Three-level multi-scale modeling of electrical contacts · 2015. 7. 28. · area of contact,...

Three-level multi-scale modeling of electrical contacts

sensitivity study and experimental validation

Vladislav A. Yastrebov, Georges Cailletaud, Henry Proudhon

MINES ParisTech, PSL Research University, Centre desMatériaux, CNRS UMR 7633

Evry, [email protected]@[email protected]

Frederick S. Mballa Mballa, Sophie Noël, Philippe Testé, Frédéric Houzé

Laboratoire Génie électrique et électronique de Paris,UMR CNRS-CentraleSupelec 8507, UPMC and Paris-Sud

Universities, Gif sur Yvette, [email protected]

[email protected]@geeps.centralesupelec.fr

Abstract—An experimental and numerical study of electricalcontact for low currents in sphere-plane set-up is presented. Athree-level multi-scale model is proposed. We use the finiteelement analysis for macroscopic mechanical and electricsimulations. It takes into account the setup geometry, elasto-plastic mechanical behavior of contacting components in thefinite-strain-plasticity framework and electrostatic properties. Asensitivity analysis with respect to the brass plastic behavior andto the thickness of coating layers is also performed. The finiteelement results are used for an asperity-based model, whichincludes elasto-plastic deformation of asperities and their mutualelastic interactions. This model enables us to simulate the realmorphology of contact spots at the roughness scale using theexperimentally measured surface topography. Finally, theGreenwood multi-spot model is used to estimate the electricalcontact resistance. This three-level model yields results which arein good agreement with experimental measurements carried outin this study.

Keywords: electrical contact, elasto-plastic material,experimental measurements, multi-scale simulations.

I. INTRODUCTIONElectrical contacts can be critical components in electronic

systems. Their electrical resistance and life duration depend onmany factors such as life cycles, interaction with theenvironment (oxidation and corrosion), surface morphology,surface chemistry and bulk material electrical properties andmechanical behavior. In this paper we study bothexperimentally and numerically the influence of the materialsproperties and the geometry of the coating layers on theelectrical contact resistance between a copper-beryllium sphereindenting a brass flat substrate coated with nickel and gold.

Electrical contacts performances and reliability criticallydepend on the characteristics and evolution of the so-called realarea of contact, composed of numerous zones whose size anddistribution are determined by the macroscopic shape ofbodies, the roughness of their surfaces, their mechanical andelectric properties as well as by mechanical loads and involvedelectric currents. Contact resistance mainly results from theconstriction of current lines at these contacting spots. A projecthas been set-up joining the efforts of two laboratories in thefields of experimental and numerical studies to clarify the

interplay of multiscale and multiphysics mechanisms in thiscomplex interfacial region.

Phenomenon of electrical contact combines mechanical andelectric effects, as the electric current can pass mainly throughcontact spots which result from mechanical deformation ofcontacting solids. Thus, if the coupling between the mechanicaland electric effects may be assumed unidirectional (which isoften the case for low currents and moderate loads) then thiscoupled problem can be split in two sub-problems: first, themechanical contact problem, second, the electrostatic problemfor the geometry obtained from the solution of the first. Withthis consideration in hand, one can combine independentmodels for mechanical and electrical contacts. For the latter,the simplest model consists in approximation of the realconducting zone by a single circular conducting spot at theinterface between semi-infinite planes [1]; for the case of twofinite cylinders contacting at a circular spot, first terms in theTaylor's expansion of the solution were obtained in [2]. Ageneralization of these simple models is the multi-spotGreenwood model [3], which approximates the contactinterface by a cluster of conducting circular spots at theinterface between two conducting half-spaces. Some furthergeneralizations of this model are also available in the literature,including models which take into account aging of contactspots prone to the growth of oxide films [4], modelsconsidering the current density distribution inside contact spots[5], models adapted for saturated contacts, when the electricalcontact area approaches the apparent contact area while thenumber of contact spots remains small [6], models includinginterface film resistance [7] and smoothed version ofGreenwood model, which represents the discrete sums byintegrals with specific kernels [8].

