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Analytical durability modeling and evaluation— complementary techniques for physical testing of automotive components M. Firat, U. Kocabicak* University of Sakarya, Faculty of Engineering, Esentepe Kampusu, Adapazari, Turkey Received 3 February 2003; accepted 17 May 2003 Abstract As a result of the commercial pressure new methods of durability evaluation have to be explored, automotive suppliers are now being asked to develop new components and subsystems in shorter times and using fewer physical prototypes. The need for the verification of the existing methods for the durability assessment have been increasing and this turns out to be the only way to propose new computational models to validate the final product within these reduced time scales and resources. The paper reviews some of the computational aspects of fatigue damage analysis and life prediction, and a practical fatigue evaluation tool is presented to meet this challenge. The computational methodology based on the local strain approach is described in detail for the fatigue damage assessment of metallic components under general multiaxial fatigue loads. The application of the proposed methodology is illustrated with an industrial example; the numerical simulation of biaxial cornering tests of light-alloy wheels is conducted, and correlations between the cornering test cycles and predicted cycles using different damage models are provided and comparisons in terms test failure locations and estimated crack initiation sites are given. # 2003 Elsevier Ltd. All rights reserved. Keywords: Fatigue testing; Fatigue damage; Wheels; Automotive design; Cornering tests 1. Introduction There is increasing pressure in the automotive industry to reduce the time taken to bring new designs to production. At the same time it is necessary for the vehicles developed to have the right attributes such as durability and low weight in order to remain competitive. It is generally recognized that these objectives cannot be met within the development program time and cost goals by the traditional route of testing and modifying a series of physical prototypes. As a result, the use of analytical CAE tools to predict the performance of vehicle, subsystem and component designs is replacing the use of physical testing for design optimization. However, the state of the art in analytical prediction of durability is such that it cannot safely 1350-6307/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2003.05.018 Engineering Failure Analysis 11 (2004) 655–674 www.elsevier.com/locate/engfailanal * Corresponding author. Tel./fax: +90-264-346-0367. E-mail address: [email protected] (U. Kocabicak).

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Page 1: Analytical durability modeling and evaluation—complementary techniques for physical testing of automotive components

Analytical durability modeling and evaluation—complementary techniques for physical testing of

automotive components

M. Firat, U. Kocabicak*

University of Sakarya, Faculty of Engineering, Esentepe Kampusu, Adapazari, Turkey

Received 3 February 2003; accepted 17 May 2003

Abstract

As a result of the commercial pressure new methods of durability evaluation have to be explored, automotive

suppliers are now being asked to develop new components and subsystems in shorter times and using fewer physicalprototypes. The need for the verification of the existing methods for the durability assessment have been increasing andthis turns out to be the only way to propose new computational models to validate the final product within thesereduced time scales and resources. The paper reviews some of the computational aspects of fatigue damage analysis

and life prediction, and a practical fatigue evaluation tool is presented to meet this challenge. The computationalmethodology based on the local strain approach is described in detail for the fatigue damage assessment of metalliccomponents under general multiaxial fatigue loads. The application of the proposed methodology is illustrated with

an industrial example; the numerical simulation of biaxial cornering tests of light-alloy wheels is conducted, andcorrelations between the cornering test cycles and predicted cycles using different damage models are provided andcomparisons in terms test failure locations and estimated crack initiation sites are given.

# 2003 Elsevier Ltd. All rights reserved.

Keywords: Fatigue testing; Fatigue damage; Wheels; Automotive design; Cornering tests

1. Introduction

There is increasing pressure in the automotive industry to reduce the time taken to bring new designs toproduction. At the same time it is necessary for the vehicles developed to have the right attributes such asdurability and low weight in order to remain competitive. It is generally recognized that these objectivescannot be met within the development program time and cost goals by the traditional route of testing andmodifying a series of physical prototypes. As a result, the use of analytical CAE tools to predict theperformance of vehicle, subsystem and component designs is replacing the use of physical testing for designoptimization. However, the state of the art in analytical prediction of durability is such that it cannot safely

1350-6307/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfailanal.2003.05.018

Engineering Failure Analysis 11 (2004) 655–674

www.elsevier.com/locate/engfailanal

* Corresponding author. Tel./fax: +90-264-346-0367.

E-mail address: [email protected] (U. Kocabicak).

Page 2: Analytical durability modeling and evaluation—complementary techniques for physical testing of automotive components

be relied upon as the sole means of verifying the durability of a product prior to production tooling. Therisk is that some unpredicted durability problems may become apparent at a late stage in the developmentprocess, leading to re-design, re-tooling and possible delay to the product launch. For this reason, abalanced approach involves a combination of analytical and physical methods.In addition, understanding of fatigue failure under multi-axial loading becomes more important then

