On oblique contact of creeping solids

Journal of the Mechanics and Physics of Solids50 (2002) 2029–2055

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On oblique contact of creeping solidsJoachim Larsson, Bertil Stor)akers∗

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

Received 19 November 2001; received in revised form 25 February 2002; accepted 12 March 2002

Abstract

Oblique indentation of power-law creeping solids by a rigid die is analysed in three dimensionswith perfectly plastic behaviour emerging as an asymptotic case. Indenter pro2les are prescribedto be axisymmetric for simplicity but not by necessity. Invariance and generality is aimed at, asthe problem is governed by only four essential parameters, i.e. the die pro2le, p, the indentationangle, �, the power-law exponent, n, and the coe4cient of friction, �. The solution strategy isbased on a self-similar transformation resulting in a reduced problem corresponding to 6at dieindentation of complete contact. The reduced auxiliary problem, being independent of loading,history and time, was solved by a three-dimensional 2nite element analysis characterized by highaccuracy. Subsequently, cumulative superposition was used to resolve the original problem andglobal and invariant relations between force, depth and contact area were determined. Detailedresults are given for the location and shape of the contact region and stick=slip contours as wellas for local states of surface stresses and deformation at 6at and spherical indenters. Due to theasymmetry prevailing, it was found that in the spherical case, contact contours proved to be ovaland shifted, although with normal and tangential forces only weakly coupled. Finite friction ascompared to full adhesion proved to have only a minor e9ect on global relations. The frameworklaid down may be applied to the contact of structural assemblies subjected especially to elevatedtemperatures and also to various issues such as compaction of powder aggregates, 6attening ofrough surfaces and plastic impact. ? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: A. Indentation and hardness; A. Creep; B. Ideally plastic material; B. Contact mechanics;B. Friction; C. Finite elements

1. Introduction

The theory laid down by Hertz (1882) for normal frictionless contact between twononconforming bodies stands as a landmark in linear elasticity. More than half a century

∗ Corresponding author. Tel.: +46-8-790086410; fax: +46-8-4112418.E-mail address: [email protected] (B. Stor)akers).

0022-5096/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S0022 -5096(02)00020 -0

2030 J. Larsson, B. Stor'akers / J. Mech. Phys. Solids 50 (2002) 2029–2055

later, Cattaneo (1938) and Mindlin (1949) supplemented Hertz theory by includingoblique contact and taking frictional e9ects into account. From this foundation a wealthof basic and technological applications has emerged concerning members such as gears,rollers, bearings, cams, rails, joints and seals to name a few and also further issuespertaining to surface roughness 6attening, wear, fretting fatigue and compaction ofgranular media. In many circumstances, however, inelastic behaviour has to be takeninto account and accordingly there is also a steady stream of analyses to include suchissues. Initiation of plastic 6ow does not occur at the surface of Hertz contact, butoccurs below. Thus, at increasing load, plastic 6ow will be contained until it reachesthe contact surface and fully plastic 6ow progresses. These matters were delineated byJohnson (1970) and subsequently, Hill et al. (1989) laid down a general theory forspherical indentation of fully plastic deformation at power-law strain-hardening solids.

Although oblique contact of elastic solids has been thoroughly analysed in the lit-erature, corresponding contributions for inelastic material behaviour seem very rare.Recently, oblique impact has been studied by Lim and Stronge (1999) for elastic–plastic solids and creep indentation by Larsson and Stor)akers (2000) in both casescon2ned though to plane strain situations. The present objective is to carry out ageneral three-dimensional analysis of oblique indentation of pure power-law creepingsolids which include the asymptotic cases of Newtonian 6ow on one hand and per-fectly plasticity on the other. The outcome may be immediately applied, e.g. to powdercompaction, Stor)akers et al. (1999), 6attening of rough surfaces, Larsson et al. (1999)and inelastic impact, Stor)akers and Larsson (2000); these earlier investigations werebased solely on normal frictionless contact.

A general theory for normal frictionless indentation of creeping solids was laiddown by Stor)akers and Larsson (1994). It was based on self-similarity and incre-mental loading combined with subsequent cumulative superposition. This fundamentalapproach was then followed by the analyses of strain-hardening plastic 6ow by Biwaand Stor)akers (1995), viscoplasticity by Stor)akers et al. (1997) and inclusion of 2nitefriction by Carlsson et al. (2000). In a recent analysis by Larsson and Stor)akers (2000)of oblique indentation of creep at plane strain, some additional background was givento the relevant progress of self-similarity analysis, obtained by various writers, e.g.Spence (1975), Hill (1992), Borodich (1993) and Bower et al. (1993).

In the three-dimensional analysis to be performed, the indenter pro2le is taken tohave any shape save for being describable by smooth homogeneous functions. Geomet-ric axisymmetry is assumed for practical reasons though not by necessity. The problemis governed by essentially four parameters, i.e. the indenter pro2le, the angle of impres-sion, the power-law exponent and the coe4cient of friction. Self-similarity is adopteda priori and the original contact problem is then reduced to an intermediate auxiliaryproblem consisting of a rigid 6at punch indenting a nonlinear elastic half-space. Theproblem of a moving boundary is then eliminated and a stationary 2nite element meshmay be constructed. Detailed and high accuracy 2eld values are then obtained based onthe intermediate analysis, and a subsequent simple cumulative superposition is appliedto solve the original problem completely. The present fundamental di4culty derivesfrom an iteration procedure necessary to determine the location, shape and size of theunknown contact contour as incomplete contact initially prevails in the originally posed

J. Larsson, B. Stor'akers / J. Mech. Phys. Solids 50 (2002) 2029–2055 2031

problem. This procedure, which is not required in the plane strain situation, Larssonand Stor)akers (2000), is to be iterated over a fully three-dimensional 2nite elementanalysis which is quite demanding. By an appropriate assumption regarding the contactcontour, the numerical labour was, however, overcome. Results may then be generatedfor di9erent 2eld values including stick and slip zones and relative slip vectors. Re-sulting contact regions will be shown and global variables which concern the relationsbetween loading and contact depth and area will be given as well as surface values ofstresses and deformation.

