Cumulative microslip at the interface of mechanical assemblies under cyclic loading

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Cumulative microslip at the interface of mechanical assemblies under cyclic loading

Nicolas Antoni, Jean-Louis Ligier, Philippe Saffré

To cite this version:

Nicolas Antoni, Jean-Louis Ligier, Philippe Saffré. Cumulative microslip at the interface of mechanical assemblies under cyclic loading. International Congress ”Engineering Mechanics Today 2004”, Aug 2004, Ho Chi MInh City, Vietnam. �hal-02013176�

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Cumulative microslip at the interface of mechanical assemblies under cyclic loading

N. ANTONI^a, J.-L. LIGIER^b, P. SAFFRE^c

a LOCIE (MMSC), ESIGEC, 73376 Le Bourget du Lac, France nicolas.antoni-renexter@renault.com

b RENAULT S.A., Powertrain Engineering Division, 92508 Rueil Malmaison, France jean-louis.ligier@renault.com

c LOCIE (MMSC), ESIGEC, 73376 Le Bourget du Lac, France philippe.saffre@univ-savoie.fr

Abstract

Mechanical contact problems involving friction are widely encountered in many industrial applications. Despite the recent progress in non-smooth mechanics and tribology, a number of these problems are still open as they cannot be properly simulated with current numerical tools. The cumulative microslip phenomenon at the interface of mechanical assemblies subject to thermal or mechanical cyclic loading may represent one of these issues. In practice very few mechanical components are designed to prevent that phenomenon from appearing and it is possible to observe significant relative displacements at the interface of affected assemblies leading to a dysfunction.

1 Introduction

The cumulative microslip phenomenon occurs in mechanical assemblies subject to thermal or mechanical cyclic loading and it is characterized by the accumulation at the contact interface of tangential microslip of one element relative to another assembly element in a preferred direction. These last features give it a determinist nature which resembles the Ratchet effect in plasticity (see for example [7]). In practice, such cumulated microslip may cause the failure of some part assemblies in reciprocating engines with internal combustion (see [8]). Some of them are particularly subject to this phenomenon ; this is the case of the connecting rod big end associated with its bearing and the connecting rod small end associated with its bushing, for which the cumulative microslip phenomenon has been found fairly often.

For a case of cumulative microslip phenomenon induced by asymmetric friction conditions at the contact interface of mechanical assemblies submitted to uniform thermal loading cycles, we show, in this article, some of the results obtained. First, due to the quasi- lack of literature on this subject, it was necessary to underscore theoretically the existence of such a phenomenon, well-known in practice. For this, a discrete analytical model of a simplified assembly involving a suitable friction law is proposed. This is special in that it extends Coulomb’s law of friction which, in its classical form, does not allow a

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reproduction of this phenomenon. The analysis of this assembly behaviour under thermal cyclic loading show under which conditions the cumulative microslip phenomenon may occur. Secondly, we analyze an anisotropic friction model that allows the determination of the law typical parameters and shows that the arbitrary assumptions previously used for the coefficient of friction are realistic. As a friction law may not be ordinary, we have bought a highly special attention to demonstrating its thermodynamic admissibility.

2 Theoretical underscoring

For the sake of simplicity, we shall first concentrate our attention on a discrete model to present the phenomenon and we shall extend it with a continuous one.

2.1 One-dimensional discrete model

Let us consider the discrete model of a simplified assembly made of a contact elastic spring (coefficient of linear thermal expansion α_T and original length l₀) with longitudinal stiffness ( )k , initially at the reference temperature T₀, for which both nodes are kept in contact on a fixed rigid support under the action of a constant normal nodal force N >0. The degree of freedom in translation of node i is denoted as u_i (i=1,2), see Figure 1.

Figure 1 : One-dimensional discrete model.

