Oblique indentation of creeping solids

Eur. J. Mech. A/Solids 19 (2000) 565–584

2000 Éditions scientifiques et médicales Elsevier SAS. All rights reservedS0997-7538(00)00189-3/FLA

Oblique indentation of creeping solids

Joachim Larsson1, Bertil Storåkers2

Department of Solid Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden

(Received 25 October 1999; accepted 28 February 2000)

Abstract – Oblique indentation of power law creeping solids at plane strain conditions is examined with rigid-perfectly plastic material behaviouremerging as an asymptotic case. Indenter profiles are dealt with in general circumstances and represented by homogeneous functions. The core of themethod developed draws on self-similarity and is based on an intermediate flat die solution. By this approach the problem of a moving contact boundarymay be suppressed and the ensuing procedure becomes independent of loading, geometry, history and time. A computational method, based on thereduced procedure, is developed to obtain high accuracy solutions based on finite elements and applicable to non-linear elasticity. The originally statedproblem is then solved subsequently by simple cumulative superposition and results given as a function of impression depth. The relation betweencontact depth and area is found to be invariant and only dependent on the power law exponent, the amount of friction, the profile and the angle ofinclination of the indenter. Detailed results are given for local states of stress and deformation for flat and cylindrical dies at variation of the remainingthree stated parameters. The presence of local stick and slip is given due attention and global relations between loading and indentation depth and contactarea discussed for practical applications. The fundamental framework laid down may be applied to structural assemblies, joints and seals and diverseapplications as flattening of rough surfaces, compaction of powder aggregates and ice-offshore structure interaction. 2000 Éditions scientifiques etmédicales Elsevier SASoblique indentation / creeping solids / self-similarity

1. Introduction

The contact theories of Hertz (1882) for normal loading and Cattaneo (1938), Mindlin (1949) for obliqueloading stand as landmarks for linear elastic analysis of local stress, strain and displacement states. From thesetheories a wealth of basic and technological applications have emerged concerned with members such as gears,rollers, cams, rails, joints and seals to name a few and also issues such as surface roughness flattening, wear andpowder compaction. To analyse matters involved is indeed demanding as non-linearities due to moving contactboundaries and frictional effects have to be appropriately mastered.

When inelastic contact must be taken into account, non-linear material behaviour is added to the difficultiesand a computational approach is imperative. Finite element methods were, however, applied early to inelasticcontact problems and an analysis of indentation by a cylindrical punch of elastic-perfectly plastic solids in planestrain was carried by Akyus and Merwin (1968) and was subsequently generalized by numerous followers. Themany material parameters present, however, cause difficulties to condense results of any generality. Whenelastic effects are insignificant though, an efficient procedure based on self-similarity has recently been broughtforth by Storåkers et al. (1997) for general visco-plastic material behaviour at normal contact.

In the present analysis oblique contact of creeping solids at plane strain conditions confined to proportionalloading is examined. The material behaviour is specified by a power law relation between stress and strain

1 E-mail: [email protected] E-mail: [email protected]

566 J. Larsson, B. Storåkers

rate reducing at one extreme to Newtonian behaviour and on the other to perfect plasticity. The approach restsfundamentally on self-similarity and incremental loading combined with cumulative superposition as originallyworked out for normal frictionless indentation by Storåkers and Larsson (1994).

In an early paper, Matthews (1980) hypothetically showed that, in some circumstances, self-similarity willprevail at power law solids of plastic and creeping materials in contact. Matthews used prototype modelsof spherical indentation at normal contact to determine hardness properties and densification of powderaggregates. Similar ideas of a semi-empirical origin was also put forward by Rubenstein (1981). The issue wasput on a rigorous foundation as setting a boundary value problem by Hill et al. (1989) who showed that in thecase of strain-hardening materials, analysis of spherical (Brinell) indentation could be formulated as a stationaryproblem and well fit for high accuracy finite element solutions which were also subsequently obtained. Theanalysis by Hill et al. (1989) was based on deformation theory of plasticity or alternatively non-linear elasticityand further progress was made by Biwa and Storåkers (1995) by an analysis based on plastic flow theory. Thesewriters took full account of an intermediate incremental approach followed by cumulative superposition. Thisissue is since long well known as regards contact problems within linear elasticity as perhaps first introducedby Mossakovski (1963) and with additional contributions by, e.g., Spence (1968, 1975b) and Hill and Storåkers(1990). It was found, however, by Storåkers and Larsson (1994) that the superposition procedure available forlinear elasticity may be drawn upon directly as well for non-linear creeping materials. This point of attackwas then pursued for plasticity and viscoplasticity by Biwa and Storåkers (1995) and Storåkers et al. (1997),respectively. The influence of finite friction was taken into account by Spence (1973, 1975a) in the case oflinear elastic materials and by Borodich (1993) for non-linear solids. For power law solids perfect bonding wasconsidered by Bower et al. (1993) and finite friction by Carlsson et al. (2000). All these efforts are, however,based on normal loading.

