NANO-SCALE CONTACT OF ROUGH SURFACES CONSIDERING STRAIN GRADIENT PLASTICITY EFFECT S Mohamed Ali1, P Sahoo2, D Misra2 and K N Saha2* Department of Mechanical Engineering, Pondicherry Engineering College, Pondicherry– 605 014, India. 2 Department of Mechanical Engineering, Jadavpur University, Kolkata – 700 032, India. *Email: [email protected], Telefax: +91 33 2414 6890 1

Abstract Recent advances in nano- and micro-scale mechanics of materials have found that the material properties of some materials change significantly with scale. For example the flow stress or yield strength of material can increase dramatically with smaller length scale. These effects are due to scale-dependent material features such as grains. Experimental methods such as nano-indentation tests and small scale wire tests originally showed these effects and it is observed that indentation hardness of metallic materials typically increases by a factor of two or three as the indentation depth decreases into micrometers and even smaller. The phenomenological theory that describes this effect is known as strain gradient plasticity. In the present study, an analysis of small-scale contact of rough surfaces is considered incorporating the effect of small scale roughness and surface energy that becomes inevitable with decreasing length scale of systems as applicable in micro-/ nano-systems. The strain gradient plasticity effects are also considered in order to elucidate its effect on the contact behavior of small scale rough surfaces. The elastic-plastic adhesive contact analysis is also extended to consider a variable asperity contact radius to evaluate if the strain gradient model will affect it differently. The model produce predictions contact load and surface separation. It is observed that strain gradient effect significantly changes the load-separation behavior in comparison to the model without these effects. The strain gradient model seems to have greater influence on the predictions of contact load than does considering a variable asperity radius. Keywords: adhesion, strain gradient, scale effect

1.

Introduction Adhesion at the contact of rough solids has been studied analytically in great detail by

Chang et al. (1988) [CEB model] and Roy Chowdhury and Ghosh (1994) [RG model] using Greenwood and Williamson’s (1966) rough surface model (Fig. 1). The CEB model was based

on the DMT (Derjaguin et al., 1975) adhesion model while the RG model used the JKR (Johnson et al., 1971) adhesion model. In RG model, it is considered that some contacting asperities remain purely elastic and other purely plastic and the model overlook a wide intermediate range of interest where elastic-plastic contact prevails. In CEB model an attempt was made to bridge this gap. In this model each contacting asperity remains in elastic Hertzian contact until a critical interference is reached, above which volume conservation of the asperity tip is imposed and a uniform average contact pressure is assumed. It was indeed a very good approximation to solve the elastic-plastic contact problem some twenty years ago particularly due to the fact that finite element method was not well developed at that time. However, the simplifying assumption introduces a discontinuity in the contact load at the transition from elastic to elastic-plastic contact. This gave rise to several modifications to the original model, e.g., Evseev et al. (1991) and Zhao et al. (2000). Unfortunately, all these modifications are based on mathematical, rather than physical, considerations to smooth the discontinuity in the CEB model and do not provide a solution to the basic problem of lacking accuracy in the elastic-plastic contact regime. Such accurate solution calls for the use of a Finite Element Method (FEM). Kucharski et al. (1994) used FEM to solve elastic-plastic contact of a single asperity and provided empirical expressions for the contact load and area. Surprisingly, the mean contact pressure in some cases was higher than the indentation hardness and therefore unreasonable. Recently, Kogut and Etsion (2002) presented an accurate elastic-plastic finite element solution for the contact of a deformable sphere pressed by a rigid flat [KE model]. Their solution provides convenient dimensionless expressions for contact load and area covering a large range of interference from yielding inception to fully plastic contact of the sphere. The more accurate FEM solution (Kogut and Etsion, 2002) provides a solution to the basic problem in the intermediate elastic-plastic contact regime. More recently, Sellgren et al. (2003) have presented a finite element-based model of contact between rough surfaces taking into account the properties of engineering surfaces. Fleck and Hutchinson (1993) have found that the flow stress or yield strength of material can increase dramatically with smaller length scales. These effects are due to scale-dependent material features such as grains. Experimental methods such as nano-indentation tests and smallscale wire tests originally showed these effects and confirmed their existence. Polonsky and Keer (1996) studied the fundamental mechanics of contacting asperity considering scale effects by using a material dislocation-based method. They observe theoretically the same trend as Fleck and Hutchinson (1993). One phenomenological theory that describes this effect is known as strain gradient plasticity. Many subsequent works have studied strain gradient plasticity and derived models for this effect. Hutchinson (2000) describes the implication of these effects and