ever at component level, since complex cyclic deformations are realized under constant amplitude or vari-able amplitude block type of loading. Hence, the mathematical modeling based on multiaxial damagefunctions using material stress–strain response, as the basic input, are employed in design iteration studiesduring product development and refinement processes. The computational fatigue analysis may followdifferent paths depending on the assumptions and the available data; though a practical methodologybased on local strain approach may be composed of three main modules. First a multiaxial constitutivemodel is used to calculate the material response at a structural point under a given loading or measuredstrain history. While applications of numerical methods such as finite element method or boundary elementmethod provide the stress–strain histories at any structural point under multiaxial loads, stress–strainanalysis coupled with a notch correction algorithm may serve as a feasible alternative solution for relativelylong loading histories. Second a multiaxial damage parameter is employed to assess damage for individualloading cycles and to transform in to an equivalent uniaxial fatigue damage process usually characterizedby a strain–life curve determined with smooth specimens. Finally, a summation rule is employed todetermine the total damage and to estimate the fatigue life.In this paper, a computational methodology is presented for fatigue damage analysis of metallic

components under general proportional and nonproportional loadings. The details of approach areillustrated through an industrial application, and the biaxial wheel-cornering tests are simulated. The fati-gue test cycles of an aluminum alloy wheel are predicted using effective strain and critical plane parameters,and the predicted cycles and estimated fatigue critical locations are compared with the tests. Additionallycorrelations with ASME effective strain, Fatemi–Socie and Smith–Watson–Topper parameters based oncritical plane approach are discussed. Furthermore, a decrease in the computed damage per cycle is inves-tigated with reference to notch stress–strain analysis approach capable of modeling ‘‘stress relaxation like’’behavior and cyclic strain accumulation.

2. Multiaxial fatigue and damage assessment

In various studies it has been shown that the local state of stress or strain influence the fatigue strength ofa component, and the local strain approach is a practical engineering approach as long as the crackinitiation plays a dominant role in durability assessment of structural components. A number of multiaxialfatigue failure criterion based on effective strain or stress measures have been used under proportionalmultiaxial loads and satisfactory results are reported in the literature [1]. Furthermore, for nonpropor-tional loadings, effective parameters based on stress or strain ranges are implemented in engineering designcodes [2,3]. In addition, energy-based approaches are proposed such as the plastic-work hypothesis intro-duced by Garud [4,5] or total hysteresis energy concept of Ellyin [6]. Beside a number of critical planedamage models are proposed based on physical observations that fatigue cracks initiate and grow on certainmaterial planes [7,8]. In critical plane approaches, the existence of a fatigue function involving normal andshear stresses or strains is postulated, and assumed to assist fatigue damage on all material planes. The planeon which damage is maximized is defined the critical plane at a given material point [9,10]. Smith et al. [11]introduced a stress–strain function to predict fatigue damage for materials whose damage development wastensile (normal) strain dominated. Accordingly, the critical plane is defined as the material plane on whichthe normal strain amplitude function is maximized, and damage accumulation is postulated to occur on thisplane under multiaxial loading conditions. The proposed stress–strain function may be expressed as,

656 M. Firat, U. Kocabicak / Engineering Failure Analysis 11 (2004) 655–674

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SWT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�a þ �mð Þ"aE

pð1Þ

where �a and "a are the amplitude of stress and strain respectively acting normal direction of the critical plane,and �m represent the mean normal stress. Depending upon strain amplitude and state of stress, fatigue cracksmay also initiate in the plane of maximum shear strain amplitude, which is the plane of maximum damage.According to Brown andMiller [12], the crack parameters governing fatigue life are themaximum shear strainand the strain normal to the plane of maximum shear strain, and critical plane is the material plane on whichthe following damage function is maximized,

�a þ k"n;max ¼ cons tan t ð2Þ

where the constant k is used to fit the experimental life curve obtained from reversed tension–compressionand torsion fatigue tests, and �a is the amplitude of engineering shear strain in the critical plane while"n,max represents the amplitude of normal strain on the same plane. Fatemi and Socie [7] suggested adamage formulation that includes the effect of mean stress. The critical plane for this damage model isidentified as the plane experiencing the maximum shear strain amplitude and the fatigue life are estimatedbased on the accumulated damage on this plane.

�a 1þ k�að Þmax

�y¼

�0f2G

2Nfð Þbþ� 0

f 2Nfð Þc

� �ð3Þ

where k is a material parameter. (�a)max and �y are the amplitude of normal tensile stress, and yield stress,respectively. In recent years, several other damage models using critical plane concept are proposed andsuccessful results are reported for various proportional and nonproportional loading conditions, see Youand Lee [13] for example, for an extensive review.

3. Rate independent plasticity

Although most components are designed not to exceed the yield stress, most of the fatigue tests at thecomponent level include load levels beyond normal service conditions in order to reduce the developmentcosts and accelerate experimental investigation. Additionally, even under loads causing nominally elasticstresses elastic–plastic deformations are observed at the geometrical discontinuities and stress raising sections.Therefore a constitutive model is necessarily employed to describe the material response beyond recoverabledeformations. While real material behavior under general multiaxial loads is time-dependent, history depen-dent, and temperature dependent; under isothermal conditions at or near room temperature, the pathdependent factors such as the history of accumulated plastic strain govern dominantly the stress–strainresponse, and the material is homogenous at the macroscopic scale, and initially isotropic. Furthermore, theexistence of an initial yield stress is assumed to distinguish the limit of recoverable and irrecoverable defor-mation, while other formulations without a yield surface notion are also presented in literature, see forexample the excellent paper of Watanabe and Atluri [14].Inelastic deformations are indicated as hysteresis loops under cyclic loads, and for metals obeying

