Finite Elements in Analysis and Design 60 (2012) 57–63

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Finite Elements in Analysis and Design

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Fatigue life prediction and optimum topology design of EPDM weather strip

J.R. Cho a,b,n, K.C. Han a, J.S. Kim b, S.B. Lee a, O.K. Lim a

a School of Mechanical Engineering, Pusan National University, Busan 609-735, Koreab Research & Development Institute of Midas IT., Gyeonggi 463-400, Korea

a r t i c l e i n f o

Article history:

Received 7 July 2011

Received in revised form

16 May 2012

Accepted 2 June 2012Available online 27 June 2012

Keywords:

Fatigue life

Optimum topology design

Nonlinear finite element analysis

EPDM weather strip

4X/$ - see front matter & 2012 Elsevier B.V.

x.doi.org/10.1016/j.finel.2012.06.001

esponding author at: School of Mechanical

ity, Busan 609-735, Korea. Tel.: þ82 51 510 3

ail address: jrcho@pusan.ac.kr (J.R. Cho).

a b s t r a c t

One of the major performances required for weather strip is the high durability against the frequent

opening and closing operations of automobile door. In this paper, the fatigue life of automobile weather

strip manufactured with EPDM rubber is predicted by making use of the nonlinear finite element

analysis and an e–N curve of EPDM rubber. Furthermore, for the sake of the fatigue life enhancement,

the local region within the weather strip cross-section which is vulnerable to the fatigue failure is

designed by means of a density-based topology optimization method. Through the numerical

experiments, it is confirmed that the topologically optimized weather strips show fatigue lives much

longer than the initial weather strip design.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

To protect the interior of automobile from rainwater, soundand dust, the specially designed rubber parts called weather stripare attached along the edges of automobile doors [1,2]. Weatherstrips are manufactured through the rubber extrusion formingprocess [1], where the raw rubber material experiences asequence of phase changes from the solid-state rubber at inletto the final liquefied rubber flow at outlet where the forming diesare installed. Even though both the overall configuration and thecross-section shape are quite different according to the car modeland the attachment location, the common requirements to be metby weather strips are the dimension precision and the highdurability. The former is for the tight assembling while the latteris to prevent the undesired structural failure caused by thefrequent opening and closing operations of automobile door.

Besides the complicated cross-section geometry, the wallthickness of weather strip is relatively small to maximize thesealing performance by increasing the flexible large deformationwhile the door is closing or opening, which does in turn maximizethe sealing performance. Naturally, weather strip experiences aremarkably large change in the stain field during the closing andopening operations of automobile door. Accordingly, the assur-ance of sufficient durability, at least up to the level equivalent tothe life span of automobile, is the first and most importantrequirement for weather strip. Furthermore, this subject is

All rights reserved.

Engineering, Pusan National

080; fax: þ82 51 514 7640.

becoming more crucial to both car makers and part manufac-turers in proportion to the worldwide regulation intensificationon the life span announcement for key automobile parts [3]. It isbecause the insufficient durability, that is a fatigue life shorterthan the desired level, may lead to the loss of the fundamentalfunction of weather strip which is triggered mostly by theunexpected crack occurrence.

The fatigue life estimation methods introduced so far could beclassified into two basic approaches, the crack nucleation approachand the crack growth approach [4]. In the former approach whichtraces back to the earliest Wohler work on railroad axles [5], thefatigue life estimation is formulated in terms of stress, strain orstrain energy density at a point within the material domain underconsideration. Since the fatigue life parameters are easily defined inthe sense of continuum mechanics, this approach is easy to imple-ment and widely used by engineers. However, restricted to hyper-elastic materials like rubber, stress is rarely adopted as a fatigue lifeparameter [6,7]. It is because not only the fatigue test is usuallyconducted under the control of displacement, but the accuracy ofstress calculated from strain is not so high owing to the highlynonlinear relation between stress and strain. On the other hand, thelatter approach based on the fracture mechanics predicts the growthof a particular initial crack or flaw which leads to the final failure.However, in rubber materials, both the up-front information aboutthe state and location of the initial crack causing the final failure andthe geometry change of the problem are not available in most cases.In this context, the robust and unified numerical implementation ofthis approach still leaves several problems to be resolved [4,8,9].