The displacement of each contacting node is obviously controlled by friction conditions at the contact interface. Let F_t_i be the nodal tangential force applied by node i on the support (i=1,2). To include frictional effects let us assume that regularized asymmetric Coulomb’s law of friction holds pointwise on the contact interface (see Figure 2). It takes two phases into account:

A reversible phase corresponding to the elastic deformation of asperities at the contact interface. Also mentioned is the adherence phase with elastic microslip. Let k_t denote the contact tangential stiffness which is the typical parameter of this phase ;

An asymmetric irreversible phase corresponding to an asymmetric macroscopic slip : the value of the friction coefficient is different depending on the direction of slip considered. It is assumed, for the one-dimensional model suggested here, that the value of the friction coefficient in the x>0-direction is higher than the one in the x<0-direction, i.e. :

− + >µ

µ (1)

u1

k

N

∆T

x u2

N

1 2

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Figure 2 : Regularized asymmetric Coulomb’s law of friction.

Let T be the uniform temperature forced by the outside world onto the system. Next, we define the temperature difference ∆T =T−T₀. Within the framework of small displacements (implying small slip), analytic developments of this assembly response under quasi-static thermal cyclic loading ∆T∈

[

∆Tmin,∆Tmax

]

show that, if condition (1) holds, the cumulative microslip phenomenon occurs as soon as ∆T_max −∆T_min >2∆T_c (see Figure 3). In such an inequality, ∆T_c denotes a critical temperature variation corresponding to the contact tangential elasticity limit of the system and is defined as :



 



 +

=

∆ ⁻

t T

c l k k

T N 1 2

α 0

µ (2)

Figure 3 : Cumulated microslip in the x<0-direction.

The cumulative microslip phenomenon is characterized by the accumulation of small displacements for each node in the x<0-direction and for each loading cycle, i.e. :





<

∆

<

∆ 0 0

2 1

u

u (3)

where ∆u_i is the incremental nodal microslip of node i for each cycle (i=1,2).

With such assumptions, it can be showed that the incremental nodal microslip is identical for both nodes, constant during (stabilized) cycles and strictly negative, as :

( _min _max 2 ) 0

0 2

1=∆ = ∆ −∆ + ∆ <

∆u u α_Tl T T T_c (4)

Tmax

∆

Tmin

∆ Tc

∆

Tc

T − ∆

∆ _max 2 0

∆u1

∆u2

∆T

ti

F

+N µ

−N

−µ

ui

0 kt

u1, u2

Tc

T + ∆

∆ _min 2

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2.2 One-dimensional continuous model

There should be now considered the continuous equivalent of the previous discrete model. A thin (h,b<<l) elastic (E is Young’s modulus, α_T is the thermal expansion coefficient) strip kept in contact on a fixed rigid support, under the action of a uniformly distributed normal surface traction p (see Figure 4), may be assimilated, for a tension only behaviour, with a serial assembly comprising an infinity of such discrete systems.

Similarly, each particle on the contact interface ∂_CΩ⁰ is assumed to obey regularized asymmetric Coulomb’s law of friction previously introduced.

Analytic developments enable again showing the occurrence of the phenomenon under thermal cyclic loading ∆T∈

[

∆Tmin,∆Tmax

]

if condition (1) holds and if

∆ −

>

∆

−

∆T_max T_min 2 T_c . In the previous inequality, ∆T_c⁻ denotes a first critical temperature variation corresponding to the original tangential contact elastic limit of the system and is defined by:











 −























=

∆

−

1 ch

sh

Eh l k Ehk

Eh l k p T

t t

T

t

c

α µ

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When this is reached, the end x=0 starts slipping in the x<0-direction whereas the rest of the strip is in the adherence phase with elastic microslip. The cumulative microslip phenomenon is characterized by the accumulation in the x<0-direction of small displacements of the strip for each particle on the contact interface ∂_CΩ⁰ and for each loading cycle, i.e. :

( ).ˆ <0 ∀ ∈∂ Ω⁰

∆ur x e_x x _C (6)

where ∆ur is the incremental small displacement for each cycle.

Figure 4 : One-dimensional continuous model.

The calculated response of the strip under thermal cyclic loading clearly exhibits, in a continuous sense, all the expected similarities with the one of the previous discrete model.

