0

Multiaxial Fatigue and Life Prediction of Elastomeric

Components

Touhid Zarrin-ghalami, Graduate Research Assistant

and

Ali Fatemi, Professor (Corresponding Author)

Mechanical, Industrial and Manufacturing Engineering Department

The University of Toledo

2801 West Bancroft Street

Toledo, Ohio 43606, USA

Tel: 419-530-8213, Fax: 419-530-8213

Email: afatemi@eng.utoledo.edu

Submitted to the:

International Journal of Fatigue

October 2012

1

Abstract

Elastomeric components have wide usage in many industries. The typical service

loading for most of these components is variable amplitude and multiaxial. In this study a

general methodology for life prediction of elastomeric components under these typical

loading conditions was developed and illustrated for a passenger vehicle cradle mount.

Crack initiation life prediction was performed using different damage criteria. The

methodology was validated with component testing under different loading conditions

including constant and variable amplitude in-phase and out-of-phase axial-torsion

experiments. The optimum method for crack initiation life prediction for complex

multiaxial variable amplitude loading was found to be a critical plane approach based on

maximum normal strain plane and damage quantification by cracking energy density on

that plane. Rainflow cycle counting method and Miner’s linear damage rule were used for

predicting fatigue life under variable amplitude loadings. The fracture mechanics

approach was used for total fatigue life prediction of the component based on specimen

crack growth data and FE simulation results. Total fatigue life prediction results showed

good agreement with experiments for all of the loading conditions considered.

Keywords: Multiaxial fatigue, Elastomers, Fatigue failure, Crack initiation, Crack

growth

2

1. Introduction

Many elastomeric components and structures are typically subjected to variable

amplitude or random multi-axial loading histories. The extreme values for the strain

and/or stress components are not usually coincide with each other under complex loading

conditions. This indicates the complex nature of strain history in applications like

vibration isolators. Therefore, the study of multiaxial fatigue and deformation of

elastomers under variable amplitude loading is an important issue for their proper design

and life prediction analysis.

Life prediction methodologies for uniaxial loading have been relatively well

developed. These methodologies for the more complex case of multiaxial variable

amplitude loading are not yet well established, particularly when the loads are

non-proportional. Very limited papers can be found in the literature related to fatigue life

prediction of elastomeric components under variable amplitude multiaxial loading.

Multiaxial fatigue life prediction methodologies for rubber can be classified into

four main approaches. These consist of the equivalent strain approaches, energy

approaches, equivalent stress approaches, and the more recent critical plane approaches.

Critical plane models can be used for both proportional and non-proportional loading

conditions and are based on the physical process of the damage process. Critical plane

approaches could be strain-based (such as the maximum normal strain), stress-based

(such as the maximum principal Cauchy stress), or energy-based (such as cracking

energy density). Energy-based critical plane approaches, which use both stress and strain,

can reflect the constitutive behavior of the material, while strain-based critical plane

approaches do not require the constitutive behavior of the material.

3

In general, a good multiaxial fatigue model should be robust, sensitive to load

phasing and mean stress, and applicable to variable amplitude loading. Another important

characteristic is the ability to consider crack closure effects.

Crack initiation is normally related to the continuum mechanics quantities (e.g.

stress or strain) which are macroscopic. The most widely used parameters for crack

initiation are the maximum principal strain, maximum principal Cauchy stress and strain

energy density (SED). Approaches such as the cracking energy density (CED) or based

on configurational mechanics (Eshelby stress tensor) are more complex and recent

approaches dealing with multiaxial fatigue analysis of elastomers.

The maximum principal strain criterion was introduced by Cadwell et al. [1] for

unfilled vulcanized natural rubber in 1940 and still remains one of the most commonly

used criteria for rubber. For incompressible materials, this criterion always gives a

positive value for maximum principal strain. Octahedral shear strain is also another

strain-based predictor which also always gives a positive value for rubber-like

materials. In addition, since rubber is incompressible, the hydrostatic stress is

independent of strain tensor.

Ayoub et al. [2] established a relationship between the fatigue life of

styrene-butadiene rubber and the stretch amplitude. They conducted multiaxial constant

and variable amplitude tests on cylindrical (axisymmetric) specimens. The specimen

curvature radius was large enough to minimize the stress triaxiality effect. Their results

showed a good agreement between predicted and experimental fatigue lives. The

predictions were based on continuum damage mechanics improved by incorporating CED

criterion. The developed model is both a fatigue damage criterion and an accumulative

damage rule.

4

Rivlin and Thomas first applied strain energy density (SED) criterion to rubber

material under static loading. This criterion has some drawbacks including inability to

differentiate between simple tension and compression, inability to account for crack

closure and the fact that all of the stored energy in the material would not release due to

crack growth [3]. Strain energy density usually uses a hyperelastic formulation which is

defined in terms of strains. If strain energy density and maximum principal strain would

be applied as scalar criteria, the fact that cracks are usually observed to orient in specific

orientation could not be predicted.

Mars and Fatemi [4] designed a novel specimen for investigating the mechanical

behavior of elastomers under multiaxial loading conditions. By utilizing this specimen

and based on their experimental observations [5] they concluded that the maximum

principal strain as a fatigue damage parameter gave the best prediction of fatigue life,

while the traditionally used strain energy density gave the worst correlation of

experimental data.

Mars and Fatemi [6] discussed the observations of crack initiation and small flaw

growth in filled natural rubber under multiaxial loading conditions. They used their

designed ring specimen with axial, torsion and both proportional and non-proportional

axial-torsion loadings. They suggested that crack nucleation in rubber starts from existing

flaws in the virgin material such as voids, surface cavities and non-rubber particles.

Fatigue crack initiation and growth were observed to occur on preferred failure planes.

