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Finite element analysis of an elastic-plastic two-layer half-space: sliding contact
Wear, 148 (1991) 261-285
261
Finite element analysis of an elastic-plastic half-space: sliding contact
two-layer
Hong Tian* and Nannaji Sakat DePatiment of Mechanical Engineering, Massachusetts Institite of Technoiogy, Cambridge, MA &?I39 (u&4.) (Received
July 24, 1990; revised X3ecember 13, 1990, accepted
February
26, 1991)
Abstract A two-dimensional
finite element stress and strain analysis of the sliding contact of a twolayer elastic-plastic half-space for various friction coefficients was conducted. First the contact pressure of the two-layer half-space under normal indentation was determined. Then the normal and tangential loadings were applied proportionally and incrementally on the surface. It was found that surface deformation, location of initial yielding, stresses and strains along the interfaces between layers strongly depend on the friction coefficient. When the friction coefficient is small (tess than 0.3), yielding initiates in the subsurface region near the leading edge of the contact, provided that the normal load is large enough. When the friction coefficient is large (greater than 0.3), yielding initiates on the surface at the trailing edge of the contact. The surface strains, especially the shear strains, are large owing to the unconstrained deformation. The magnitudes of shear stresses and strains along both interfaces are signi~cant~y large for high friction. On the basis of the analysis, the implications for interface failures are qualitatively addressed.
1. Introduction Thin films are widely used in modern mechanical, electronic, magnetic, and optical devices. In electrical connectors, in particular, thin composite layers of gold, nickel, and copper or copper alloys are widely used to meet multi~nctionai requirements [I]. A minimum normal force is also required to maintain low electrical contact resistance and low noise; but the high friction forces’ generated by high friction coefficients during frequent insertion and withdrawal cause contact failures. To better underst~d the mechanisms of the contact failures of thin film systems, it is necessary to have a comprehensive knowledge of the sliding contact stress and strain fields of the thin film systems. Contact stress analyses of homogeneous media have been made for many decades. Johnson [2] has recently reviewed the mechanics of nonconforming contacts. The hvodimensional quasistatic frictional or sliding contact of a homogeneous elastic halfspace was investigated by M’Ewan [3], Poritsky [4], and Smith and Liu [S]. It has been found that the effect of tangential force is to bring the point of maximum shear stress closer to the surface. Johnson and Jefferis [6] used both the Mises and Tresca *Present Address: Hoya Electronics ~~oratjon, U.S.A. tAutbor
to whom correspondence
0043-1648/P1/$3.50
960 Rincon Circle, San Jose, CA 95131,
should be addressed.
0 1991 -
Elsevier Sequoia,
Lausanne
262
yield criteria to determine the location of initial yield in a sliding contact. They found that yielding initiates in the subsurface region when the friction coefficient y is small (,u
2. Statement
of the problem
In the normal contact of two non-conforming bodies, friction at the contact interface affects the stress field only if the elastic constants of the two materials are different 1133. Under combined normal and tangential loading, the friction at the contact interface has different effects on the stress field, depending on whether the contact is static or sliding. In a static contact, Mindlin 1141 showed that, under a constant normal load N even the smallest tangential force F will result in some siip at the contact edge.
263
The annular slip zone spreads into the contact region as F increases until F = ,FrN, when gross slip occurs. Owing to the “irreversibility” implied by the frictional contact, the final contact stress field depends not only on the final values of the normal and tangential forces, but also on the history of loading. If the normal and tangential loads are applied simultaneously and proportionally, i.e. by oblique loading, the slip zone does not exist when the angle of the oblique loading p (p= tan-’ (FIN)) is less than the angle of friction (tan-’ EL) [15, 161. In sliding contacts, it is usually assumed that the contact pressure distribution is independent of the tangential loading, and that Amonton’s Law of sliding friction holds both macroscopically and microscopically, i.e. F=&&N
(1)
and 4(x) = &V(x)
(2)
where q(x) andp(x) are the tangential and normal surface traction distributions within the contact region, respectively. In the elastic analysis of homogeneous half-spaces, the normal pressure is usually specified by the Hertzian pressure and the tangential traction by eqn. (2) on the contact surface [4-6]. However, in the case of layered media [17] or when plastic deformation is involved 1181, the contact pressure distribution deviates from the Hertzian distribution. In a finite element analysis of the normal indentation of a two-layer elastic-plastic half-space by the present authors [12], it was shown that the contact pressure no longer obeys the Hertzian pressure distribution and that the deviation from the Hertzian distribution depends on the layer thickness and the extent of pfastic deformation. In sliding contact, the tangential loading has an insignificant effect on the contact width and the contact pressure distribution, especially if y < 1 1191. Thus for the analysis of sliding contacts of layered elastic-plastic media, a two-step procedure can be used. First the contact pressure distribution under normal indentation is determined, and then the tangential loading is specified by eqn. (2). Since the final contact stress field depends on the history of loading, only the simultaneous and proportional loading scheme of normal and tangential forces is considered in the present analysis. Consider a rigid cylinder of radius R indenting a two-layer half-space under a normal force per unit length L and a tangential force per unit length T. Figure 1 shows the coordinate system and the dimensions of the layers. The contact edge (~=a) is defined as the leading edge, and x= -a as the trailing edge of the contact. The thicknesses of the top layer (layer 1) and the interlayer (layer 2) are h1 and hz, respectively. Perfect bonding is assumed at the interfaces between layers, i.e. displacements across the interfaces are continuous u(‘)(x, h,) =24(2)(x7 h,)
@a)
7Jqq h,) = ?J@)(x,h,)
(3b)
d2)(x, hl + h2) = d3)(x 7hl + h,)
(44
TJyq hlfh,)=?P(x, hl+h2)
(4b)
U-V=0
asx,y-
00
(5)
where the superscripts l-3 stand for the top layer, the interlayer, and the substrate, respectively. A two-step procedure is followed to obtain the stress and strain fields.
264
Fig. 1. Schematic of sliding contact of a two-layer half space. TABLE 1 Mechanical properties of contact materials Material
E (GPa)
I,
oY (MPa)
a
Top layer (Au 4 O.S%Ni) Interlayer (Ni) Substrate (brass: Cu-35%Zn)
79 205 125
0.32 0.32 0.32
298 811 405
0.1 0.1 0.1
First the contact pressure of frictionless indentation of the two-layer determined. The boundary conditions for the first step are as follows. On the contact surface
7&,
0) = 0
half-space
is
lulaa
W
Pl>a
(6b)
--<
(64
The penetration depth 6 is assumed to be smaller than the radius of the indenter R or the half contact width u. After the contact pressure p(x) is obtained from the first step, the normal and tangential loadings are applied proportionaily. The boundary conditions for the second step are as follows (7a)
(‘W a,(x,
0) = 0
T&x, 0) = 0
Ixl>n
(7c)
kl>a
(7d)
The primary emphasis of this study is on the siiding contact of multi-layer systems employed in electrical contacts. That is, the top layer is gold, which is hardened with
26.5
0.5% nickel, the interlayer is nickel, and the substrate is copper or All materials are assumed to possess a bilinear elastic-plastic behavior modulus Et, is equal to a constant cy times the elastic modulus E). The and plastic properties of the layers and the substrate are listed in values fall in the typical range of electrical contact materials.
a copper (i.e. the relevant Table 1.
alloy. plastic elastic These
3. The finite element model The contact pressure distribution and contact width of a two-layer half-space under normal indentation have been obtained in a previous work [12]. The results are directly used in the present analysis. Since the combined normal and tangential loading is no longer symmetric, the whole layered half-space has to be modeled. Figure 2(a) shows the finite element mesh and the boundary conditions, and Fig. 2(b) shows the details of the mesh near the contact. It was assumed that the bottom plane CD is far away from the contact center, and that the vertical displacements are equal to zero. The horizontal displacements in the bottom plane (CD) were not restricted, except that of point C which stays put. The normal and tangential loads were applied on the top plane AB within the contact region. The horizontal and vertical dimensions of the mesh are 200 pmX200 pm, both being large enough to allow the stresses to be insignificant at the boundaries. The whole mesh is an arrangement of 630 quadrilateral, eight-node, isoparametric elements, and the total number of nodes is 2023. The mesh was refined in the region below the contact. The multiple node constraints in the multi-purpose finite element program (ABAQUS) were used to connect the elements of different sizes for mesh refinement. The horizontal and vertical spacing of nodes on and below the contact surfaces are 0.5 Ir;m and 0.3125 pm, respectively. The twolayer half-space was modeled by assigning different elastic and pfastic properties to the elements of the top rows of the mesh. The first two rows were assigned to the
Fig. 2. (a) The finite element mesh.
mesh and the boundary
conditions,
and (b) the detail of the fine
top layer, and the next two rvws immediately below the top layer were assigned to the interlayer. The interfaces between the layers were assumed to be perfectly bonded, i.e. the displacements at the interfaces were continuous (eqns, (3)-(S)). These interface constraints were satisfied by using common nades at the interfaces, and the comman nodes belong to the elements on both sides of the interface. The normal and tangential loads were applied node by node with a consistent loading scheme within the contact region, and they were in one executing step of the ABAQUS code so that the incremental and proportional loading could be undertaken. The parameter settings of the executing programs were detailed in the previous work [12]. The cumputat~o~s were performed on a Micro VAX If computer, and the typical CPU time for one run was about 90 min. To verify whether the boundary condjt~o~s and the finite eIement mesh in Fig. 2 are fair rep~ese~~at~ons of a sern~-~~~~~te s&id, ikst the sGding contact of a ~ornoge~~us eIastic ha&space was analyzed. The twv layers and the substrate were assigned the same Young’s modulus and Poisson’s ratio. An eltipticaf pressure distribution (Hertz pressure distribution) was applied on the surface along with a proportional distributed shear traction. The differences between the calculated stresses and those of the analytical solutions presented by Poritsky [4], and by Smith and Ciu [S], were negligible. Thus the present finite element model and mesh are assumed to be an acceptable representation of an elastic half-space in sliding contact.