When at present analysing oblique loading in plane strain, the indenter is taken to have any shape savefor being describable by homogeneous functions. Geometric symmetry is assumed for simplicity thoughnot by necessity. Self-similarity is then adopted a priori and the original contact problem is reduced to anintermediate one consisting of a flat rigid punch indenting a non-linear elastic half-space. The problem of amoving contact boundary is eliminated and a stationary finite element mesh is then constructed. Detailed andaccurate field values are obtained based on the intermediate analysis and subsequently the original problemmay be completely solved by simple cumulative superposition. Finite friction is taken into account and resultspresented depend, in essence, solely on the indenter geometry, the angle of obliquity, the creep exponent andthe coefficient of friction. The findings are given for global variables such as relations between the loadingand the depth and area of contact as well as local states for stresses, displacements and sliding. Besides theirintrinsic interest the fundamental findings may be used for different applications such as powder compaction(Akisanya and Cocks, 1995), moving ice floes on off-shore structures (Larsson, 1997), and surface roughness(Larsson et al., 1999).

2. Formulation of the problem

The problem to be examined concerns a creeping half-space, obliquely indented by a rigid punch in planetransverse strain as shown infigure 1. In a quasi-static analysis and in the absence of body forces, equilibriummay be expressed as

∂σij

∂xj= 0, (1)

Oblique indentation of creeping solids 567

Figure 1. Plane oblique contact between a rigid indenter and a deformable half-space.

and strains as

εij = 1

2

(∂ui

∂xj+ ∂uj∂xi

), (2)

in obvious notation with linear kinematics assumed.

The half-space is modelled as an isotropic power law creeping solid with the standard constitutive equation

εij = 3

2

(σe

σc

)n−1sij

σc, (3)

whereσc andn are material parameters and the effective stress is given by

σe =√

3

2sij sij , (4)

according to von Mises and wheresij is the deviatoric stress.

The local punch shape, according tofigure 1, is given by a smooth homogeneous positive function

f (x1)= |x1|pDp−1

, (5)

whereD is a characteristic length of the profile and punch symmetry is assumed for simplicity though not bynecessity. The range of profiles are encompassed by a wedge and a flat die corresponding top= 1 andp→∞,respectively. The punch is indented obliquely by an angleγ .

Starting first with the case of full adhesion between the punch and the creeping solid, in the so far unknowncontact region, the inhomogeneous boundary conditions follow as

u1= h tanγ ; u2= h− |x1|pDp−1

; −a26 x16 a1, x2= 0, (6)

with notation as infigure 1.


Should frictional slip occur, it is conjectured a priori that stick will prevail only in an internal region,−a2+ s26 x16−s1+a1, and in the remaining region slip will be governed by Coulomb friction. The in-planeboundary conditions are then to be supplemented by

|σ12| = µ|σ22|; u1− h tanγ =−λσ12; u2= h− |x1|pDp−1

;(7)

−a26 x16−a2+ s2, a1− s16 x16 a1, x2= 0,

whereλ is a positive multiplier specifying that the direction of slip opposes the direction of the frictional shearstress.