Masing’s hypothesis a stable symmetric loop is formed and described with power law expressions amongwhich Ramberg–Osgood equation is a popular one. The isotropic and kinematic hardening rules forms thebasis of classical plasticity models, and yield surface change during plastic deformations are modeled as auniform expansion or movement in stress space in the direction of yield surface normal [15,16]. Armstrongand Frederick [17] modified the yield surface translation proposed by Prager and Ziegler, and introduced a

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recovery term in the backstress evaluation to model strain memory effect. Mroz [18] introduced a ‘‘field ofwork hardening modules’’ concept based on the piecewise linear representation of uniaxial stress–straincurve. Mroz’s multi-surface model uses the concept of discrete yield surfaces in stress space together withan image point concept on nested yield surfaces, and models a elastic or plastic shakedown under propor-tional nonbalanced loading paths or a constant ratcheting rate under nonproportional loads. Garud [4]modified the image point definition of Mroz to realize a non-intersecting translation of nested yield sur-faces. Dafalias and Popov [19], Krieg [20], and McDowell [21,22] developed the two-surface theories basedon the Mroz’s multi-surface model in order to enhance the computational efficiency. Bower [23] modifiedthe nonlinear kinematic hardening rule of Armstrong and Frederick by using an additional internalvariable to introduce a decaying ratcheting rate under nonbalanced loading paths. Chaboche and hiscoworkers [24–28] postulated an additive decomposition of total backstress each following a nonlinearbackstress evaluation. Onno and Wang [29,30] forwarded a threshold concept for modeling accurateratcheting rates under multiaxial nonproportional loading paths. Jiang [31] introduced a modified hard-ening rule following the additive composition of total backstress introduced by Chaboche, and classifiedother nonlinear kinematic hardening rules using a limiting surface concept for backstress parts andsimplified determination of material parameters in backstress evaluation equation.The modeling of cyclic material behavior has been an active research area in deformation analysis of

metallic structures, and new constitutive models are introduced to simulate the transient material behavioras well as other topics such as nonproportional hardening or cross-hardening behavior under complexloading conditions.The application of various plasticity models have been employed in the evaluation of damage parameters

in the multiaxial damage assessment of structural components in various studies under complex loadingconditions [4,13,32–38]. Moreover, a number of studies indicate plasticity algorithms based on nonlinearkinematic hardening rule as a promising alternative in the modeling of stress–strain response under generalmultixial unbalanced loading paths [31,39,40].

4. Notch analysis methods

The geometrical discontinuities on structural components such as fillets, welds, shoulders, and notchesare generally the critical failure sides at which fatigue cracks mostly initiate [41]. While computation ofstresses and strains is a boundary value problem that can be solved by satisfying the equilibrium equations,the material constitutive equation, and compatibility equations together with the boundary conditions,there are several approximation schemes introduced to estimate notch stress and strain for relatively longloading histories. Peterson [42] compiled stress concentration factors for various geometries to computeelastic stresses at the critical sections based on the elasticity or photoelasticity. Neuber [43] studied a semi-infinitely prismatic body with various flank angles and proposed an expression relating the theoreticalstress concentration factor to the elastic–plastic stress and strain concentration factors. Glinka [44] pro-posed a uniaxial approximation formula for a notched body under plane stress conditions based onequivalents of strain energy density of two identical bodies made of ideally elastic and elastic–plasticmaterials. Hoffmann [45] and Hoffmann and Seeger [46,47] proposed a multiaxial extension of Neuber’srule by replacing the notch and nominal stress–strain quantities with respective equivalent forms, andreported successful notch root stress–strain calculations for round bars with mild, sharp and sharp-deepnotches. Moftakhar [48] analyzed two materials, one ideally elastic and the other elastic–plastic, andshowed that the total strain energy density at the notch tip of an elastic body is greater than that of elastic-plastic body if the deviatoric stress components are kept constant under a monotonically increasingmultiaxial loading condition. Barkey [49] introduced the notion of structural yield surface in relatingnominal stresses to the notch strains based on the assumption that a constitutive relation may describe the

658 M. Firat, U. Kocabicak / Engineering Failure Analysis 11 (2004) 655–674

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material response of a structural element with a geometrical discontinuity, and proposed an approximationmethod by employing a multiple-surface plasticity model together with an anisotropic yield surface usingthe elastically calculated nominal stresses as inputs. Barkey [49] measured the notch root strains of roundedbars with a mild notch and a tension plate with a hole under proportional and nonproportional tension–torsion loads, compared the computed notch root strains with the experiments and as well as with FEanalysis results, and reported successful results. Koettgen [50] introduced the notion of pseudo stress–strainfor the sake of simplified nominal stress or strain definitions for arbitrary geometries, and replaced theBarkey’s anisotropic structural yield surface with the matrix of elastically-calculated scaling factors. Theelastically computed stresses are input to the structural plasticity algorithm to compute the real notchstrains via a pseudo stress–notch strain curve. Next the calculated notch strains are input to the cyclicplasticity model to compute the notch stresses. Koettgen reported good correlations of the notch rootstrains for various shaft geometries when compared with elastic–plastic FE analyses. Singh [51] extendedthe equivalent strain energy density approach for nonproportional cyclic loading conditions using amulti-surface plasticity model.