The expression for the incremental microslip is not given here due to its complexity.

zO

y ∆T

eˆx

p

l Ω0

∂_C

x y

z b

x h eˆy

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Results have been extended to the asymmetric case of two fitted tubes (one being fixed) but the analytical expression is too complicated to be presented in this paper.

At this stage, the cumulative microslip phenomenon may occur when the contact interface behaviour is governed by asymmetric friction conditions. The physical validity of such assumptions is given in the following paragraph.

3 Sufficient condition for existence

In order to justify the physical admissibility of regularized asymmetric Coulomb’s law of friction we have proceeded with two steps. First, a natural approach based on micromechanical observations provide an anisotropic friction model enabling, via averaging, a determination of the law characteristics (in this respect see [10], [11]). Then its thermodynamic admissibility is proven.

3.1 Anisotropic friction model

Let us consider the idealized surface texture (in practice, there are always dispersions, see Figure 5a) of a fixed rigid body denoted as (2) on which are periodically distributed long wedge-shaped asperities (see Figure 5b). Each of them is characterized by geometrical properties (α,β,h) as well as mechanical

(

µα,µβ

)

. This last couple represents the real friction coefficients for each wedge of one asperity (see Figure 5c).

Figure 5 : A surface texture model.

An elastic body denoted as (1) on which are isotropically and uniformly distributed asperities modelled as identical elastic springs (i.e. same original length, same longitudinal stiffness k_n and same flexural stiffness k_f ), is kept in contact on the previous one (see Figure 6).

Figure 6 : A type of contact model.

kn

Sliding direction (1)

tˆb tˆ_a

(2)

tˆa Real rough surface

(2) a..

α h β

c.

b..

nˆ

tˆb

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The steady macroscopic sliding of body (1) over body (2) is then considered, assuming first an infinite flexural stiffness : the springs contract and release while sliding along wedge-shaped asperities so that the body (1) motion only occur in the nominal contact plane

( )

tˆa,tˆb . It is then possible to express, averaging over a period, the average friction coefficient in the principle direction tˆ of the contact interface. Its expression varies _a according to the considered path direction and it is independent from k_n and from h :

( )









+

−

= +

+

−

= +

− +

β α

β α µ α β µ µ

β α

α β µ β α µ µ

α β

β α

tan tan

tan , tan

, tan tan

tan , tan

,

H H

a a

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where the superscripts “+” and “-” denote the tˆ_a and

( )

−tˆa directions, respectively, and the function H is defined as :

( ) x y

y y x

x

H 1 tan

, tan

−

= + (8)

From equations (7), it is shown that it is possible to reach a macroscopic friction asymmetry in the case of a complete asymmetry of the body (2) surface texture (geometrical and mechanical properties), i.e. :

(α ^≥β) and

(

µ_α ^≥µ_β

)

^⇒µa⁺ ^≥µa⁻ (9) One of the features of equations (7) is that they do not depend on the height h of wedge asperities. So, they remain valid for a body (2) surface texture that is physically more realistic, with asperities of a constant shape (i.e. (^α,^β) constant) but with a Gaussian distribution of their heights (this is the case met most often in practice but other distributions may also be considered, for instance Normal log. distribution). Note that the height h for each wedge asperity must be measured from the same mean line.

An infinite flexural stiffness has been assumed and thus a rigid body motion of the springs during the steady sliding of body (1). If this stiffness is assumed to be large but finite, there is then a bending of the springs prior to their sliding along wedge asperities indeed in relation with a linear elastic behaviour which allows a reversible microslip u_t^e_a of body (1) relative to body (2) in the principle slip direction tˆ_a of the nominal contact plane.