Equivalently many models exist to solve the mechanicalcontact between rough surfaces, whose solution determines themorphology of the mechanical contact area, which may beconsidered as the upper boundary for the electrical contact area[9]. Among these models there are analytical asperity-basedmodels [10-13], which approximate the effective roughness oftwo contacting surfaces by a series of spherical or ellipsoidalnon-interacting asperities, with specific distribution of theirgeometrical properties, which in turn follow the randomprocess roughness model developed in [14,15]. A multiscale

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generalization of asperity based models is suggested in [16],which assumes however a kind of scale separation in thesurface roughness. A different approach is developed in [17],which considers a full contact for varying spectral content as astarting point, and next extends the result for the case of partialcontacts [18]. These models, associated approximations,limitations and possible extensions are summarized in [19,20].Apart from these analytical and semi-analytical models,various numerical methods may be used to solve themechanical boundary value problem with contact constrainsand explicitly integrated surface roughness for elastic [21-23],visco-elastic [24] and elasto-plastic material models [25,26].Different approaches and results of these full-scale analyses aresummarized in [27].

In weakly coupled electro-mechanical contact problems,the multiscale and multiphysics model is essentially acombination of one of aforementioned mechanical andelectrical models. It is worth mentioning that the mechanicalpart is sometimes simply neglected and the spot distribution iseither simulated by a random model or is obtained as a cross-section of a rough surface (geometrical overlap model), see e.g.[8,28,29]. We refer to following references [30-34], as to thestate of the art works, in which more complete and accuratemodels are used for mechanical and electrical contacts.Moreover, in [34] a link of aforementioned models with arather different, incremental stiffness approach of Barber isdiscussed [35]. In conclusion of the bibliographical review, itshould be remarked that the electrical contact problem isessentially similar to the one of thermal conductance throughthe same configuration of contact spots, under condition thatthe thermal expansion of solids and convective heat exchangeare neglected. Hence, a rich bibliography on thermal contactconductance [36,37] is partly relevant to our topic.

Our work combines within a unified framework severalnumerical models for mechanical and electrical contacts. Amechanical finite element model, which uses accurateconstitutive equations to capture elasto-plastic materialbehavior, allows us to predict the structural response forcomplex loading including cyclic loads in contact. Animproved elasto-plastic asperity model with elastic interactionsbetween asperities is then used to capture the roughness effectusing the real topography of ours samples measured withatomic force microscopy. This model provides us withstatistically meaningful results and allows us to estimate thedata dispersion . Finally, the original Greenwood model is usedto analyze the electrical resistance of contact clusters. Thismodel is validated on a series of fine experiments conducted inthe project.

In Section II, the general methodology is described. SectionIII presents the experimental setup and correspondingmeasurements. The numerical approach (finite elementanalysis) as well as the mechanical constitutive models whichare used in the simulations are discussed in Section IV. Theroughness of the contacting surfaces is analyzed in Section V;the multi-scale electro-mechanical model and the associatedresults are presented in Section VI, followed by a discussion.

II. METHODOLOGYThe general objective of the study is to predict the

experimentally observed variation of the electrical contactresistance for “real” sphere-plane contacts under cyclic

loadings using multiscale electro-mechanical simulations. Toreach this objective, we integrate in the numerical model (1)realistic constitutive material models (elasto-plastic withisotropic and kinematic hardening), (2) an accuraterepresentation of the roughness morphology of the contactingbodies as measured by atomic force microscopy (AFM) andinterferometric profilometry. The multiscale nature of theelectric and mechanical contact manifests itself in a separationof scales between the macroscopic geometry of the contactingsolids and the microscopic roughness on a smaller scale (Fig.1). In our study we use this scale separation in the followingway. First, we conduct the mechanical simulation of theindentation for various materials and coating thicknessesassuming perfectly smooth surfaces (with no roughness). Atevery load step, a complementary electrostatic simulation isconducted to evaluate the electrical contact resistance. Bothsimulations are carried out using a finite element softwarewith implicit integration [38,39]. The effect of the roughnessand of the microscopic deformation of asperities is then takeninto account in a semi-analytical model of interactingasperities, which uses the real roughness topography. Thismodel provides us with the exact morphology of the contactspots, which is then used to estimate the contact resistance viathe Greenwood model (Eq. (4) in [3]). The sensitivity of themechanical model with respect to material properties (initialyield stress) and to the thickness of coating layers is thenanalyzed. The numerical results are compared with theexperimental data.

Fig. 1. Separation of scales in the electric and mechanical contact betweenrough solids: (a) the macroscopic scale is characterized by nominally flat(smooth) surfaces, (b) at certain magnification the discrete nature of thecontact is revealed. The real contact area is considerably smaller than thenominal contact area predicted at macroscopic scale with the Hertz contacttheory; the real contact area can be approximated by a set of a-spots (c).