2. Formulation of the problem

The problem to be analysed refers to a creeping half-space, obliquely indented by arigid punch. A fully three-dimensional analysis is aimed at and common punch pro2leswill be dealt with.

In a quasi-static analysis and in the absence of body forces, equilibrium equationsmay be expressed as

9�ij9xj

= 0 (1)

and strain–displacement relations as

ij =12

(9ui9xj

+9uj9xi

)(2)

in obvious notation with assumed linear kinematics.The half-space is subjected to nonlinear creep with a power-law (Norton) constitutive

equation

ij =32

(�e

�c

)n−1 sij�c; (3)

where a dot denotes di9erentiation with respect to natural time and �c and n arematerial parameters.

The e9ective stress is given by

�e =√

3sijsij=2 (4)

according to von Mises, where sij is the deviatoric stress.The punch may have a general shape, f = f(x1; x2), save for being smooth and

convex. For practical reasons and simplicity, the pro2le, as shown in Fig. 1, is presentlyrestricted to obey axisymmetry and homogeneity as given by

f(r) =rp

Dp−1 ; (5)

where r is the polar radius and D is a characteristic length. The range of pro2les isencompassed by a blunt cone and a 6at cylindrical die corresponding to p = 1 andp→ ∞, respectively. The rigid punch is indented obliquely by an angle �.


Fig. 1. Oblique contact between a rigid indenter and a deformable half-space.

Referring to Fig. 1 and 2rst considering the case of fully adhesive contact betweenthe indenter and the half-space, in the so far unknown contact region, the inhomoge-neous boundary conditions read

u 1 = h tan �; u 2 = 0; u3 = h− rp

Dp−1 ; r6 a�(�); x3 = 0; (6)

where the contact contour, given by �(�) and a, is to be determined. Normalizing �(�)to unity at �= 0, in the special case of purely normal indentation, �= 0, the parametera will reduce to the radius of the contact contour.

In the case of frictional slip, it is conjectured a priori that stick will prevail onlyin a single internal contact region, r6 a�(�), where �(�)6 �(�) and the slip in theremaining contact region is governed by Coulomb friction. The boundary conditions,Eq. (6), in the slip zone are then to be replaced by√

(�231 + �2

32)=|�33| = �; �3i=√�2

31 + �232 = −si=

√s21 + s22;

u3 = h− rp

Dp−1 ; a�(�)6 r6 a�(�); x3 = 0; (7)

where si is the slip velocity in the i-direction being opposed to the direction of thefrictional shear stress.

Outside the contact region at the free surface, the remaining boundary conditionsreduce to

�31 = �32 = �33 = 0; r ¿a�(�); x3 = 0: (8)

Formulation of the remaining remote boundary conditions will be discussed in detail,below.

As the governing equations are to be formulated in rate form, it proves advantageousin tentatively introducing reduced variables according to

axi = xi; (9)


Fig. 2. Oblique 6at punch indentation corresponding to the reduced problem.

hu i(xk) = u i(xk ; a; t); (10)

hij(xk)=a= ij(xk ; a; t); (11)

(h=a)1=n�c�ij(xk) = �ij(xk ; a; t): (12)

The original 2eld equations (1)–(3) then reduce to the normalized form9�ij9xj

= 0; (13)

ij =12

(9u i9xj

+9u j9xi

); (14)

ij =32�n−1

e sij : (15)

The boundary conditions for the fully stick case, Eq. (6), now read by the aid ofEq. (10)

u 1 = tan �; u 2 = 0; u 3 = 1; r6 �(�); x3 = 0: (16)

In a combined stick–slip situation, the corresponding supplementary slip conditions(7) reduce by Eqs. (10) and (12) to√

(�231 + �2

32)/

|�33| = �; �3i

/√�2

31 + �232= − si

/√s21 + s22;

u 3 = 1; �(�)6 r6 �(�); x3 = 0: (17)

It may then be observed from Eqs. (16) and (17) that the reduced problem laiddown formally constitutes that of a 6at punch obliquely indented by an angle � to aunit vertical depth and with an unknown contact contour r6 �(�), as depicted in Fig. 2.

In the originally stated problem, Eqs. (6) and (7), there is one eigenvalue, thecharacteristic radius a, and two eigenfunctions �(�) and �(�) to be determined. In ageneral case of 2nite friction, � and � are coupled. Leaving for a moment the solution


of this combined problem to be further explained in detail below, in the case of fullyadhesive behaviour, � ≡ �, a solution may be obtained in a straightforward way.

As the indentation depth, h, has been a priori assumed to be a single function ofthe characteristic radius, a, by Eqs. (6), (9) and (10), then∫ a

0u 3(r)

dhds

ds= h− rp

Dp−1 : (18)

As the impression of the intermediate 6at punch problem is uniform, integration inparts by Eq. (18), then yields a Volterra integral equation of the second kind,

h(r=�(�)) −∫ r=�(�)

0u 3(r)

dhds

ds=rp

Dp−1 : (19)

By a variable transformation, the solution is

h(a) =ap

cp(�)Dp−1 (20)

with the parameter

cp(�) =(

1�(�)

)p− p

∫ ∞

�(�)

u 3(r)

rp+1 dr; (21)

where �(�) is to be determined such that

cp(�) = const: (22)

The parameter cp accordingly depends only on the pro2le, p, the power-law index,n, the angle of obliquity, �, and the coe4cient of friction, �, in the case of 2nitefriction.