5. Mathematical modeling

The computational methodology for the fatigue life prediction of metallic components is composed of amultiaxial stress–strain analysis model coupled with a notch correction algorithm and a damage assessmentmodel together with a summation rule. Due to the better correlations with experimental trends undergeneral unbalanced loading paths, the cyclic plasticity model proposed by Chaboche and his coworkers isintegrated with the pseudo stress approach of Koettgen and his coworkers to account the geometry con-straint effects, and is employed in the stress–strain analysis. The local loads are described with elasticstresses as input to the stress-controlled notch correction method, and the local elastic–plastic strains arecomputed. Next, the real stresses are computed with the constitutive model using calculated notch strains.The ASME effective strain parameters, Smith–Watson–Topper and Fatemi–Socie critical parameters areemployed in the fatigue damage assessment together with linear damage accumulation rule are used in thefatigue life predictions. In all computations, the validity of Palmgren–Miner damage accumulation isassumed.

5.1. Modeling local loads

The fatigue loads are modeled as pseudo stresses, which are nothing but fictitious quantity computedwith the theory of elasticity for the same boundary value problem [50]. Assuming small deformations,moreover, the anisotropy of yield surface is modeled with a matrix of scaling constants. Considering a setof M different external loads, L, acting on a given component, the pseudo-stress tensor e�ij is the super-position of a set ofM stress tensors equal to the elastic stress tensor calculated for each external load actingon the component separately.

e�ij ¼XMm¼1

Cij� �

mLm ð4Þ

where (Cij)m are scaling coefficients that are equal to the elastic stress tensor calculated for each singleexternal load Lm with unit magnitude. To relate the local loads to the elastic–plastic response at a struc-tural point, a pseudo stress–notch strain, or load–notch strain curve is employed. The pseudo stress–notchstrain curve may be generated by FE analyses or by a uniaxial approximation formula such as Neuber’s

M. Firat, U. Kocabicak / Engineering Failure Analysis 11 (2004) 655–674 659

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rule [43] or equivalent strain energy density method [44]. The details of the approach may be found inrespective works in literature [49,50,52].

5.2. Multiaxial stress–strain analysis

A rate-independent plasticity model using nonlinear kinematic hardening rule is employed to calculatethe stress–strain history. A brief description of the model is given below, and the detailed mathematicalformulation can be found in the papers of Chaboche and his coworkers [24,25,27].Small deformations and additive decomposition of total strain as elastic and plastic parts are assumed.

Elastic deformations followHooke’s law until the yield condition is satisfied. The yield function is expressed as:

f ¼ J2 � � X�

� k ð5Þ

where J2 is the second invariant of deviatoric relative stress, and k is the yield stress in simple shear. Xrepresents the total backstress composed of m parts. The shape and the orientation of yield surface in stressspace are assumed not to change, and the size of yield surface may be changed to account transient effects.The evaluation equation for the increment of backstress parts is expressed as,

dX ið Þ ¼2

3C ið Þd"p � � ið ÞX ið Þdp ð6Þ

where C and � are material parameters, and dp is the increment of accumulative plastic strain. Fadingmemory effect of strain path is introduced with the second term into the evaluation of total backstress. Thenormality hypothesis and the consistency conditions leads to the expression of hardening modulus h as thesum of hardening modulus from each backstress parts. Assuming the von Mises Criterion,

h ¼Xmi¼1

h ið Þ ð7Þ

and,

h ið Þ ¼ C ið Þ �3

2� ið ÞX ið Þ :

�0 � X0

kð8Þ

Using the hypothesis of normal dissipativity, a flow potential F is introduced and the expression ofincrement of plastic strain tensor is derived,

F ¼ J2 � � X�

þ3

4

Xmi¼1

� ið Þ

C ið ÞX ið Þ : X ið Þ ð9Þ

d"p ¼@F

@�: dp ¼

1

h

@f

@�: d�

* +@f

@�ð10aÞ

d"p ¼2

3

�0 � X0

J2 � � X� dp ð10bÞ

660 M. Firat, U. Kocabicak / Engineering Failure Analysis 11 (2004) 655–674

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e material parameters C(i) and g(i) are computed using the cyclic stress–strain curve of the material,

Thand are mainly influenced by the cyclic deformation characteristics. Depending on the strain and stressamplitudes, either Masing or non-Masing behavior may be observed under cyclic balanced loading condi-tions. Jiang [31] proposed a general method in the computation of material parameters in the Armstrong–Frederich type of backstress evaluation, and this approach is employed in this study. A simple proceduremay be outlined in the following way; first m points are selected from the uniaxial cyclic stress–strain curvedetermined with cyclic, reversed tension–compression tests of smooth specimens. The material parametersC(i) and g(i) are calculated via the following two equations,

� ið Þ ¼

ffiffiffi2

3

r1

"pa ið Þ

i ¼ 1; 2; . . . . . .mð Þ ð11Þ

C ið Þ ¼ H ið Þ �H iþ1ð Þ i ¼ 1; 2; . . . . . .mð Þ ð12Þ

where H(1) is the slope between points (I�1) and (i) at the stress–strain curve, and the following conditionsare taken.

