Fatigue life analysis and predictions for NR and SBR under
variable amplitude and multiaxial loading conditions
Ryan J.Harbour
a,1
,Ali Fatemi
b,*
,Will V.Mars
c,2
a
Corporate Rearch,Goodyear Tire and Rubber Company,142Goodyear Boulevard,Akron,OH 44316,USA
b
Mechanical,Industrial and Manufacturing Engineering Department,The University of Toledo,2801West Bancroft Street,Toledo,OH 43606,USA
c
Rearch Department,Cooper Tire and Rubber Company,701Lima Avenue,Findlay,OH 45840,USA
Received 25March 2007;received in revid form 22August 2007;accepted 28August 2007
西藏能去旅游
Available online 5September 2007
Abstract
This paper investigates the effects of variable amplitude loading conditions on the fatigue lives of multiaxial rubber specimens.Two filled rubber materials were ud and compared to investigate the effects of strain-crystallization on crack development NR,which strain crystallizes,and SBR,which does not.The applicability of Miner’s linear damage rule for predicting fatigue lives of variable amplitude tests in rubber and the u of both scalar and plane-specific equivalence parameters to characterize fatigue life results were also inves-tigated.A fatigue life prediction approach that utilizes normal strain to find the critical plane and the cracking energy density on that plane to determine fati
gue life is introduced and compared to other approaches.The effects of load quence and temperature on fatigue life,as well as differences in fatigue lives using both stiffness and critical crack length failure criteria are discusd.Ó2007Elvier Ltd.All rights rerved.
Keywords:Multiaxial fatigue of rubber;Variable amplitude loading;Fatigue life prediction;Equivalence parameters
1.Introduction
The nature of rubber to withstand large strains without being permanently deformed has made it a popular mate-rial choice for many manufactured products such as tires.This wide range of product usage means that rubber under-goes a large variety of loading conditions that need to be analyzed in order to fully understand the failure process of rubber.Realistic rvice histories for the rubber com-ponents almost always involve variable amplitude loading conditions.In order to improve the durability and the ana-lytical methods ud for predicting fatigue lives of rubber components,a better understanding of the effects of vari-
水土不服长痘痘怎么治
able amplitude loading conditions on the fatigue behavior of rubber is necessary.
While various rearchers,including Mars and Fatemi [1,2],have studied fatigue lives for rubber,the majority have focud on constant amplitude loading conditions.The rearch by Klenke and Beste [3]on metal–rubber components subjected to single step test signals reprented one of the first applications of Miner’s linear damage rule [4]to predict fatigue lives involving rubber.Sun et al.[5]investigated the effects of load quencing on fatigue life by using step-up and step-down experimental signals and found that Miner’s linear damage rule was not applicable under tho conditions.A rainflow filtering process has been investigated by Steinwegger et al.[6]as an attempt to reduce test times for both uniaxial and multiaxial vari-able amplitude test signals.
The goal of the fatigue experiments described in this work was to investigate the effects of variable amplitude loading conditions on fatigue behavior in filled rubbers.Since producing a component with a long rvice life is
0142-1123/$-e front matter Ó2007Elvier Ltd.All rights rerved.doi:10.1016/j.ijfatigue.2007.08.015
*
Corresponding author.Tel./fax:+14195308213.
E-mail address:afatemi@eng.utoledo.edu (A.Fatemi), (W.V.Mars).1
Formerly Graduate Student at the University of Toledo.2
Tel.:+14194231321;fax:+14194244305./locate/ijfatigue
Available online at
International Journal of Fatigue 30(2008)
1231–1247
International Journalof Fatigue
the ultimate goal of the analysis and design process,an understanding of the effects of variable amplitude loading conditions on fatigue life is important bad on the fact that most actual rvice load histories are variable in nature.The variable amplitude test signals ud in this study were lected to simulate some common aspects from actual load histories.The results from the variable amplitude fatigue experiments were ud to determine the applicability of Miner’s linear damage rule for fatigue life prediction in rub-ber.While this paper focus on the analysis of the fatigue life results from the experiments,other articles have been published that detail the obrvations with regards to con-stitutive behavior[7]and crack development[8].