Once the reduced problem has been solved, 2eld values in the original problem maybe found by simple cumulative superposition through integration by radial paths inanalogy with the method proposed by Hill and Stor)akers (1990) for the axisymmetriclinear elastic case. By integration of Eq. (10), the displacement 2eld follows as

ui(xk ; a) =∫ t

0u i(xk ; a; t) dt =

∫ a

0u i(r)

dhdw

dw; (23)

where w = r=r and r is the distance from the origin to an arbitrary point, xk , in thehalf-space.

Using Eq. (20), Eq. (23) may be transformed to read

ui(xk ; a) = ph(a)( ra

)p ∫ ∞

r=au i(r)

1

rp+1 dr (24)

and likewise for the strain 2eld

ij(xk ; a) = ph(a)rp−1

ap

∫ ∞

r=aij(r)

1rp

dr: (25)

The stress 2eld may be read directly from Eq. (12).It is of immediate interest to investigate when the theory laid down is applicable in

realistic situations. The in6uence of elasticity has been assumed negligible and in thepresent asymptotic case of perfect plasticity, some features are previously known. In


Fig. 3. Finite element mesh used at computations: (a) complete mesh and (b) mesh details in the vicinityof the contact region (top view).

the current context and notation, Johnson (1970) introduced a representative parameter� = aE=(D�y), where E is Young’s modulus and �y is the yield stress in simplecompression. Johnson (1985) found that at spherical indentation, the elastic–plasticregime was entered at �=1:5 roughly. At increasing load, plastic 6ow is still containedbut when the plastic zone reaches the free surface, the material can extend more freely.Elastic e9ects may then be neglected locally and the contact is said to be fully plastic.Based on experimental 2ndings, Johnson (1985) proposed that for �¿ 15 full plasticitywill prevail in the spherical case. This was later con2rmed by Biwa and Stor)akers(1995) from computations based on elastic–plastic theory. Still higher values of � wererecently suggested by Mesarovic and Fleck (1999). At the other limit of the similarityregime, i.e. at large � values, the assumption of linear kinematics is no longer wellapplicable. A 2nite deformation description has to be adopted, where especially largerotations at the contour have to be taken into account. In a general case, a self-similarityanalysis starts to deteriorate when a=D is ¿ 0:1, say.

3. Solution procedure

In order to explicitly solve the originally stated boundary value problem, 2rst a2nite element procedure was developed to solve the intermediate 6at die problem inreduced variables. The method adopted was essentially based on an earlier analysisof normal frictionless indentation as has been explained in detail by Stor)akers et al.(1997). The in6uence of 2nite friction was subsequently taken into account by Carlssonet al. (2000). A 2rst e9ort to deal with the present problem of oblique loading wasmade by Larsson and Stor)akers (2000) analysing a plane strain situation which is tobe followed here in appropriate circumstances.

One half of a half-space needs to be modelled by 2nite elements as symmetry pre-vails with respect to the plane x2 = 0, according to Fig. 3a. A 2nite remote boundaryhas to be introduced and this was chosen as 50 times the characteristic contact radius a.At a distance, displacements vary as r1−2n and the lowest decay is for the linear case,n=1. For this case, r=50a was found su4ciently large to obtain a three-digit accuracy.At the outer boundary, the displacements were set to zero at all nodes, for simplicity.Some trials with alternative constraints were tested but did not in6uence the resultssigni2cantly. The mesh deployed here consists of 21, 600 eight-node isoparametric


trilinear brick, hybrid (constant pressure) elements and 63, 590 nodes resulting in 104,130 degrees of freedom. In Fig. 3b, mesh details are shown in the vicinity of the con-tact region. When modelling 2nite frictional contact a more dense and less progressivedivision in the contact region was used. For high n values, pronounced piling-up wasfound and the use of linear elements proved to be too sti9 due to “parasitic shear”in the regions of local bending. This is a well-known phenomenon and described, e.g.by Cook (1989). Quadratic brick elements were deployed instead, with the hydrostaticpressure interpolated linearly, to avoid this spurious shear. Since combining quadraticelements with 2nite friction is very demanding from a numerical point of view, ele-ments in the contact zone were kept linear in both models. The quadratic model usedfor high n values (n¿ 10) had almost the same number of degrees of freedom as thefully linear model.

The nonlinear behaviour in the reduced problem was modelled by a Ramberg–Osgood material, i.e.

=�E

+�′

E

(��0

)n−1

� (26)

here shown in its uniaxial form. In the 2nite element model, the commercial codeABAQUS (2001) was employed and the material parameters E; �′ and �0 in Eq. (26)were chosen in such a way that the in6uence of the linear term is suppressed resultingin the desired power-law constitutive relation. The accuracy of the approximation madewhen simulating nonlinear elasticity by a Ramberg–Osgood material has been earlierinvestigated in a contact problem by Larsson (1997) and it was shown that with propervalues of the material parameters inserted in Eq. (26), the approximation is quitesatisfactory.

The fundamental numerical issue to be solved is to determine the unknown contactcontour, �(�), and the associated interior stick–slip contour, �(�), cf. Fig. 2. In the FEmodel, the punch was constructed by rigid elements, and the contact was dealt with bya master=slave formulation and friction by Coulomb’s law, allowing a small amountof relative “elastic slip” (10−7). The stick–slip contour, �(�), is given, without furthere9orts, once the solution to the reduced problem has been found.