�a 0ð Þ ¼ �y ¼ffiffiffiffiffi3k

p; "a 0ð Þ ¼ 0;H Mþ1ð Þ ¼ 0 ð13Þ

The incremental stress–strain relations are discretized in time using implicit backward Euler method [53],and following the solution method proposed by Chaboche and Cailletaud [54], a nonlinear scalar functionproposed by Hartmann and Haupt [55] is used to describe a convergence condition for increment ofaccumulated plastic strain for a given time step. The nonlinear scalar equation is solved iteratively by asuccessive substitution and updating the total backstress and yield function. Once the convergence toincrement of accumulated plastic strain for a given time step is found, the stress and strain tensors at theend of time step are updated.

5.3. Fatigue damage analysis

Two groups of fatigue damage parameters are used in the multiaxial fatigue assessment. First, strainamplitudes based on Salt and Seqa effective range measures are employed as multiaxial damage para-meters, following the procedure given in design codes [2,3]. Assuming that the strain history is given, thesteps in the calculation of Salt and Seqa parameter may be outlined as follows:

1. A reference history point t* in a given cycle is chosen, a fictitious relative total strain �ij with respect tothis history points is defined, and calculated as,

�ij ¼ "ij� �ti

� "ij� �t�

ð14Þ

2. The principal strains �1, �2, and �3 of fictitious relative total strain are determined, and the principalstrain differences for the given cycle is computed.

�12 ¼ �1 � �2ð Þ ð15aÞ

�23 ¼ �2 � �3ð Þ ð15bÞ

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�13 ¼ �1 � �3ð Þ ð15cÞ

The steps 1 and 2 are repeated by switching t* point for all history points in the cycle, and largestprincipal strain difference is chosen. The Salt and Seqa parameters are defined in the following equations,respectively.

SALT" ¼ max1

2�12

; 12 �23

; 12 �23

� �

ð16Þ

SEQA" ¼1ffiffiffi

2p

1þ �ð Þ�12ð Þ

2þ �23ð Þ

2 �13ð Þ2

� �12 ð17Þ

The fatigue cycles corresponding to the computed strain amplitude is calculated by solving strain–lifeequation iteratively, and the increment of fatigue damage associated with the cycle is calculated to be,

DDSALT"¼1

Nfð18Þ

In the second group, the Smith–Watson–Topper and Fatemi–Socie parameters using the critical planeconcept are employed. The critical planes according these damage parameters are defined as the materialplanes on which the amplitude of normal strain, e0xx, and shear strain becomes maximum respectively(Fig. 1). Considering the strain–life curve from cyclic reversed tension–compression and torsion tests of

Fig. 1. Schematic showing the components of strain vector, ", on a material plane.

662 M. Firat, U. Kocabicak / Engineering Failure Analysis 11 (2004) 655–674

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smooth specimens as the baseline data, the Smith–Watson–Topper and Fatemi–Socie parameter–lifeequations are given as;

"a�max ¼�0

f

� �E

2Nfð Þ2bþ�0

f"0f 2Nfð Þ

bþcð19Þ

�a 1þ k�að Þmax

�y

� �¼

�0f2G

2Nfð Þbþ� 0

f 2Nfð Þc

ð20Þ

The computation of maximum damage Dmax and the determination of the critical plane angles, criticaland critical, in fatigue damage assessment with Smith–Watson–Topper parameter for a given stress–strainhistory may be outlined in the following basic steps:

1. The material planes that are candidate for the maximum damage critical plane is determined bycomputing the transformation matrices for a given set of y and f angles.2. The stress and strain tensors at each history point is transformed into the ith material plane, describedby angles i and i.3. A history point t* is chosen in a jth cycle, and a relative normal strain &"0xx for all remaining historypoints in the current cycle is computed as,

D"0xx ¼ "0xx� �ti

� "0xx� �t�

ð21Þ

0 j

and the maximum value of (&E xx)max is determined for the jth cycle by sweeping all other history pointsfor t*. Once (&E0xx)jmax is determined, the maximum amplitude of normal strain (&E0xx)jmax in the jth cyclefor the ith material plane is calculated as,

i "0xx� �ja

¼D"0xx� �jmax

2ð22Þ

and, multiplied with the corresponding value of tensile normal stress (�0xx)max to compute the SWT-para-

meter for the cycle.4. The increment of damage associated with the ith material plane for the jth cycle is calculated as,

iDDj ¼1

Nfð23Þ

where Nf is determined by solving the parameter–life equation given in (19) iteratively.5. The total damage for the ith material plane iD is determined by calculating of increment of damage forall cycles in the stress history, by repeating the steps 3–4.

Total damage for all material planes are calculated by repeating the steps 2–5, and the maximum damageDmax computed with linear accumulation rule is selected as the largest total damage among all materialplanes and the corresponding normal vector defined by angles critical and critical is defines the critical planeat the material point for the given strain history.The computation of the critical plane angles, critical and critical, and maximum damage Dmax according

to Fatemi–Socie parameter is performed following a similar procedure as described below.