Outside the contact region at the free surface, the remaining boundary conditions reduce to

σ21= σ22= 0; x1<−a2, a1< x1, x2= 0. (8)

As the governing equations are to be formulated in rate form it proves advantageous to tentatively introducereduced variables according to

axi = xi, (9)

hui(xi)= ui (xk, a1, a2, t), (10)

hεij(xi)/a= εij (xk, a1, a2, t), (11)

(h/a)1/nσcσij(xi)= σij (xk, a1, a2, t), (12)

where 2a = a1+ a2 according tofigure 1.

The original field equations, (1)–(3), then reduce to the normalized form

∂σij

∂xj= 0, (13)

εij = 1

2

(∂ui

∂xj+ ∂uj∂xi

), (14)

εij = 3

2σ n−1e sij . (15)

Further, the boundary conditions in rate form for the fully stick case read by equation (6) and (10)

u1= tanγ ; u2= 1; α − 26 x16 α, x2= 0, (16)

with α = a1/a.

In a combined stick-slip situation, the corresponding supplementary slip conditions read by equations (7),(10) and (12)

|σ12| = µ|σ22|; u1− tanγ =−λσ12; u2= 1,(17)

α− 26 x16 α− 2+ β2, α− β16 x16 α, x2= 0,

with β1= s1/a, β2= s2/a.

Summarizing so far it may first be observed from equations (16) and (17) that the reduced problem obtainedconstitutes that of a flat punch with a contact width of 2 resulting from oblique indentation at angleγ to a unit


Figure 2. Flat punch oblique indentation corresponding to the reduced problem.

vertical depth as depicted infigure 2. It is evident already at this stage that the resulting reduced stress state isindependent of the original punch profile but depends only on the relative indentation rate.

In the original problem there are four initially unknown eigenvalues to be determined i.e.a1, a2 whichdefines the asymmetrical contact region ands1, s2 the corresponding stick-slip region, cf.figure 1. By aid of thereduced problem, however, they can be solved pair-wise and consecutively. The impression of the intermediateflat punch problem is homogeneous and a solution may be obtained from an arbitrarily chosen origin. Thestick-slip region withβ1, β2 defined in equation (17) may then be solved first and independently of the originalcontact regimea1, a2. Once this solution is obtained it remains to satisfy the original displacement boundaryconditions, equations (6) and (7). As the indentation depthh has a priori been assumed to be a single functionof the contact regionsa1, a2 by equations (6), (9) and (10), then∫ a1

0u2(x1)dh

dsds = h− x

p1

Dp−1; x1> 0, (18)

and likewise fora2 whenx16 0.

Integration in parts then yields an integral equation

h(x1/α)−∫ x1/α

0u2(x1)dh

dsds = x

p1

Dp−1; x1> 0. (19)

The solution is

h(a)= ap

c+p Dp−1, (20)

where the parameter

c+p =(

1

α

)p− p

∫ ∞α

u2(x1)

xp+11

dx1 (21)

depends only on the profilep and the power law indexn. Obviously the relative range of slip,β1, β2 isindependent of the die shape.


Determining c−p in an analogous way by replacinga1 by −a2 in equation (18), the eigenvalueα inequation (19) and accordinglya1, a2, may finally be determined by

c+p = c−p . (22)

Displacements and strains in the original problem may now be found from cumulative superposition throughintegration by radial paths as explained by Hill and Storåkers (1990) for linear elasticity. Thus by use ofequation (10) the total displacements may be determined as

ui(xk, a1, a2)=∫ t

0ui(xk, a1, a2, t)dt =

∫ a

0ui(r)dh

dsds, (23)

wheres = r/r andr is the distance from the origin to the point of interest given by the coordinates,xk , in thehalf-space,figure 1. By a variable transformation froms to r and using equation (20) the total displacementscan be written as

ui(xk, a1, a2)= ph(a)(r

a

)p ∫ ∞r/a

ui(r)1

rp+1dr , (24)

and likewise strains as

εij (xk, a1, a2)= ph(a)rp−1

ap

∫ ∞r/a

εij (r)1

rpdr , (25)

analogously.