As to the contact contour, it may 2rst be remembered that Mindlin (1949) foundthat for incompressible linear elastic materials the contour is a circle precisely at spher-ical oblique indentation. It seems reasonable then for nonlinear materials to adopt anelliptical contour as a 2rst approximation and the resulting accuracy to be examineda posteriori. In Fig. 4, the major and minor axes together with the shift are de2nedby the parameters b and d. The eigencondition demands for spherical indentation thatc2(�), Eq. (21), should be constant along any ray on the surface from the origin. Here,this condition is enforced merely along three directions, corresponding to c+

2 ; cm2 and

c−2 in Fig. 4, in order to 2nd the best choice of b and d.An introduced error measure |c+

2 − cm2 | + |c+2 − c−2 | was minimized by the variation

of the parameters b and d through an iterative procedure. As every iteration requires aremeshing of the 2nite element procedure and accordingly can be very time consuming,the choice of an iteration procedure should be carefully contemplated. A successfulscheme was found to start with a circular contour and change the axes (1 + b)=2 and


Fig. 4. De2nition of ellipticity parameters in the iteration scheme.

d, Fig. 4, in turn. In Eq. (21), where c2(�) is de2ned, in this case for instance c−2 isgiven by

c−2 =1

b2 + 2

∫ −b

−∞

u 3(x1)|x1|3 dx1 (27)

and likewise for c+2 and cm2 .

At a typical iteration, k, the origin is shifted and a new b value is determined by

bk+1 =

[c+

2k − c−2; k +(

1

bk

)2]−1=2

(28)

and consecutively for d

dk+1 =

(c+

2k − cm2; k +(

1

dk

)2)−1=2

−(

1 − bk2

)2

1=2

: (29)

This procedure of alternating iterations is then continued until the error falls shortwithin a speci2ed tolerance (0.001 presently). Convergence proved to be quite fast andfor most cases computed, a solution with the tolerance prescribed was achieved afterat most 15 iterations.

The approximation introduced by assuming an elliptically projected contact regionmay now be examined a posteriori. By determining c2 values for intermediate direc-tions, typically 30 directions in the fan-shaped grid in Fig. 3b, it was found that therelative variation of c2 was ¡ 2% for all the cases analysed (�6 "=3).

For large angles of inclination, 2nite friction analyses are very time consuming, dueto signi2cant relative slip, and present results in this context are con2ned to fullyadhesive contact for �¿"=6 and n¿ 3. The time consumption increases both with


Fig. 5. The shape and shift of the contact region for normal spherical indentation and at severe obliquity(� = "=3; n = 5; 100 and � → ∞).

increasing angle of inclination and creep exponent. Typically, a single FE analysistakes about 20 h in the case of fully adhesive contact and at least twice as much inthe case of 2nite friction. The contact region iteration procedure further requires abouta tenfold increase. In Fig. 5, the shape and the shift of the contact contour are shownfor the most extreme case analysed, � = "=3; n = 1; 5 and 100, in comparison withthe normally indented spherical case. As may be seen, the shift is about 16% and thelength of the horizontal axis exceeds the vertical one by 6% for n = 100, while thecontact contour is independent of the level of inclination for n= 1.

The cumulative superposition used to determine c2, Eq. (21), and displacement 2elds,Eq. (23), reduces to simple integration along radial paths. It was readily carried outusing Gauss quadrature.

4. Results

The reduced stationary problem laid down is governed essentially by the creep expo-nent, n, the angle of inclination, �, and the coe4cient of friction, �. Solving the originalcontact problem further depends on the pro2le exponent, p. In a complete investigation,there is an abundance of combinations to analyse and consequently, no internal 2elddistributions will be reported here, and the coe4cient of friction is restricted basicallyto the two cases � = 0:3 and full adhesion. The global output variables which are ofprimary interest are contact forces, stress compliances and regions of contact with as-sociated stick and slip contours together with the kinematic invariant c2, governing the


Fig. 6. Normalized vertical 6at die surface displacements u3=h at the symmetry plane for selected n valuesand various angles of inclination at the reduced problem: (a) normal indentation (�=0); �=0:3 (frictionless,dashed line); (b) � = "=6; � = 0:3; and (c) � = "=3; � → ∞.

relation between contact area and depth. Selected stress and displacement 2elds willbe shown for the surface.

First 6at indentation of a cylinder, p→ ∞, is analysed implying complete contact. InFig. 6, vertical surface displacements are shown in a plane x2 =0 for di9erent values ofinclination angle, �, the creep exponent, n, and for friction coe4cients �=0:3 (Figs. 6aand b) and �→ ∞, i.e. full adhesion (Fig. 6c). In general, it was found that there wasonly a slight deviation in vertical surface displacements comparing 2nite friction andfull adhesion. This leads to the invariant relation, Eq. (21), between indentation depthand contact area being only weakly dependent on the friction coe4cient. In the obliquecase, it may 2rst be seen that substantial piling-up will prevail up-stream, increasingwith the power-law exponent, while down-stream the piling-up is much less pronouncedand even sinking-in will occur for lower n values as vertical displacements tend tobecome axisymmetric. Further, it is evident that in the case of nonlinear creep, surfacedeformations are very localized and essentially con2ned within a region correspondingto twice the contact radius. In Fig. 7, the corresponding horizontal surface displacements


Fig. 7. Normalized horizontal 6at die surface displacements u1=h at the symmetry plane for selected n valuesand various angles of inclination at the reduced problem: (a) normal indentation (�=0); �=0:3 (frictionless,dashed line); (b) � = "=6; � = 0:3; and (c) � = "=3; � → ∞.

are depicted for the same set of parameters. In Figs. 7a and b the coe4cient of frictionequals 0.3 and regions of stick and slip may be discerned while in Fig. 7c full adhesionis prescribed, and accordingly, the horizontal displacements in the contact region areuniform.