The results presented here are for a rigid cylinder of radius R (CL75 mm) which is much larger than the expected haff contact width CI. The normal bads were 5 N mm-’ and IO N mm-‘. It was shown in ref. 12 that the 5 N mm-’ normal bad is loading. an elastic loading, while the 10 N mm-’ normal Xoad is an elastic-plastic The ratio of the tangential force to the normal force, ix. the friction coefficient, was from 0 to 0.5. (Tha finite element of analysis for a larger friction coefficient (pm 1.0) was tried, but no converged solution was obtained in the allowed number of increments. The strain increment was larger than fifty times the yield strain owing to relatively small work hardening of the contact materials considered in the present study. Thus it may not be possible TV obtain converged solutions in a reasonably short period of time.) The thicknesses of the top layer and the interkyer were chosen as I,25 pm and 2.5 I*_rn.The ~lc~lated stresses are normalized by the yield strengths of the top or rY.top, and of the substrate, @lf,& or ry,L,subrdepending on the approfayer, ~~~~~~ priateness. The positive values indicate tensile stresses and strains, and the negative values compressive: stresses and strains. 4.1. Sur$~e defmrnation and penetration depth To qualitatively illustrate the effect of friction on deformation in the layers and the substrate, especially in the top layer, the deformed meshes are shown in Fig. 3 in for the 5 N mm-’ normal load at various friction coefficients. The displacements the deformed mesh are magnified 20 times for clarityy. The top two rows of elements are the top layer, the next two rows are the interiayer, and the rest of the elements are the substrate, For normal indentation (p==(I) displacements of the layers and the substrate are, as expected, symmetric about the y-axis. When the friction coefkient is greater than zero, however, the symmetry is Iost and the materials deform large& along the direction of the ta~~e~t~a~ force. fn the finite element ana@& of an
267
mp
layer
rnterlaye Substrate
(21)
Fig. 3. Deformed meshes under the S N mm-’ normal bad at various frj~t~on coefficients. (a) L=5 N mm-’ , s=O, (b) L-S N mm-‘, ~~.1-0.1;fc} t-5 N mm”, &=0.3; (d) L-5 N mm-‘, p = 0.5. Original magnification factor = 20. elastic-plastic half-space by Tangena and Hurkx @], a hump was found at the leading edge of the sliding contact as the friction coefficient increases. In the present case, a two-layer half-space, the displacements in the top layer are cIeariy much larger than those of the interlayer and the substrate, but no hump is found at the leading edge of the contact. Figure 4 shows the deformed meshes for the 10 N mm-’ normal load at various friction coefficients. As expected, the contact width and displacements are greater than in the ease of the S N mm”’ normal load. As the friction coefikient increases, the meshes of the top layer are deformed aloo@ the sliding direction. At p=O.S, a small hump is observed at the leading edge of the contact. Table 2 iists the penetration depth uf the two-layer half-space under combined normal and tangential loadings. The penetration depth increases with the normal load as well as the friction coefficient. Thus the tangential force contributes ncn only to the horizontal displacement, but also to the vertical indentation. The same observation
268
was added by Johnson &33] for a ~u~~~~~~~~s &a&c ~~~f-~~a~e under normaI and tangential loadings. The penetration depth 6, however, is stiII small compared with tho thickness of the top layer hI. The ratio S/h1 in Table 2 shows that the penetration depth is about 10% of the thickness of the top layer under the 5 N mm-’ normal load, and less than 20% under the 10 N mm-’ normal foad, for the whole range of friction coe~c~e~fs co~s~d~red hereTo i~~ustrate the effect of the fr~ct~o~ coefikient OR the surface ~efo~at~o~~ the strains on the contact surface are plotted in Fig. 5. The horizontal strain c-~ under 5 N mm-’ and 10 N mm-’ normal loads with various friction coefficients are plotted in Figs. s(a) and S(b), respcctiveiy. As the friction coe~cie~t increases, the materiat
269
TABLE
2
width and penetration tangential loading
Contact
Normal load, L (N mm)-’ 5
10
depth of a two-layer half-space under combined normal and
Friction coefficient
Half-width of contact a (pm)
Penetration 6 (pm)
depth,
6
0.0
5.S
0.1 0.3 0.5
5.S S.S 5.S
0.106 0.107 0.109 0.113
0.0848 0.0856 0.0872 0.0904
0.0 0.1 0.3 0.5
8.5 8.S 8.5 8.5
0.197 0.199 0.207 0.216
0.1576 0.1592 0.1656 0.1728
7;;
in the trailing side of the contact (x/u ~0) is stretched horizontally (e=>O), and the material in the leading side (X/U>O) is squeezed (e,OS (Figs. 5(c) and S(d)). The surface shear strains are plotted in Figs. 5(e) and 5(f). For p=O, the contact surface is free from shear traction. Thus the shear strain is zero everywhere on the surface. When p>O, the surface shear strain is generated within the contact region I$x,+z/~ 1). The direction of the shear strain is along the sliding direction, and its magnitude increases significantly with the friction coefficient. At y=O.S, the maximum shear strain on the surface is about 1.5% for the 5 N mm-’ normal load, and 8.5% for the 10 N mm-’ normal load. 4.2. Location of initial yielding It is well known that yielding occurs when the Mises equivalent stress flM&s is equal to the yield strength in simple tension WY.In plane strain, oh((iscsis given by
In the normal contact, i.e. without friction, of a homogeneous elastic half-space, yielding initiates in the subsurface region at roughly~=O.7~ below the contact center. Under combined normal and tangential loadings, the location of maximum Mises equivalent stress or maximum shear stress is brought closer to the surface, or even to the surface when the friction coefficient is large (about 0.25 for the Tresca yielding criterion, and 0.3 for the Mises criterion) [3-6]. King and O’Sullivan (lo] show that this conclusion is also true when a stiff or compliant layer bonded on an elastic halfspace is under normal and tangential loading. For sliding contact of a two-layer halfspace comprising materials of different yield strength, however, the location of the maximum Mises equivalent stress is not necessarily the location of the initial yielding. Figure 6 shows the Mises equivalent stress and the equivalent total plastic strain contours under the 5 N mm-’ normal load for various friction coefficients. Only the stress and plastic strain contours in the region near the contact are plotted for clarity. The horizontal dashed lines in the plots represent the approximate locations of the interfaces between layers. For normal indentation (Fig. 6(a), p==O), the top layer, the
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Pig. 6. The Mises equivalent stress and equivalent total plastic strain contours for the 5 N mm-’ normal load. The value of the Mises stress contour numbers: 1 - 50 MPa; 2 - 100 MPa; 3 - 150 MPa; 4 - 2fJo MPa; 5 - 250 MPa; 6 - 300 MPa; 7 - 350 MPa; 8 - 400 MPa; 9 -450 MPa; 10 - 500 MPa; 11 - 550 MPa. (a) L=S N mm-‘, p=O; (b) L=S N mm-‘, ~~‘0.1; (c) L-5 N mm-‘, ~=0.3; (d) L-5 N mm-‘, p=O.S.