This paper begins with an overview of the experimental program including the materials and type of specimen ud,the scope of load signals tested under both constant and variable amplitude loading conditions,and the criteria ud to determine failure.Next,equivalence parameters ud to correlate multiaxial fatigue data are prented,fol-lowed by the constant amplitude fatigue life behaviors. Then,variable amplitude fatigue test results are prented, including investigation of the applicability
of Miner’s linear damage rule and comparison of experimental and predicted fatigue lives bad on this rule.A computationally efficient model for fatigue life analysis that combines maximum
normal strain and cracking energy density criteria is also propod and its predictive capability evaluated.Finally, discussions of the effects of load quence and interaction on fatigue life,the choice of using a stiffness drop versus a specified crack length failure criterion,differences between NR and SBR fatigue behaviors,and effects of tem-perature are provided.
2.Experimental program
2.1.Materials and test specimen
The experiments ud specimens molded from both filled natural rubber(NR)andfilled styrene–butadiene rubber(SBR)compounds(recipes provided in Table1). Thefiller loadings were lected to produce roughly the same compound stiffness levels.NR strain crystallizes,which refers to a pha transformation that some elasto-mers experience due to the application of strain,while SBR does not.Strain crystallization[9]can significantly affect both the strength of the material and its fatigue properties.
The experiments ud the multiaxial ring specimen designed by Mars and Fatemi[10]for a wide range of mul-tiaxial loading conditions.The specimen consists of a rub-ber ring bonded between two steel mounting rings,as shown in Fig.1.This test specimen experiences normal and shear strains during the application of simultaneous axial and twist displacements to the specimen.
2.2.Test load signals
The test signals ud with the multiaxial ring specimen consisted of both constant and variable amplitude signals.
Nomenclature
N i number of cycles comprising block i
N15fatigue life bad on15%load drop
N f,N f i number of cycles to failure,for block i
N f,q number of quences to failure
N FCG fatigue life bad on crack length criterion ~r unit normal vector to candidate failure
plane
R d,R h axial displacement ratio,twist displace-
ment ratio
t time W,W max strain energy density,maximum
W c,W c,max cracking energy density,maximum
a angle of crack orientation from horizontal
plane
d,d max,d min axial displacement,maximum,minimum
e strain tensor
e1,e1,max principal strain,maximum
e n,e n,max,D e n normal strain,maximum,range
h,h max,h min twist angle,maximum,minimum
x,x i frequency,for block i
白衣送酒Table1
Recipes forfilled NR andfilled SBR compounds
Ingredient Filled NR Filled SBR
PHR a%of
weight
PHR a%of
weight NR10053.7
SBR rubber cold,dry,NST(SBR1502)10050.5 Carbon black,N2347537.9 Carbon black,N6506032.2
Aromatic petroleum hydrocarbon oil2 1.1157.6 Zinc oxide8 4.33 1.5 Resorcinol donor3 1.6
Stearic acid,rubber grade2 1.110.5 Sulfur,elemental 1.80.9 tert-Butyl benzothiazole sulfenamide
(TBBS)
1.40.7
Polymerized1,2-dihydro-2,2,4-
trimethylquinoline(TMQ)
10.510.5 CO Neodeconaoate0.50.3
Sulfur20%naphthenic ba oil 4.5 2.4
Melamine formaldehyde resin on a silica
carrier
4.2 2.2
DCBS0.80.4
N-(Cyclohexylthio)phthalimide(PVI)0.20.1
Total parts per hundred rubber186.0100.0198.2100.0 a Parts per hundred rubber,by weight.
1232R.J.Harbour et al./International Journal of Fatigue30(2008)1231–1247
Test paths graphically characterizing the test signals in terms of the applied axial displacements and rotational twists applied during the test are shown in Fig.2.The con-stant amplitude tests(paths A–E)consist of a single load-ing condition,while the variable amplitude tests(paths F–I)consist of blocks of constant amplitude cycles that are combined to form a test quence.It is important to note that the letter designations for test paths in this study do not correspond to the test path designations of Mars and Fatemi[2].
During the application of all torsional cycles during this investigation,the axial displacement was constrained to zero,producing a non-zero axial load in the specimen.This method of applying torsional cycles was utilized to be con-sistent with the experiments conducted by Mars and Fatemi[2].It was concluded that this axial load developed during torsion cycles was not significant with regards to the experiments bad on the obrvation by Mars and Fatemi that torsion tests constrained to have zero axial load produced similar results to torsion tests constrained to have zero
axial strain[2].