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1. The material planes that are candidate for the maximum damage critical plane is determined bycomputing the transformation matrices for a given set of and angles.2. The stress and strain tensors at each history point are transformed into the ith material plane,described by angles i and i.3. A history point t* is chosen in a jth cycle, and a magnitude of relative shear strain i(D�)j for allremaining history points in the current cycle is computed with,

D� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"0xy

� � "0xy

� �h i2þ "0xy

� � "0xz� ��h i2r

ð24Þ

and the maximum value of D� is determined for the jth cycle by sweeping all other history points for t*.Once i D�

jmax

is determined, the amplitude of shear strain in the jth cycle for the ith material plane is

calculated as,

i �að Þj¼

i D� j

max

2ð25Þ

In order to compute the tensile normal stress on the ith material plane for the jth cycle, i(�max)j, relative

normal stress i(&�0xx)j is computed using the same time points at which the amplitude of shear strain i(�a)

j

is calculated. Then mean normal (tensile) normal stress for ith material plane at jth cycle, i(�n)j, is calcu-

lated with,

i �nð Þj¼

D�0xx

� �j2

ð26Þ

4. The increment of damage associated with the ith material plane for the jth cycle is calculated as,

iDDj ¼1

Nfð27Þ

where Nf is determined by solving the parameter–life equation given in (20) iteratively.5. The total damage for the ith material plane iD is determined by calculating of increment of damage forall cycles in the stress history, by repeating the steps 3–4.

Total damage for all material planes are calculated by repeating the steps 2–5, and the maximum damageDmax is selected as the largest total damage among all material planes and the corresponding normal vectordefined by angles critical and critical is defines the critical plane at the material point for the given stress–strain history.

6. Case study: fatigue analyses of biaxial cornering fatigue test of light-alloy wheels

The proposed computational methodology is employed to predict the test cycles of a light-alloy wheel inbiaxial cornering fatigue tests. Four different wheel loads are considered for 16 tests in total, and both the

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fatigue damage and damage-critical locations on the wheel are estimated. The computations and testresults are compared in terms of number of cycles and damaged sides on the wheels.

6.1. Biaxial cornering fatigue test simulation of light-alloy wheels

The biaxial cornering test of wheel-hub assemblies is one of the commonly applied durability tests amongthe wheel manufacturers to validate the operational strength. Exerting a combined vertical and lateralforces in a static manner, while the dynamics is introduced by wheel rotation in the test machine, theservice loading is simulated [56]. The vertical and lateral components of test loads are intended as roadloads acting on the tire-wheel assembly during straight line driving and cornering maneuvers, and duringthe test, a constant or variable amplitude bending moment together with a shear force is applied to thewheel via the hub assembly (Fig. 2). Since the service life of a typical wheel is over 10 million cycles,cornering test loads are magnified with factors given in engineering standards such as SAE, DIN or ISO toaccelerate fatigue process. There are four set of cornering loads applied for each load level in total 16biaxial fatigue tests, and the mean of corresponding number test cycles are given on Table. 1.

6.2. Computational issues and idealization of biaxial cornering fatigue tests

Regarding cornering fatigue tests, while probable failure sides may be estimated roughly, by experience itis known that test failure locations in cornering tests may change from one design to another as well aswith increasing test loads for the same design. Additionally, estimation of number of cornering test cycles isequally important while shaping a brand-new design. Therefore, an engineering analysis indicating test

Fig. 2. Schematic for biaxial cornering fatigue loads.

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failure locations and estimating number of test cycles is a practical need in design development iterations.The stresses on the wheel may be classified as stresses developed due to the mechanical loads and asresidual stresses induced by the manufacturing and heat treatment operations. While it is known that heattreatment processes may influence material properties and fatigue strength depending on the degree ofporosity induced during the casting process, no attempt is undertaken to model such uncertainties due tolack of direct quantitative measure in the fatigue damage analysis. The mechanical stresses developed onthe wheel during cornering tests may be grouped as pre-stresses mainly on the hub region due to boltpretension and as the dynamic stresses due to the centrifugal forces and cornering forces. The wheel ismounted to the test clamp plate via four bolts, with a 35 Nm assembly moment, to the flange connection ofrotation shaft, and the wheel rim is secured to the clamp plate at six locations. At the start of tests, lateraland vertical loads on the wheel are set to the test levels and kept static throughout the test, then therotation of the wheel starts and the rotational speed reaches a constant value of 600 rpm approximately in100–200 cycles.Due to the relatively complex geometry of the wheel and loading conditions, the FE method is used in

the calculation of the local loads. Moreover, the transient effects during start-up are ignored, and cen-trifugal force acting on the wheel is modeled as a distributed body force, with a constant angular speed ofvalue 10 Hz, acting statically. Therefore the total stress at any point of the wheel is assumed to be the sumof the stress due to the bolt pretension, the stress due to centrifugal force and the stresses due to the verticaland lateral cornering forces. The linear elastic FE analyses are employed to calculate the scaling constantsof pseudo stress tensor history at each material point on the wheel, and pseudo stress tensor history e�ij(t)at a material point is described as,

e�ij tð Þ ¼ Cij� �0

bolt-pretensionþ Cij� �0

centrifugalþ Cij� �

lateralFL þ Cij

� �out-of-phase

vertigalFvf tð Þ þ Cij

� �in-phase

vertigalFvg tð Þ ð28Þ

where (Cij)0bolt pretension is a set of six scaling constants corresponding to a constant bolt pretension,