For axial, torsion and in-phase axial-torsion loading the cracking plane was transverse to

the maximum principal strain direction. For more complex loading they still observed

preferred nucleation planes but their relation to the principal strain directions was

sometimes different. Crack closure affected the nucleation plane and more closure was

5

observed in cyclic torsion under static compression. This also happened in fully reversed

axial-torsion experiments due to friction during the torsion-compression part of the

loading.

Andre et al. [7] performed torsion experiments on axisymmetrical notched sample

geometry specimens and concluded that the maximum principal Cauchy stress could be

related to multiaxial fatigue damage mechanisms. They proposed that the maximum

principal Cauchy stress could be used as a fatigue damage quantification parameter and

observed that crack orientation is perpendicular to its direction. Brunac et al. [8] extended

Haigh’s diagram to arbitrary 3D loadings by considering the smallest sphere containing

the closed path of the positive part of the Cauchy stress tensor. They concluded that this

approach results in successfully predicting fatigue life.

Saintier at al. [9] investigated the micro-mechanisms leading to crack initiation

under multiaxial fatigue loading conditions and suggested that cracks initiate from

inclusions or large carbon black agglomerates. They observed decohision and cavitation

as damage mechanisms and suggest that cyclic loading does not produce a new damage

mechanism in rubber, in contrast to metallic materials. They observed that the maximum

first principal stress reached during the cycle defined the orientation of the crack in all of

the fatigue loading conditions they considered. In another study, Saintier et al. [10]

proposed a critical plane approach for fatigue crack initiation based on the

micro-mechanisms of crack initiation. In their study, under non-proportional multiaxial

loading, dependent on the material, fatigue crack growth orientation was found to be on

the plane of maximum shear stress amplitude (shear cracking), maximum normal stress

plane (tensile cracking), or even mixed-mode shear and tensile cracking.

6

Mars and Fatemi [11] concluded that scalar equivalence criteria were not capable

of predicting the fatigue initiation life in natural rubber. They suggested using cracking

energy density which is the portion of the strain energy density that is available to cause

crack growth on a particular plane. The application of this method involves knowledge of

the constitutive behavior of the material. Zine et al. [12] applied the cracking energy

density criterion in a FE code and found good agreement between numerical and

analytical results for common strain states. Their experiments results also showed the

efficiency of this criterion to explain fatigue life of elastomers under multiaxial loading

conditions.

Harbour et al. [13] used the multiaxial ring specimen in [4] and performed both

constant and variable amplitude axial-torsion experiments. They used two rubber

materials, one which strain crystallizes (natural rubber) and one which does not (SBR).

They concluded that both cracking energy density and normal strain approaches were

able to predict the dominant crack orientations for some of the test signals in each

material (i.e. axial and multilevel axial tests), but not successful for some other loadings

such as for fully reversed torsion experiments. In another study Harbour et al. [14] used

the maximum normal strain to find the critical plane and the cracking energy density on

that plane to determine fatigue life. Their results showed that this criterion produced

similar fatigue life results compared to other approaches such as cracking energy density,

strain energy density and maximum principal strain. Because maximum normal strain is

independent of constitutive behavior, the critical plane can be identified more easily than

for CED. They [14] also studied the effect of variable amplitude multiaxial loading and

concluded that Miner’s linear damage rule gave reasonable predictions of their

experimental results. Verron et al. [15] proposed a multiaxial criterion for crack

7

nucleation based on the local properties of the Eshelby second-order tensor [16] and

defined in terms of continuum mechanics parameter.

Wang et al. [3] evaluated fatigue life prediction approaches by using experimental

results from proportional and non-proportional loading paths applied to small

axisymmetric diabolo specimens made of vulcanized NR. They conclude maximum

principal strain and octahedral shear strain provide good predictions of the fatigue life.

They also found that compared to traditional criterion of strain energy density, the

cracking energy density model gives better life prediction results.

Mars [17] investigated the duty cycle on each material plane along with its

corresponding damage to transform the multiaxial loading into the localized flaws. After

identifying the damaging events, the original duty cycle is simplified and reconstituted to

a new duty cycle including the number of the most damaging events of the original duty

cycle. The new shortened duty cycle maintains those features of the original duty cycle

corresponding to the original mode of failure and shortens the time scale of the test. For

multiaxially loaded rubber parts, Flamm et al. [18] proposed discretizing the continuous

signal by a level crossing cycle counting method for each loading channel. Then they

constituted the stress amplitude history and performed Rainflow cycle counting on these

alternating points.

The objective of this study was to evaluate a robust methodology for fatigue life

prediction of elastomeric components under complex loading conditions and validate it

by using an automobile cradle mount as an illustrative example. In this paper, first the life

prediction methodology for general random and multiaxial loading is discussed. Both

crack initiation and crack growth approaches are included. Then applications to constant

amplitude axial-torsion in-phase and out-of-phase loading as well as variable amplitude

8

loading are demonstrated using experimental results from a vehicle cradle mount made of

natural rubber. Material deformation and fatigue behavior of the mount are characterized

in [19]. The component FEA and fatigue life analysis and predictions for uniaxial

constant amplitude and variable amplitude loadings are presented in [20] and [21],

respectively.

2. Life prediction methodology for general random and multiaxial

loading

Component fatigue damage is typically localized at points of high strains or

stresses. For complex component geometry, finite element analysis is often performed to

identify the critical location and obtain the stress and strain states for that location. A

cycle counting method and a cumulative damage rule are then used for damage

calculations. The cycle counting procedure relates the damage effect of variable

amplitude loading to constant amplitude material fatigue data and fits. The most popular

method is the Rainflow method proposed by Matsuishi and Endo [22], which is also used

here. Linear damage rule is the simplest form of cumulative damage rule, which is used

in this work. However, load sequence effect or interaction between cycles is not

accounted for in this rule.