Weather strip may lose the sealing performance when thecrack occurred according to the frequent opening and closingoperations of automobile door. Once this situation happens, the

J.R. Cho et al. / Finite Elements in Analysis and Design 60 (2012) 57–6358

only way to recover the sealing performance is to replace thedamaged weather strip with a new one. The goal of the currentstudy is to predict and enhance the fatigue life of weather stripmanufactured with the filled EPDM (ethylene-propylene-diene-monomer) rubber by making use of the finite element analysisand the topology optimization. Weather strip experiences thestress relaxation [10] and creep during the closing and opening ofautomobile door for long periods of time. However, this compli-cated viscoelastic behavior is beyond the scope of this study,rather a single opening–closing cycle of automobile door isconsidered for the numerical prediction of the fatigue life ofweather strip.

For the current study, the crack nucleation approach in whichthe maximum principal strain serves as a fatigue life parameter isemployed. The material behavior of EPDM weather strip ismodeled by a five-term Moonley–Rivlin strain energy densityfunctional. A cyclic variation of strain within the weather stripduring a single opening–closing cycle of automobile door isapproximated by the large deformation frictional contact analysis[11,12], and the fatigue life against the cyclic opening and closingoperations of door is predicted with the help of an e–N curve ofthe filled EPDM rubber. In order to enhance the fatigue life of theinitial weather strip design, the density-based topology optimiza-tion [13,14] is employed to optimally tailor the weather stripcross-section by restricting the deformation of weather stripwithin a linear elastic range. The numerical methodology isillustrated through the numerical experiments, from which it isconfirmed that the fatigue lives of the topologically optimizedweather strip designs are significantly improved.

2. Problem description

2.1. Weather strip for automobile door

A typical configuration of the front door of modern automo-biles is schematically depicted in Fig. 1(a), where the weatherstrip under consideration in this study is one attached along thetop edge of door. This weather strip plays a leading role inpreventing the invasion of rain water, when compared with theother weather strips attached to the front door. The undeformedcross-section shape of the weather strip under the considerationis represented in Fig. 1(b), together with the detailed section viewof the metallic holding part which is welded to the front doorchassis. The weather strip is assembled into the front door chassisin the manner of being fit into the holding bracket without anybonding treatment.

Fig. 1. Schematic view: (a) front door chassis of a

The front door is to be closed and opened along the pathshown in Fig. 1(b), where the automobile body frame which is incontact with the weather strip moves away from the weatherstrip at the time when the front door is completely opened. Theweather strip is configured such that the contact with the bodyframe occurs at two different regions, the upper and lowerregions, for the sake of the complete and safe protection ofexternal rain water, dust and noise. Furthermore, the shape ofthe weather strip cross-section is characterized by two distinctparts, the upper lever-type extruder and the lower elliptic-typetube. The upper part plays a leading role in preventing rain waterand dust, while the lower part shuts out the external noise. Fromthe mechanics point of view, the weather strip experiences thelarge deformation and strain while the front door is closing,which may cause the unexpected fatigue failure.

2.2. Material characteristics

The weather strip is manufactured with the solid-state EPDMrubber by the special rubber extrusion forming machine [1]. Thesolid-state raw material at inlet of the forming machine changesto the melted thermal rubber flow by the compressing screw andthe external heat during the extrusion forming process, but thefinal product extruded out from the dies at outlet of the formingmachine becomes to be solidified again by the relatively lowerroom temperature. The final solidified weather strip exhibits theincompressible hyperelastic behavior, so a five-term Moonley–Rivelin material model is adopted in which the strain energydensity functional W(I1,I2,I3) is given by

W ¼ C10ðI1�3ÞþC01ðI2�3ÞþC11ðI1�3ÞðI2�3ÞþC20ðI1�3Þ2

þC30ðI3�3Þ3þ1

2k ðI3�1Þ2 ð1Þ

here, Ii are the invariants of Green–Lagrange strain tensor and Cij

are the material dependent Moonley–Rivlin constants. The para-meter k is introduced to enforce the incompressibility of rubbersuch that the higher k is the more incompressible is the material.