stresses how important it can be for these effects to be considered in modeling mechanics at the micro-scale. The measured indentation hardness of metallic materials typically increased by a factor of two or three as the indentation depth decreases into the micrometers (Nix, 1989; Oliver and Pharr, 1992; De Guzman et al., 1993; Stelmashenko et al., 1993; Suresh et al., 1999; Zagrebelny et al., 1999). The indentation size effect has also been studied using spherical indenters (Tymiak et al., 2001; Swadener et al., 2001, 2002) for which the indentation hardness depends not only on the indentation depth (or equivalently the contact radius) but also on the indenter radius. The size dependent material behavior has been attributed to geometrically necessary dislocations associated with non-uniform plastic deformation in small volumes (Nye, 1953; Ashby, 1970; Gao and Huang, 2003). Discrete dislocation simulations (Cleveringa et al., 1997; 1998; 1999a; 1999b) have also shown the size-dependent behavior of metallic materials at the micron scale. The indentation size effect in spherical indentation experiments has been studied recently by Qu et al. (2006) via the conventional theory of mechanism-based strain gradient plasticity (CMCG) established from the Taylor dislocation model (Taylor, 1934,1938). A simple, analytic indentation model was given by Qu et al. (2006), which relates bulk hardness to indentation hardness considering strain gradient effects in terms of the radius of the spherical indenter. Moreover asperities of contacting rough surfaces vary in size and shape in reality and span the size range of strain gradient plasticity effects. The present study considers an analysis of smallscale contact of rough surfaces in presence of adhesion considering the strain gradient plasticity effects and a variable asperity radius.

2.

Loading Analysis Following the works of Mukherjee et al. (2004) [EP model], which is an extension of RG

model and also hardness H and asperity radius R are considered as constant, the load expression for EP model may be expressed in terms of the following governing parameters, viz., elastic adhesion index θ and plasticity index ψ . Thus the non-dimensional load for EP model may be written as P EP = ∫

∆c 1

∆0

1.03  3 / 2 4.34 3 / 4  ∆c 2 1.425 −  3 / 2 4.34 3 / 4  −  ∆ − 1 / 2 ∆  φ ( ∆ )d∆ + 1.425  ∆c 1 − 1 / 2 ∆c1  ∫∆ ∆ φ ( ∆ )d∆ θ θ ∆c1     c1

− ∞  4.75 1.40  3 / 2 4.34 3 / 4  ∆c 3 6.28  −   φ ( ∆ )d∆ + 1.263  ∆c 1 − 1 / 2 ∆c 1  ∫ ∆1.263 φ ( ∆ )d∆ + ∫ ∆− 110 ∆c θ θ  ∆c1   ∆c 2  ψ

(1)

where

P EP =

P KNR σ 1/ 2

3/ 2

,∆ =

δ , σ

∆c =

δc , σ

∆c1 =

δ c1 , σ

∆c 2 =

δ c2 δ d , ∆c 3 = c 3 , h = , σ σ σ

−

ψ = (E * / H ) σ / R , θ = Kσ 3 / 2 R 1 / 2 /( γR ) , and φ ( ∆ ) is the normalised asperity height distribution function. First and last part of the equation (1) shows the non-dimensional load of elastic asperities and plastically deformed asperities and second and third parts correspond to elastic-plastic regime of deformation. Here δ c is the critical deformation of an asperity at yielding inception, i.e., for the transition from elastic to elastic-plastic deformation that takes place when mean contact pressure exceeds 0.6H (Tabor, 1951). For δ < δ c and for δ > 110δ c the contact is perfectly elastic and perfectly plastic, respectively. δ c1 is the apparent critical deformation corresponding to real critical displacement δ c while δ c 2 and δ c 3 are the apparent displacements corresponding to real displacements 6δ c and 110δ c respectively (Kogut and Etsion, 2002). ∆0 is the non-dimensional apparent displacement corresponding to actual displacement δ = 0 . Considering the effect of changes in the hardness of the material (Qu et al., 2006) during loading, the bulk hardness (H) of the deforming body may be related to indentation hardness ( H ' ) as H' = H . S