3. Solution procedure

In order to solve the originally stated boundary value problem, first a finite element procedure was developedto solve the intermediate flat die problem in reduced variables. The method adopted was essentially based onearlier analysis of normal contact loading and has been explained in detail earlier, cf., e.g. (Storåkers et al.,1997) and in particular for plane strain (Larsson, 1997). The procedure will only be briefly summarized here.As no symmetry prevails at oblique contact, a complete half-plane has to be modelled. In the axisymmetric caseit proved sufficient to abbreviate the half-space by 50 times the contact radiusa. In the present plane problemdisplacements decline at a slower rate and to maintain high accuracy it was found necessary to extend the finiteelement mesh to 700a, a being half the contact width.

Half the finite element mesh designed is depicted infigure 3consisting of 5000 nine-node isoparametricelements and 25 331 nodes resulting in 45 662 degrees of freedom. In the present analysis of the intermediatereduced problem an impression depth of 0.01 was consistently used and the results appropriately scaled. At theouter boundary the vertical displacements were set to zero. The horizontal displacements were likewise set tozero at the interface with the free surface. This was done to satisfy equilibrium with a minimum of constraints.Some trials with further constrained displacements were tested but did not influence the results significantly.

The non-linear material behaviour in the reduced problem was modelled by the commercial code ABAQUS(1999) by a Ramberg–Osgood material, i.e.

ε = σE+ α

′

E

(σ

σ0

)n−1

σ, (26)


Figure 3. Finite element mesh used at computations. (a) Complete mesh; (b) mesh details in the vicinity of the contact region.

here shown in its uniaxial form. The material parametersE, α′ and σ0 are chosen in such a way that theinfluence of the linear term is suppressed resulting in the desired power-law constitutive relation. The accuracyof the approximation made when simulating non-linear elasticity by a Ramberg–Osgood material has beeninvestigated by Larsson (1997) and it is shown that with proper values of the material parameters inserted theapproximation is very good.

In the commercial code ABAQUS, the equation defining the direction of slip to be opposite the direction ofthe frictional shear stress is given by

du1

ds− tanγ =−λσ12, (27)

wheres is a loading parameter. At a first glance this condition appears different from the one specified inthe reduced problem, equation(17)2. However, integrating equation (27), remembering thatσ12 andu1 vanishwhens = 0, will result in the slip boundary condition of the reduced problem, as further explained by Carlssonet al. (2000).

The reduced stationary problem is governed essentially by the creep exponent,n, the angle of inclination,γ ,and the coefficient of friction,µ, according to Coulombs law. The output variables of interest; contact forces,regions of contact with associated asymmetric slip, stress compliances and displacement fields are eitherevaluated directly from the solution to the reduced problem or determined by cumulative superposition, inthe order to a posteriori justify the procedure just laid down.

4. Results

Explicit results are now presented to shed light on the characteristics of the solution to the oblique contactproblem. As has already been forecast only stress distributions and the relative regions of slip are independentof the die shape and for obvious reasons only selected results will be depicted below.

For flat die indentation,figure 4shows the vertical surface displacement for three values of inclination,γ ,and a set of values for the creep exponent at a given friction coefficient(µ= 0.3). As may be seen there will besubstantial piling-up upstream at higher values ofn while smaller piling-up or even sinking-in prevails in thewake of the contact region. Infigure 5the corresponding horizontal surface displacements are depicted for thesame friction coefficient. Results from other values of friction were also analysed but it was found in generalthat their influence on the strain state well outside the contact area was small. At highern-values,n > 10 say,


Figure 4. Normalized vertical surface displacements in the vicinity of the contact at the reduced flat die problem for various angles of inclination andcreep exponents,µ= 0.3. (a)γ = 0; (b) γ = π/6; (c) γ = π/4. (- - -) Frictionless case,n= 100.


Figure 5. Normalized horizontal surface displacements in the vicinity of the contact at the reduced problem for various angles of inclination and creepexponent,µ= 0.3. (a)γ = 0; (b) γ = π/6; (c) γ = π/4. (- - -) Frictionless case,n= 100.


essentially full adhesion prevails while at lowern-values there is stick as well as slip. As a comparison, thefrictionless case is shown forn= 100 at symmetrical indentation. Further, specifically atn 6 2 andγ = π/4macrosliding will occur.