The relation between principal tangential stress σ_t_a at the nominal contact interface and this reversible displacement is thus linear, i.e. :

t e t

t k

u_a σ^a

= (10)

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One generalizes on the one hand equation (10) from a two-dimensional point of view, assuming an isotropic bending behaviour of the springs, which then becomes :

t e t

t k

u_r σ^r

= (11)

The springs start sliding in an elastically deformed state whenever the macro slip condition has been satisfied, which allows on the other hand a justification of the well- known decomposition (see [2], [3], [6], [11] and [16] for example) of the total relative tangential displacement (observable), into one reversible elastic part denoted as u^r_t^e (however small it is), and one irreversible inelastic part denoted as ur_tⁱⁿ and corresponding to the macro slipping :

in t e t

t u u

u^r = ^r +^r (12)

Equation (11) is always adopted in the literature but is not basically very realistic if whatever a surface texture is taken into account. A reversible orthotropic phase could be thus considered, characterized by two contact tangential stiffness k_t_a and k_t_b respectively associated with the principle slip directions tˆ and _a tˆ . _b

3.2 Thermodynamic admissibility

Generally speaking, a mechanical constitutive law cannot be any, and it must comply with some principles for it to have a physical sense. It must, in particular, be such that the two fundamental principles of thermodynamics, thus the Clausius-Duhem basic inequality, be satisfied. Starting from experimental observations, a method based on the concept of generalized standard materials is applied in order to rebuild a law valid from a thermodynamic point of view. Such approach is commonly used to derive mechanical constitutive models for material non-linearities (see [7] for example). A friction law may be considered as a mechanical constitutive law for a of continuous medium region that is not of volume any more and only existing in a contact situation. The general and natural framework for writing friction laws is thus that of continuum thermodynamics.

Since the method previously quoted may not be directly particularized for contact interface, it must therefore be adapted. An implementation has been proposed by several authors. There may be quoted, among others, Moreau (see [9]) for his work on non-smooth phenomena like plasticity and friction, Felder (see [4]) for his work on surface interactions applied to metal forming, or also Strömberg et al (see [16]) for studying fretting (wear process arising when contacting surfaces undergo oscillatory small displacements).

The general approach consists first in using the Clausius-Duhem inequality deducted from the expressions in local form of the first and second principles of thermodynamics at the contact interface. In order to prove the thermodynamic validity of regularized asymmetric Coulomb’s law of friction, the three following steps are taken successively into consideration :

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Choice of a set of independent state variables, internal representation of the state ;

Derivation of state laws for reversible interactions from a specific free energy ;

Derivation of evolution laws for irreversible interactions from a specific dissipation potential which must be taken as convex (but not necessarily differentiable) in order to ensure satisfaction of the dissipation inequality.

Set of independent state variables : The following elementary set is selected :

(

t tⁱⁿ

)

C = ur,ur

χ (13)

These two state variables (observable and internal, respectively) are assumed to comply with the decomposition in equation (12). The associated thermodynamic forces are respectively YC =(σ^rt,σ^rt).

Specific free energy and state laws : The thermodynamic potential to be taken into account is written as:

( ) ( )

²

2

, _tⁱⁿ 1 _t _t _tⁱⁿ

t

C ur ur = k ur −ur

ψ (14)

Taking into account the decomposition in equation (12), the state law which derives from this thermodynamic potential is given by:

e t t t kur

r =

σ (15)

which is simply the reversible phase in equation (11).

Specific dual pseudo-potential and evolution laws : Let us consider the closed convex set R2

K ⊂ as follows:

( )

{

^∈ ²^/ ^, ^≤⁰

}

= _t g _t _n

K σ^r R σ^r σ (16)

where g is a convex differentiable function, so-called friction criterion. The proposal dual pseudo-potential, from which derive the complementary laws, is defined as :

( )t K( )t C σ^r =I σ^r

Φ^∗ (17)

where I_K denotes the indicator function of the set K.

As this is convex, I_K is also convex. This function is also minimum at σr_t =0r. The pseudo-potential Φ^∗_C defined above being convex with respect to the dual variable ( )σ^r_t and minimum at

(

σ^rt ⁼⁰^r

)

, the Clausius-Duhem inequality is therefore satisfied.