(2)

where, S is hardness scale factor given by (Qu et al., 2006)  141α' 2 G 2 b  S = 1 + H 2 R  

1/ 2

(3)

where, α ' the empirical coefficient around 1/3 in the Taylor dislocation model (Taylor, 1934), G is the shear modulus, b is the magnitude of the Burgers vector and R is the average radius of the spherical tip of the asperity. Using the isotropy relation G = E / [2( 1 + υ )] and plasticity index ψ = (E * / H ) σ / R , where E* is composite elastic modulus given by E* = E /( 1 − υ 2 ) for the present case of contact of a rough deformable surface with a rigid flat surface, the hardness scale factor may be rewritten as  ( 1 − υ )2 2  b   S = 1 + 141α' 2 ψ   4  σ  

1/ 2

(4)

Then the condition for plasticity that yields the non-dimensional critical value of apparent asperity displacement indicating the transition from elastic to elastic-plastic deformation for the case of varying hardness may be expressed (Mukherjee et al.,2004) for the present case as

∆c13 / 4 −

1.42

ψ

S ∆c1

1/ 4

−

4.34

θ 1/ 2

≥0

(5)

Now the asperity variation with height needs to be considered. The simplified model considering changes in the radius of curvature of asperities used here is similar to the work by Adams and Muftu (2005) and Jackson (2006). The current work assumes that the effective asperity radius R’ can be described as a function of the height z by R' = R( 1 − Dz )

(6)

where R is the average radius of curvature of the asperities and D is a constant describing how they vary. R is the average asperity radius since the Gaussian distribution is symmetrical about the mean height, z = 0. If D is greater than zero then the higher asperities are sharper or have a smaller radius of curvature than the shorter ones. If D is less than zero, then the higher asperities are rounder and have a larger radius of curvature. If D is equal to zero then the asperity radius does not vary. If a desired asperity radius R1 , is desired at a certain height, z0 , the following formula can be used D=

1 R  1 − 1  z0  R

(7)

Fig. 1 Contact between a rough deformable surface and a rigid flat plane

For the current work, z0 is set to 10σ . At this value there are very few asperities and if the surfaces are separated by this distance very few asperities will be in contact. To incorporate the

varying asperity radius in the statistical model, equation (6) can be written in non-dimensional form as R' = RFR , where, FR may be termed as radius scale factor and is given as   1  R  FR = 1 −  1 − 1 (h + ∆ ) R  10   where h = d / σ

and

(8)

∆ = δ / σ , d and δ being mean separation and asperity deformation

respectively. Incorporating the radius variation in equation (5) yields

∆c1

3/ 4

1.42

−

ψ

FR

1/ 2

 141α' 2 ( 1 − υ )2 ψ 2 1 + 4 FR 

 b     σ 

1/ 2

∆c11 / 4 −

4.34

θ

1/ 2

FR

1/ 4

≥0

(9)

If there is no radius variation, i.e., for FR = 1 , equation (9) will be reduced to equation (5). The relation between the real critical displacement ( ∆c ) and apparent critical displacement ( ∆c1 ) (Mukerjee et al., 2004) , for the present case may be expressed as

∆c = ∆c1 −

2.89 FR

θ

1/ 4

1/ 2

∆c11 / 4

(10)

(

)

Similarly ∆0 may be written as ∆0 = 4.125 / θ 2 / 3 FR

1/ 3

. The total non-dimensional load for the

present case considering the effect of scale-dependent hardness and height dependent asperity radius as  3 / 2 4.34 1 / 4 3 / 4  1 / 2 −  ∆ − 1 / 2 FR ∆  FR φ ( ∆ )d∆ ∆0 θ   − 1.03  3 / 2 4.34 1/ 4 3 / 4  ∆c 2 1/ 2 + 1.425  ∆c 1 − 1 / 2 FR1 ∆c 1  ∫ ∆1.425 FR φ ( ∆ )d∆ θ ∆c1   ∆c 1