Fig. 2 represents the stress–strain curves of the filledEPDM rubber specimen which were obtained by the uniaxialtension and compression tests at temperature 25 1C with threedifferent strain rates. The curves were taken at the tenth cycle ofuniaxial tension and compression loadings. The five materialconstants Cij determined by the least square curve fitting of thestress–strain curves at the strain rate of 10�2/s are as follows:C10¼0.116422, C01¼�0.042809, C11¼0.003753, C20¼�0.000635and C30¼0.000038 kgf/mm2.

utomobile, (b) cross-section of weather strip.

Fig. 2. Stress–strain curves of the filled EPDM rubber: (a) tension, (b) compression.

Fig. 3. The e–N curves of the filled EPDM [16].

J.R. Cho et al. / Finite Elements in Analysis and Design 60 (2012) 57–63 59

Fig. 3 represents the e–N curves of the filled EPDM rubberwhich are given in a paper by Abraham et al. [15], for which thedumbbell test specimens were cycled in tension and compressionunder uniaxial loading. The dynamic fatigue tests have beenperformed with a sinusoidal load of 1 Hz under the control ofload at room temperature. Among three curves in the figure, weuse the e–N curve obtained by the load amplitude equal to 500N

for the current study in order to reflect the severe closingoperation of automobile door. In order to employ the e–N curveobtained by the uniaxial tension/compression test to predict thefatigue life of the material subject to multiaxial cyclic loadings,the maximum principal strain e1, max is used to represent thecombined effect of multiaxial strain components occurred withinsuch a material. In other words, the uniaxial strain on the verticalaxis of the e–N curves shown in Fig. 3 is replaced with themaximum principal strain to derive a mathematical fatigue lifemodel for the weather strip exhibiting the multiaxial straindistribution. Thus, the fatigue life N of the weather strip can beexpressed by

em1,maxN¼ C ð2Þ

with the material-dependent power index m and constantC [15–16]. The two material constants m and C which aredetermined using the test data at two end points of the e–N

curve labeled by rectangles are �2.947 and 2,554, respectively.

3. Fatigue life evaluation

A finite element model for the large deformation frictionalcontact analysis is shown in Fig. 4, where the weather strip rubberis discretized with the total of 14,811 elements (3- and 4-node

elements), while the body frame, door chassis and the holdingbracket are modeled as rigid bodies. Both the door chassis and theholding bracket are clamped while the body frame is forced tomove along the opening and closing path by the forced displace-ment. The simulation model is assumed to be in the plane straincondition and the parameter k for enforcing the incompressibilityof rubber is set by 1.0�109 kgf/mm2 [11]. The frictional dynamiccontact with the frictional coefficient m of 1.0 is specified to thefive contact interfaces indicated by the dotted rectangles inFig. 4(a). The friction coefficient is determined by referring to apaper by Cho et al. [17]. Two incremental dynamic contactanalyses are carried out in sequence, the pre-contact betweenthe weather strip and the upper/lower ends of the holding bracketin 20 displacement increments and the main contact between thebody frame and the weather strip in the remaining 140 displace-ment increments.

Fig. 4(b) shows the time histories of the resultant force acting onthe body frame in the opposite direction of the door closing pathand the peak equivalent strain that occurred in the weather strip.The reaction force exhibits a hyperelastic increase with the incre-ment of forced displacement, while the peak equivalent strainrapidly increases during the pre-contact, due to the compressionof the weather strip by the lower sharp end of the holding bracket,but it decreases during the second contact between the weatherstrip and the body frame. Distribution of the maximum principalstain at the final increment is shown in Fig. 5(a), where the peakmaximum principal strain e1,max ¼ 0:3088 occurs at the lower sharpcut indicated by A within the weather strip. The fatigue life profilepredicted by Eq. (2) expressed in terms of the maximum principalstrain at the final increment is represented in Fig. 5(b), where theregions with the fatigue life greater than 5.0721�107 are treated asinfinity. The lowest fatigue life corresponding to the locationexhibiting the peak maximum principal strain turns out to be8.150�104 cycles.

