÞA0βtð1�e�βðγRðrÞþ rϕÞ�dr : ð4:4Þ

For the proposed linear residual strain field this has ananalytical solution, similar to (3.21), although it is too unwieldyto reproduce here. To estimate the yield strain we assumeγY �

ffiffiffi3

pðσUTS=EÞ. As the strain is very large it is reasonable to

assume that the stresses approach the UTS of 1175 MPa. Themodulus is about 209 GPa for Nimonic 90 giving a rough valueof γY ¼1%. This is a large value, and is therefore expected to have asignificant effect. In reality it is unlikely that such a high residualstrain exists after heat-treatment, but taking into account theadditional interaction with the residual strains due to bending(approximately 10 times larger) that are of a similar form, this maynot be unreasonable. Fig. 14 compares the results for Haynes 282at 700 1C with a residual torsional stress state and without anyresidual stress (as in Fig. 11b). It is clear that the inclusion of aresidual stress field can significantly reduce the δ-dependence ofthe stress relaxation profile with a relatively small change in theshape of the curve. In the example in Fig. 14, the residual strainmodel now demonstrates a variance which is smaller than theexperimental data. This fits with the prediction that the Haynes282 springs are unlikely to contain substantial residual stressesdue to their high temperature two-stage heat treatment. Theobserved variance is, however, much closer to that observed inthe experiment for Nimonic 90 in Fig. 10 and René 41 in Fig. 12.

5. Summary

The stress relaxation of three nickel-based superalloy springshas been investigated at temperatures between 600 1C and 700 1C.Experimental results indicate that the springs relax very quickly,typically degrading to 50% of their initial spring force within 1–1000 h. The best performing material is Haynes 282, followed byRené 41 and then Nimonic 90. The stresses near the surface of thespring wire are calculated to be approaching the yield stress of the

alloy (if residual stresses are neglected). A model based on thetorsional creep of a cylindrical wire using a modified version of theDyson creep model has been used to predict the stress relaxationresponse of the springs. It is found that the expected performanceof the materials is between one and three orders of magnitudehigher than the observed response. The origin of this poorperformance of nickel-based superalloys in spring applications isexplored within the context of the model. Coarsening of theprecipitate-strengthening gamma-prime phase is found to be anegligible effect, as is the evolution of damage, the transfer ofstress from the matrix to the particles, and the increase in thedislocation density with strain. The remaining phenomena that areexpected to significantly enhance the stress relaxation of thematerials are the high density of geometrically necessary disloca-tions in the spring material due to the bending and torsioninduced in a coiled spring wire during the manufacturing process,the resistance of these dislocation populations to heat treatments,the resulting persistent residual stresses derived from thesedislocations, and the inability of these materials to effectivelyprecipitation harden during service due to the very rapid initialstress relaxation rate.

Acknowledgements

The contribution of the final year project students Anna Kareer,Alex Harris, Alan Holliday and Vijay Anumula is gratefullyacknowledged.

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tive

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