FACTA UNIVERSITATIS
Series: Mechanical Engineering Vol. 15, No 2, 2017, pp. 295 - 306
DOI: 10.22190/FUME170503007D
© 2017 by University of Niš, Serbia | Creative Commons Licence: CC BY-NC-ND
Original scientific paper
SIMULATION OF FRICTIONAL DISSIPATION
UNDER BIAXIAL TANGENTIAL LOADING
WITH THE METHOD OF DIMENSIONALITY REDUCTION
UDC 531.4
Andrey V. Dimaki1, Roman Pohrt2, Valentin L. Popov2,3,4
1Institute of Strength Physics and Materials Science SB RAS, Tomsk, Russia
2Berlin University of Technology, Germany
3National Research Tomsk Polytechnic University, Russia
4National research Tomsk State University, Russia
Abstract. The paper is concerned with the contact between the elastic bodies subjected
to a constant normal load and a varying tangential loading in two directions of the
contact plane. For uni-axial in-plane loading, the Cattaneo-Mindlin superposition principle
can be applied even if the normal load is not constant but varies as well. However, this is
generally not the case if the contact is periodically loaded in two perpendicular in-plane
directions. The applicability of the Cattaneo-Mindlin superposition principle guarantees the
applicability of the method of dimensionality reduction (MDR) which in the case of a uni-
axial in-plane loading has the same accuracy as the Cattaneo-Mindlin theory. In the
present paper we investigate whether it is possible to generalize the procedure used in the
MDR for bi-axial in-plane loading. By comparison of the MDR-results with a complete
three-dimensional numeric solution, we arrive at the conclusion that the exact mapping is
not possible. However, the inaccuracy of the MDR solution is on the same order of
magnitude as the inaccuracy of the Cattaneo-Mindlin theory itself. This means that the
MDR can be also used as a good approximation for bi-axial in-plane loading.
Key Words: Friction, Dissipation, Tangential Contact, Biaxial In-plane Loading,
Circular Loading, Cattaneo, Mindlin, MDR
1. INTRODUCTION
Friction is a dissipative process transforming mechanical energy into heat and material
changes of the contacting partners. The energy dissipation may be connected with material
dissipation (wear) [1] or utilized for structural damping [2]. Studying both wear and
Received May 03, 2017 / Accepted June 20, 2017
Corresponding author: Andrey V. Dimaki
Institute of Strength Physics and Materials Science SB RAS, Akademicheskii av. 2/4, 634055 Tomsk, Russia
E-mail: dav@ispms.tsc.ru
296 A.V. DIMAKI, R. POHRT, V.L. POPOV
damping requires the solution of a tangential contact problem. The simplest case of a
tangential loading is an increasing uni-axial tangential loading at a constant normal force.
This problem has been solved first by Cattaneo [3] and later independently by Mindlin [4].
They have shown that a tangential stress distribution can be represented as a superposition
of two solutions for the normal contact problem of the same geometry, only multiplied with
the coefficient of friction. This reduction to the normal contact problem is exactly the
feature which allows the application of the method of dimensionality reduction (MDR) [5],
(see also Chapter 5 devoted to tangential contact in [6]). However, Cattaneo and Mindlin
have not noticed a small inconsistency in their solution. In their theory, it is assumed that
the frictional stresses in the slip domain are all directed in the direction of the applied
tangential force. With the exception of the unrealistic case where both the contacting
materials have Poisson ratio zero, this assumption violates the condition that at every
position in the slip domain, the slip is directed in the direction opposing the tangential
stresses. The reason for this is the presence of an additional slip motion perpendicular to the
direction of the applied force. This was first pointed out by Johnson [7] who showed that
the maximum inclination of slip angle is on the order of magnitude ν/(4-ν) which is equal to
0.09 for ν=1/3 and 0.14 for ν=1/2. He concluded that the error is not large and that the
Cattaneo-Mindlin solution is a good approximation. Later comparisons with numerical
solutions have shown that the above mentioned inconsistency may have an important
influence on the distribution of wear but has almost no impact on the macroscopic force-
displacement relations [8]. A detailed analysis can be found also in [9].
In the present paper we consider a more complicated problem of bi-axial oscillating
loading (superimposed loading in two in-plane directions). The aim of the paper is twofold:
on one hand, we are interested in a better understanding of the energy dissipation in bi-
axially loaded contacts; on the other hand, we would like to check the applicability of the
dimensionality reduction method to this class of problems. At present, there are only a few
numerical studies providing the dependencies of dissipated friction energy on the
parameters of loading [10]. The applicability of the MDR would provide a simple tool for
simulating arbitrary loading histories with applications in the dynamics of structures with
frictional contacts.
2. ENERGY DISSIPATION IN A SINGLE-POINT CONTACT FOR CIRCULAR MOVEMENT
Let us start by considering a single isotropic linearly elastic massless element which
can deform in normal direction as well as in two tangential directions. We will call this
element a “spring”. The spring should have out-of-plane stiffness kz and isotropic in-
plane stiffness kx=ky. It is first pressed against a rigid half-plane with a normal force Fz
and then moved in the direction of the x-axis. We will assume that at the immediate
contact point between the spring and the substrate, there is friction characterized by a
constant coefficient of friction μ. When the free end of the spring is moved horizontally,
it first deforms elastically until the in-plane displacement achieves the critical value
0/
zx
l F k
. (1)
After this, the lower contact point starts sliding and the force remains constant.
