a r X i v :0706.2762v 1 [c o n d -m a t .s o f t ] 19 J u n 2007
Contact mechanics:relation between interfacial paration and load
B.N.J.Persson
IFF,FZ-J¨u lich,52425J¨u lich,Germany
I study the contact between a rigid solid with a randomly rough surface and an elastic block with
a flat surface.I derive a relation between the (average)interfacial paration u and the applied normal squeezing pressure p .I show that for non-adhesive interaction and small applied pressure,p ∼exp(−u/u 0),in good agreement with recent experimental obrvation.
When two elastic solids with rough surfaces are squeezed together,the solids will in general not make con-tact everywhere in the apparent contact area,but only at a distribution of asperity contact spots[1,2,3,4].The paration u (x )between the surfaces will vary in a nearly random way with the lateral coordinates x =(x,y )in the apparent contact area.When the applied squeez-ing pressure increas,the average surface paration u = u (x ) will decrea,but in most situations it is not possible to squeeze the solids into perfect contact corresponding to u =0.The space between two sol
ids has a tremendous influence on many important ,heat transfer[5],contact resistivity[6],lubrication[7],aling[8],optical interference[9],....In this paper I will prent a very simple theory for the (average)para-tion u as a function of the squeezing pressure p .I will show that for randomly rough surfaces at low squeezing pressures p ∼exp(−u/u 0)where the reference length u 0depends on the nature of the surface roughness but is in-dependent of p ,in good agreement with experiments[9].We consider the frictionless contact between elastic solids with randomly rough surfaces.If z =h 1(x )and h 2(x )describe the surface profiles,E 1and E 2are the Young’s elastic moduli of the two solids and ν1and ν2
p
E 2
.
Introduce a coordinate system xyz with the xy -plane in the average surface plane of the rough substrate,and the z -axis pointing away from the substrate,e Fig.1.The paration between the average surface plane of the block and the average surface plane of the substrate is de-noted by u with u ≥0.When the applied squeezing force p increas,the paration between the surfaces at the in-terface will decrea,and we can consider p =p (u )as a function of u .The elastic energy U el (u )stored in the sub-strate asperity–elastic block contact regions must equal to the work done by the external pressure p in displacing the lower surface of the block towards the ,
∞
u
du ′A 0p (u ′)=U el (u )(1)
or
p (u )=−
1du
,
(2)
where A 0is the nominal contact area.Equation (2)is
exact.Theory shows that for low squeezing pressure,the area of real contact A varies linearly with the squeezing force pA 0,and that the interfacial stress distribution,and the size-distribution of contact spots,are independent of the squeezing pressure[11,12].That is,with increasing p existing contact areas grow and new contact areas form in such a way that in the thermodynamic limit (infinite-sized system)the quantities referred to above remain un-changed.It follows immediately that for small load the elastic energy stored in the asperity contact region will increa linearly with the load ,i.e.,U el (u )=u 0A 0p (u ),where u 0is a characteristic length which depends on the surface roughness (e below)but is independent of the squeezing pressure p .Thus,for small pressures (2)takes the form
p (u )=−u 0
dp
圆柱体的表面积怎么算
FIG.2:The parametersαandβas a function of the Hurst exponent H for three different values of the ratio q1/q0.
or
p(u)∼e−u/u0(3) in good agreement with experimental data for the contact between elastic solids when the adhesional interaction be-tween the solids can be neglected[9].We note that the result(3)differs drastically from the prediction of the Bush et al theory[13],and the theory of Greenwood and Williamson(GW)[14],which for low squeezing pressures (for randomly rough surfaces with Gaussian height dis-tribution)predict p(u)∼u−a exp(−bu2),where a=1in the Bush et al theory and a=5/2in the GW theory. Thus the theories do not correctly describe the interfa-cial spacing between contacting solids.
The elastic energy U el has been studied in Ref.[15] and[12],and in the simplest approximation it takes the form
U el≈A0E∗π
(2π)2 d2x h(x h(0) e−i q·x,
where .. stands for enmble average.The parameter γ=1in the simplest ca but in general one expectγ<1 (but of order unity)to take into account that the elastic energy stored in the contact region(per unit surface area) is less than the average elastic energy(per unit surface area)for perfect contact,e Ref.[12].We will u the contact mechanics theory of Persson,where for elastic non-adhesive contact the function[17,18]
P(q)=
2
∂u
=
2
du
e−s2p2(7) Substituting(4)and(7)in(2)gives
刻板p(u)=−
√πγ q1q0dq q2C(q)w(q)e−[w(q)p/E∗]2dp
p Integrating this from u=0(complete contact,corre-sponding to p=∞)to u gives
u=√πγ q1q0dq q2C(q)w(q) ∞p dp′1
with
log w =
q1q0dq q2C(q)w(q)log w(q)
π h2
观察日记绿豆q0
−2(H+1)
(13)
where H=3−D f,where D f is the fractal dimension. The mean of the square of the substrate surface height profile is h2 =h2rms.Substituting(13)in(10)gives
u0=h rms/α(14) where
α−1= 2H(1−H)
2(1−H) x2(1−H)−1
be U el+U ad.The theory can also be applied to study how the spacing u(ζ)depends on the magnification.Here u(ζ)is the(average)spacing between the solids in the apparent contact areas obrved at the magnificationζ. The quantity u(ζ)is of crucial importance for lubricated als[22].The results of the generalizations of the the-ory will be prented elwhere.
Finally we note that the obrvation of an effective exponential repulsion has important implications for tri-bology,colloid science,powder technology,and materials science[9].For example,the density or volume of gran-ular materials has long been known to have a logarith-mic dependence on the externally applied isotropic pres-sure or stress,as found,for example,in the compression stage during processing of ceramic materials[23].Recent work on the confinement of nanoparticles have also in-dicated an exponential force upon compression[24],sug-gesting that this relationship could be prevalent among quite different types of heterogeneous surfaces.
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