It should be underlined that forn= 10 and 100, no results are recorded forγ = π/6 in figures 4and5 andahead. In fact the computational procedure did not converge and elements close to the contact boundary becameseverely distorted. This comes as no surprise though as forn→∞, implying perfectly plastic behaviour,particle velocities in the associated slip-line field are indeterminate forγ 6 π/4, from a slip-line field solutionby Green (1954). This was emphasized by Fleck et al. (1992) in their analysis of bonded isolated contactsof powders. For anyγ 6 π/4, the symmetrical Prandtl field, cf. (Hill, 1950), will apply. Atγ = π/4 thecomputational procedure did converge and was well-behaved.

The limiting case ofn→∞ is equivalent to the case of perfect plasticity andn= 1 corresponds to a linearelastic material. Forn-values higher than unity the behaviour of the solution is regular. In linear elasticity,however, the vertical displacement generally contains a logarithmic singularity. For a flat rigid punch indentinga linear elastic half-space normally and assuming frictionless contact, the vertical displacements outside thecontact area are given by

u2= δ2− 2(1− υ2)P

πEln(x1

a+((

x1

a

)2

− 1)1/2)

, (28)

cf., e.g. (Johnson, 1985), whereδ2 is arbitrary. In the present analysis accordingly the eigenvaluecp inequation (20) which governs the relation between the contact area and the indentation depth formally vanishesfor a half-space. In order to assess the accuracy of the finite element procedure designed, the arbitrary constantin equation (28) was determined by setting the displacement equal to zero at 350a. The boundary conditionsfor the reduced problem are accordinglyu2(x1 = ±350) = 0 andu2(x1 = ±1) = 1, c2 resulting in 0.162 forthe frictionless case (0.164 for full adhesion) for a cylinder. Based on the analytical solution, equation (28), theresulting value is 0.153 which conforms well with the present numerical solution.

Once the reduced flat die problem has been solved, the solution for any die shape,p, may be found by simplecumulative superposition as explained above, equations (21)–(25). The most common cases refer to a bluntwedge,p = 1, and a parabolic die,p = 2. In the Hertz approximation it should be noted that a parabolic diemay not be distinguished from a cylindrical one. Concentrating now on cylindrical profiles, infigure 6valuesof the invariantc2= a2/(hD) are plotted versus the creep exponent for three different values of obliquity andconstant friction,µ= 0.3. It may first be seen that the results for the relation between normal indentation depthand contact area are remarkably independent of the angle of inclination despite the severe asymmetry of theloading. Further, since it has been found that the influence of friction on the vertical surface displacement issmall the value ofc2 is also not much affected by the amount of friction. A good approximation forc2 is givenasc2= 1.4e−1.5/n for n > 2.

The integration in equation (24) is performed along radial paths and infigures 7 and 8 the surfacedisplacements in vertical and horizontal directions, respectively, are shown for a cylindrical indenter,p = 2.Essentially the deformation modes show the same features as regards piling-up and sinking-in as for flat dieimpression, according tofigure 4. As may be seen, the asymmetry of the contact region is not very pronounced.At an inclination angle ofπ/4 andn= 100, the unbalance ofa1/a does not exceed 1.15.

As has already been emphasized, the reduced stress distribution in power law creeping solids is independentof the shape of the indenter. Infigures 9and10 the surface stresses are shown for a coefficient of friction equalto 0.3 and various degrees of inclination. There is only a small difference in normal stresses between variousangles of indentation while the difference in shear stress is naturally more pronounced. It should be noticed,


Figure 6.The eigenvaluesc2= a2/(hD) as a function of 1/n at indentation by a cylindrical die for various inclination,µ= 0.3. (∗) γ = 0; (◦) γ = π/6;(+) γ = π/4.

however, that in many cases there is a tensile stress in the wake of the indenter which might be importantfor initiation of fracture. The distributions of surface stresses are shown forn = 3 andn = 100 (asymptoticperfectly plastic behaviour), but similar qualitative features are found for interjacentn-values. Infigure 9(c)and10(c) it may be seen that the contact pressure distribution is almost independent of the impression obliquity.The Prandtl slip-line field solution, cf., e.g. (Hill, 1950), predicts the contact pressure distribution to be uniformfor the case of perfect plasticity.