interlayer, and the substrate are all in the elastic regime. For p=O.l (Fig, 6(b)), the Mises stress contours are only slightly different from those of normal indentation. The maximum Mises equivalent stress moves slightly closer to the surface. However, no
272
plastic deformation occurs either in the layers or in the substrate. At p=O.3 (Fig. 6(c)), however, the Mises equivalent stress in the top layer has reached the yield strength, and plastic deformation initiated on the surface near the trailing edge of the contact. However, the m~imum Mises stress occurs inside the interlayer. No plastic deformation occurs in the interlayer because of the high yieid strength of the interlayer. Figure 6(d) shows that at y=OS the Mises stress contours are further distorted toward the leading side, i.e. in the direction of the tangential force. The top surface is subject to more severe plastic deformation, while the interlayer and the substrate are in the elastic regime. The layers and the substrate are in the elastic regime under normal indentation at the 5 N mm-’ load. When an oblique load is applied, the top layer is in the plastic regime only if the ratio of the tangential load to the normal load is larger than a critical value, about 0.3 for the present case. The plastic deformation initiates on the surface near the trailing edge of sliding contact, which is little different from the case of a homogeneous half-space. Yielding initiates on the surface near the leading edge of the contact for two-dimensional sliding contacts of homogeneous media [3-6]. In the cylindrical sliding contact of a single-layer half-space &>0.3), the maximum Mises stress is on the surface near the leading edge if the layer is stiff, and near the trailing edge if the layer is compliant [lo]. In other words, the soft surface layer causes the possible surface failure to move to the back edge of the contact. Thus for a two-layer haIf-space with a soft top layer, yielding is initiated near the trailing edge at ~3 0.3. Moreover, in the elastic analysis of the homogeneous half-space and the single-layer half-space, the maximum Mises stress is always on the surface when ~30.3. For the two-layer elastic-plastic half-space of the present case, however, the maximum Mises stress is always inside the interlayer owing to the high elastic modulus and the high yield strength of the interlayer. To see the effect of the tangential force more extensively, a normal load of 10 N mm”‘, which can cause plastic deformation of the two-layer half-space in normal indentation [12], was applied along with various friction forces. Figure 7 shows the Mises equivalent stress and equivalent total plastic strain contours. For normal indentation (CL= 0), yielding initiates in the substrate. The plastic zone is fully contained by the material, which still remains elastic, so that the equivalent total plastic strain (0.07%) is of the same order of magnitude as the surrounding elastic strains. As a small tangential force is applied on the surface (Fig. 7(b), p=O.l), the Mises stress in the top layer has reached its yield strength and yielding has taken place in the top layer at the top layer-interlayer interface. Notice that the location of initial yielding in the top Iayer is in the subsurface on the right-hand side of the contact, i.e. the region near the leading edge of the contact. Another plastic zone is in the substrate. Its location moves slightly towards the top right-hand side and the growth of the plastic zone is blocked by the hard interlayer. For ~=0.3 (Fig. 7(c)), the Mises equivalent stress on the surface reaches the yield strength of the top layer first, and thus yielding initiates on the surface near the trailing edge of the contact. Plastic strain on the surface is larger (about 1.2%) owing to the uncontained deformation mode. The location of initial yielding in the substrate moves further right. The extent of the plastic deformation in the plastic zone of the substrate is greater than in the case of p=O.l. Figure 7(d) shows that yielding takes place in both the top layer and the substrate when p=OS. The plastic zones and the extent of the plastic deformation are greater than in the previous case (~=0.3). Although the Mises equivalent stress is high in the interlayer, still no plastic deformation has taken place in the interlayer owing to the high yield strength of the interlayer.
273 424a
t
b-1
424a
(b)
Fig. 7. The Mises equivalent stress and equivalent total plastic strain contours for mm-” normal load, The value of the Mises stress contour numbers: 1 - 50 MPa; MPa; 3 - 150 MPa; 4 - 200 MPa; 5 - 250 MPa; 6 - 300 MPa; 7 - 350 MPa; MPa; 9 - 450 MPa; 10 - 500 MPa; 11 - 5.50 MPa. (a) L-10 N mm-‘, p=O; (b) mm-’ , w-0.1; (c) L-10 N mm-l, i~~0.3; (d) L-10 N mm-I, ,u-0.5.
the 10 N 2 - 100 8 - 400 L=lO N
274
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Fig. 8. Normalized a9 for various friction coefficients: (a) and (b) along the top layer-interlayer interface; (c) and (d) along the interlayer-substrate interface. It is apparent ‘from the above analysis that the tangential force is the primary cause of the uncontained plastic deformation of the top layer. For a small normal load (5 N mm-‘), no plastic deformation takes place in the top layer when p=O.l (Fig. 4(b)). For a larger normal load (10 N mm-‘), plastic deformation does take place in the top layer when ~~-0.1 (Fig. 5(b)). When paO.3, yielding initiates on the surface (and in the substrate also if the normal load is larger). For the sliding contact of a two-layer half-space of the present case, surface yielding initiates near the trailing edge, which is different from the case of a homogeneous half-space. Whenever the yielding initiates in the subsurface, however, it occurs at the region near the leading edge of the contact. 4.3. Interfacial stresses The magnitude and the distribution of subsurface stresses along the interfaces are of great interest in assessing the mechanical reliability of multi-layer systems. Figures 8(a) and 8(b) show the normal stress aw along the top layer-interlayer interface,
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and Figs, 8(c) and S(d) along the interlayer-substrate interface. aw is normalized by the yield strength of the top layer for the stress at the top layer-interlayer interface, and by the yield strength of the substrate for the stress at the interlayer-substrate interface. ow is compressive along both interfaces for the range of friction coefficients considered in the present study except for the case of the 10 N mm-’ normal load with p-0.5, when a small tensile stress is generated at x/a> 1. Thus the friction coefficient has an insignificant effect on the normal stress along the interfaces. The distributions move slightly to the right for /.L>O. Figure 9 shows the a, stress along the top layer-interlayer interface for various friction coefficients. Because a, is discontinuous across the interfaces, two sets of o$, curves appear in Figs. 9(a)-9(d). The solid lines are for the stresses in the top layer dzp, whereas the dashed lines are for the stresses in the interlayer &*. u.!.!$is greater than a’,“P along the interface. The friction coefficient has a significant effect on the magnitude and the distribution of u--. When p>O, tensile stress a,, both in the top layer and in the interlayer, is found at the trailing side of the contact, and the
276
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-2.5
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magnitudes of the tensile stress increase with the friction coefficient. The location of the maximum compressive stress a, is shifted to the Leading side (x/u >O), and the magnitude of the compressive stress aiso increases with the friction coefficient. At the interlayer-substrate interface, again two sets of u= curves appear in Figs. 10(a)-10(d). The solid lines are for the stresses in the substrate dzb. whereas the dashed lines are for the stresses in the interlayer c?$. For normal indentation (~==0), o;, is mostly compressive. The magnitude of a”zb is greater than that of o$’ at b/al < 1, and smaller at k/a/ > 1. For an oblique loading (p> 0), the maximum compressive o, occurs at aboutxla = land the magnitude of a, increases with the friction coefficient. a, on the left-hand side ‘of the contact, i.e. x/a ~0, is tensile, and the magnitude of the tensile stress also increases with the friction coefficient. For a compression-dominated stress field such as the present one, shear stresses along the interfaces are particularly important. Figure 11 shows shear stress TV aiong the top layer-interlayer interface (Figs. 11(a) and 11(b)) and along the interlayer-substrate interface (Figs. 11(c) and 11(d)). The shear stresses are normalized by the local shear yield strength ry, l0P or ry, sub fry = cry/G). At the top layer-interlayer
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Fig. 11. Normalized z+,for various friction coefficients: (a) and (b) along the top layer-interlayer interface; (c) and (d) along the interlayer-substrate interface. interface, the magnitude of rV increases considerably with the friction coefficient at the leading side (x/a >O), and decreases with the friction coefficient at the trailing side of the contact (x/a ~0). The same trend is found at the interlayer-substrate interface, but the changes in the magnitudes are much smaller than at the top layer-interlayer interface. Thus as the friction coefficient increases, the interfacial shear stresses at both interfaces become greater. 4.4. Interfacial strains As discussed in a previous paper [12], the strain field, especially the interfacial strains, is important in determining the initiation and propagation of interface cracks in the contact mechanics of layered media. Strain plots are especially important when the contact is in the elastic-plastic regime. Also, because strain is a dimensionless number it requires no normalization by arbitrarily chosen parameters. Thus in the present study attention is also focused on the interfacial strains. Figures 12(a)-12(d) are the normal strains e_ along the top Iayer-interlayer interface for various friction
278
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”
pO.1 -
x/a
-1
”
z
0 I
1
”
2 ’
”
I
2 I
I EgQ
/
_+ 0.20 i
0.00
B -Om20 _ -0.M ~-0.60 13 -0.80 -1.00
I 1
-1.20 (b)
-2 -1 I I I, _ PO.5 0.40 - a2
0.60
0.60 0.40 c,
0.20
i
0.00
-3
b I, I I t I
1 I
I
0.20 z yj
0.00
g -0.20 a -0.40
$-o*2o -0.4a I
e
9-0.60 w -0.80
~-0.60 w -0.80 -1.w -1.20
(4 Fig. 12. e,,, along
the top layer-interlayer
interface
for various friction coefficients.