4. Topology optimization for enhancing the fatigue life

4.1. Density-based linearized topology optimization

Topology optimization is classified into the parameter andshape optimization methods, in the aspect of seeking the opti-mum distribution of material within a design domain OdDOARN(N¼2, 3) for a given material boundary G¼GD [GN underthe specific loading and constraint conditions. In the finiteelement-based topology optimization, the design domain iscommonly divided into a finite number of elements OK and anoptimum material topology is sought by selectively excluding theelements which do not play an important role in bearing the

Fig. 4. (a) Finite element model, (b) time history of load and peak equivalent strain.

Fig. 5. Initial design at the final increment: (a) maximum principal strain distribution, (b) fatigue life profile.

Fig. 6. Density and homogenization approaches for topology optimization.

Fig. 7. Definition of topology optimization problem: (a) for design domain I,

(b) for design domain II.

J.R. Cho et al. / Finite Elements in Analysis and Design 60 (2012) 57–6360

applied load. But, the material model employed for the finiteelements within the design domain depends on the topologyoptimization approach, and it also influences the type of designvariable [18]. In the homogenization design method (HDM)introduced by Bendsøe and Kikuchi [19], each finite element isassumed to be a periodic microstructure composed of infinitenumber of holes, as represented in Fig. 6. The material propertiesare homogenized and the design variables become the sizes andthe rotational angles of each hole. Meanwhile, in the densityapproach (DA) [20,21], only the relative density ratios XK of eachhomogenized finite element with respect to the base materialdensity r0 become the design variables and the intermediatedensity ratios are to be appropriately penalized. The latterapproach is more popular thanks to its simplicity for the numer-ical implementation.

But, the most significant feature of the topology optimizationis that it is restricted to linear problem, regardless of the topology

optimization approach. Because of this limitation, we linearizethe stress–strain curve shown in Fig. 2 of the weather strip byrestricting the weather strip compression within the linear elasticrange. The linearized Young’s modulus of the weather strip iscalculated by the basic relation of 2(C10þC01)¼E/2(1þn) with theMoonley–Rivlin constants in Eq. (1) and the Poisson’s ratio n(ffi0.49). The design domains OdCOAR2 defined for the linear-ized topology optimization are represented in Fig. 7, which arechosen somewhat subjectively from the fact that the strainconcentrations shown in Fig. 5(a) are caused by the over-stiffenedinterior rubber lump except for the elliptic-like tube and theupper extruder. In consideration of the assembling with theholding bracket, the bottom region of the weather strip is

Fig. 8. Topology optimization results: (a) optimum topology I, (b) optimum

topology II.

Fig. 9. Finite element meshes: (a) for improved design I, (b) for improved

design II.

J.R. Cho et al. / Finite Elements in Analysis and Design 60 (2012) 57–63 61

completely excluded from the design domain I and partiallyincluded into the domain II (particularly, the region near thepoint A). The total of 3,606 and 5,043 elements (3- node and4-node elements) are used to discretize the design domains I andII, but the total numbers of elements to discretize the entireweather strip including the design domain are equally 15,864 forboth cases. The clamped boundary condition is specified to thebottom surface of the weather strip, and one-tenth of the contactforce Fc acting on the weather strip when the door is completelyclosed are applied to the linearized topology optimization pro-blem because the large deformation contact analysis is notsupported by commercial topology optimization code.

The design variables XiA[0,1] for the topology optimization ofthe weather strip cross-section are defined by

Xi ¼ri

r0

, i¼ 1,2, � � � ,NDT ð3Þ

in which NDT denotes the total number of finite elements withinthe design domain. The linearized Young’s moduli EH

i of eachfinite element with the modified homogeneous density ri arecalculated by the following power-law material model given by

EHi ¼ E0

ri

r0

� �p

¼ ðXiÞpE0, p41 ð4Þ

According to Bendsøe and Sigmund [13], the penalizationpower index p is recommended to be equal or greater than 3 inview of the Hashin–Strikrikman bounds for the two-phase com-posites [22,23]. The total strain energy U(u) stored within theweather strip under the action of external load is computed by