III. SAMPLES AND EXPERIMENTAL SET-UPCyclic indentation tests were performed on nominally flat

substrates by a copper beryllium ball of radius 1.75 mm. Theflats were one millimeter thick brass alloy CuZn30 [CW505L,30 wt% Zn] planes coated with electrodeposited nickel andgold layers both 1µm thick. Bare CuBe balls were used.Particular attention was devoted to their surface. Several typesof experiments involving different surface finishes of the ballswere performed. The balls were either rinsed, thoroughlycleaned or mechanically polished before the measurements.The experiments were conducted on a special test benchdepicted in Fig. 2. The ball is mounted in a holder and pressedagainst the plate with a stepping actuator (1/10 μm by step),which is controlled by a displacement capacitive sensor (apreliminary calibration procedure permits to convert the

vertical displacement into the value of the normal force via thestiffness of the guiding elastic rings). A water-cooling circuitinsures the thermal stability of the set-up, specifically thesensor’s measurements. The elements of the contact areconnected to the electrical setup following the four-terminalmethod. The electrical circuit includes a DC voltage/currentsource, an electrometer and a digital voltmeter, all controlledover the IEEE-488 bus. Once the required force is reached, aDC current I = 10mA is imposed to the contact; oppositepolarities are used to eliminate thermoelectric voltages(current-reversal method) [40]. Measurements of the potentialdifference (U) enable us to calculate the total electricresistance

(1)where Rc is the contact resistance and Rs is an additionalresistance due to the bulk parts of ball-holder and brass planebetween contact area and potential measurement terminals. Inthe considered system Rs is estimated to be about 0.09 m.Under assumption of elastic Hertzian contact and Holm'selectrical contact, the first term can be estimated as follows:

(2)

where R is the ball radius, a is the contact radius, * theeffective resistivity, E* the effective modulus, and * isusually taken as a mean resistivity of CuBe and CuZn. Theeffective elastic modulus is considered for the pair CuBe andCuZn

(3)

where v is the Poisson's ratio and E is the Young's modulus.

Fig. 2. (a) General layout of the contact resistance measurement setup: 1 –rigid frame, 2 – capacitive force sensor, 3 – elastic washers, 4 – insulatingblocks, 5 – ball holder, 6 – CuBe ball, 7 – plane CuZn substrate coated withNi and Au, 8 – displacement stepping motor. (b) Mounted CuBe ball, twowires (grey: current feeding, red: voltage measurement) and the water coolingsystem (tube); (c) sample holder for the plate (7) with connections.

Series of three loading-unloading cycles were performed invarious conditions (ball finish and indentation zone).Experimental data for a representative test as well as the therange of experimental data obtained for different runs (shadedarea) are depicted in Fig. 3 which shows the evolution of themeasured resistance with respect to the applied load as well asthe analytical estimation from Eq. (1,2). The first loading isobserved to be distinct from the subsequent loading-unloadingcurves because of the plasticity onset in the CuZn substrate.

After the first hardening the system follows a stabilized cycle:the loading and unloading curves follow the same trajectories.The hysteresis is attributed to the kinematic hardening inCuZn. The variability in the experimental results can beexplained by differences in the surface states of the balls(including roughness and contamination) and on the localroughness of the plane at the location of the indentation.

Fig. 3. Experimental data and analytical estimation: left – normal scale, right– logarithmic scale. Shaded area encloses all the experimental data points forthe first three loading-unloading cycles for different ball finishes and atdifferent locations on the plate. The data points show three cycles of arepresentative experiment; the dashed line shows a reference analyticalestimation for elastic Hertzian contact Eq. (1,2) for Rs=0.09 m. Arrows withthe color code indicate the load direction.

IV. NUMERICAL SIMULATIONS AND MATERIAL MODELSThe macroscopic calculation is made on an axisymmetric

model, as shown in Fig. 4. A convergence study has beenperformed in order to choose the optimum element size in thecontact zone. Linear elements are used in the finite elementmesh to ensure an optimal contact treatment. The nickel andgold layers are discretized by two element layers in thickness.Elements in the vicinity of the contact have a size of 500 nm.The number of elements in contact reaches more than 80 at thepeak load. A prescribed displacement is imposed on theequatorial plane of the half sphere (A in Fig.4), which remainsflat due to a multi-point constraint condition, while the bottomof the mesh representing the substrate (B in Fig.4) is fixed invertical direction. The mechanical simulation allows us toobtain the evolution of the radius of the contact surface withrespect to the applied force. At each load step, the two initiallyseparate meshes are fused in the contact zone for thecomputation of the electrical problem. Doing so, there is nodiscontinuity in electric potential at the boundary between thesphere and the plane. The electrical model at macroscopicscale thus assumes a perfect contact between the two bodies(no resistive interfacial film).