coefficients. Figures 13(a)-13(d) are the normal strains h along the interIayer-substrate interface for various friction coefficients. Because of the constraints of displacement continuity at the interface, two sets of strain curves are plotted. The solid lines represent the interfacial strains in the top layer, and the dashed lines the strains in the interlayer. For ~=0, E,,, is mostly compressive, though a very small tensile strain is found at b/al> 1. The magnitude of E,,. in the top layer is much larger than that in the interlayer. For oblique loading, the magnitude of the tensile strain at x/a > 1 increases slightly with the friction coefficient, and the magnitudes of compressive strain at /x/u1
-3
0.60
-2 I _ PO.0
0.40
-
0.20 _
a
c
Q)
I
-1 I
x/a I
-ep _____*
F
0 I 1 I I I I I I
I
1 I
2 I,
3 I
z e
0.00
e
0.00
Q) -0.20 a -0.40 _
E-Oe20 -0.40 _
&0.60
g-0.60 w -0.60
I I I I I I I I
0 -0.60 -1.00
-, -1.20 '
-1.00 I
-1.20
(4 -3
-2 -1 I, I 0.60 _ PO.3 0.40 - ey a
0.20
i
0.00
x/a I
b 1 I,,,,, 1 I I I
I I I I b)
2
3
-0.20 2 a -0.40
-1.00 -1.20
I I I 1 (4
Fig. 13. G along the interlayer-substrate
interface
for various friction coefficients.
normal load. At the top layer-interlayer interface, E, on the left-hand side of the contact (x/u0) also increases with the friction coefficient. Thus the tangential loading causes a larger tensile strain E, on the trailing side and a larger compressive strain E, at the leading side of the contact. The same effects are also observed at the interlayer-substrate interface (Figs. 14(c) and 14(d)). The shear strain along the interfaces is plotted in Figs. 15(a)-15(d) for the top layer-interlayer interface, and in Figs. 16(a)-16(d) for the interlayer-substrate interface. Again, two sets of shear strain,curves appear in the plots because of the constraints of the displacement continuity at the interfaces. The solid lines represent the interfacial strains in the top layer or in the substrate, whereas the dashed lines represent the interfacial strains in the interlayer. For p=O, the shear strains on the trailing side (x/u CO) are in the opposite direction to the shear strain on the leading side of the contact. The maximum interfacial shear strain occurs at about k/ul= 1 along both
280
-3 -2 -1 0.40 1 i ! ’ ’ j L=tON/mm
0.40
xfa *
0 z ;
3
1 ’
’
2 ’
’
2 I
a
0.30 2
0.20
s
O.lc1
5
2
l/c
0.10
k fL--0.00
g&-0.00
g--a.1o X
X
id
i
i
-v__
t ____I
R
/
Q -0.20 -0.30 -0.M
fb)
0.40
-3
-2
-1
x/a 0
1
2
3
s
-3 ’
0.30
0.30 z
0.40
-2 -1 ’ ! ’ ’ C=lON/mm
c
b f-” I ,
1 I
I
0.28 0.10
k a-0.00
______”
$ -0.10 0
-0.20 -0.30
-0.30
-0.40
-0.40
/
Fig, 14. Interfacial l= for various frictian coefficients: (a) and @) along the top layer-interlayer interface;
(c> and (d) along the inter&w-substrate
interface.
interfaces. The shear strainsin the interiayer are smaller than those in the top layer or in the substrate. For p> 0, the shear strains tend to be in the direction of &ding. As the friction coefficient increases, the interfacial shear strains in the top layer along the top layer-interlayer interface increase significantty, while- the magnitude of the interfacial shear strains in the interlayer increases only slightly. Along the interlayer-substrate interface, the magnitudes of the shear strain in the substrate also increase considerably with the friction coefficient, while the shear strains in the interlayer are not greatly affected. At ~=OS1 almost all the interfacial shear strains are along the sliding direction. The maximum magnitude of the inter-facial shear strain in the top layer is about 7%, and 1.5% in the substrate. Thus friction has a significant effect on the magnitude and distribution of shear strains along both interfaces,
It is evident that the tangential force, or the friction coefkient, has a significant effect on the Ioeation of yiefding, surface deformation, interfacial stresses and strains,
281
-3 0.60""
-2
-1 '
x/a I
0 I
1 ""I
2 PO.1
0.60
b)
6.0
6.0
7.0
7.0
c1 6.0
_+ 6.0
i
5.0
i
2 a
4.0
&I 4.0 a 3.0 _
" 2 0
3.0 2.0
3 w
1.0
5.0
2.0 1.0
0.0
0.0
-1.0
-1.0
4
(4 Fig. 1.5. .+ along the top layer-interlayer
I
(d) interface for various friction coefficients.
etc. especially the shear stresses and strains along the interfaces. In the present study, the interfaces are assumed to be infinitely strong. In reality, the interface can sustain only a certain level of stress or strain. As discussed before [12], three criteria have been developed for the initiation of interfacial cracks: energy criterion, the local strain criterion, and the local stress criterion. Among the three, the local shear strain criterion seems to be relevant to the compression-dominated contact problems. When the interfacial shear strain exceeds a criticalvalue, it is likely that debonding or microcracking occurs. Thus the interfacial stresses and strains should be minimized as much as possible. It is apparent from the FEM analysis that when the friction coefficient is less than 0.3, the interfacial shear stresses and strains are not much different from the case of normal indentation. However, they become significantly larger when the friction coefficient is larger than 0.3, and interfacial failure is likely. Thus the sliding friction of a multi-layer contact system should be kept below 0.3. One obvious solution is to use lubricants. When lubricants are undesirable, one should consider the compatibility
282
0.80
-3
0.60 -
-2 I
I
-1 I,
X/Q 0 i I ,
I: I
/
1 t
*
2 2% po.0
-
-3
3
ey
-2
-1
x/a 0
1
2
3
0.60
I I , ,
-0.60 -0.80
(b)
1.5
-r : f
1.0
zi >; 0.5 0" 0.0
(4 Fig. 16. eT along the interlayer-substrate
interface
for various friction coefikients.
of sliding contact materials [21]. Low sliding friction can be obtained with incompatible sliding pairs. Unfortunately the contact stress and strain fields of multi-layer media cannot be generalized in a concise and elegant form as in the case of homogeneous elastic media [S, 223, Thus the results obtained here may not be strictly valid for other ~mbinations of multi-layer contact systems. In addition, the analysis presented here is actually for oblique loading on a two-layer half-space. The friction force is modeled by a tangential force acting on the surface. Whether the sliding friction between the contact interfaces can be precisely represented by the surface tractions needs closer examination. Nevertheless, the effect of friction on subsurface stresses and strains is well represented by this study. Moreover, the present study considers only one-time loading. In practice the contact failure usually takes place after more than one sliding cycle. Merwin and Johnson 1231 showed that the plastic shear strain accumulates under repeated loading. Thus the history of loading and accumulation of plastic shear strain is also important in assessing the sliding life of the multi-layer contact systems.
283
5. Conclusions A two-dimensional finite element analysis of the sliding contact of a two-layer elastic-plastic half-space was conducted. The following conclusions can be drawn from the analysis. (a) Surface deformation and profile are strongly affected by the friction coefficient. The surface material is deformed along the sliding direction. A small hump is observed at the leading edge of the contact when the friction coefficient is large. (b) The location of initial yielding strongly depends on the friction coefficient as well as the normal load. Yielding initiates in the substrate for a small friction coefficient if the normal load is large enough. For the “soft” surface layer yielding initiates on the surface (and in the substrate as well if the normal load is larger) when ~>0.3. Whenever yielding initiates on the surface, it always occurs near the trailing edge of the contact. Whenever yielding initiates in the subsurface, however, it occurs at the leading region of the contact. The tangential force is the primary cause of the uncontained plastic deformation of the top layer. (c) Normal stresses and strains along the interfaces are not greatly affected by friction. The horizontal shear stresses and strains along the interfaces are strongly influenced by the friction coefficient. For frictional loading, a, in the trailing region of the contact is tensile, and is compressive in the leading region of the contact. The magnitudes of shear stress and strain along both interface increase significantly with the friction coefficient.
Acknowledgments We are grateful to the Teradyne Connection Systems, Inc. of Nashua, NH, for supporting this work. Thanks are due to Messrs. Mark Gailus, Lou Spiridigliozzi and Phil Stokoe of Teradyne for their interest in this work. Thanks are also due to Professor Ernest Rabinowicz of MIT for many helpful suggestions. The ARAQUS finite element code is made available to MIT by Hibbitt, Karlsson and Sorenson, Inc., Providence, RI, under the terms of an academic licence agreement.