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Several choices of g are possible. One friction limit criterion, differentiable and convex, directly issued from the previously described anisotropic friction model, is defined by :

( )















≤

≥

 −

















 +









≤

 −

















 +









≥

≤

 −

















 +









≥

 −

















 +









=

− +

−

+

−

+ +

0 et 0 if

, ,

2 / 2 1 2

b a

t t

n b

t a

t

t t

n b

t a

t

t t

n b

t a

t

t t

n b

t a

t

n t

g t

σ σ

µ σ σ µ

σ

σ σ

µ σ σ µ

σ

σ σ

µ σ σ µ

σ

σ σ

µ σ σ µ

σ

σ σ

σ (18)

where

(

µa⁺,µa⁻

)

and

(

µb⁺,µb⁻

)

denote the friction coefficients associated with the principle slip directions tˆ and _a tˆ , respectively, according to the path direction considered. _b

Attention must be paid to the physical interpretation of these last two coefficients formally entered. The previous anisotropic model only shows basically a single friction coefficient µ_b =µ_b⁺ =µ_b⁻ (generally different from the two other ones) in the principle direction tˆ . A write-up that is so general assumes the existence of a surface texture _b exhibiting such macroscopic friction properties, which is not obvious. The criterion expression has been built from that of the usual orthotropic friction limit criteria, shown as ellipses (see [1], [5], [11], [15]).

An example of representation

(

µa⁺ ⁼⁰^,²^,µa⁻ ⁼⁰^,⁰⁸^,µb⁺ ⁼⁰^,¹²^,µb⁻ ⁼⁰^,¹⁵

)

, together with that of a Coulomb’s isotropic friction criterion

(

µa⁺ ⁼µa⁻ ⁼µb⁺ ⁼µb⁻ ⁼⁰^,⁰⁸

)

to compare, is given on Figure 7. It should be noted that the representation for this anisotropic criterion is made of four truncated ellipses.

Denoting the component of the rate of the internal variable ur_tⁱⁿ in the tˆ_i-direction as

in t_i

u& (i=a,b), the evolution law that derives from the convex (but not differentiable) dual

pseudo-potential Φ^∗_C is expressed as :

, , 0 , 0 , 0

, g g i a b

u g

i i

t in

t ≥ ≤ = =

∂

= ∂ λ λ

λ^& σ ^& ^&

& (19)

A sliding rule, associated with the friction limit criterion, has been selected here. It is however possible to introduce a family of sliding potentials that are not associated with this criterion (in this respect, see [11]).

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Figure 7 : Anisotropic friction limit criterion: example of representation.

4 Conclusion

Asymmetric friction conditions at the contact interface which characterize regularized asymmetric Coulomb’s law of friction and represent a sufficient condition to obtain the cumulative microslip phenomenon have been achieved from micromechanical considerations on surface textures. This law, for which the thermodynamic validity has been proven, is thus associated with a given surface texture. In practice, we must underline that the available surface texture parameters in the industrial tools do not accurately enable the detection and characterization of surface textures of the type presented in this article.

Also, usual FEA do not allow the direct integration of regularized asymmetric Coulomb’s law of friction whereas it should be easy for a given situation, to define asymmetry friction levels to be avoided so the phenomenon does not occur.

A comparison of these results with experiments is planned in order to validate this type of approach in spite of the simplifying assumptions used.

Acknowledgements – This work is supported by Renault S.A.. Pr. J. Pastor, responsible for MMSC research team, is also gratefully acknowledged here.

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n aσ µ⁺

n bσ µ⁻

−

n aσ µ⁻

−

n bσ µ⁺

ta

σ

Isotropic Coulomb’s friction limit criterion

Anisotropic friction limit criterion

tb

σ

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Contact Friction for Large Deformation Analysis, Journal of Applied Mechanics, 52, pp. 639-648, 1985.

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Académie des Sciences, Paris, t. 271, série A, pp. 608-611.

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[16] STRÖMBERG N., JOHANSSON L., KLARBRING A., Derivation and Analysis of a Generalized Standard Model for Contact, Friction and Wear, Int. J. Solids Structures, Vol. 33, 13, pp. 1816-1836, 1996.