Pl = ∫

∆c 1

− 1.40  3 / 2 4.34 1/ 4 3 / 4  ∆c 3 1/ 2 + 1.263  ∆c 1 − 1 / 2 FR1 ∆c 1  ∫ ∆1.263 FR φ ( ∆ )d∆ θ ∆c1   ∆c 2

(11)

 4.71 1 / 2 6.28 1 / 2  1 / 2 −  S FR ∆ − FR  FR φ ( ∆ )d∆ 110 ∆c θ   ψ

+∫

∞

where P l = FR 1 = 1 −

P

KNR

l 1/ 2

σ

3/ 2

, ∆=

δ δ δ δ δ d , ∆c = c , ∆c 1 = c1 , ∆c 2 = c 2 , ∆c 3 = c 3 , h = σ σ σ σ σ σ

,

− 1  R   1 − 1 (h + ∆c1 ) and φ ( ∆ ) is the normalised asperity height distribution function. 10  R

The ∆c 2 and ∆c 3 to be used in equation (11) may be obtained from equation (10) by replacing ∆c

−

by 6 ∆c with ∆c 1 = ∆c 2 and ∆c by 110 ∆c with ∆c1 = ∆c 3 respectively. For Gaussian surfaces φ ( ∆ ) −

(

)

2

is written as φ ( ∆ ) = 1 / 2π e −( h+∆ )

3.

/2

.

Results and Discussion Equations established in the previous section are evaluated numerically. The load-

separation behaviour is investigated for typical combinations of θ and ψ . In the present analysis, typical values of θ between 2 and 30 are considered in order to analyse the effect of adhesion. The value of the plasticity index, ψ determines the nature of contact. For ψ < 0.6 , the contact is predominantly elastic and for ψ > 1 , the contact is predominantly plastic. Thus ψ values are considered in the range of 0.5 to 2.5 in order to consider the whole range of deformation from predominantly elastic to predominantly plastic including elastic-plastic. The non-dimensional mean separation ( h = d / σ ) is considered between +1 to -1. At larger separations, h > 1 the number of asperities in contact is very small, as it can be evidenced from the magnitude of external load. To study the effect of radius variation the ratio Rr ( Rr = R1 / R ), i.e., desired asperity radius to average asperity radius ratio is taken as 0.1 and 5 in this model. First the asperity radius of curvature is held constant and the model considering strain gradient effects is compared to the model without these effects [EP model]. Fig. 2 shows the plots of dimensionless applied load against dimensionless mean separation for different values of ψ (ψ = 0.5, 0.8 and 2.5) and for varying elastic adhesion index θ ( θ = 5, 10 and 15). Fig. 2(a) is for ψ = 0.5 which corresponds to predominantly elastic contact, Fig. 2(b) is for ψ = 0.8 which corresponds to elastic-plastic contact and Fig. 2(c) is for ψ = 2.5 which corresponds to predominantly plastic contact. The general trend of variation is that at low θ (high adhesion), applied load is negative (tensile) for all ψ . For higher θ (low/no adhesion cases) applied load is positive (compressive). This trend remains the same for both with and without scale effect. For elastic (ψ =0.5) and elastic-plastic (ψ =0.8) cases there is no significant difference between loadseparation plots with scale effect and without scale effects. It is due to the fact that in the present model of scale effects, strain gradient effect is considered in terms of the variation of hardness. Hardness plays an insignificant role in elastic and elastic-plastic regime. This leads to the same load-separation plots for with and without scale effect in elastic and elastic-plastic regime. However, in predominantly plastic contact situation (ψ = 2.5) as seen in Fig. 2(c), the scale effect model yields higher load than the same without scale effect.