References 1 D. P. Seraphim, R. La&y and C-Y. Li (eds.), Principles of Electronic Packaging, McGraw-
Hill, New York, (1989). 2 K. L. Johnson, One hundred years of Hertz contact, Proc. Ins. Mech. Eng., I96 (1982) 363-378. 3 E. M’Ewan, Stresses in elastic cylinders in contact along a genratrix, Phil. Mug., 40 (1949) 45U59. 4 H. Poritsky, Stresses and deflections of cylindrical bodies, J. Appl. Mech., 17 (1950) 191-201. 5 J. L. Smith and C. K. Liu, Stresses due to tangential and normal loads on an elastic solid with application to some contact stress problems, .I. Appl. Mech., 20 (1953) 157-166. 6 K. L. Johnson and J. A. Jefferis, Plastic flow and residual stresses in rolling and sliding contact, Proc. Inst. Mech. Eng. Symp. on Rolling Contact Fatigue, London, 1963, pp. 54-77. 7 H. S. Nagaraj, Elastoplastic contact of bodies with friction under normal and tangential loading, J. Tribal., Trans. ASME, 106 (1984) 519-526. 8 A. G. Tangena and G. A. M. Hurkx, Calculations of mechanical stresses in electrical contact situations, IEEE Trans. Components, Hybrid, and Manufact. Technol., CHMT-8 (1) (1985) 13-20.
284 9 T. Ihara, M. G, Shaw and B. Bhushan, A finite clement analysis of contact stress and strain in an elastic film on a rigid substrate - Part II: With friction J. ?X&, Trans. ,&!%fE, IO&’ (1986) 534-539. 30 R. B. King and T. C. O’Sullivan, Sliding contact stresses in a two-dimensional layered elastic half-space, Int. J. Solids Strut., 2,7 (1987) 581-597. 11 K. Komvopoulos, N. Saka and N. P. Sub, The role of hard layers in lubricated and dry sliding, J. Tribal, Trans. ASME, 109 (1987) 223-231. 12 H. Tian and N. Saka, Finite element analysis of a two-layer elastic-plastic half-space: normal contact, Wear, 148 (1991) 47. 13 K. I... Johnson, Contact Mechanics, Cambridge University Press, Cambridge, (1985), pp. 119-124. 14 R. D. MindIin, Compliance of elastic bodies in contact, J. Appr &&ch., AS&fE Truns., 16 (1949) 259-268. 15 R. D. kindles and N. Deresicwicz, Elastic spheres in contact under varying oblique forces, J. Appl. Me&., ASME Tram, 20 (1953) 327-344. 16 W. Deresiewicz, Oblique contact of nonspherical elastic bodies, 1. Appt. Mech., ASME Trans., 24 (1957) 623-624. 17 P, K. Cupta, J. A. Walowit and E. F. Finkin, Stress distributions in plane strain layered elastic solids subjected to arbitrary boundary loading, J. Lube. Technol, Tram ASME, 9.5 (1973) 427-433. 18 K. L. Johnson, Contact Mechnnics, Cambridge University Press, Cambridge, (1985), pp. 171-184. 19 K. L. Johnson, Contac: Mechanics, Cambridge University Press, Cambridge, 1985, pp. 202-210. 20 K, L, Johnson, Aspects of friction, &IX. 7th Leeds-Lyon Syrup. on Tribofogv, University of Leeds, 1980, pp. 3-12. 21 E. Rabinowicz, The inRuence of ~rn~a~bil~~ on different tr~bo~og~&a~phenomena, ASLE Tratts. 14 (1971) 204-212. 22 G. M. ~am~iton, Explicit equations for the stresses beneath a sliding spherical contact, Pmt. Inst. Mech. Erg., 197C (3983) 53-59. 23 J, E. Merwin and K. L. Johnson, An analysis of plastic deformation in roIIing contact, Prac. Inst. Me& Eng,, 177 (1963) 676-690.
Appendix A: Nomenclature half contact width Young’s modulus plastic modulus tangential force thickness of top layer thickness of interlayer normal force per unit length normal force pressure tangential traction radius of cylindrical indenter tangential force per unit length components of displacement Greek symbols constant angle of oblique loading ; S pe~e~at~on depth
285 e,, Ed, E+, components of strain friction coefficient p V Poisson’s ratio Mises equivalent stress UMises ff!., =w* components of stress %, Gy yield strength CY shear yield strength TY
261
Finite element analysis of an elastic-plastic half-space: sliding contact
two-layer
Hong Tian* and Nannaji Sakat DePatiment of Mechanical Engineering, Massachusetts Institite of Technoiogy, Cambridge, MA &?I39 (u&4.) (Received
July 24, 1990; revised X3ecember 13, 1990, accepted
February
26, 1991)
Abstract A two-dimensional
finite element stress and strain analysis of the sliding contact of a twolayer elastic-plastic half-space for various friction coefficients was conducted. First the contact pressure of the two-layer half-space under normal indentation was determined. Then the normal and tangential loadings were applied proportionally and incrementally on the surface. It was found that surface deformation, location of initial yielding, stresses and strains along the interfaces between layers strongly depend on the friction coefficient. When the friction coefficient is small (tess than 0.3), yielding initiates in the subsurface region near the leading edge of the contact, provided that the normal load is large enough. When the friction coefficient is large (greater than 0.3), yielding initiates on the surface at the trailing edge of the contact. The surface strains, especially the shear strains, are large owing to the unconstrained deformation. The magnitudes of shear stresses and strains along both interfaces are signi~cant~y large for high friction. On the basis of the analysis, the implications for interface failures are qualitatively addressed.
1. Introduction Thin films are widely used in modern mechanical, electronic, magnetic, and optical devices. In electrical connectors, in particular, thin composite layers of gold, nickel, and copper or copper alloys are widely used to meet multi~nctionai requirements [I]. A minimum normal force is also required to maintain low electrical contact resistance and low noise; but the high friction forces’ generated by high friction coefficients during frequent insertion and withdrawal cause contact failures. To better underst~d the mechanisms of the contact failures of thin film systems, it is necessary to have a comprehensive knowledge of the sliding contact stress and strain fields of the thin film systems. Contact stress analyses of homogeneous media have been made for many decades. Johnson [2] has recently reviewed the mechanics of nonconforming contacts. The hvodimensional quasistatic frictional or sliding contact of a homogeneous elastic halfspace was investigated by M’Ewan [3], Poritsky [4], and Smith and Liu [S]. It has been found that the effect of tangential force is to bring the point of maximum shear stress closer to the surface. Johnson and Jefferis [6] used both the Mises and Tresca *Present Address: Hoya Electronics ~~oratjon, U.S.A. tAutbor
to whom correspondence
0043-1648/P1/$3.50
960 Rincon Circle, San Jose, CA 95131,
should be addressed.
0 1991 -
Elsevier Sequoia,
Lausanne
262
yield criteria to determine the location of initial yield in a sliding contact. They found that yielding initiates in the subsurface region when the friction coefficient y is small (,u
2. Statement
of the problem
In the normal contact of two non-conforming bodies, friction at the contact interface affects the stress field only if the elastic constants of the two materials are different 1133. Under combined normal and tangential loading, the friction at the contact interface has different effects on the stress field, depending on whether the contact is static or sliding. In a static contact, Mindlin 1141 showed that, under a constant normal load N even the smallest tangential force F will result in some siip at the contact edge.
263
The annular slip zone spreads into the contact region as F increases until F = ,FrN, when gross slip occurs. Owing to the “irreversibility” implied by the frictional contact, the final contact stress field depends not only on the final values of the normal and tangential forces, but also on the history of loading. If the normal and tangential loads are applied simultaneously and proportionally, i.e. by oblique loading, the slip zone does not exist when the angle of the oblique loading p (p= tan-’ (FIN)) is less than the angle of friction (tan-’ EL) [15, 161. In sliding contacts, it is usually assumed that the contact pressure distribution is independent of the tangential loading, and that Amonton’s Law of sliding friction holds both macroscopically and microscopically, i.e. F=&&N
(1)
and 4(x) = &V(x)
(2)
where q(x) andp(x) are the tangential and normal surface traction distributions within the contact region, respectively. In the elastic analysis of homogeneous half-spaces, the normal pressure is usually specified by the Hertzian pressure and the tangential traction by eqn. (2) on the contact surface [4-6]. However, in the case of layered media [17] or when plastic deformation is involved 1181, the contact pressure distribution deviates from the Hertzian distribution. In a finite element analysis of the normal indentation of a two-layer elastic-plastic half-space by the present authors [12], it was shown that the contact pressure no longer obeys the Hertzian pressure distribution and that the deviation from the Hertzian distribution depends on the layer thickness and the extent of pfastic deformation. In sliding contact, the tangential loading has an insignificant effect on the contact width and the contact pressure distribution, especially if y < 1 1191. Thus for the analysis of sliding contacts of layered elastic-plastic media, a two-step procedure can be used. First the contact pressure distribution under normal indentation is determined, and then the tangential loading is specified by eqn. (2). Since the final contact stress field depends on the history of loading, only the simultaneous and proportional loading scheme of normal and tangential forces is considered in the present analysis. Consider a rigid cylinder of radius R indenting a two-layer half-space under a normal force per unit length L and a tangential force per unit length T. Figure 1 shows the coordinate system and the dimensions of the layers. The contact edge (~=a) is defined as the leading edge, and x= -a as the trailing edge of the contact. The thicknesses of the top layer (layer 1) and the interlayer (layer 2) are h1 and hz, respectively. Perfect bonding is assumed at the interfaces between layers, i.e. displacements across the interfaces are continuous u(‘)(x, h,) =24(2)(x7 h,)
@a)
7Jqq h,) = ?J@)(x,h,)
(3b)
d2)(x, hl + h2) = d3)(x 7hl + h,)
(44
TJyq hlfh,)=?P(x, hl+h2)
(4b)
U-V=0
asx,y-
00
(5)
where the superscripts l-3 stand for the top layer, the interlayer, and the substrate, respectively. A two-step procedure is followed to obtain the stress and strain fields.