The constant amplitude tests focud on pure axial and torsion constant amplitude tests,since Mars and Fatemi[2] found the loading conditions to be the bounding cas in terms of fatigue life.The results from proportional and non-proportional test signals generally fell between the pure axial and torsion tests on fatigue life plots.Path A designates a pure axial test with an R d of zero and a twist angle of zero.Paths B and C reprent torsion tests for dif-ferent values of R h and axial displacements maintained at zero.R d and R h reprent the ratio of the minimum to the maximum displacement for axial and torsion cycles, respectively.Path B was for torsion with a minimum twist angle of zero(R h=0)and path C was for fully reverd torsion(R h=À1).A limited number of fully reverd tor-sion tests were also conducted with static axial displace-ments to investigate the effects of static axial displacements on the fatigue behavior during torsion tests. In path D a tensile static displacement was applied,while in path E a static compressive displacement was applied.
The variable amplitude signals consisted of multilevel and multiaxial tests.The multilevel test signals(paths F and G)consisted of two blocks with different peak strain levels for the same type of loading:axial or torsion.The signals investigated the effects of variable amplitude load-ing on material behavior when the planes of crack growth for each component of the signal were the same.
The num-ber of cycles in each block depended on the predicted level of damage of each component bad on the constant ampli-tude test results.
The multiaxial signals(paths H and I)consisted of alter-nating blocks of axial and torsion cycles designed to study the effects of variable amplitude loading in multiaxial experiments.The fact that pure axial and torsion tests gen-erally produced cracks on different planes was the reason for using the loading conditions in the multiaxial test sig-nals.Path H consisted of axial cycles alternating with tor-sion cycles at an R h ratio of zero.Path I alternated axial cycles with fully reverd torsion cycles.For some multiax-ial paths,the number of cycles for each block differed between tests in order to vary the dominant axis of loading. Test conditions and resulting fatigue lives for load paths in Fig.2are shown in Table2for constant amplitude tests and in Table3for variable amplitude tests.
2.3.Fatigue life failure criteria
The determination of the fatigue life for an experiment requires the u of a criterion to define when failure occurs. This investigation utilized two approaches to determine
临时装片general design of the multiaxial ring test specimen
bonded between two steel mounting rings.
δ
θ
θ
θδ
θ
δ
= Displacement
R.J.Harbour et al./International Journal of Fatigue30(2008)1231–12471233
fatigue life.A stiffness-bad approach defined specimen failure in terms of a load or torque drop.A fatigue crack length failure approach defined failure in terms of crack length on the surface of the specimen.
The stiffness approach defines failure of the specimen as the point at which the axial load amplitude
or torque amplitude reaches85%of the respective amplitude for a specified reference cycle(128th cycle in this study).This method relates the failure of the specimen to the deteriora-tion of the rubber material and the resulting loss in stiffness associated with the nucleation and growth of cracks.In general,an initial transient softening occurs for both mate-rials during thefirst100cycles,followed by an extended period of gradual softening until afinal period of rapid softening.The15%amplitude drop was chon since it gen-erally coincides with the ont of the rapid softening that occurs as the cracks become large.This amplitude drop level is also preferable since it is greater than the amplitude drop that occurs during the gradual softening period that immediately precedes the period of rapid softening at fail-ure.A reference cycle beyond thefirst100cycles was lected to prevent the large initial softening associated with the Mullins effect[11]from affecting the results.The 128th cycle was chon specifically since it reprents the first cycle recorded after the initial100cycles using a factor of2logarithmic data acquisition scheme.
In order to apply the load amplitude drop criterion for the different loading conditions in a variable amplitude sig-nal,it is important to compare similar cycles when comput-ing amplitude drops to yield appropriate results.For variable amplitude tests involving repeated quences of cycles,the analysis method lects the entire test quence that contains the128th cycles as the reference values.Since
cyclic quencing can produce variability in load and tor-que levels within a block of a test quence,the amplitude for each cycle in a quence should be compared to the amplitude for the corresponding cycle from the reference quence.Since the15%load or torque amplitude drop generally coincides with the sharp drop in amplitude at fail-ure,slightly altering the lection of the reference cycle or quence does not produce significant changes in the fati-gue life.