In Fig. 8, the transverse horizontal surface displacements are depicted for a repre-sentative case of oblique indentation and 2nite friction for various n values. Accordingto the analytical solution of the linear, incompressible case, transverse displacementsshould vanish. The deviation between the present numerical solution for n= 1 and theanalytical one is due to the discretization errors and the fact that there is macroslidingin the depicted case.

For the particular case of n = 100, implying asymptotically perfect plasticity, thecontact pressure and aligned shear stress distributions are depicted in Fig. 9. FromFig. 9a it may be seen that despite rather severe indentation inclinations, pressuredistributions are still fairly axisymmetric. Fig. 9b shows that shear stresses are a decadelower but still at �="=6, nearly antisymmetric. There are some 6uctuations in the stressdistributions shown and likewise in Fig. 14. These are due to the fact that the stresses


Fig. 8. Normalized transverse 6at die surface displacements u2=h at x1 = 0 for selected n values and� = "=6; � = 0:3.

Fig. 9. Reduced 6at die surface stresses at the symmetry plane for n = 100; � → ∞ and various angles ofinclination: (a) contact pressure and (b) aligned shear stress.

have been evaluated at Gauss points just below the surface, where it is known thatsuch numerical oscillations might occur, in contrast to that at the very surface.

For a general indenter of pro2le, p, 2rst the reduced problem has to be properlysolved in accordance to the shape and shift of the contact region, Eqs. (27)–(29).Subsequently, the solution to the original problem may be determined by cumulativesuperposition as explained above, Eqs. (23)–(25). The most common cases refer to a(blunt) cone, p= 1, and a parabolic die, p= 2, which in the latter case is equivalentto a spherical indenter within the Hertz approximation. Focusing attention on spheres,in Fig. 10, the normalized contact region is shown for indentation angles �= "=6; "=3and for a set of n values. In general, it may be seen that the contact regions all havea closely circular shape. For instance, for �= "=6, the magnitude of the major axis inthe elliptical approximation di9ers by ¡ 1% from the minor axis. It is clear, though,that the centre of the contact region is shifted from the origin and the shift increases


Fig. 10. The shape and shift of the contact region together with ellipticity parameters for various n values(1; 2; 5 and 100): (a) � = "=6; � = 0:3 and (b) � = "=3 and full adhesion.

with the value of n. For n= 1, though, the contact contour is precisely circular and itscentre coincides with the point of maximum indentation depth according to Mindlin(1949).


Further, in Fig. 10, the amount of shift and shape of the contact region as a functionof n is shown by the use of the principal axes, (1 + b)=2 and d, as de2ned in Fig. 4.It may be seen that both the shift and the deviation from a circle increases with thelevel of inclination save for the linear case, n= 1. From Fig. 10b, it is evident thoughthat there is still only a relative di9erence between the major and minor axes of about6% for the most extreme case.

In Fig. 11a, the kinematic invariant c2 relating indentation depth to contact area, Eq.(20), is shown as a function of n for three di9erent angles of inclination. The di9erencebetween a full adhesion and a 2nite friction solution was found to be small. In contrast,comparing frictionless and fully adhesive normal contact yields a signi2cant di9erence(up to 12%) for high n values. Consequently, when choosing between fully adhesionalor frictionless contact analysing normal indentation, modelling of full adhesion is ingeneral in better agreement with the intermediate case of 2nite friction, say at drymetal to metal contact.

Only in the case of normal contact does c2 yield an exact relation between indentationdepth and true contact area, since for the oblique case the contact area also dependson the contact shape. As in the elliptical shape approximation, the true contact area isproportional to d(1+b)=2, cf. Fig. 4, a more physically related measure when analysingoblique contact would be

c2 = a2d(1 + b)=(2hD): (30)

This new invariant c2 is plotted versus n in Fig. 11b. It may be seen that the relationbetween contact area and depth is remarkably independent of the inclination angle forall the values of n.

To determine the displacement 2eld for the spherical case, integration in Eq. (24)has to be performed along radial paths. In Figs. 12 and 13, respectively, vertical andhorizontal surface displacements are shown for various � and n values evaluated for aspherical indenter, p=2. Essentially, the deformation modes show the same features asregards piling-up and sinking-in as for 6at die impression, according to Figs. 6 and 7.

In Fig. 14, contact pressure and shear stress distributions are shown at sphericalindentation for the same parameters as given in Fig. 9, i.e. the 6at die case. This beingthe asymptotic perfectly plastic case, it may be seen that the two cases show verysimilar characteristics.

Distributions of reduced stresses in three dimensions are shown in Fig. 15, wherethe contact pressure and the horizontal shear stress are plotted in the contact zone fora representative case when n = 3; � = "=6 and � = 0:3. The contact pressure is againclose to being axisymmetric and the stick–slip contour may be detected by the gradientdiscontinuity of the contact pressure and even more so for the shear stress distributionin Fig. 15b. In general, the trends shown in Fig. 15 are also found for other n values.The singular behaviour in stresses closer to the contact contour (Fig. 15 being truncatedat 97% of the contact radius) decreases with increasing creep exponent. For very highn values approaching perfect plasticity, the singularity vanishes and the contact pressureis close to being uniform.