264
Fig. 1. Schematic of sliding contact of a two-layer half space. TABLE 1 Mechanical properties of contact materials Material
E (GPa)
I,
oY (MPa)
a
Top layer (Au 4 O.S%Ni) Interlayer (Ni) Substrate (brass: Cu-35%Zn)
79 205 125
0.32 0.32 0.32
298 811 405
0.1 0.1 0.1
First the contact pressure of frictionless indentation of the two-layer determined. The boundary conditions for the first step are as follows. On the contact surface
7&,
0) = 0
half-space
is
lulaa
W
Pl>a
(6b)
--<
(64
The penetration depth 6 is assumed to be smaller than the radius of the indenter R or the half contact width u. After the contact pressure p(x) is obtained from the first step, the normal and tangential loadings are applied proportionaily. The boundary conditions for the second step are as follows (7a)
(‘W a,(x,
0) = 0
T&x, 0) = 0
Ixl>n
(7c)
kl>a
(7d)
The primary emphasis of this study is on the siiding contact of multi-layer systems employed in electrical contacts. That is, the top layer is gold, which is hardened with
26.5
0.5% nickel, the interlayer is nickel, and the substrate is copper or All materials are assumed to possess a bilinear elastic-plastic behavior modulus Et, is equal to a constant cy times the elastic modulus E). The and plastic properties of the layers and the substrate are listed in values fall in the typical range of electrical contact materials.
a copper (i.e. the relevant Table 1.
alloy. plastic elastic These
3. The finite element model The contact pressure distribution and contact width of a two-layer half-space under normal indentation have been obtained in a previous work [12]. The results are directly used in the present analysis. Since the combined normal and tangential loading is no longer symmetric, the whole layered half-space has to be modeled. Figure 2(a) shows the finite element mesh and the boundary conditions, and Fig. 2(b) shows the details of the mesh near the contact. It was assumed that the bottom plane CD is far away from the contact center, and that the vertical displacements are equal to zero. The horizontal displacements in the bottom plane (CD) were not restricted, except that of point C which stays put. The normal and tangential loads were applied on the top plane AB within the contact region. The horizontal and vertical dimensions of the mesh are 200 pmX200 pm, both being large enough to allow the stresses to be insignificant at the boundaries. The whole mesh is an arrangement of 630 quadrilateral, eight-node, isoparametric elements, and the total number of nodes is 2023. The mesh was refined in the region below the contact. The multiple node constraints in the multi-purpose finite element program (ABAQUS) were used to connect the elements of different sizes for mesh refinement. The horizontal and vertical spacing of nodes on and below the contact surfaces are 0.5 Ir;m and 0.3125 pm, respectively. The twolayer half-space was modeled by assigning different elastic and pfastic properties to the elements of the top rows of the mesh. The first two rows were assigned to the
Fig. 2. (a) The finite element mesh.
mesh and the boundary
conditions,
and (b) the detail of the fine
top layer, and the next two rvws immediately below the top layer were assigned to the interlayer. The interfaces between the layers were assumed to be perfectly bonded, i.e. the displacements at the interfaces were continuous (eqns, (3)-(S)). These interface constraints were satisfied by using common nades at the interfaces, and the comman nodes belong to the elements on both sides of the interface. The normal and tangential loads were applied node by node with a consistent loading scheme within the contact region, and they were in one executing step of the ABAQUS code so that the incremental and proportional loading could be undertaken. The parameter settings of the executing programs were detailed in the previous work [12]. The cumputat~o~s were performed on a Micro VAX If computer, and the typical CPU time for one run was about 90 min. To verify whether the boundary condjt~o~s and the finite eIement mesh in Fig. 2 are fair rep~ese~~at~ons of a sern~-~~~~~te s&id, ikst the sGding contact of a ~ornoge~~us eIastic ha&space was analyzed. The twv layers and the substrate were assigned the same Young’s modulus and Poisson’s ratio. An eltipticaf pressure distribution (Hertz pressure distribution) was applied on the surface along with a proportional distributed shear traction. The differences between the calculated stresses and those of the analytical solutions presented by Poritsky [4], and by Smith and Ciu [S], were negligible. Thus the present finite element model and mesh are assumed to be an acceptable representation of an elastic half-space in sliding contact.
The results presented here are for a rigid cylinder of radius R (CL75 mm) which is much larger than the expected haff contact width CI. The normal bads were 5 N mm-’ and IO N mm-‘. It was shown in ref. 12 that the 5 N mm-’ normal bad is loading. an elastic loading, while the 10 N mm-’ normal Xoad is an elastic-plastic The ratio of the tangential force to the normal force, ix. the friction coefficient, was from 0 to 0.5. (Tha finite element of analysis for a larger friction coefficient (pm 1.0) was tried, but no converged solution was obtained in the allowed number of increments. The strain increment was larger than fifty times the yield strain owing to relatively small work hardening of the contact materials considered in the present study. Thus it may not be possible TV obtain converged solutions in a reasonably short period of time.) The thicknesses of the top layer and the interkyer were chosen as I,25 pm and 2.5 I*_rn.The ~lc~lated stresses are normalized by the yield strengths of the top or rY.top, and of the substrate, @lf,& or ry,L,subrdepending on the approfayer, ~~~~~~ priateness. The positive values indicate tensile stresses and strains, and the negative values compressive: stresses and strains. 4.1. Sur$~e defmrnation and penetration depth To qualitatively illustrate the effect of friction on deformation in the layers and the substrate, especially in the top layer, the deformed meshes are shown in Fig. 3 in for the 5 N mm-’ normal load at various friction coefficients. The displacements the deformed mesh are magnified 20 times for clarityy. The top two rows of elements are the top layer, the next two rows are the interiayer, and the rest of the elements are the substrate, For normal indentation (p==(I) displacements of the layers and the substrate are, as expected, symmetric about the y-axis. When the friction coefkient is greater than zero, however, the symmetry is Iost and the materials deform large& along the direction of the ta~~e~t~a~ force. fn the finite element ana@& of an
267
mp
layer
rnterlaye Substrate
(21)
Fig. 3. Deformed meshes under the S N mm-’ normal bad at various frj~t~on coefficients. (a) L=5 N mm-’ , s=O, (b) L-S N mm-‘, ~~.1-0.1;fc} t-5 N mm”, &=0.3; (d) L-5 N mm-‘, p = 0.5. Original magnification factor = 20. elastic-plastic half-space by Tangena and Hurkx @], a hump was found at the leading edge of the sliding contact as the friction coefficient increases. In the present case, a two-layer half-space, the displacements in the top layer are cIeariy much larger than those of the interlayer and the substrate, but no hump is found at the leading edge of the contact. Figure 4 shows the deformed meshes for the 10 N mm-’ normal load at various friction coefficients. As expected, the contact width and displacements are greater than in the ease of the S N mm”’ normal load. As the friction coefikient increases, the meshes of the top layer are deformed aloo@ the sliding direction. At p=O.S, a small hump is observed at the leading edge of the contact. Table 2 iists the penetration depth uf the two-layer half-space under combined normal and tangential loadings. The penetration depth increases with the normal load as well as the friction coefficient. Thus the tangential force contributes ncn only to the horizontal displacement, but also to the vertical indentation. The same observation