UdðuÞ ¼

1

2

ZOeðuÞ : EHeðuÞdO

¼1

2uT

ZO

BT EHBdO� �

u� �¼

1

2uT K½ � u

� �ð5Þ

in which e(u), EH, B and [K] denote the (6�1) strain matrix, the(6�6) elasticity constant matrix, the gradient matrix of basisfunctions and the global stiffness matrix, respectively. The rela-tive mass fraction mf of the design domain, which is desired to beminimized by the topology optimization, to the total mass M0 ofthe initial design domain is expressed by

mf ¼M

M0¼

PNDTi ¼ 1 ViXiPNDT

i ¼ 1 Vi

¼V

V0ð6Þ

Then, the constrained optimization problem for seeking anoptimum topology of the weather strip cross-section is formu-lated as follows:

Find X¼ XiNDTi ¼ 1, Xi ¼ ri=r0 ð7Þ

Minimize V ¼mf V0 ð8Þ

Subject to ½K �u¼ f ð9Þ

0rXmini rXir1 ð10Þ

ð1�bÞ ðuiniI Þjr ðuIÞjrð1þbÞðu

iniI Þj, I¼ x,y, j¼ 1,2,3 ð11Þ

In which, uini denotes the deformed configuration of the initialdesign model shown in Fig. 4(a) by the displacement and loadingconditions shown in Fig. 7. The constraints (11) on the nodaldisplacements at points 1, 2 and 3 indicated in Fig. 7 are includedso as to constrain the topology optimization models shown inFig. 7 to be deformed like uini. Meanwhile, the lower bound Xmin

i

in Eq. (10) is introduced to prevent the finite element stiffnessmatrix from being a singular matrix [24].

4.2. Optimum topology and fatigue life enhancement

The topology optimization is carried out with the simulationparameters set by b¼0.1 and Xmin

i ¼ 0:01, and the optimumtopologies obtained with the two design domains are representedin Fig. 8(a) and (b). Topology optimizations are completed in 12iterations in both cases. Except for the void produced at thebottom below the elliptic-type tube within the design domain II,the cross-section topologies for two cases are almost similar toeach other. From the comparison with the initial weather stripcross-section shown in Fig. 4(a), it is observed that the bothoptimum topologies produce more voids in the region on theright of a central big rectangle void in the vicinity of the elliptic-like tube. According to the detailed numerical results, the cross-section areas A of each optimum cross-section topology are171.92 and 170.53 mm2, which are smaller than the initial sectionarea Aini of 174.95 mm2.

In order to evaluate the fatigue life of the topologicallyoptimized weather strip designs, the finite element modelsshown in Fig. 9(a) and (b) are generated, for which the weatherstrip is discretized with the total of 27,344 and 3030,724elements (3- and 4-node elements). As in the previous largedeformation frictional contact analysis, both the body frame andholding bracket are treated as rigid bodies and the same boundaryconditions and the friction coefficient are used. Furthermore, thecontact analysis is composed of two steps in sequence, the pre-contact in 20 displacement increments and the main contactbetween the body frame and the weather strip in the remaining140 displacement increments.

The distributions of the maximum principal strain of twoimproved design models I and II at the final step are representedin Figs. 10(a) and 11(a) respectively. It is observed from thecomparison with the initial design shown in Fig. 5(a) that theoverall strain distributions are almost similar and the peakmaximum principal strains occur at the same position indicatedby A. But, the peak strain levels are found to be remarkablyreduced from 0.3088 to 0.2697 and 0.2665. Figs. 10(b) and11(b) show the fatigue life profiles which are evaluated with

J.R. Cho et al. / Finite Elements in Analysis and Design 60 (2012) 57–6362

the maximum principle strains at the final displacement incre-ment. The critical fatigue lives of the weather strip are influencedby the local contact region A between the weather strip and theholding bracket as in the initial design, but the critical fatigue lifeis improved from 8.150�104 to 1.215�105 and 1.258�105

cycles respectively.The increment-histories of the peak equivalent strain are

comparatively represented in Fig. 12(a), from which it is clearlyobserved that the improved design models exhibit the increment-histories with the overall magnitudes remarkably lower than theinitial design. Nevertheless, the increment-histories of the totalreaction force acting on the door frame do not show the remark-able decrease, which implies that the sealing performance of theimproved design models is kept at the same level as the initialdesign. It is because the fatigue life of the weather strip isdominated by the local concentration of the maximum principalstrain and the improved design models were sought in the

Fig. 11. Improved design II: (a) distribution of the maximum principal strain,

(b) fatigue life profile.