Non linear constitutive equations were used for the threematerials of the substrate (CuZn, Au and Ni), while an elastic

behavior is considered for the ball CuBe material model. Ineach case, the material data are taken from literature(respectively [41,42] and [43] for brass, gold and nickel). Thematerial models incorporate either only non-linear kinematichardening (Au, Ni) or a combination of isotropic and severalkinematic hardening (CuZn), in order to correctly representcyclic responses of the materials. The expressions aresummarized below, Eq.(4-9). The yield function, f, in Eq.(4)uses the von Mises invariant, denoted by J, of the effectivestress (the stress minus the kinematic variable), as specified inEq.(5). The size of the elastic domain is defined by the sum ofthe initial yield stress, y, and the isotropic hardening variable,R. Two material parameters, namely the possible amount ofhardening Q and the parameter characterizing the saturationrate, b, are present in Eq.(6) to define isotropic hardening. Theexpression of kinematic hardening has a driving termproportional to the plastic strain rate, and a fading memoryterm proportional to its actual value. The product D(p)characterizes the non linearity of the stress evolution inside acycle. Its initial value is D and the final D, that allowsrepresenting a sharper hysteresis loop after a few cycles. Thevariable p brought into play in both Eq.(6) and Eq.(7) is thecumulated plastic deformation, defined by its rate, as shown inEq.(9).

(4)

(5)

(6)

(7)

(8)

(9)

where s is the deviatoric part of the stress tensor σ, the dotrepresents the time derivative and the colon represents tensorcontraction.

A simple sensitivity analysis was carried out with respectto the yield stress of the brass: 53 MPa < σy < 550 MPa and tothe thickness of both coating layers in the range 0.5

orthogonal directions and by the height distribution; both aredepicted in Figs. 7-8, respectively.

Fig. 5. Results of finite element analysis: contact radius (left) and electricresistance (right) evolution in cycling loading. The shaded area spans allexperimental results and the dashed line is the reference analyticalestimation for elastic Hertzian contact Eq. (1,2) computed for Rs=0.09m.

Fig. 6. Example of roughness measurements (AFM) on the coated brasssubstrate.

The roughness is observed to be anisotropic (mainlybecause of the rolling) with a particular scaling in terms of thePSD; the height distribution is not Gaussian, with a ratherpronounced tail, which arises due to numerous high asperities(see Fig. 8). Rigorously, such a surface cannot be analyzed bystandard methods of the random process model [14,15,44]. Itshould be noted that the surface anisotropy does not imply astrong anisotropy of asperities, equivalently the isotropy of thesurface does not imply the isotropy of asperities. As wasshown in [45], according to the random process model, themean ratio of asperity principal curvatures is approximatelythree and the probability to find a circular asperity is zero.

The surface roughness was analyzed numerically. First thedata were filtered in Fourier space by a low pass cut-off filterat |k|=512. Next the data were processed to identify allasperities and their relevant properties: in-plane coordinatex,y, peak height z and principal curvatures κ1,κ2. These datawere then used in the following section to obtain the realisticcontact morphology (real contact area) for different loads and

various indentation zones. Note that the roughness of the ballis not included here.

Fig. 7. Power spectral density of the surface roughness in the direction X(left) and Y (right): gray triangles represent all the measured surfaces, redcircles correspond to the average data.

Fig. 8. Distribution of surface heights: gray triangles represent all themeasured surfaces, red circles correspond to the average data. In the inset anexample of a high asperity; whose population changes significantly thedistribution tail.

VI. THREE-LEVEL MULTISCALE MODELA realistic estimation of the electrical contact resistance

needs to take in consideration the roughness of the contactingsurfaces. The approach in this work is based on theconsecutive use of three computational tools:

(1) a finite element analysis to solve the indentationproblem for smooth coated substrate within finite strainplasticity framework;

(2) a semi analytical iterative tool based on elasto-plasticdeformation of elastically interacting asperities;

(3) Greenwood's model [3] to estimate the electrical contactresistance through a localized cluster of a-spots:

(10)

where ai is the radius of the i-th contact spot and sij is thedistance between centers of spots i and j.