268
was added by Johnson &33] for a ~u~~~~~~~~s &a&c ~~~f-~~a~e under normaI and tangential loadings. The penetration depth 6, however, is stiII small compared with tho thickness of the top layer hI. The ratio S/h1 in Table 2 shows that the penetration depth is about 10% of the thickness of the top layer under the 5 N mm-’ normal load, and less than 20% under the 10 N mm-’ normal foad, for the whole range of friction coe~c~e~fs co~s~d~red hereTo i~~ustrate the effect of the fr~ct~o~ coefikient OR the surface ~efo~at~o~~ the strains on the contact surface are plotted in Fig. 5. The horizontal strain c-~ under 5 N mm-’ and 10 N mm-’ normal loads with various friction coefficients are plotted in Figs. s(a) and S(b), respcctiveiy. As the friction coe~cie~t increases, the materiat
269
TABLE
2
width and penetration tangential loading
Contact
Normal load, L (N mm)-’ 5
10
depth of a two-layer half-space under combined normal and
Friction coefficient
Half-width of contact a (pm)
Penetration 6 (pm)
depth,
6
0.0
5.S
0.1 0.3 0.5
5.S S.S 5.S
0.106 0.107 0.109 0.113
0.0848 0.0856 0.0872 0.0904
0.0 0.1 0.3 0.5
8.5 8.S 8.5 8.5
0.197 0.199 0.207 0.216
0.1576 0.1592 0.1656 0.1728
7;;
in the trailing side of the contact (x/u ~0) is stretched horizontally (e=>O), and the material in the leading side (X/U>O) is squeezed (e,
In the normal contact, i.e. without friction, of a homogeneous elastic half-space, yielding initiates in the subsurface region at roughly~=O.7~ below the contact center. Under combined normal and tangential loadings, the location of maximum Mises equivalent stress or maximum shear stress is brought closer to the surface, or even to the surface when the friction coefficient is large (about 0.25 for the Tresca yielding criterion, and 0.3 for the Mises criterion) [3-6]. King and O’Sullivan (lo] show that this conclusion is also true when a stiff or compliant layer bonded on an elastic halfspace is under normal and tangential loading. For sliding contact of a two-layer halfspace comprising materials of different yield strength, however, the location of the maximum Mises equivalent stress is not necessarily the location of the initial yielding. Figure 6 shows the Mises equivalent stress and the equivalent total plastic strain contours under the 5 N mm-’ normal load for various friction coefficients. Only the stress and plastic strain contours in the region near the contact are plotted for clarity. The horizontal dashed lines in the plots represent the approximate locations of the interfaces between layers. For normal indentation (Fig. 6(a), p==O), the top layer, the
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interlayer, and the substrate are all in the elastic regime. For p=O.l (Fig, 6(b)), the Mises stress contours are only slightly different from those of normal indentation. The maximum Mises equivalent stress moves slightly closer to the surface. However, no
272
plastic deformation occurs either in the layers or in the substrate. At p=O.3 (Fig. 6(c)), however, the Mises equivalent stress in the top layer has reached the yield strength, and plastic deformation initiated on the surface near the trailing edge of the contact. However, the m~imum Mises stress occurs inside the interlayer. No plastic deformation occurs in the interlayer because of the high yieid strength of the interlayer. Figure 6(d) shows that at y=OS the Mises stress contours are further distorted toward the leading side, i.e. in the direction of the tangential force. The top surface is subject to more severe plastic deformation, while the interlayer and the substrate are in the elastic regime. The layers and the substrate are in the elastic regime under normal indentation at the 5 N mm-’ load. When an oblique load is applied, the top layer is in the plastic regime only if the ratio of the tangential load to the normal load is larger than a critical value, about 0.3 for the present case. The plastic deformation initiates on the surface near the trailing edge of sliding contact, which is little different from the case of a homogeneous half-space. Yielding initiates on the surface near the leading edge of the contact for two-dimensional sliding contacts of homogeneous media [3-6]. In the cylindrical sliding contact of a single-layer half-space &>0.3), the maximum Mises stress is on the surface near the leading edge if the layer is stiff, and near the trailing edge if the layer is compliant [lo]. In other words, the soft surface layer causes the possible surface failure to move to the back edge of the contact. Thus for a two-layer haIf-space with a soft top layer, yielding is initiated near the trailing edge at ~3 0.3. Moreover, in the elastic analysis of the homogeneous half-space and the single-layer half-space, the maximum Mises stress is always on the surface when ~30.3. For the two-layer elastic-plastic half-space of the present case, however, the maximum Mises stress is always inside the interlayer owing to the high elastic modulus and the high yield strength of the interlayer. To see the effect of the tangential force more extensively, a normal load of 10 N mm”‘, which can cause plastic deformation of the two-layer half-space in normal indentation [12], was applied along with various friction forces. Figure 7 shows the Mises equivalent stress and equivalent total plastic strain contours. For normal indentation (CL= 0), yielding initiates in the substrate. The plastic zone is fully contained by the material, which still remains elastic, so that the equivalent total plastic strain (0.07%) is of the same order of magnitude as the surrounding elastic strains. As a small tangential force is applied on the surface (Fig. 7(b), p=O.l), the Mises stress in the top layer has reached its yield strength and yielding has taken place in the top layer at the top layer-interlayer interface. Notice that the location of initial yielding in the top Iayer is in the subsurface on the right-hand side of the contact, i.e. the region near the leading edge of the contact. Another plastic zone is in the substrate. Its location moves slightly towards the top right-hand side and the growth of the plastic zone is blocked by the hard interlayer. For ~=0.3 (Fig. 7(c)), the Mises equivalent stress on the surface reaches the yield strength of the top layer first, and thus yielding initiates on the surface near the trailing edge of the contact. Plastic strain on the surface is larger (about 1.2%) owing to the uncontained deformation mode. The location of initial yielding in the substrate moves further right. The extent of the plastic deformation in the plastic zone of the substrate is greater than in the case of p=O.l. Figure 7(d) shows that yielding takes place in both the top layer and the substrate when p=OS. The plastic zones and the extent of the plastic deformation are greater than in the previous case (~=0.3). Although the Mises equivalent stress is high in the interlayer, still no plastic deformation has taken place in the interlayer owing to the high yield strength of the interlayer.
273 424a
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Fig. 7. The Mises equivalent stress and equivalent total plastic strain contours for mm-” normal load, The value of the Mises stress contour numbers: 1 - 50 MPa; MPa; 3 - 150 MPa; 4 - 200 MPa; 5 - 250 MPa; 6 - 300 MPa; 7 - 350 MPa; MPa; 9 - 450 MPa; 10 - 500 MPa; 11 - 5.50 MPa. (a) L-10 N mm-‘, p=O; (b) mm-’ , w-0.1; (c) L-10 N mm-l, i~~0.3; (d) L-10 N mm-I, ,u-0.5.