Fig. 12. Comparison of increment-histories: (a)

Fig. 10. Improved design I: (a) distribution of the maximum principal strain,

(b) fatigue life profile.

direction of reducing the local strain concentration by thetopology optimization.

The comparison between the initial design and two improveddesign models is given in Table 1, where the values in parenthesisindicate the relative differences with respect to the initial design.The cross-section area of the weather strip is slightly reduced by1.74% and 2.53%, and the peak equivalent strain and the peakmaximum principal strain show the remarkable decrease accord-ing to the topology optimization. The fatigue life of the initialweather strip design is improved by 49.08% and 54.36%, but onthe contrary the peak reaction force shows the slight increase.Thus, it has been justified that the topology optimization success-fully enhances the fatigue life while not deteriorating the sealingperformance. When the maximum opening/closing times ofautomobile door per day are assumed to be 10, the fatigue livesof the improved weather strip designs reach 33.3 and 34.5 years,which significantly improves the lifespan equal to 22.3 years ofthe initial design.

5. Conclusion

A numerical technique for predicting and improving thefatigue life of weather strip manufactured with the filled EPDMrubber has been proposed. For the fatigue life estimation, an e–N

curve expressed in terms of the maximum principal strain wasapplied to the Green–Lagrangian strain field within the weatherstrip which was obtained by the large deformation frictionaldynamic contact analysis. And, the density-based topology opti-mization method with the help of the linearized static finiteelement analysis was employed to seek an optimum topology ofthe weather strip cross-section which can improve the fatiguelife. A linearized topology optimization problem is defined byrestricting the rubber deformation within the linear elastic range,by applying one-tenth of the contact forces acting on the weatherstrip when compressed completely, and by constraining thetopology optimization model to be deformed like the initialweather strip design.

The validity of the proposed numerical technique has beenjustified such that the optimum topology of the weather stripcross-section which remarkably improves the fatigue life wassuccessfully sought without any numerical instability. From thenumerical results, the following main observations are drawn.First, the fatigue life of the initial weather strip design is improvedby 49.08–54.36% according to the topology optimization. Second,the total weights per unit length of the improved weatherstrip designs I and II are reduced by 1.73 and 2.53% respectively.

peak equivalent strain, (b) reaction force.

Table 1Comparison of the initial and two improved design models.

Items Design models

Initial Improved I Improved II

Cross-section area A (mm2) 174.95 171.92 (�1.74%) 170.53 (�2.53%)

Peak equivalent strain emax

Final increment 0.1722 0.1586 (�8.71%) 0.1556 (�11.61%)

At the increment showing the peak 0.2018 0.1586 (�21.41%) 0.1556 (�22.89%)

Peak reaction force FR (N) 0.05494 0.05613 (þ2.17%) 0.05524 (þ0.55%)

Peak maximum principal strain e1, max

Final increment 0.3088 0.2684 (�13.1%) 0.2665 (�13.7%)

Fatigue life Nf (cycles) 0.815�105 1.215�105 1.258�105

(þ49.08%) (þ54.36%)

The values in parenthesis indicate the relative differences with respect to the initial design.

J.R. Cho et al. / Finite Elements in Analysis and Design 60 (2012) 57–63 63

Third, the peak reaction force acting on the body frame whichdefinitely influences the sealing performance is almost keptunchanged even though the fatigue life is improved and the totalweight per unit length is reduced by the topology optimization.

It is concluded, through the numerical experiments, that theproposed numerical technique successfully leads the designerto an optimum topology which can improve the fatigue life ofrubber-like materials.

Acknowledgment

This work was supported by a 2-Year Research Grant of PusanNational University.

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