A damage quantification parameter is also needed to relate the component

multiaxial stresses and strains to uniaxial specimen test and data. Harbour et al. [14]

evaluated maximum normal strain as a critical plane approach. Normal strain can be

defined from:

1ˆ..ˆ)ˆ(

NCNeN

(1)

9

where C = FT F is the Green’s deformation tensor, F is the deformation gradient tensor

and N̂ is the unit vector normal to the plane in space. By using a MATLAB script, the

maximum value of normal strain and its direction could be defined on all planes in space

in spherical coordinates. Then, the normal strain history on maximum normal strain

(MNS) plane is calculated.

The next step is using the strain-life equation obtained from uniaxial fatigue tests,

as shown in Figure 1(a) and represented by:

3.30max,,1

4 )(1086.6 fN (2)

The strain R ratio effect during each loading cycle at the critical location can be

accounted for by using maximum equivalent R = 0 engineering strain ( 0max,,1 ) based on

Mars-Fatemi R ratio material model [23] discussed in [19] and given by:

)74()741(max,,10max,,1

33

25.6 RRR

(3)

For variable amplitude loading, cycle counting is performed on the critical plane

(i.e. MNS) and damage is calculated for each cycle using the linear damage rule (LDR).

Another successful damage quantification parameter mentioned previously is

cracking energy density (CED). The increment in cracking energy density dWc in the

spatial description is defined in terms of the traction vector T

and the unit displacement

vector

d on a given plane in the instantaneous deformed configuration and is given by:

dTdWc . (4)

The normal to the plane is defined by the unit vector r

. By converting from the spatial

description to the material description, the final expression for CED increment is given

by [24]:

10

RIER

RdESIER

RCR

RdESCRdW

T

T

T

T

c

)2(

~)2(

~

00

(5)

where /0 is the ratio of the deformed mass density (volume) to the undeformed mass

density (volume), F is the deformation gradient, S~

is the 2nd

Piola-Kirchhoff stress

tensor, and E is the Green-Lagrange strain tensor. The relationship between unit vector in

the current configuration r

and the unit vector in the undeformed configuration R

is then

given by:

RF

RFr

(6)

Also C = 2E + I, where C is the Green deformation tensor. Equation 5 gives the cracking

energy density in terms of the stress and strain measures of the material description and

in terms of a unit vector in the undeformed configuration.

Two series of calculations were made, one for calculating cracking energy density

(CED) on maximum normal strain (MNS) plane, and another for calculating CED on

CED critical plane with the increment of 5° in spherical coordinates. Crack closure effect

is considered in each approach since the normal traction on each plane is used in the

calculation. Calculation of CED history on MNS plane is only on a single plane, while

calculation of CED history on the critical CED plane is performed on 1296 (= 36 × 36)

planes to find the critical plane with the highest damage value. Therefore, significant

difference in computation time exists between the two approaches.

SED (W) and CED values are the same for uniaxial loading. The SED or CED life

Equation is given by:

47.0)(148 fNW (7)

11

This Equation is obtained by uniaxial crack initiation experiments [19].

Fatigue crack initiation approach is typically related to the fatigue life to grow a

crack from a natural flaw to a length on the order of about a millimeter. Crack growth

approach then considers fatigue life from this crack length to failure. Crack initiation is

often used for applications where micro-cracks dominate the total fatigue life. Crack

growth approach is usually considered when a large portion of the total life is involved

with macro-crack growth. In this study both approaches were considered.

To calculate total component fatigue life, the value of maximum principal strain

can be used as the fatigue damage parameter. The strain energy release rate, associated

with this value, depends on the square of the maximum principal strain [25]. The strain

energy release rate (Tε,max,0) is computed from:

aCT 20max,,10max,, )( (8)

where C is the material’s initial Young’s modulus and is 2.34 MPa for the natural rubber

in this investigation and a is crack length or depth, as appropriate. The constant C is not

strictly consistent with the classical form of the strain energy release rate for simple

tension frequently used for rubber strength and fatigue analyses [21]. However, this

equation still holds the key feature of linear variation of energy release rate with crack

length. Recent work by Ait-Bachir et al. [26] also confirms the form of Equation 8 by

showing energy release rate to depend on crack length and one far-field property,

independent of the state of loading.

Crack growth rate equation obtained from planar tension specimen crack growth

rate data as shown in Figure 1(b), is represented by [19]:

40max,,1

22520max,

5 104)(104 aCTdN

da (9)

12

By utilizing maximum equivalent R = 0 engineering strain as a damage parameter along

the crack path, the relationship between strain and crack depth is then obtained.

Predictions based on the damage parameter along the critical dominating crack element

for each individual event is obtained from integrating Equation 9 and is given by:

104

0max,,1

114566

aaN f

(10)

where a0 and a1 are natural flaw size and critical element’s depth, respectively (in mm).

For the natural rubber material of the component used in this study the effective initial

flaw size was 0.02 mm [19].

3. Applications to multiaxial constant amplitude loading

3.1 Experimental program and results

Component level fatigue tests were conducted in order to validate the fatigue life

prediction methodology discussed above. Figure 2(a) shows the component used, which

is a vehicle cradle mount. The finite element model of the component is shown in

Figure 2(b). Axial loading with amplitude of 1100 N and R ~ 0 and torsion loading with

amplitude of 30 N.m. and R = -1 were used as the experimental loading conditions. Both

in-phase and 90 out-of-phase tests were conducted with two tests for each condition.

One test of each condition was stopped in regular intervals to monitor crack initiation and

growth. Fatigue tests were conducted with a sinusoidal waveform of 1 Hz using a

servo-hydraulic axial-torsion load frame and axial as well as rotational displacement

responses of each test specimen were periodically recorded.