A cond failure criterion for determining the fatigue lives of specimens involves the analysis of crack growth on the surface of the specimen.This approach defines the failure of the specimen as the point at which the largest crack reaches a specified critical length.The initial critical length ud to define failure in this investigation was 1mm.The crack lengths for each test were obtained bad on the analysis of digital images of the specimen surface captured at various points throughout the test using a pixel analysis method.The nsitivity of the fatigue life results to the critical crack length was found to vary depending on the material.The fatigue life results bad on the crack length failure criterion for SBR exhibited less nsitivity to the lected critical crack length than the NR results. By interpolation the crack growth data,the fatigue life that corresponded to the specified crack length could be estimated.
3.Fatigue life equivalence parameters
家常炒饭In order to analyze the fatigue behavior of a material,an equivalence parameter is ud to characterize all types of loading conditions in terms of a single variable.The tradi-tional equivalence parameters are usually defined bad on scalar magnitudes and do not refer to a specific material plane.Recent rearch has introduced plane-specific itical plane approaches)that do account
Table2
Constant amplitude test parameters and fatigue life results for NR and SBR
Path code x
(Hz)
d max
(mm)
d min
(mm)
R d h max
(°)
h min
(°)
R h N15
(cycles)
N FCG
(cycles)
Natural rubber(NR)
A 2.5 2.540.0000.00.0–284,199141,000
A 1.0 3.800.0000.00.0–113,84071,150
A 1.0 5.000.0000.00.0–44,50617,625
A 1.0 5.000.0000.00.0–74,58454,200
A 1.07.440.0000.00.0–23,19216,360
A 1.07.440.0000.00.0–29,90126,750
A a 1.0 5.000.0000.00.0–37,996–
B 1.00.000.00–15.00.0090,25040,500
B 1.00.000.00–15.00.0097,26045,050
B 1.00.000.00–15.00.0094,95661,720
C 2.00.000.00–7.9À7.9À1268,26875,500
C 2.00.000.00–7.9À7.9À1151,150–
C 1.00.000.00–10.0À10.0À1104,50042,845
C 1.00.000.00–10.0À10.0À169,57639,660
C 1.00.000.00–12.0À12.0À146,314–
D 1.0 3.14 3.09112.0À12.0À139,88842,600
宁静的小村庄E 1.0À1.23À1.27112.0À12.0À123,908–
SBR
A 2.5 2.540.0000.00.0–>400,000
A 2.0 3.200.0000.00.0–143,228–
A 1.0 3.800.0000.00.0–52,485–
A 2.0 3.800.0000.00.0–46,370–
A 2.0 3.900.0000.00.0–31,39022,915
A 1.0 5.000.0000.00.0–38,19216,805
A 1.0 5.000.0000.00.0–36,1447300
A 1.0 6.500.0000.00.0–6,714–
A b 1.0 5.000.0000.00.0–56,45627,250
B 1.00.000.00–15.00.0054,280–
B 1.00.000.00–15.00.0075,54024,550
B 1.00.000.00–15.00.0074,59824,990
C 1.00.000.00–10.0À10.0À1144,86440,560
C 1.00.000.00–11.3À11.3À158,75217,040
C 1.00.000.00–11.3À11.3À158,2888745
C 1.00.000.00–12.5À12.5À111,0232755
C 1.00.000.00–12.5À12.5À118,0004900
C c0.20.000.00–12.5À12.5À123,3077875
C c0.20.000.00–12.5À12.5À129,755–
D 1.0 1.52 1.48111.3À11.3À123,908–
D 1.0 2.02 1.98111.3À11.3À118,890–
E 1.0À0.98À1.02111.3À11.3À111,155–
a Static dwell test for20h at5.00mm of displacement prior to beginning
高士其简介axial fatigue test.
b Test included10-s dwell period after everyfive applied cycles.
c Fan u
d to further reduc
e temperature o
f specimen alon
g wit
h re-
duced frequency.
1234R.J.Harbour et al./International Journal of Fatigue30(2008)1231–1247
for material plane orientation.Since plane-specific param-eters have directions associated with each value,they can be ud to predict crack orientations.Mars and Fatemi [1]investigated the applicability of veral common equiv-alence parameters for multiaxial loading conditions.The included the maximum principal strain,strain energy den-sity,and cracking energy density criteria.The equiva-lence parameters were also ud in the prent study.