(Cij)0centrifugal is a set of six scaling constants corresponding to a constant centrifugal force due to the wheel

rotation, and (Cij)lateral is a set of six scaling constants corresponding to a unit lateral force on the wheelduring biaxial cornering test. The scaling constant sets (Cij)

out-of-phasevertical and (Cij)

in-of-phasevertical are intended to

account the vertical wheel force of unit magnitude during one revolution of wheel, and the time functionsf(t) and g(t) are simple harmonic functions with a phase difference of 90. The scaling constant set(Cij)

out-of-phasevertical out-of-phase vertical for the whole wheel is computed with a linear elastic FE analysis in

which only F1 is acting on the wheel. The scaling constant set (Cij)in-phasevertical iscomputed similarly while only

F2 is active. In both cases, unit magnitudes are considered. Using the wheel rotation angle j as a reference,the functions f(t) and g(t) may be expressed as,

f tð Þ ¼ cos ’ð Þ ð29Þ

g tð Þ ¼ sin ’ð Þ ð30Þ

Table 1

The cornering tests loads and corresponding mean fatigue test cycles

Test ID

Wheel load (kg) Vertical force FV (N) Lateral force FL (N) Number of test cycles (Nf)

1

500 7550 5660 2.27E+06

2

600 9060 6800 1.03E+06

3

650 9810 7350 8.55E+05

4

700 10 570 7920 5.26E+05

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converged finite element mesh, composed of 114 564 linear tetrahedral elements, for the wheel-hub

Aassembly is generated (Fig. 3). Scaling constant sets (Cij)

0bolt-pretension, (Cij)

0centrifugal are the elastic stress

components on the wheel corresponding to separate FE analyses considering the bolt pretension of 35 Nmand constant angular speed of 10 Hz, respectively. The maximum effective values for each case are com-puted to be 42 and 1.2 MPa, respectively. The maximum elastic stress is observed in both cases on thetransition section from hub to the hub-to-rim connection arms due to the notch constraint effects.Regarding the other set of coefficients in pseudo stress tensor computations for each point, three additionallinear elastic analyses are done. The aluminum alloy used in the production of wheel is designated asG-AlS1llMg whose mechanical properties together with strain–life parameters are given in Table 2.

6.3. Simulation of cornering fatigue tests

The pseudo stress–notch strain curve employed for the stress–strain analysis of complete wheel is calculatedfor each material point, equivalent in this content to a node in the FE mesh of the wheel based on theNeuber’s rule as a uniaxial approximation formula, and the total backstress tensor is composed of fivebackstress parts by using pseudo stress–notch strain curve at each material point. In fatigue life predic-tions, the strain life curve determined from the strain-controlled fatigue tests of uniaxial smooth specimendescribed using Coffin–Manson strain–life equation is employed. The computational fatigue analysis ofwheel biaxial cornering fatigue tests are conducted in two steps. First, a global analysis is performed in thatall material points on the surface of wheel are analyzed for a single test cycle. The fatigue damage distribu-tion on the surface of the wheel is predicted, and the fatigue cycles at a material point, equivalently the FEnode in this content, is determined estimated by using multiaxial damage parameters described in pervious

Fig. 3. (a) Wheel FE mesh, (b) schematic showing boundary conditions.

Table 2

The material properties and fatigue life parameters of the aluminum alloy wheel

Young’s modulus (GPa)

74

Cyclic strength coefficient (MPa)

430

Cyclic yield stress (MPa)

40

Fatigue strength coefficient (MPa)

205

Fatigue strength exponent

�0.1182

Fatigue ductility coefficient

0.068

Fatigue ductility exponent

�0.409

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sections. Next, the most critical point candidate for test failure location is analyzed, and the characteristicsof both stress–strain history and variation of fatigue damage per cycles are evaluated. In the globalapproach, the stress–strain history of all wheel surface nodes is analyzed for a single cycle following a pre-loading step including the bolt pre-tension and centrifugal forces. Next, assuming a cyclically stable beha-vior, the fatigue damage is calculated. A compilation of predicted lives with effective strain and criticalplane damage parameters are presented in Figs. 4 and 5 respectively. The correlations with ASME effectivestrain parameters are non-conservative for low wheel loads and become conservative up to factor 10 athigh loads. Compared with Seqa parameter Salt strain parameter performs better in the whole range, andprovides estimates with fewer shifts for higher cycles. The fatigue damage distributions with effective strainparameters are same for all tests regardless the load level, and the most critical locations are found to be onwheel hub-to-arm transition regions.On the other hand, fatigue test cycles predicted with critical plane based parameters using stress–strain

functions lead significantly better in the whole life range (Fig. 5). Fatemi–Socie parameter predicts all tests

Fig. 4. Test cycles predicted with effective parameters.

Fig. 5. Test cycles predicted with critical plane parameters.