The loading conditions chosen resulted in fatigue lives between about 21,500

cycles and 67,500 cycles. The maximum axial and rotational displacements increased

13

initially due to the Mullin’s effect and then stabilized after several hundred cycles. When

a fatigue crack initiated and then grew to a critical size, the displacement and rotation

amplitudes increased rapidly due to reduced stiffness. Failure was defined as the lower

number of cycles at which the axial or rotational displacement amplitudes increased

drastically from its gradual linear change, as illustrated for a typical test in Figure 3.

The experimental fatigue lives at different load levels are tabulated in Table 1.

Test results from uniaxial fatigue tests reported in [20] and [21] are also included in this

table. The life comparisons show shorter life for 90° out-of-phase loading, as compared

to in-phase loading. Experimental life scatter between duplicate tests was within a factor

of three for in-phase tests and within a factor of two for out-of-phase tests. This table also

shows the first crack observations for the tests which were stopped at regular intervals for

monitoring. The crack nucleation life for a duplicate test for which initiation was not

monitored was calculated based on the ratio of crack nucleation life to total life of the

monitored test of the same loading condition. Crack growth life was a significant portion

of the total fatigue life (about 75%) for in-phase test, while for out-of-phase test, this ratio

was about 30%.

Crack length as well as its depth and their changes were measured at critical

(failure) location by periodic test interruptions and visual inspection. For in-phase

loading, crack length did not have a significant effect on fatigue life, as even at a length

of about 4 cm, the component was still at only about half of its total life. Crack depth had

a more dominant effect on life than crack length. When crack depth was on the order of

3 cm, the component was near failure. Stiffness drop also correlated better with crack

depth than with crack length. Crack length and depth versus cycles are shown in

Figure 4(a) for in-phase loading. For out-of-phase loading, crack length was a key factor,

14

rather than crack depth and this is shown in Figure 4(b). In out-of-phase loading, the ratio

of crack initiation life to total life increased about 3 times of the same loading level for

in-phase condition, therefore, less life is spent in crack growth for OP loading. Since this

ratio was much higher in OP tests compared to what was observed for uniaxial load

cases, phasing had much more effect on the ratio of initiation to total life. This is shown

in Figure 5.

To evaluate any difference between load control (LC) and displacement control

(DC) mode, IP and OP displacement-controlled tests were also performed based on

midlife displacements of the LC tests. In DC tests failure was defined based on 40% load

or torque drop, whichever occurred sooner. The midlife load-displacement and

torque-rotation was similar in both LC and DC tests. Cracks initiated more rapidly under

LC than under DC. This is the same as what was observed for CA uniaxial tests [20]. The

results are tabulated in Table 1. Crack initiation location was the same for all cases of LC

and DC mode tests, however.

3.2 Life predictions and validation of methodology

The nonlinear FE program ABAQUS was used to simulate the nonlinear and large

deformation hyper-elastic behavior of the mount. The Marlow strain energy density form

was used to model deformation of incompressible isotropic elastomers. The mount and

the FE mesh are shown in Figure 2. To adequately model the geometry, a relatively fine

mesh with about 30,000 elements was used. The element type used was the eight node

brick (hexahedral) element. For cyclic loading, the amplitude tool of the FE software was

used, so that a loading cycle could be exactly simulated. Stable deformation curve was

applied as an input to the model, therefore initial softening (Mullin’s effect) was

inherently considered. FE simulations assume non-linear perfectly elastic constitutive

15

behavior, therefore, the same deformation curve for loading and unloading. Quasi-static

simulation was performed for constant amplitude axial loading.

Load and/or displacement are applied to the reference point (located on the inner

axis of the mount), which is coupled with inner FE nodes of the mount at the interface

with metallic part by utilizing the rigid body constraint capability of the software. To

prevent rigid body motion the outer portion of the mount was fixed. More details of FE

model can be found in [20].

Maximum principal strain distributions of the mount under IP loading is shown in

Figure 6, which is similar to that for OP loading. The maximum principal strain occurs at

the interface between the inner bushing and the bulk rubber material at a location of

abrupt geometry change. Locations, at which the maximum principal strain occurred, as

indicated in Figure 6 for in-phase loading, were also observed to be the site(s) of fatigue

cracking during all fatigue tests of the component. The maximum principal strain at the

critical location determined from FEA was, therefore, one of the parameters used for

evaluating the fatigue damage.

By using the data from critical element (defined as the location with the highest

maximum principal strain) of the mount in each loading cycle, life predictions were

performed. Values of the maximum normal strain (MNS), critical cracking energy

density (CED), and CED on MNS plane are listed in Table 2. Through a tensor

transformation program developed in MATLAB, the maximum normal strain value and

direction was defined. Equations 2 and 3 based on uniaxial fatigue properties were then

used to perform life predictions. The correlation of the predicted and experimental results

is shown in Figure 7(a). The predictions are within a factor of two for all experiments by

16

using the maximum normal strain parameter. Numerically it is an efficient parameter for

determining the critical or failure plane(s) through a 3-D search process.

Maximum normal strain plane direction and Cauchy stress history were used in

another MATLAB program to obtain cracking energy density on the MNS plane.

Equation 7 for CED based on uniaxial fatigue properties was used to perform life

predictions. The correlation of predicted results with experimental results for CED is

shown in Figure 7(b) and the same plot for CED on MNS plane is shown in Figure 7(c).

The results show that using CED on MNS gives similar predictions to the critical CED

criterion. As discussed earlier, this is a computationally efficient approach (by a factor of

about 45), to predict crack initiation life of the component, since CED history is

calculated on a single plane.

Total life of the component was also predicted based on Equation 10.

The predicted and experimental results are tabulated in Table 2. The correlation of

predicted versus experimental total lives are shown in Figure 8. The predictions are all

within a factor of three of the experiments.