By computing the variable|σ12|/|µσ22| in the contact region, the influence of various parameters on the stickand slip regions may be studied in some detail.Figure 11 illustrates for a representative set of parameters,n= 5, γ = π/6, the influence of the friction coefficient. As may be seen, slip is initiated at the wake side ofthe indenter and the region grows with decreasing coefficient of friction. Possible macrosliding occurs whenthe shear stress at the other end finally reaches its maximum value according to Coulomb’s law of friction. Inthe special case of normal contact, two symmetrical regions of slip arise, but in this particular case of obliquityslip occurs in region−a26 x16−a2+ s2 at the wake side while the other potential slip region is not present.

In figure 12 reduced values,β2 = s2/a, for the slip region are given as a function of the coefficient offriction and for two different values of the creep exponent, at the specific angles of inclination,γ = π/6and γ = π/4. At low values ofµ macroslip occurs but with increasing values the stick region grows untilthe case of full adhesion prevails. As a general conclusion it is found that for largern-values and moderateinclination(γ < π/4) there is very little slip for coefficients of friction corresponding to ordinary metal surfacesin contact(µ > 0.3).

In the present approach the displacement of the indenter is prescribed and given by the angle of inclination,γ ,corresponding to proportional indentation. Determining the corresponding force, its angle of obliquity will,however, be different. The reduced contact force can be given in dimensionless variables as


Figure 7. Normalized vertical surface displacements in the vicinity of the contact at the indentation of a cylinder for various angles of inclination andcreep exponents,µ= 0.3. (a)γ = 0; (b) γ = π/6; (c) γ = π/4.


Figure 8. Normalized horizontal surface displacements in the vicinity of the contact at the indentation of a cylinder for various angles of inclination andcreep exponents,µ= 0.3. (a)γ = 0; (b) γ = π/6; (c) γ = π/4.


Figure 9. Reduced surface stresses in the vicinity of the contact for various angles of inclination, (–· –) γ = 0; (—) γ = π/6; (- - -) γ = π/4.n= 3 andµ= 0.3. (a) In-plane stress; (b) Shear stress; (c) Normal stress (Contact pressure).


Figure 10. Reduced surface stresses in the vicinity of the contact for various angles of inclination, (–· –) γ = 0; (—) γ = π/4. n= 100 andµ= 0.3.(a) In-plane stress; (b) Shear stress; (c) Normal stress (Contact pressure).


Figure 11.The slip indicator variableσ12/(µσ22) is plotted in the contact region forn= 5 andγ = π/6. (—)µ= 0.15; (- - -)µ= 0.20; (–·–)µ= 0.30.

Fn = Fn

abσc

(a

h

)1/n

=∫ α

α−2σ22dx1, (29)

Ft = Ft

abσc

(a

h

)1/n

=∫ α

α−2σ12dx1, (30)

for the normal and tangential directions, respectively, which are independent of the geometry of the indenter.These components form a resulting force of indentation where infigure 13the reduced normal contact force isplotted versus the tangential force for four material exponents forµ= 0.3. As may be expected the force in thenormal direction is only slightly affected by the obliquity. The deviation between the directions of impressionand resulting force is strong and increases with increasingn-value. Results from rigid-plastic slip-line theoryfor the case of metal junctions as analysed by Green (1954), are also depicted infigure 13. As may be seen thecomparison with the present results is quite satisfactory forn= 100.

In the spirit of Tabor (1951) it is advisable to relate a dimensionless loadF to a representative strain ratea/D.Introducing

F = F

2abσc

(D

a

)1/n

, (31)

according to equations (20), (29) and (30) this relation reduces to

F =(

2

c2

)1/nF

2, (32)

for the cylindrical case,p = 2.


Figure 12.Normalized slip regions2/a as a function of the coefficient of friction for various creep exponents. (a)γ = π/6; (b) γ = π/4.

In the case of purely normal loading at plane strain and a cylindrical die, Larsson (1997) found that within avery good approximation

Fn = κ1(κ2)1/n, (33)

whereκ1, κ2 are universal constants independent ofn and explicitlyκ1= 3 (approximately the Prandtl value,(2+ π/√3)) andκ2= 2/3.