In model (1) (described in detail in Section IV) for eachvalue of the load F we obtain the contact radius a0 and thepressure distribution in the contact zone p(r), r

hysteresis, comes from a kinematic hardening in the brasssubstrate. To capture the non-linear material behavior, ourmechanical finite element analysis uses adequate materialmodels, which include kinematic and isotropic hardening forisotropic J2-plasticity [48]. In agreement with experiments, itenables us to distinguish between the first loading and thesubsequent cycles, however, the stabilized hysteresis, found inexperiments, was not reproduced in simulations for variousmaterial parameters. The electrical contact resistanceevaluated at macroscopic scale under assumption of perfectlyconducting interface does not display a stabilized cycle for therange of tested material and geometrical parameters (see Fig.5). Such a strong hysteresis should be associated withkinematic hardening of the brass at macroscopic scale and ofCuBe and Au at the scale of asperities. Frictional dissipationand adhesion may also contribute to this hysteresis. For thesubsequent studies, the frictional contribution has to beconsidered and all material models have to be more properlyadjusted especially in terms of the kinematic hardening.However, in the framework of the currently used asperitybased model, the cyclic elasto-plastic deformation at asperityscale cannot be properly taken into account, which is also thecase for most elasto-plastic models.

Fig. 10. a-spots in the multiscale model for different contact spot radius (a)a=10 μm, (b) a=16 μm, (c) a=22 μm.

The roughness of the studied surface (rolled brass coatedwith nickel and gold), while being frequently encountered inreal world applications, is not typical for theoretical models, inwhich roughness is often assumed to be fractal, Gaussian andisotropic. The surface in this work obeys none of theseassumptions and requires an accurate characterization andinterpretation. For example, the contact spots, aligned in therolling direction, form contact (conducting) bands separatedby non-contact regions (see Fig. 10).

Comparison between the three-level computational modelproposed in the paper and experiments shows that the modelcaptures quantitatively the evolution of the electrical contactresistance during the first loading. However, it cannot predictthe large variability of experimentally obtained results (this isprobably because of too different finishes of the ball used inexperiments). On the contrary, the model predicts thenarrowing of the resistance values dispersion with increasingload, while in the experiments the variability remains large forhigh loads (compare dots with shaded area in Fig. 9). Thereare several reasons for this discrepancy:

1. apart from the contact roughness, the contactresistance is also associated with possible presence of oxide

films or other contamination on contacting surfaces: theelectrical contact area is not equivalent to the mechanicalcontact area and can be significantly smaller in presence ofpoorly conducting spots [9];2. the roughness and the surface state of the CuBe ballwas neglected (probably it is the most criticalapproximation);3. the passage from the finite element macroscopicresult to the asperity based model requires a lot ofassumptions, which in realistic case are only partly satisfied(see Section IV);4. the asperity based model cannot take into account themerge of contact zones associated with different asperities[49] and the curvature variation of asperities;5. Hertz contact theory even being extended to elasto-plastic material behavior cannot take into account thelayered composition of the substrate and the finite strainhardening as well as it cannot capture a cyclic loading withkinematic hardenings.

To avoid the oversimplification of the elasto-plastic Hertzcontact used for asperities, one may use a properly tunedheuristic model based on a series of finite element simulationsof a single asperity on a layered substrate [26]. To overcomethe ensemble of the aforementioned difficulties of themultiscale model a full scale finite element [26] or boundaryelement model for elasto-plastic material [50] (or theircombination) with an accurate representation of the surfaceroughness should be used. This mechanical model should becomplemented with a subsequent full scale finite elementsimulation of the current flux through the contact interfacesimilar to what was done in [31]. However, to solve properlyboth the mechanical and electric problems at the roughnessscale, a significantly finer meshes would be needed, seediscussions in [22,51]. This task appears realistic in thepresent state of hardware and finite element software.

Note that such a weak electro-mechanical coupling ispossible only for low electric currents as the Joule heating inthe interface is negligibly small and thus does not affectmaterial properties of contacting elements. However, whenconsidering high electric currents a strongly coupled thermo-electro-mechanical model with roughness should beintroduced similar to [31].

Finally, regardless all inherent drawbacks, the three-levelmultiscale and multiphysics model for electrical contactsyields reasonable results in good agreement with experiments.The model uses the real roughness topography and has noadjustable parameters, only material mechanical and electricalproperties.

ACKNOWLEDGMENTThis work has benefited from the financial support of the

LabeX LaSIPS (ANR-10-LABX-0040-LaSIPS) managed bythe French National Research Agency under the"Investissements d'avenir" program (n°ANR-11-IDEX-0003-02).

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I. IntroductionII. MethodologyIII. Samples and Experimental Set-UpIV. Numerical simulations and material modelsV. Surface roughnessVI. Three-Level Multiscale modelVII. Conclusion

Three-level multi-scale modeling of electrical contacts · 2015. 7. 28. · area of contact,...

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