By computing the variable√�2

31 + �232=(�|�33|) in the contact region, the in6uence

of various parameters on the stick and slip regions may be studied in some detail. Since


Fig. 11. The invariant parameter, c2, determining the relation between depth and contact area for a sphericalpunch as a function of 1=n, for �=0; "=6 and "=3. Also the normal frictionless case is shown: (a) c2=a2=(hD)and (b) relation between depth and true area assuming elliptical contact area.

all the stress components are equally scaled with respect to the indentation rate, thedivision into stick and slip regions is invariant. Fig. 16 illustrates, for a representativeset of parameters, n = 3 and � = "=6, the in6uence of the coe4cient of friction on


Fig. 12. Normalized vertical surface displacements at the symmetry plane for a spherical punch plotted forselected n values and various angles of inclination: (a) normal indentation (� = 0); � = 0:3 (frictionless,dashed line); (b) � = "=6; � = 0:3; and (c) � = "=3; � → ∞.

the stick and slip regions. As may be seen, the stick zone is neither circular nor sym-metrically located as it is for linear materials. In this case, stick dominates up-streamand macrosliding occurs for � = 0:15 when the stick zone vanishes as in Fig. 16d.Also, relative surface slip vectors are shown in Fig. 16 and in general, it can be seenthat even at a low level of inclination, local slip is mainly opposed to that of thehorizontal displacements prescribed. It was conjectured initially that a single interiorstick zone would prevail in all the circumstances before macrosliding. As may be seenin Fig. 16, however, this assumption is not exactly true since there exists a very smallindependent stick region close to the outer contour characterized by the occurrence ofa minute region of opposite slip.

In the present analysis, the displacement of a rigid indenter has been prescribedand given by the angle � corresponding to proportional indentation. Obviously, it isof major interest to determine the resulting forces. To normalize the forces in a ra-tional way it may 2rst be observed that the nominal pressure may, by Eq. (12), be


Fig. 13. Normalized horizontal surface displacements at the symmetry plane for a spherical punch plottedfor selected n values and various angles of inclination: (a) normal indentation (�= 0); �= 0:3 (frictionless,dashed line); (b) � = "=6; � = 0:3; and (c) � = "=3; � → ∞.

expressed as

FnA

=1A

∫A

(−�33) dA=�cA

(ha

)1=n ∫A

(−�33) dA; (31)

where the projected contact area is given by

A= a2A=a2

2

∫ 2"

0�(�)2 d�: (32)

Accordingly, the normal contact force may be expressed in a reduced form as

Fn =FnA�c

(a

h

)1=n

: (33)

In analogy, the corresponding reduced tangential component F t is determined byintegration from Eq. (31) but with the reduced shear stress component �31 introduced.


Fig. 14. Reduced surface stresses at the symmetry plane for n= 100; � → ∞; p= 2 and various angles ofinclination: (a) contact pressure and (b) aligned shear stress.

First, the problem of a 6at cylindrical indenter will be dealt with for the whole rangeof inclination angles, 06 �6 "=2, cf. Fig. 2. Obviously, the normalization with respectto h in Eq. (33) will not be appropriate as h → 0 for � → "=2. Instead, the resultingoblique impression, q, cf. Fig. 1, is employed and the reduced normal contact force is


Fig. 15. Reduced surface stresses at the contact zone for n = 3; � = "=6; � = 0:3 and p = 2: (a) contactpressure and (b) aligned shear stress.

given by

UFn =FnA�c

(aq

)1=n

(34)

and analogously for the corresponding shear force UFt .


Fig. 16. Stick (dark) and slip (light) regions and corresponding slip vectors for � = "=6 and n = 3:(a) � = 0:30; (b) � = 0:25; (c) � = 0:20 and (d) � = 0:15.

Fig. 17. Normalized normal indentation load versus normalized tangential load for 6at die indentation forvarious creep exponents and full adhesion with inclination angles indicated.


Fig. 18. Normalized normal indentation load versus normalized tangential load for 6at die indentation of per-fectly plastic solids for fully adhesive contact with inclination angles indicated. Two- and three-dimensionalcomparison.

In Fig. 17, the so-reduced normal force UFn is plotted versus the tangential forceUFt for the three-dimensional 6at die case at full adhesion when n = 1; 3; 5 and 100.It may 2rst be noted that n = 1 corresponds to oblique indentation of a linear elasticincompressible solid. The analytical solution is well known, cf., e.g. Johnson (1985),and reads in the current notation

Fn =8 cos �

3qa�c; Ft =

16 sin �9

qa�c (35)

or in the reduced form by Eq. (34)

UFn = 8 cos �=(3"); UFt = 16 sin �=(9"): (36)

The results shown in Fig. 17 for n= 1 are indistinguishable from those of Eq. (36).At the other end, when n= 100, approaching perfectly plastic behaviour, the curve

shown in Fig. 17 is compared with an empirically based formula for metallic junctionsproposed by Tabor (1958). The curves are similar in shape although the present resultsimply a somewhat sti9er response for the shear force.

In particular, for the case of n= 100, it is of interest to compare the 3D results withthose of plane strain in particular as a slip-line theory solution is available, cf. Green(1954). The force distribution is reproduced in Fig. 18 with the same normalization asin Eq. (34), where a now denotes half the contact width in the two-dimensional case.Also, in Fig. 18, 2D results are given for n = 100 by Larsson and Stor)akers (2000)


Fig. 19. Normalized normal indentation load versus normalized tangential load for spherical indentationdepicted for various creep exponents and levels of friction: full adhesion (∗); 2nite friction; �= 0:3 ( ) andfrictionless (5).

supplemented with some additional values at higher angles of inclination. It may beseen that all the curves show a similar behaviour although the shear force response isslightly sti9er in the three-dimensional case.