4. Applications to variable amplitude multiaxial loading

For evaluating more complex loading conditions, variable amplitude proportional

and non-proportional axial-torsion tests were also conducted on the elastomeric

component. Two in-phase and two out-of-phase tests were conducted. For in-phase

loading the peaks and valleys in both axial and torsion load signals are reached at the

same time, while for out-of-phase loading they are not (see Figure 9). Experiments were

conducted with the load range of 3,680 N in comparison with minimum load of near zero

and torque range of 50 N.m. with mean value of near zero. For one duplicate test of each

17

condition, the test was stopped at regular intervals (every 10% of the expected life) to

monitor crack initiation and growth.

Test results for variable amplitude tests are given in Table 1. Experimental life

scatter between duplicate tests was within a factor of two for both in-phase and

out-of-phase loading. Crack initiation range is also tabulated in Table 1. Crack growth

life constitutes a significant portion of the total life in all tests. In out-of-phase tests the

ratio of crack initiation life to total life increased about 1.5 times, as compared to

in-phase loading. The same observation was made in constant amplitude loading. This is

shown in Figure 5. Crack length and depth measurement results showed that crack depth

was key factor in component fatigue life for both in-phase and out-of-phase tests. This

was also observed in constant amplitude uniaxial tests.

The maximum principal strain occurs at the interface between the inner bushing

and the bulk rubber material at a location of abrupt geometry change, similar to all other

loading cases and corresponding to that observed during the fatigue tests. This crack

initiation site remained the same for all of the simulations, either IP or OP.

The same methodology used for constant amplitude axial-torsion loading

condition was used to define crack initiation life as well as total fatigue life of the

component. The only difference for variable amplitude loading is the use of Rainflow

cycle counting method for event identification and then using Miner linear rule to

accumulate the fatigue damage. Figures 10(a) and 10(b) show normal strain history on

maximum normal strain plane for in-phase and out-of-phase loadings, respectively. As

can be seen from this figure, the maximum normal strain history for both IP and OP

conditions at critical location is different from the nominal loading history applied to the

component (see Figure 9). Cycle counting was performed on the shown MNS history.

18

After event identification, by utilizing Miner rule, the fatigue life prediction was

performed.

Life prediction results, shown in Figure 7 for different damage parameters,

indicate that MNS criterion is not accurate for complex variable amplitude multiaxial

loading. Both critical CED and CED on MNS criteria give better predictions. Most of the

total fatigue life predictions are within a factor of three of experiments in this case

according to Figure 8.

5. Discussion of Results

Predicted crack nucleation locations based on the developed methodology

matched observed failure locations for all loading conditions. Crack initiation lives were

predicted well (mostly within a factor of two) based on the maximum normal strain and

the developed methodology for most of the loading conditions. Because only strain is

involved in this approach, maximum normal strain can be a very efficient parameter to

determine critical or failure plane. Most of the predictions were on non-conservative side

(i.e. predicted life longer than experimental life). This is, at least partly, because of the

fact that in the FE model the strain and stress at critical location were lower than in the

component since a finer mesh could not be used due to practical considerations.

Of the two CED criteria, the CED on MNS plane criterion gives quite acceptable

predictions with much less calculation time, compared to the critical CED plane criterion.

This is because CED history is calculated on just one MNS plane and not on many planes

in space. Calculation of CED requires knowledge of the constitutive behavior of the

material, however. Both of these approaches resulted in better life predictions than MNS

for multiaxial variable amplitude loading conditions.

19

Two criteria of Smith-Watson-Topper (SWT) commonly used for metallic

materials and strain energy density (SED) traditionally used for elastomers were also

evaluated. All of the predictions based on SWT criterion were overly conservative

predictions by up to a factor of 23. Predictions based on SED resulted mostly in life

within a factor of two for this particular component and the loading conditions

considered. SED criterion gives close predictions compared to CED criterion because of

the particular component geometry where based on the FE results the maximum principal

strain at the critical location is much higher than the other two principal strains. This

indicates essentially uniaxial strain state, although the loading of the component was

multiaxial. In addition, the principal strain at the critical location is tensile, while for

compression strain there would be a large deviation between SED and CED criteria

predictions. The difference between SED and CED criteria is also large when torsion

loading is dominant compared to axial loading, which was not the case for the loading

considered in this study. Differences between CED and SED criteria based on different

loading conditions are also discussed in [15]. Overall, of all fatigue crack initiation

criteria studied here, those which use critical plane approaches work better than the scalar

damage parameters or those damage parameters which do not take constitutive behavior

into account.

Crack growth constitutes a significant portion of the component total fatigue life.

The ratio of initiation to total fatigue life varied between 6 and 69 % for different loading

cases. For constant amplitude or variable amplitude uniaxial tests, by increasing the load

amplitude, the ratio of crack initiation life to total life decreased (i.e. more life spent in

crack growth). This is expected, since higher load level initiates a crack quicker and

fatigue life is dominated by crack growth.

20

Multiaxial load phasing had an important effect on the ratio of crack initiation to

total life. For constant amplitude out-of-phase loading, this ratio increased about 3 times

compared to in-phase loading at the same level. This means more life was involved in

crack initiation for out-of-phase loading, compared to in-phase loading. Total life

comparison showed shorter life for out-of-phase loading compared to

in-phase loading for both constant and variable amplitude loadings, indicating much

faster crack growth rate once a crack has initiated in out-of-phase loading.

The fracture mechanics approach was used for total fatigue life prediction for

each loading condition based on specimen crack growth data and FE simulation results.

The methodology used for component total fatigue life resulted in reasonable predictions

for nearly all the loading conditions. The prediction results are sensitive to initial crack

length used for analysis, but not very sensitive to the final crack length used. This was

also observed in [20] where the critical element of the component had the highest effect

in the total life prediction.

6. Conclusions

1. Shorter total life was observed for out-of-phase loading compared to

in-phase loading at the same level for both constant and variable amplitude loadings.