Relating tofigures 6and13, it is seen that equation (33) holds also at oblique indentation. Thus for inclinationanglesγ < π/4, equation (33) is within 1.5% accuracy for alln > 3.

Hill (1992) analysed normal frictionless axisymmetrical creep indentation and determined the depth variationwith time for constant loading. For the present case of oblique loading in plane strain, from equations (29)and (30) the total load is qualitatively given by

F ∼ a(n−1)/nh1/n, (34)

for n > 1.


Figure 13. Normalized normal indentation load versus normalized tangential load for various creep exponents,µ = 0.3. (- - -) Slip-line solutionaccording to Green (1954).

By equation (20), then for constant load, the depth varies with time as

h∼ tp/(n+p−1) (35)

and the contact region as

a ∼ t1/(n+p−1) (36)

for a given die profilep.

If the indentation rate, dh/dt , is prescribed and constant, then the loading varies as

F ∼ t (n−1)/(np) (37)

and the contact region as

a ∼ t1/p. (38)

5. Concluding remarks

Oblique contact of creeping solids indented by a rigid die was examined under fairly general conditions asregards die shape and inclination and also the influence of finite friction. Under proportional indentation it wasshown that the depth of indentation and resulting contact area possess invariance and self-similarity properties.It was shown that a fundamental solution to the specific problem of a flat die of fixed geometry contains all theinformation necessary for the assemblage of the solution to any die shape and may be carried out by simplecumulative superposition. Although the problem was analysed in plane strain, there are four essential input


parameters and a full parameter study was not attempted. Some significant and detailed results were obtainedthough for flat and parabolic die shapes at inclination anglesγ 6 π/4.

For parabolic profiles it was found that the level of inclination has only a small influence on the relationbetween the indentation depth and the area of contact and also on the distribution of contact pressure. This isknown to be so both for linear elastic and perfectly plastic material behaviour but proved to prevail for anycreep exponent. Likewise the influence of friction did not have a large effect. The resulting contact zone did notexhibit pronounced asymmetry. It did increase with the creep exponent but was at most of the order of 0.15 inthe limiting case of perfect plasticity. The phenomenon of piling up showed severe asymmetry, though, whereit was very pronounced upstream the indenter forn-values larger than three, say. It should be noticed that inmany cases there is a tensile stress present in the wake of the indenter which might have consequences for theinitiation of fracture.

At normal contact and linear elastic behaviour, it was proved by Spence (1973, 1975a) that the relativeregions of slip and stick were independent of the indenter shape in case of finite friction. This was found to beso also for oblique indentation of creeping solids. In case of normal contact, slip will be initiated symmetricallyin the outer contact regions. In the oblique case, at least for an inclination angle larger thanπ/6, slip is initiatedat the wake side and the region grows with increasing angle of inclination, decreasing creep exponent andcoefficient of friction. Macrosliding occurs when the stress at the forward end finally reaches its maximumvalue according to Coulomb’s law of friction. In general it was found that for largen-values very little microslipoccurs for coefficients of friction corresponding to ordinary metal surfaces in contact,µ = 0.3 say, and atmoderate inclination,γ = π/4. Accordingly this issue may be dealt with as one of full adhesion and this goesalso for the global relations between loading and indentation depth and contact area as well as for the resultingstress and strain fields well outside the contact region.

No attempt was made at present to include the case of non-proportional indentation. This appears to bea fairly straightforward procedure though, at least for the case of full adhesion, as the current constitutiveequation is truly history independent. The stress distribution at any stage may be solved directly from theintermediate incremental field and subsequent superposition of displacements and strains, as in equations (19),(24), (25) may be solved in analogy as above. It must only be remembered that proper attention must be paidto the changing incremental field during the superposition procedure.

Acknowledgements

The writers are indebted to Dr Shiro Biwa for pertinent and kind advice. This investigation was supportedby the Swedish Research Council for Engineering Sciences (TFR) which is gratefully acknowledged.