(a)

(b)

(c)

Fig. 2 Plots of non-dimensional applied load against mean separation for varying θ with and without scale effect for different ψ

However, the difference is not so clearly visible in Fig. 2(c) as the relative magnitude of difference in load from the two models is small. This can be seen from the separate plot of the difference in non-dimensional load from the two models (with scale effect and without scale effect) against mean separation shown in Fig. 3. With scale effect, the effective hardness increases leading to a higher load capacity at a particular mean separation compared to the same without scale effects in the predominantly plastic contact situations. It can also be seen from Fig. 3 that with increase in θ (i.e., for low/no adhesion contacts), the difference in load is higher and at low θ (i.e., highly adhesive contacts) this difference in load is insignificant. Thus consideration of scale effects is a must for non-adhesive contacts and predominantly plastic

contacts. However, for elastic/elastic-plastic contacts and for adhesive contacts, scale effect may be neglected.

Fig. 3 Difference in non-dimensional load against mean separation (Scale effect load – No scale effect load) for varying θ and at ψ = 2.5.

Next the effect of varying asperity radii of curvature is considered where strain gradient scale effect is not included, i.e., hardness is considered to be constant. Fig. 4 shows the plots of non-dimensional applied load against mean separation for typical combinations θ ( θ = 5 and 15) and ψ (ψ = 0.5, 0.8 and 2.5) with varying values of asperity radius ratio, Rr ( Rr = R1 / R ) . Three different values of Rr are considered here: Rr = 0.1, 1 and 5. Rr = 0.1 means R1 is set to 0.1R. This value models a surface whose taller asperity peaks have a smaller radius of curvature than the lower peaks. Rr = 1 means no radius asperity variation with height of asperity. Rr =5 represents a surface whose taller asperity peaks have a larger radius of curvature than the lower peaks.

It can be seen from these plots that the effect of varying asperity radius is more

prominent for low θ , i.e., high adhesive contact situations while at high θ , the effect is relatively less. At low θ ( θ = 5 ) it is seen that at a particular mean separation, the applied load for Rr = 5 is maximum while the same at Rr = 0.1 is the minimum. This trend remains the same for all ψ values at low θ case. Since Rr = 0.1 represents a surface whose taller asperity peaks have a smaller radius of curvature and this contributes to high adhesion leading.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4 Plots of non- dimensional load against mean separation for asperity radius variation at different combinations θ and ψ .

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 5 Plots of non- dimensional load against mean separation for scale effect and asperity radius variation at different combinations θ and ψ .

to a smaller applied load. For Rr = 5 the reverse is true leading to a higher applied load. At high

θ ( θ = 15 ) as can be seen in Fig. 4(b), (d) and (f) the effect of varying asperity radius is not uniform. It depends on nature of contact, i.e., on ψ . For elastic (ψ = 0.5 ) and plastic (ψ = 2.5 ) contact situations Rr = 5 yields higher load at a particular mean separation compared to Rr = 0.1. However, at ψ = 0.8 (elastic-plastic) the reverse trend is observed. Next the strain gradient effects and asperity radius variation effects are considered simultaneously. Fig. 5 shows the plots of non-dimensional applied load against non-dimensional mean separation for typical combinations of θ ( θ = 5 and 15) and ψ (ψ = 0.5, 0.8 and 2.5) with simultaneous considerations of scale effect and varying asperity radius. It may be seen from figures that load-separation plots for no scale effect with no radius variation are almost coincident with plots for scale effect with no radius variation at all combinations of θ and ψ . It indicates that the effect is less prominent compared to asperity radius variation effects. With varying asperity radius, the plots significantly differ from each other showing that asperity radius variation alters the load capacity to a significant level particularly at high adhesive contact (low θ ) and at low mean separation.

4.

Conclusions An analysis of adhesive contact is presented taking into account two new considerations; a

varying hardness effect due to scaling effects and a variable asperity radius of curvature. The strain gradient model accounts for changes in hardness due to scaling effects. The variable asperity radius of curvature is due to varying asperity height. The scale effects are considered for normal adhesive contact between two rough surfaces with small-scale asperities. The strain gradient scale effects affect adhesive contact insignificantly while the asperity radius variation alters the load capacity to a significant degree particularly at highly adhesive contact situations

5.

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