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within a band of factor 5, and all predicted lives are conservative except Test 3, while Smith–Watson–Topper parameter shown a similar trend, nonetheless the predicted lives becomes non-conservativeexcluding Test 4. Regarding the test failure sites identified on the surface of the wheel with dye penetrantinspection, the backside of hub-to-rim connection arm close to the hub side is observed for all tests. Fortests with 650 kg wheel loads, either one or two arms are determined that have fatigue cracks approxi-mately on the same side close to hub notch radius, while the number of failed arms increases to three for alltests under 700 kg load (Fig. 6). A comparison of estimated fatigue critical locations and the test failureside shows that none of the predictions have indicated the wheel arm backside form as the most criticalpoint, and Fatemi–Socie parameter indicated this point, shown on the FE mesh (Fig. 7), with the thirdhighest damage, and located approximately on the path of final fatigue crack observed in tests. Similarly,the SWT parameter indicated the same point with fifth highest damage. Furthermore, only Fatemi–Socieparameter tracks the test trends accurately in the given load range.Due to the better correlations in both computed cycles and fatigue failure sites, the local analyses are

conducted using Fatemi–Socie damage parameter based on critical plane approach to evaluate the localstress–strain response as well as the variation of fatigue damage with loading cycles. The analysis parameters

Fig. 6. The failed arms in biaxial cornering tests under 700 kg wheel load, and developed fatigue crack on the cross section.

Fig. 7. The three most fatigue critical locations predicted with critical plane Fatemi–Socie damage parameter.

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and loading conditions are retained with those employed in the global analyses conducted formerly. Thestress–strain response at node 16 786 is computed for cycles up to 100 000, and the variation of fatiguedamage per cycle are determined (Fig. 8). Examining the stress and strain component histories, a limitedamount of stress relaxation like behavior is observed at this point, while a significant shifts in the totalstrain response are observed in the direction of mean stress for all components. The variations of normalstress and strain components along with pseudo stress components are shown in Fig. 9(a) and (b) respec-tively from an initial ramp loading to the 100 000th cycle. While no change is observed in the elliptical pathof pseudo stress components, there a slight shift coupled with a rotation of response ellipse, as expecteddue to unbalanced loading path that includes a constant mean stress in local stress history. Also therotation in normal stress plot proves the effect of elastic–plastic coupling modeled of normal and shearcomponents with the notch analysis method. Considering the normal strain components, a similar variationis observed, indicating the cyclic strain accumulation with increased number of cycles in the direction of

Fig. 8. FE node for local analyses under 700 kg wheel load.

Fig. 9. (a) Stress response at node 16 786, (b) strain response at node 16 786.

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normal mean stress components, and this trend is seen all other stress–strain components continuously dueto the constant amplitude out-of-loading conditions.The effect of variations in stress and strain tensor components are examined by comparing the fatigue

damage per cycle based on Fatemi–Socie parameter as the number of cycles increases up to 100 000. Asshown in Fig. 10, the damage per cycle is determined to be a monotonically decreasing function, indi-cating that the estimation of fatigue life using the damage predicted with the first cycle after monotonicloading is an appropriate strategy resulting in an conservative results for this particular case. Thereduction in damage per cycle is determined to be approximately 13%, so that the fatigue cycles areincreased from 390 000 to 440 000. The critical plane on which a fatigue crack initiation is predicted ischanging in a few degrees range, and observed to correlate sufficiently with the propagation direction of thefatigue crack on the surface of the wheel (Fig. 11).

7. Conclusion

The main objective of this paper was to propose a computational scheme for anti-fatigue design ofstructural components made of typical engineering metals. The proposed fatigue life prediction methodology

Fig. 10. The variation of average damage per cycle and the predicted fatigue life.

Fig. 11. (a) Fatigue crack on wheel arm, (b) the normal vector of the critical plane according to Fatemi–Socie damage parameter.

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was based on the local strain-life approach, and used available models for multiaxial cyclic plasticity, notchanalysis, multiaxial damage estimation. A cyclic plasticity model using nonlinear kinematic hardening ruleis integrated with an approximation method for notch stress–strain analysis, and the four multiaxialdamage parameters are employed for fatigue damage assessment analyses along with linear accumulationrule. Following the mathematical description of computational methodology, the fatigue damage simu-lation of an industrial application is conducted. Fatigue cycles and crack initiation locations of an alumi-num alloy wheel in biaxial cornering tests are estimated. In global analyses, first the critical locations aredetermined using the computed damage per cycle, and various points on the wheel hub–arms connectionare identified as the fatigue critical locations, and best correlations are obtained with critical plane para-meters, while effective parameters performed relatively poor. Fatigue test cycles predicted using Fatemi–Socie damage parameter based on critical plane concepts are considerably close to test cycles for all wheelloads, and falls in to a band within a factor of 3–5. In order to investigate the trends predicted withincreasing number of cycles, the numerical simulation for the wheel failure location is conducted up to atest cycle of 100 000 using Fatemi–Socie parameter, and the variation of damage per cycle is determined tobe monotonically decreasing function, indicating that the estimation of fatigue life using the damage pre-dicted with the first cycle after monotonic loading is an appropriate strategy resulting in an conservativeresults for this particular case. The reduction in damage per cycle is determined to be approximately 13%,increasing from 390 000 to 440 000.Finally, it is worth to note that the critical plane parameters involving mean stress terms perform equally

similar predictions under nonproportional unbalanced loading case considered here, and Fatemi–Socie andSmith–Watson–Topper models based on critical plane concept constitute a pair of damage parametersapplicable in both cases within the margin of acceptable accuracy from engineering point of view in thedesign of wheels with conformance to anti-fatigue requirements in biaxial cornering tests. Furthermore, adecrease in the computed damage per cycle is investigated with reference to notch stress–strain analysisapproach capable of modeling ‘‘stress relaxation like’’ behavior and cyclic strain accumulation.

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