In Fig. 19, the reduced normal component is plotted versus the tangential one forthe spherical case, these two form a resulting contact force and are normalized withrespect to Eq. (33). The results are shown for � = 0:3 and for adhesive behaviourwith inclination levels, �, indicated. For normal contact, �= 0, also frictionless contactresults are shown.

It should be emphasized that in contrast to Figs. 17 and 18, where a solution is soughtfor indentation by a rigid die of a given circular contour, in Fig. 19, the indenter is ofa spherical shape where the contact region is to be determined as to shift and contourfor di9erent values of �; n and �, cf. Fig. 10. It is well known, however, that for linearincompressible materials, n=1, normal and tangential forces are uncoupled in the sensethat purely tangential tractions do not give rise to any vertical displacements and reci-procally so. As may be seen in Fig. 19 this is so presently and also approximately forother n values in the considered range of inclinations. Since the reduced forces are nor-malized also with respect to the contact area, Eq. (33), the decrease in the normal forcedue to increasing inclination is counteracted by the decrease of the area, cf. Fig. 10.These two e9ects almost balance each other and there is virtually no coupling for�6 "=3 for all the values of n. It is also evident that at 2nite friction, � = 0:3, the


in6uence is not very signi2cant compared to the fully adhesive case but predicts aslightly lower shear sti9ness generally. Only close to the limit of macrosliding is thedi9erence in tangential force appreciable. The results depicted in Fig. 19 for the casesof 2nite friction with the associated, signi2cant, microslip, are admittedly sparse due toheavy computational demands. Also results are not given for inclination angles �¿"=3.This is, however, mainly due to the deteriorating assumption of an elliptical contactregion.

The results presented in Fig. 19 and combined with the depth-to-area relation in Fig. 11may now be applied to oblique contact problems where only normal contact has beenconsidered earlier. As was forecast in the introduction, a consistent theory for pow-der compaction based on a4ne particle motion was recently laid down by Stor)akerset al. (1999). A basic assumption was made because the interaction between particleswas considered frictionless. The present results may now be applied to include alsofrictional behaviour.

In Fig. 19, also earlier results based on slip-line theory (Green, 1954), correspondingto 2D parabolic indentation for creep at n = 100 (Larsson and Stor)akers, 2000), areshown. In contrast to the shape of the two-dimensional yield surface at perfect plasticity,the relation between the normal and shear force is virtually independent for the sphericalcase, for �6 "=3.

Remembering the invariance relations and similarity principles established above,the qualitative behaviour of some indentation histories may be simply illustrated. Thus,from Eq. (31) the total load is given proportionally by

F ∼ a(2n−1)=nh1=n: (37)

By use of Eq. (20), under prescribed constant load the indentation depth varies withtime as

h ∼ tp=(2n+p−1) (38)

and the contact area as

A ∼ t2=(2n+p−1) (39)

for a given creep exponent, n, and pro2le, p.If instead the indentation rate, dh=dt, is prescribed and constant, the resulting load

varies as

F ∼ t(2n−1)=np (40)

and the contact area simply as

A ∼ t2=p (41)

5. Concluding remarks

A general procedure was established to analyse three-dimensional oblique die contactat power-law creeping solids, n, with special attention given to the in6uence of angle


of indentation, �, die pro2le, p, and 2nite friction, �. The solution strategy was partlybased on an earlier similar analysis of plane strain indentation. It was found thatinvariance relations and similarity principles resulting in a reduced problem of completecontact, may be carried out also in a three-dimensional situation, though with somesigni2cant modi2cations. Solving the reduced problem by a 2nite element analysisproved to be very demanding as both the contact contour and stick–slip contour haveto be determined as part of the solution procedure especially in the case of 2nitefriction. Assuming a contact contour of elliptical shape proved to be a reasonableapproximation. Solving the problem initially for complete contact and subsequently, bycumulative superposition that established the growth of a moving boundary was foundto be very advantageous.

Explicitly, for spherical indentation, p= 2, governed by the remaining set n; �; �, itwas found that an invariant relation, c2, between the true contact area and indentationdepth was only weakly in6uenced by the presence of 2nite friction, �. The level ofinclination, �, a9ected the shift and the shape of the contact contour but only minutelyfor the c2 relation. The c2 relation depends strongly on the creep exponent varyingroughly proportionally from 1=2 for n = 1, Hertz solution, to about 1:3 for n → ∞,perfect plasticity. In the latter case, pronounced piling-up and signi2cant asymmetry insurface deformations prevail.

For linear incompressible materials it is since long well known, Cattaneo (1938),Mindlin (1949), that irrespective of the indentation angle, �, the contact contour isprecisely circular and no shifting of the contact area is present. The present resultsshow that at nonlinear materials, n¿ 1, shifting occurs in the forward direction, andthe shape of the contact contour was found to be oval rather than circular. Whenapproximating the deviation from a circle by an ellipse, it was found that at �¡"=6the shape was quite similar to the circular one and the major (horizontal) and minoraxes di9er by ¡ 1% for all values of n, while for � = "=3 the deviation is about 6%(full adhesion).

At proportional indentation and 2nite friction, the relative stick and slip regions werefound to be invariant at progressing contact at a given indenter pro2le. However, theywere neither circular nor symmetrically located as at a normal contact. As Coulomb’slaw sets a limit for macro-sliding to occur, for all values of n, save for n = 1, thereexists an interval of friction coe4cients exhibiting micro-slip and this interval increaseswith the creep exponent. When micro-slip occurs, no signi2cant e9ect on the resultingcontact forces was found.