In addition, for out-of-phase loading the ratio of crack initiation life to total life was

about 3 times higher than for in-phase loading. This indicates more life was involved

in crack initiation but much faster crack growth rate once a crack initiated in out-of-

phase loading, as compared to in-phase loading.

2. Crack initiation location was observed to remain the same for all of the loading

conditions used for fatigue tests of the component. This location corresponded to the

21

point at which the maximum principal strain occurred. Therefore, maximum normal

strain (MNS) is an effective parameter to determine the critical or failure plane.

3. Although the maximum normal strain (MNS) parameter correlated the constant

amplitude fatigue life data within mostly a factor of about two, it could not correlate

the data satisfactorily for the more complex case of variable amplitude loading.

4. Amongst the different approached evaluated, a computationally efficient and

relatively accurate fatigue crack initiation life prediction approach for complex

loading was found to be a critical plane approach based on MNS and quantifying

damage on this plane using the cracking energy density (CED) parameter.

5. For variable amplitude loading the rainflow cycle counting of the maximum normal

strain (MNS) history was found to be an efficient method. Miner linear damage rule

was then used to accumulate damage on the MNS plane based on constant amplitude

fatigue data, resulting in satisfactory life predictions for variable amplitude loading.

6. Crack growth constituted a significant portion of the component total fatigue life. The

fracture mechanics approach was used for total fatigue life prediction based on

specimen crack growth data and FE simulation results. The methodology used for

component total fatigue life resulted in reasonable predictions with nearly all the life

predictions being within a factor of three of the experimental lives.

22

References

[1] Cadwell, S., Merrill, R.A., Sloman, C.M., and Yost, F.L., 1940, “Dynamic fatigue life

of rubber,” Rubber Chemistry and Technology, Vol. 13, No. 2, pp. 304-315.

[2] Ayoub, G., Nait-Abdelaziz, M., Zairi, F., Gloaguen, J.M. and Charrier, P., 2011, “A

continuum damage model for the high-cycle fatigue life prediction of styrene-butadiene

rubber under multiaxial loading,” International Journal of Solids and Structures,

Vol. 48, pp. 2458-2466.

[3] Wang, Y., Yu, W., Chen, X. and Yan, L., 2008, “Fatigue life prediction of vulcanized

natural rubber under proportional and non-proportional loading,” Fatigue and Fracture of

Engineering Materials and Structures, Vol. 31, pp. 38-48.

[4] Mars, W.V. and Fatemi, A., 2004, “A novel specimen for investigating the

mechanical behavior of elastomers under multiaxial loading conditions”, Experimental

Mechanics, Vol. 44, No. 2, pp. 136-146.

[5] Mars, W.V. and Fatemi, A., 2005, “Multiaxial fatigue of rubber: Part II: experimental

observations and life predictions”, Fatigue and Fracture of Engineering Materials and

Structures, Vol. 28, pp. 523-538.

[6] Mars, W.V. and Fatemi, A., 2006, “Nucleation and growth of small fatigue cracks in

filled natural rubber under multiaxial loading,” Journal of Material Science, Vol. 41,

pp. 7324-7332.

[7] Andre, N., Cailletaud, G. and Piques, R., 1999, “Haigh diagram for fatigue crack

initiation prediction of natural rubber components,” Kaust Gummi Kunstst, Vol. 52,

pp. 120-123.

[8] Brunac, J., Gerardin, O., and Leblond, J., 2009, “On the heuristic extension of

Haigh’s diagram for the fatigue of elastomers,” International Journal of Fatigue,

Vol. 31, pp. 859-867.

[9] Saintier, N., Cailletaud, G., and Piques, R., 2006, “Crack initiation and propagation

under multiaxial fatigue in a natural rubber,” International Journal of Fatigue, Vol. 28,

pp. 61-72.

[10] Saintier, N., Cailletaud, G., and Piques, R., 2006, “Multiaxial fatigue life prediction

for a natural rubber,” International Journal of Fatigue, Vol. 28, pp. 530-539.

[11] Mars, W.V. and Fatemi, A., 2005, “Multiaxial fatigue of rubber: Part I: equivalence

criteria and theoretical aspects”, Fatigue and Fracture of Engineering Materials and

Structures, Vol. 28, pp. 515-522.

23

[12] Zine, A., Benseddiq, N., Nait-Abdelaziz, M., Ait Hocine, N. and Bouami, D., 2006,

“Prediction of rubber fatigue life under multiaxial loading,” Fatigue and Fracture of

Engineering Materials and Structures, Vol. 29, pp. 267-278.

[13] Harbour, R.J., Mars, W.V. and Fatemi, A., 2008, “Fatigue crack orientation in NR

and SBR under variable amplitude and multiaxial loading conditions,” Journal of

Material Science, Vol. 43, pp. 1783-1794.

[14] Harbour, R.J., Mars, W.V. and Fatemi, A., 2008, “Fatigue life analysis and

predictions for NR and SBR under variable amplitude and multiaxial loading conditions,”

International Journal of Fatigue, Vol. 30, pp. 1231-1247.

[15] Verron, E., Le Cam, J., and Gornet, L., 2006, “A multiaxial criterion for crack

nucleation in rubber,” Mechanics Research Communications, Vol. 33, pp. 493-498.

[16] Eshelby, J.D., 1975, “The elastic energy-momentum tensor,” Journal of Elasticity,

Vol. 5, No. 3-4, pp. 321-335.

[17] Mars, W.V., 2010, “Identifying the damaging events in a multiaxial duty cycle,”

Constitutive Models for Rubber VI, Heinrich, G., Kaliske, M., Lion, A. and Reese, S.,

Eds., Proceedings of the 6th

European Conference on the Constitutive Models for Rubber,

London, Taylor and Francis; 1st edition, pp. 261-267.