References

ABAQUS (User’s manual, Version 5.8), Hibbitt, Karlsson and Sorensen, Providence, RI, 1999.Akisanya A.R., Cocks A.C.F., Stage I compaction of cylindrical particles under non-hydrostatic loading, J. Mech. Phys. Solids 43 (1995) 605–636.Akyus F.A., Merwin J.E., Solution of the nonlinear problems of elastoplasticity by finite element methods, AIAA J. 6 (1968) 1825–1831.Biwa S., Storåkers B., Analysis of fully plastic Brinell indentation, J. Mech. Phys. Solids 43 (1995) 1303–1334.Borodich F.M., The Hertz frictional contact between nonlinear elastic anisotropic bodies (the similarity approach), Int. J. Solids Structures 30 (1993)

1513–1526.Bower A.F., Fleck N.A., Needleman A., Ogbonna N., Indentation of a power law creeping solid, Proc. R. Soc. Lond. A 441 (1993) 97–124.Carlsson S., Larsson P.-L., Biwa S., On frictional effects at inelastic contact between spherical bodies, Int. J. Mech. Sci. 42 (2000) 107–128.Cattaneo C., Sul contatto di due corpi elastici: distribuzione locale degli sforzi, Rendiconti dell’Accademia Nazionale dei Lincei 27 (1938) 342–348,

434–436, 474–478.Fleck N.A., Kuhn L.T., McMeeking R.M., Yielding of metal powder bonded by isolated contacts, J. Mech. Phys. Solids 40 (1992) 1139–1162.


Green A.P., The plastic yielding of metal junctions due to combined shear and pressure, J. Mech. Phys. Solids 2 (1954) 197–211.Hertz H., Uber die Berührung fester elastischer Körper, J. Reine Angew. Math. 92 (1882) 156–171.Hill R., The Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1950.Hill R., Storåkers B., Zdunek A., A theoretical study of the Brinell hardness test, Proc. R. Soc. Lond. A 423 (1989) 301–330.Hill R., Storåkers B., A concise treatment of axisymmetric indentation in elasticity, in: G. Eason, R.W. Ogden (Eds.), Elasticity: Mathematical

Methods and Applications, Ellis Horwood, Chichester, 1990, pp. 199–210.Hill R., Similarity analysis of creep indentation tests, Proc. R. Soc. Lond. A 436 (1992) 617–630.Johnson K.L., Contact Mechanics, Cambridge University Press, Cambridge, 1985.Larsson P.-L., On creep deformation at plane contact problems, Cold Regions Sc. Tech. 26 (1997) 67–82.Larsson J., Biwa S., Storåkers B., Inelastic flattening of rough surfaces, Mech. Mat. 31 (1999) 29–41.Matthews J.R., Indentation hardness and hot pressing, Acta Metall. 28 (1980) 311–318.Mindlin R.D., Compliance of elastic bodies in contact, J. Appl. Mech. 16 (1949) 259–268.Mossakovski V.I., Compression of elastic bodies under condition of adhesion (axisymmetric case), Prik. Mat. Mekh. 27 (1963) 418–427.Rubenstein C., A critical appraisal of static hardness measurements, J. Appl. Mech. 48 (1981) 796–802.Spence D.A., Self similar solutions to adhesive contact problems with incremental loading, Proc. Roy. Soc. A 305 (1968) 55–80.Spence D.A., An eigenvalue problem for elastic contact with finite friction, Proc. Camb. Soc. 73 (1973) 249–268.Spence D.A., The Hertz contact problem with finite friction, J. Elasticity 5 (1975a) 297–319.Spence D.A., Similarity consideration for contact between dissimilar elastic bodies, in: A.O. de Pater, J.J. Kalker (Eds.), The Mechanics of the

Contact between Deformable Bodies, Proc. IUTAM Symp., Enschede, Netherlands, 1974, Delft University Press, Delft, 1975b, pp. 67–76.Storåkers B., Larsson P.-L., On Brinell and Boussinesq indentation of creeping solids, J. Mech. Phys. Solids 42 (1994) 307–332.Storåkers B., Biwa S., Larsson P.-L., Similarity analysis of inelastic contact, Int. J. Solids Structures 34 (1997) 3061–3083.Tabor D., Hardness of Metals, Clarendon Press, Oxford, 1951.

Documents

Oblique indentation of creeping solids