The maximum principal strain is a convenient equiva-lence parameter in cas involving displacement control, since it does not require the u of a constitutive equation to predict strain levels during the analysis process. Although there is a direction associated with the loading state for the maximum principal strain at the maximum loading condition that has been shown to be a good in
dica-tor of crack orientation in some cas,the maximum prin-cipal strain has not been applied as a plane-specific approach since it is only defined on one plane.
Strain energy density is a scalar parameter and is,there-fore,independent of plane orientation.It was computed in terms of the experimental axial load and torque by numer-ically integrating the experimental stress–strain results.For multiaxial cycles,the total strain energy density for the cycle is the sum of the axial and torsional strain energy density components.For paths that pass through a fully unloaded state,the integration begins with an initial strain energy density of zero at the fully unloaded state and inte-grates along the loading direction to the maximum loading condition.For paths that do not pass through a fully unloaded paths D and E),the integration process includes calculation of the minimum strain energy density at the point of minimum loading and the integration of the data along the loading curve to the maximum loading. Since calculations of strain energy density are bad on experimental results,they account for the effects of strain and cyclic softening.
Cracking energy density is a plane specific or critical plane parameter and reprents the portion of the strain energy density available on a particular plane to be relead through crack growth[1].The cracking energy density on each plane was calculated from the strain history via the fatigue life predi
ction software HYPERFATIGUE,using a Neo-Hookean constitutive model[12].The definition and calculation of the parameter cracking energy density
Table3
Variable amplitude test parameters and fatigue life results for NR and SBR
Path code Block1x1N1d max,1or h max,1R d,1or R h,1Block2x2N2d max,2or h max,2R d,2or R h,2N15(q)N FCG(q) Natural rubber(NR)
F Ax 1.057.440Ax 1.010 5.00028531484
F Ax 1.057.440Ax 1.010 5.00030441644
F a Ax 1.057.440Ax 1.010 5.00030562063
G Tor 1.0512.0À1Tor 1.01010.0À136782188
G Tor 1.0512.0À1Tor 1.01010.0À138462022
H Ax 1.05 5.000Tor 1.01015.0045502184
H Ax 1.03 5.000Tor 1.01515.0033252128
H Ax 1.05 5.000Tor 1.01015.0026921633
H Ax 1.03 5.000Tor 1.01515.0039852311
H Ax 1.08 5.000Tor 1.0515.0022161595
H a Ax 1.05 5.000Tor 1.01015.0029991380
H Ax 1.08 5.000Tor 1.0515.0034482034
I Ax 1.05 5.000Tor 1.0510.0À157132742
I Ax 1.05 5.000Tor 1.0510.0À167663900
SBR
F Ax 2.05 3.800Ax 2.015 3.2003595–
F b Ax 2.05 3.800Ax 2.015 3.20048711075
G Tor 1.0512.5À1Tor 1.01511.3À1892392
G Tor 1.0512.5À1Tor 1.01511.3À11633–黑寡妇键盘
G Tor 1.0512.5À1Tor 1.01511.3À11781481
H Ax 1.03 3.800Tor 1.01215.003921314
H Ax 1.03 3.800Tor 1.01215.0018841000
H Ax 1.07 3.800Tor 1.0415.001192–
H Ax 1.07 3.800Tor 1.0415.0030791293
H Ax 1.05 3.800Tor 1.0815.001002491
H Ax 1.05 3.800Tor 1.0815.0023781534
I Ax 1.05 3.800Tor 1.0511.3À118351080
I Ax 1.03 3.800Tor 1.0711.3À11197–
I Ax 1.03 3.800Tor 1.0711.3À12107–
I Ax 1.05 3.800Tor 1.0511.3À12375523
I Ax 1.05 3.800Tor 1.0511.3À126221180
I Ax 1.07 3.800Tor 1.0311.3À131502331
a Cycles applied in modified order as illustrated in Fig.10rather than block form.
b Stati
c dwell test for20h at3.80mm of displacement prior to beginning axial fatigue test.
R.J.Harbour et al./International Journal of Fatigue30(2008)1231–12471235