[18] Flamm, M., Steinweger, T., and Weltin, U., 2003, “Lifetime prediction of

multiaxially loaded rubber springs and bushings,” Constitutive Models for Rubber III,

Busfield, J.J.C. and Muhr, A.H., Eds., Proceedings of the 3rd

European Conference on

the Constitutive Models for Rubber, London, Taylor and Francis; 1st edition, pp. 49-53.

[19] Zarrin-Ghalami, T. and Fatemi, A., 2012, “Material deformation and fatigue

behavior characterization for elastomeric component life predictions,” Polymer

Engineering and Science, Vol. 52, pp. 1795-1805.

[20] Zarrin-Ghalami, T. and Fatemi, A., 2012, “Rubber component deformation and

fatigue life analysis and predictions,” Journal of Automobile Engineering (In-press).

[21] Zarrin-Ghalami, T., and Fatemi, A., 2012, “Cumulative Fatigue Damage and Life

Prediction of Elastomeric Components,” Fatigue and Fracture of Engineering Materials

and Structures (In-press).

[22] Matsuishi, M. and Endo, T., 1968, “Fatigue of metals subjected to varying stress,”

Japan Society of Mechanical Engineering, Fukuoka, Japan.

[23] Mars, W.V. and Fatemi, A., 2003, “A phenomenological model for the effect of

R ratio on fatigue of strain crystallizing rubbers”, Rubber Chemistry and Technology,

Vol. 76, pp. 1241-1258.

24

[24] Mars W. V., 2001, “Multiaxial Fatigue of Rubber,” Ph.D. Dissertation, The

University of Toledo, Toledo, Ohio.

[25] Mars, W.V. and Fatemi, A., 2007, “The correlation of fatigue crack growth rates in

rubber subjected to multiaxial loading using continuum mechanical parameters,” Rubber

Chemistry and Technology, Vol. 80, Issue 1, pp. 169-182.

[26] Ait-Bachir, M., Mars, W.V., and Verron, E., 2012, “Energy release rate of small

cracks in hyperelastic materials,” International Journal of Non-linear Mechanics, Vol.

47, Issue 4, pp. 22-29.

25

Acknowledgements

Funding of this study was provided by Chrysler Group LLC. PAULSTRA

provided the cradle mount component for the study.

26

Table 1: Component experimental conditions and test results.

Test

Type

Load

Type

Load

Path

Control

Mode

Control

Amplitude

(Range)

RP /RM

or

R /R

Frequency

(Hz)

a / a

or

Pa / Ma

Fatigue Life (Cycles or Blocks)

Initiation1 Total

2

A3 CA N.A. LC 1025 N ~0 1 6.24 mm >148,000

A CA N.A. LC 1100 N ~0 1 7.16 mm 18,165<N<22,7064 69,683

A CA N.A. LC 1100 N ~0 1 7.52 mm 28,000<N<35,000 107,411

A CA N.A. LC 1300 N ~0 1 7.88 mm 36,359

A CA N.A. LC 1500 N ~0 1 10.1 mm 3,780<N<7,560 19,069

A CA N.A. LC 1500 N ~0 1 10.4 mm 4,000<N<8,000 20,180

A CA N.A. LC 1500 N ~0 3 8.85 mm 3,786<N<7,572 19,100

A CA N.A. LC 1100 N 0.2 1,0.5,0.75 8.04 mm 23,456<N<46,912 161,447

A CA N.A. LC 1100 N 0.2 1 7.39 mm 16,000<N<32,000 110,129

A CA N.A. LC 1500 N 0.2 0.75 9.44 mm N<3,826 26,034

A CA N.A. LC 1500 N 0.2 1 9.7 mm N<3,000 20,414

A VA N.A. LC (3000 N) ~0 16.5 S/B5 8.59 mm B<2,598 6,327

A VA N.A. LC (3000 N) ~0 16.5 S/B 8.77 mm B<1,500 3,653

A VA N.A. LC (3600 N) ~0 16.5 S/B 9.94 mm B<320 2,615

A VA N.A. LC (3600 N) ~0 16.5 S/B 9.87 mm B<300 2,454

A-T CA IP LC 1100 N/30 N.m ~0/-1 1 6.85 mm/19.26° 15,510 67,406

A-T CA IP LC 1100 N/30 N.m ~0/-1 1 7.04 mm/18.62° 7,200 31,290

A-T CA IP DC 6.95 mm/18.94° 0.1/-1 1 1044 N/28.2 N.m 0<N<3,600 74,230

A-T CA OP LC 1100 N/30 N.m ~0/-1 1 7.04 mm/19.68° 15,806<N<18,967 25,119

A-T CA OP LC 1100 N/30 N.m ~0/-1 1 8.14 mm/21.67° 13,500<N<16,200 21,454

A-T CA OP LC 1100 N/30 N.m ~0/-1 1 8.08 mm/21.97° >21,250

A-T CA OP DC 7.53 mm/20.68° ~0/-1 1 1085 N/30 N.m 43,750

A-T VA IP LC (3680 N/50 N.m) ~0/-1 43.5 S/B 11.32 mm/15.52° 122<B<244 1,393

A-T VA IP LC (3680 N/50 N.m) ~0/-1 43.5 S/B 9.81 mm/13.41° 150<B<300 1,710

A-T VA OP LC (3680 N/50 N.m) ~0/-1 47.6 S/B 11.22 mm/15.60° 319<B<638 1,501

A-T VA OP LC (3680 N/50 N.m) ~0/-1 47.6 S/B 11.99 mm/16.71° 170<B<340 800

A-T VA OP LC (3680 N/50 N.m) ~0/-1 47.6 S/B 7.63 mm/20.68° 340<B<510 2,000

1 1

st observed crack (with the length of 1 mm) by periodic test interruptions and visual examination

2 In LC defined by sudden increase of displacement amplitude (log scale), in DC defined by 40% load drop

3 A- Axial, T- Torsion, CA- Constant Amplitude, VA- Variable Amplitude, LC- Load-Controlled,

DC- Displacement-Controlled, IP- In-Phase, OP- Out-of-Phase 4 Bold font indicates the estimated initiation life which is calculated based on the ratio of initiation to total life of the

test with the same loading which is stopped in regular intervals for crack growth monitoring 5 S/B is seconds per block

27

Table 2: Summary of component crack initiation and total life experiments and predictions.