Also, corresponding contact forces have been determined. For the spherical case, boththe contact forces and the shape and size of the contact region change with obliquity.When normalizing the contact forces with respect to the true contact area, the normaland tangential contact forces were found to be remarkably independent. The resultingcontact pressure was very close to being axisymmetric. At moderate obliquities, therewere only small deviations between the two cases of 2nite friction, � = 0:3, and fulladhesion, �→ ∞, as found for the normal and tangential resulting forces.

The problem analysed was based on proportional indentation. As the constitutiveequation adopted is history independent and the solution procedure was based on inter-mediate incremental 2elds, a solution for nonproportional, though monotonous, loading


seems fairly straightforward to obtain at least for adhesive contact. As the frameworklaid down originates from a solution for creep at normal contact by Stor)akers andLarsson (1994), which was later supplemented to include strain-hardening plastic 6ow(Biwa and Stor)akers, 1995), and viscoplasticity (Stor)akers et al., 1997), an extensionalso in this direction of the present oblique analysis constitutes a true challenge.

Acknowledgements

The writers are indebted to Prof. Per-Lennart Larsson and Prof. Bo HVaggblad for per-tinent and kind computational advice. This investigation was supported by the SwedishResearch Council for Engineering Sciences (TFR) which is gratefully acknowledged.

References

ABAQUS, 2001. (User’s manual, Version 6.2). Hibbitt, Karlsson and Sorensen, Providence, RI.Biwa, S., Stor)akers, B., 1995. Analysis of fully plastic Brinell indentation. J. Mech. Phys. Solids 43, 1303–

1334.Borodich, F.M., 1993. The Hertz frictional contact between nonlinear elastic anisotropic bodies (the similarity

approach.) Int. J. Solids Struct. 30, 1513–1526.Bower, A.P., Fleck, N.A., Needleman, A., Ogbonna, N., 1993. Indentation of a power law creeping solid.

Proc. Roy. Soc. London Ser. A 441, 97–124.Carlsson, S., Biwa, S., Larsson, P.-L., 2000. On frictional e9ects at inelastic contact between spherical bodies.

Int. J. Mech. Sci. 42, 107–128.Cattaneo, C., 1938. Sul contatto di due corpi elastici : distribuzione locale deXgli sforzi. Rendiconti

dell’Accademia Nazionale dei Lincei 27, 342–348, 434–436, 474–478.Cook, R.D., 1989. Concepts and Applications of Finite Element Analysis. Wiley, Singapore.Green, A.P., 1954. The plastic yielding of metal junctions due to combined shear and pressure. J. Mech.

Phys. Solids 2, 197–211.Hertz, H., 1882. Uber die BerVuhrung fester elastischer KVorper. J. Reine Ange Math 92, 156–171.Hill, R., 1992. Similarity analysis of creep indentation tests. Proc. Roy. Soc. London Ser. A 436, 617–630.Hill, R., Stor)akers, B., 1990. A concise treatment of axisymmetric indentation in elasticity. In: Eason,

G., Ogden, R.W. (Eds.), Elasticity: Mathematical Methods and Applications. Ellis Horwood, Chichester,pp. 199–210.

Hill, R., Stor)akers, B., Zdunek, A., 1989. A theoretical study of the Brinell hardness test. Proc. Roy. Soc.London Ser. A 423, 301–330.

Johnson, K.L., 1970. The correlation of indentation experiments. J. Mech. Phys. Solids 18, 115–126.Johnson, K.L., 1985. Contact Mechanics. Cambridge University Press, Cambridge.Larsson, P.-L., 1997. On creep deformation at plane contact problems. Cold Regions Sci. Technol. 26, 67–82.Larsson, J., Stor)akers, B., 2000. Oblique indentation of creeping solids. Eur. J. Mech. A=Solids 19, 565–584.Larsson, J., Biwa, S., Stor)akers, B., 1999. Inelastic 6attening of rough surfaces. Mech. Math. 31, 29–41.Lim, C.T., Stronge, W.J., 1999. Oblique elastic–plastic impact between rough cylinders in plane strain. Int.

J. Eng. Sci. 37, 97–122.Mesarovic, S. Dj., Fleck, N.A., 1999. Spherical indentation of elastic–plastic solids. Proc. Roy. Soc. London

Ser. A 455, 2707–2728.Mindlin, R.D., 1949. Compliance of elastic bodies in contact. J. Appl. Mech. 16, 259–268.Spence, D.A., 1975. The Hertz contact problem with 2nite friction. J. Elasticity 5, 297–319.Stor)akers, B., Larsson, P.-L., 1994. On Brinell and Boussinesq indentation of creeping solids. J. Mech. Phys.

Solids 42, 307–332.Stor)akers, B., Larsson, J., 2000. On inelastic impact and dynamic hardness. Arch. Mech. 52, 779–798.


Stor)akers, B., Biwa, S., Larsson, P.-L., 1997. Similarity analysis of inelastic contact. Int. J. Solids Struct.34, 3061–3083.

Stor)akers, B., Fleck, N.A., McMeeking, R.M., 1999. The viscoplastic compactions of composite powders. J.Mech. Phys. Solids 47, 785–815.

Tabor, D., 1958. Junction growth in metallic friction: the role of combined stresses and surface contaminationProc. Roy. Soc. London Ser. A 251, 378–393.

On oblique contact of creeping solids

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