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and Figs, 8(c) and S(d) along the interlayer-substrate interface. aw is normalized by the yield strength of the top layer for the stress at the top layer-interlayer interface, and by the yield strength of the substrate for the stress at the interlayer-substrate interface. ow is compressive along both interfaces for the range of friction coefficients considered in the present study except for the case of the 10 N mm-’ normal load with p-0.5, when a small tensile stress is generated at x/a> 1. Thus the friction coefficient has an insignificant effect on the normal stress along the interfaces. The distributions move slightly to the right for /.L>O. Figure 9 shows the a, stress along the top layer-interlayer interface for various friction coefficients. Because a, is discontinuous across the interfaces, two sets of o$, curves appear in Figs. 9(a)-9(d). The solid lines are for the stresses in the top layer dzp, whereas the dashed lines are for the stresses in the interlayer &*. u.!.!$is greater than a’,“P along the interface. The friction coefficient has a significant effect on the magnitude and the distribution of u--. When p>O, tensile stress a,, both in the top layer and in the interlayer, is found at the trailing side of the contact, and the
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magnitudes of the tensile stress increase with the friction coefficient. The location of the maximum compressive stress a, is shifted to the Leading side (x/u >O), and the magnitude of the compressive stress aiso increases with the friction coefficient. At the interlayer-substrate interface, again two sets of u= curves appear in Figs. 10(a)-10(d). The solid lines are for the stresses in the substrate dzb. whereas the dashed lines are for the stresses in the interlayer c?$. For normal indentation (~==0), o;, is mostly compressive. The magnitude of a”zb is greater than that of o$’ at b/al < 1, and smaller at k/a/ > 1. For an oblique loading (p> 0), the maximum compressive o, occurs at aboutxla = land the magnitude of a, increases with the friction coefficient. a, on the left-hand side ‘of the contact, i.e. x/a ~0, is tensile, and the magnitude of the tensile stress also increases with the friction coefficient. For a compression-dominated stress field such as the present one, shear stresses along the interfaces are particularly important. Figure 11 shows shear stress TV aiong the top layer-interlayer interface (Figs. 11(a) and 11(b)) and along the interlayer-substrate interface (Figs. 11(c) and 11(d)). The shear stresses are normalized by the local shear yield strength ry, l0P or ry, sub fry = cry/G). At the top layer-interlayer
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interfaces. The shear strainsin the interiayer are smaller than those in the top layer or in the substrate. For p> 0, the shear strains tend to be in the direction of &ding. As the friction coefficient increases, the interfacial shear strains in the top layer along the top layer-interlayer interface increase significantty, while- the magnitude of the interfacial shear strains in the interlayer increases only slightly. Along the interlayer-substrate interface, the magnitudes of the shear strain in the substrate also increase considerably with the friction coefficient, while the shear strains in the interlayer are not greatly affected. At ~=OS1 almost all the interfacial shear strains are along the sliding direction. The maximum magnitude of the inter-facial shear strain in the top layer is about 7%, and 1.5% in the substrate. Thus friction has a significant effect on the magnitude and distribution of shear strains along both interfaces,
It is evident that the tangential force, or the friction coefkient, has a significant effect on the Ioeation of yiefding, surface deformation, interfacial stresses and strains,
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etc. especially the shear stresses and strains along the interfaces. In the present study, the interfaces are assumed to be infinitely strong. In reality, the interface can sustain only a certain level of stress or strain. As discussed before [12], three criteria have been developed for the initiation of interfacial cracks: energy criterion, the local strain criterion, and the local stress criterion. Among the three, the local shear strain criterion seems to be relevant to the compression-dominated contact problems. When the interfacial shear strain exceeds a criticalvalue, it is likely that debonding or microcracking occurs. Thus the interfacial stresses and strains should be minimized as much as possible. It is apparent from the FEM analysis that when the friction coefficient is less than 0.3, the interfacial shear stresses and strains are not much different from the case of normal indentation. However, they become significantly larger when the friction coefficient is larger than 0.3, and interfacial failure is likely. Thus the sliding friction of a multi-layer contact system should be kept below 0.3. One obvious solution is to use lubricants. When lubricants are undesirable, one should consider the compatibility
282
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of sliding contact materials [21]. Low sliding friction can be obtained with incompatible sliding pairs. Unfortunately the contact stress and strain fields of multi-layer media cannot be generalized in a concise and elegant form as in the case of homogeneous elastic media [S, 223, Thus the results obtained here may not be strictly valid for other ~mbinations of multi-layer contact systems. In addition, the analysis presented here is actually for oblique loading on a two-layer half-space. The friction force is modeled by a tangential force acting on the surface. Whether the sliding friction between the contact interfaces can be precisely represented by the surface tractions needs closer examination. Nevertheless, the effect of friction on subsurface stresses and strains is well represented by this study. Moreover, the present study considers only one-time loading. In practice the contact failure usually takes place after more than one sliding cycle. Merwin and Johnson 1231 showed that the plastic shear strain accumulates under repeated loading. Thus the history of loading and accumulation of plastic shear strain is also important in assessing the sliding life of the multi-layer contact systems.
283
5. Conclusions A two-dimensional finite element analysis of the sliding contact of a two-layer elastic-plastic half-space was conducted. The following conclusions can be drawn from the analysis. (a) Surface deformation and profile are strongly affected by the friction coefficient. The surface material is deformed along the sliding direction. A small hump is observed at the leading edge of the contact when the friction coefficient is large. (b) The location of initial yielding strongly depends on the friction coefficient as well as the normal load. Yielding initiates in the substrate for a small friction coefficient if the normal load is large enough. For the “soft” surface layer yielding initiates on the surface (and in the substrate as well if the normal load is larger) when ~>0.3. Whenever yielding initiates on the surface, it always occurs near the trailing edge of the contact. Whenever yielding initiates in the subsurface, however, it occurs at the leading region of the contact. The tangential force is the primary cause of the uncontained plastic deformation of the top layer. (c) Normal stresses and strains along the interfaces are not greatly affected by friction. The horizontal shear stresses and strains along the interfaces are strongly influenced by the friction coefficient. For frictional loading, a, in the trailing region of the contact is tensile, and is compressive in the leading region of the contact. The magnitudes of shear stress and strain along both interface increase significantly with the friction coefficient.
Acknowledgments We are grateful to the Teradyne Connection Systems, Inc. of Nashua, NH, for supporting this work. Thanks are due to Messrs. Mark Gailus, Lou Spiridigliozzi and Phil Stokoe of Teradyne for their interest in this work. Thanks are also due to Professor Ernest Rabinowicz of MIT for many helpful suggestions. The ARAQUS finite element code is made available to MIT by Hibbitt, Karlsson and Sorenson, Inc., Providence, RI, under the terms of an academic licence agreement.
References 1 D. P. Seraphim, R. La&y and C-Y. Li (eds.), Principles of Electronic Packaging, McGraw-
Hill, New York, (1989). 2 K. L. Johnson, One hundred years of Hertz contact, Proc. Ins. Mech. Eng., I96 (1982) 363-378. 3 E. M’Ewan, Stresses in elastic cylinders in contact along a genratrix, Phil. Mug., 40 (1949) 45U59. 4 H. Poritsky, Stresses and deflections of cylindrical bodies, J. Appl. Mech., 17 (1950) 191-201. 5 J. L. Smith and C. K. Liu, Stresses due to tangential and normal loads on an elastic solid with application to some contact stress problems, .I. Appl. Mech., 20 (1953) 157-166. 6 K. L. Johnson and J. A. Jefferis, Plastic flow and residual stresses in rolling and sliding contact, Proc. Inst. Mech. Eng. Symp. on Rolling Contact Fatigue, London, 1963, pp. 54-77. 7 H. S. Nagaraj, Elastoplastic contact of bodies with friction under normal and tangential loading, J. Tribal., Trans. ASME, 106 (1984) 519-526. 8 A. G. Tangena and G. A. M. Hurkx, Calculations of mechanical stresses in electrical contact situations, IEEE Trans. Components, Hybrid, and Manufact. Technol., CHMT-8 (1) (1985) 13-20.
284 9 T. Ihara, M. G, Shaw and B. Bhushan, A finite clement analysis of contact stress and strain in an elastic film on a rigid substrate - Part II: With friction J. ?X&, Trans. ,&!%fE, IO&’ (1986) 534-539. 30 R. B. King and T. C. O’Sullivan, Sliding contact stresses in a two-dimensional layered elastic half-space, Int. J. Solids Strut., 2,7 (1987) 581-597. 11 K. Komvopoulos, N. Saka and N. P. Sub, The role of hard layers in lubricated and dry sliding, J. Tribal, Trans. ASME, 109 (1987) 223-231. 12 H. Tian and N. Saka, Finite element analysis of a two-layer elastic-plastic half-space: normal contact, Wear, 148 (1991) 47. 13 K. I... Johnson, Contact Mechanics, Cambridge University Press, Cambridge, (1985), pp. 119-124. 14 R. D. MindIin, Compliance of elastic bodies in contact, J. Appr &&ch., AS&fE Truns., 16 (1949) 259-268. 15 R. D. kindles and N. Deresicwicz, Elastic spheres in contact under varying oblique forces, J. Appl. Me&., ASME Tram, 20 (1953) 327-344. 16 W. Deresiewicz, Oblique contact of nonspherical elastic bodies, 1. Appt. Mech., ASME Trans., 24 (1957) 623-624. 17 P, K. Cupta, J. A. Walowit and E. F. Finkin, Stress distributions in plane strain layered elastic solids subjected to arbitrary boundary loading, J. Lube. Technol, Tram ASME, 9.5 (1973) 427-433. 18 K. L. Johnson, Contact Mechnnics, Cambridge University Press, Cambridge, (1985), pp. 171-184. 19 K. L. Johnson, Contac: Mechanics, Cambridge University Press, Cambridge, 1985, pp. 202-210. 20 K, L, Johnson, Aspects of friction, &IX. 7th Leeds-Lyon Syrup. on Tribofogv, University of Leeds, 1980, pp. 3-12. 21 E. Rabinowicz, The inRuence of ~rn~a~bil~~ on different tr~bo~og~&a~phenomena, ASLE Tratts. 14 (1971) 204-212. 22 G. M. ~am~iton, Explicit equations for the stresses beneath a sliding spherical contact, Pmt. Inst. Mech. Erg., 197C (3983) 53-59. 23 J, E. Merwin and K. L. Johnson, An analysis of plastic deformation in roIIing contact, Prac. Inst. Me& Eng,, 177 (1963) 676-690.
Appendix A: Nomenclature half contact width Young’s modulus plastic modulus tangential force thickness of top layer thickness of interlayer normal force per unit length normal force pressure tangential traction radius of cylindrical indenter tangential force per unit length components of displacement Greek symbols constant angle of oblique loading ; S pe~e~at~on depth
285 e,, Ed, E+, components of strain friction coefficient p V Poisson’s ratio Mises equivalent stress UMises ff!., =w* components of stress %, Gy yield strength CY shear yield strength TY