Test

Type

Load

Type

Load

Path ε1,max,0

Control

Amplitude

(Range)

RP /RM

Initiation Life (Cycles or Blocks) Total Life (Cycles or Blocks)

Experimental

1.Measured

2.Estimated

Prediction

(MNS)

Prediction

(CED on

MNS Plane)

Prediction

(CED Critical

Plane)

Experimental Prediction

A CA N.A. 1.33 1025 N ~0 1: >148,000 145,752

A CA N.A. 1.39 1100 N ~0 1: 28,000<N<35,000

2: 18,165<N<22,706 20,008 20,983 20,809

1: 69,683

2: 107,411 100,516

A CA N.A. 1.76 1300 N ~0 1: 36,359 37,496

A CA N.A. 2.08 1500 N ~0

1: 4,000<N<8,000

2: 3,780<N<7,560

2: 3,786<N<7,572

6,339 6,420 6,383

1: 19,069

2: 20,180

3: 19,100

18,493

A CA N.A. 1.58 1100 N 0.2 1: 16,000<N<32,000

2: 23,456<N<46,912 12,779 10,632 10,543

1: 110,129

2: 161,447 53,687

A CA N.A. 2.43 1500 N 0.2 1: N<3,000

2: N<3,826 3,665 2,932 2,883

1: 20,414

2: 26,034 9,195

A VA N.A. 2.01 (3000 N) ~0 1: B<1500

2: B<2,598 1,951 1,338 979

1: 6,327

2: 3,653 4,268

A VA N.A. 2.47 (3600 N) ~0 1: B<300

2: B<320 932 662 865

1: 2,615

2: 2,454 1,631

A-T CA IP 1.45 1100 N,

30 N.m ~0/-1

1: 7,200

2: 15,510 11,348 10,209 10,119

1: 67,406

2: 31,290 25,646

A-T CA OP 1.42 1100 N,

30 N.m ~0/-1

1: 13,500<N<16,200

2: 15,806<N<18,967 21,585 25,389 18,677

1: 25,119

2: 21,454 55,765

A-T VA IP 2.29 (3680 N,

50 N.m) ~0/-1

1: 150<B<300

2: 122<B<244 910 513 563

1: 1,393

2: 1,710 3,680

A-T VA OP 2.30 (3680 N,

50 N.m) ~0/-1

1: 170<B<340

1: 340<B<510

2: 319<B<638

1,606 746 770

1: 1,501

2: 800

3: 2,000

4,664

28

Figure captions

Figure 1: Material fatigue properties [19]. Equivalent fatigue crack initiation life as a

function of equivalent maximum strain (a). Fatigue crack growth rate data

correlation at different R ratios based on Mars-Fatemi R ratio model (b).

Figure 2: Vehicle cradle mount used as illustrative example (a), and rubber mount FE

model (b).

Figure 3 : Definition of failure illustrated by variation of axial and rotational

displacements in constant amplitude in-phase axial-torsion tests of the cradle

mount.

Figure 4: Evolution of crack length and depth for constant amplitude axial-torsion

(a) in-phase test with Nf = 31,290, and (b) out-of-phase test with

Nf = 21,454.

Figure 5: Crack initiation to total fatigue life ratio for all types of loadings.

Figure 6: Maximum principal strain location for constant amplitude in-phase loading

simulation.

Figure 7: Experimental versus predicted component initiation life for all loading

conditions based on (a) maximum normal strain, (b) critical CED, and

(c) CED on MNS plane.

Figure 8: Experimental versus predicted component total fatigue life for all loading

conditions.

Figure 9: Variable amplitude in-phase and out-of-phase axial-torsion loading history

used for component testing.

Figure 10: Normal strain on maximum normal strain (MNS) plane history of variable

amplitude and axial-torsion loading for (a) in-phase, and (b) out-of-phase

loading.

29

(a)

(b)

Figure 1: Material fatigue properties [19]. Equivalent fatigue crack initiation life as a

function of equivalent maximum strain (a). Fatigue crack growth rate data

correlation at different R ratios based on Mars-Fatemi R ratio model (b).

30

(a)

(b)

Figure 2: Vehicle cradle mount used as illustrative example (a), and rubber mount FE

model (b).

31

Figure 3 : Definition of failure illustrated by variation of axial and rotational

displacements in constant amplitude in-phase axial-torsion tests of the cradle

mount.

32

(a)

(b)

Figure 4: Evolution of crack length and depth for constant amplitude axial-torsion

(a) in-phase test with Nf = 31,290, and (b) out-of-phase test with

Nf = 21,454.

33

Figure 5: Crack initiation to total fatigue life ratio for all types of loadings.

34

Figure 6: Maximum principal strain location for constant amplitude in-phase loading

simulation.

35

(a)

(b)

36

(c)

Figure 7: Experimental versus predicted component initiation life for all loading

conditions based on (a) maximum normal strain, (b) critical CED, and

(c) CED on MNS plane.

37

Figure 8: Experimental versus predicted component total fatigue life for all loading

conditions.

38

Figure 9: Variable amplitude in-phase and out-of-phase axial-torsion loading history

used for component testing.

39

(a)

(b)

Figure 10: Normal strain on maximum normal strain (MNS) plane history of variable

amplitude and axial-torsion loading for (a) in-phase, and (b) out-of-phase

loading.