If the spring is moved on a circle with radius R<l0, then it remains in the stick state at
any time. However, if the radius of movement exceeds critical value, R≥l0, the contact
Simulation of Frictional Dissipation under Biaxial Tangential Loading... 297
point will slip. In the stationary state, it will move in a circle with a smaller radius rc,
while the in-plane displacement of the spring remains constant and equal to l0. The
frictional force is assumed to be opposite to the elastic force and at the same time it has to
be directed opposite to the velocity vector. Therefore, the contact point between the
spring and the half-plane will move in the direction of the elastic displacement. On the
other hand, this velocity will be directed tangentially to the inner circle with radius rc,
which means that the elastic displacement of the spring is directed tangentially to this
circle, as shown in Fig. 1. The dissipation power is then obviously given by the equation
2
macro macro 0
cos 1 ( / )
zz
W v F v F l R
, (2)
where vmacro is the absolute velocity of the spring motion. For one cycle of motion with
radius R>l0 the value of the dissipated energy is
2
cycle 0
2 1 ( / )
z
W W t R F l R
, (3)
where Δtcycle is the time needed to perform one cycle of circular motion. If the initial position
of the spring does not correspond to the stationary one, it moves on a spiral asymptotically
approaching the circle with radius rc as shown in Fig. 1b.
Fig. 1 a) The scheme of a circular motion of a single spring; b) The results of the numerical
simulation: the evolution of the trajectory of a single spring during a circular motion
3. ENERGY DISSIPATION IN A CURVED CONTACT FOR CIRCULAR MOVEMENT
Generally, a non-conforming contact between elastic solids cannot be modeled with a
single spring. In the case of uni-axial in-plane loading, the contact problem can be reduced
to a contact of a rigid plane profile with a series of independent springs. This method is
known as the method of dimensionality reduction [5, 6, 11]. It replaces a contact between
two continuum bodies with an ensemble of independent one-spring problems and thus
reduces the general contact problem to the above one-spring problem (see Fig. 2).
298 A.V. DIMAKI, R. POHRT, V.L. POPOV
Fig. 2 Mapping of a three-dimensional contact into one-dimensional one
If the MDR-procedure was applicable to the bi-axial in-plane loading, then we could
compute the energy dissipation rate just by summing Eq. (2) over all effective springs of
the MDR-model. Let us assume at this point that this is indeed possible and calculate the
dissipation in a circularly moving and curved contact. Later we will check and discuss the
accuracy of this procedure.
We consider the movement of a parabolic indenter having the shape z=f(r)=r2/(2r0).
According to the MDR-rules [5, 6], in the equivalent MDR model it is to be replaced by
the plane profile
2
22 0
0
( )d
()
xf r r x
g x x r
xr
. (4)
This profile is brought into contact with an elastic foundation consisting of independent
springs, each spring having normal stiffness Δkz and equal tangential stiffnesses Δkx and
Δky for the displacements along the
x
-axis and
y
-axis (not shown in Fig. 2) which are
defined according to the rules
**
,
z x y
k E x k k G x
, (5)
where
22
12
*12
11
1
EE
E
and
12
*12
(2 ) (2 )
1
44GG
G
, (6)
with E1 and E2 being the Young’s moduli, G1 and G2 the shear moduli and ν1 and ν2 the
Poisson’s ratios of the contacting bodies. Further, throughout the paper, we assume that
the contacting materials satisfy the condition of “elastic similarity”
12
12
1 2 1 2
GG
, (7)
which guarantees the decoupling of normal and tangential contact problems [12].
If the indentation depth is d, then the vertical displacement of an individual spring at
position x is given by
,1 ( ) ( )
zD
u x d g x
(8)
and the normal force of a single spring equals to
*
( ) ( ( )) ( ( ))
zz
F x k d g x E x d g x
. (9)
Simulation of Frictional Dissipation under Biaxial Tangential Loading... 299
The dissipation power in one spring at the position x is given by Eq. (2) which we
rewrite here as
22
*
*
macro *
( ( ))
1 ( ( )) 1
z
z
x
FE d g x
W F v E x d g x
R k R
G
. (10)
Let us assume that we have a situation with partial slip. Radius c of the stick region is
determined by the condition
*
*
1
() G
d g c R E
(11)
whence
2*
*
0
1cG
dR
rE
. (12)
The whole dissipation power is thus equal to
2
*22
22
macro
22
0
2( ) 1 d
a
c
vE ax
W a x x
rac
, (13)
where
0
a r d
is the contact radius. Evaluation of the integral yields
macro
3()
2z
W v F c
, (14)
where
2
12
2
2
1
( ) (1 ) 1 d
1
c
cc
(15)
with
c c / a
. Function
()c
is shown in Fig. 3. From (14) we see that the energy
dissipation power is given by the formally calculated "nominal power" vmacroμFz multiplied
with function
3
2()c
, which only depends on the reduced radius of the stick area.
Fig. 3 Dependence
c
300 A.V. DIMAKI, R. POHRT, V.L. POPOV
4. CALCULATION OF STRESSES IN THE FRAMEWORK OF MDR
The above MDR-solution is based on the Coulomb criterion for sticking and sliding
for the springs of the effective one-dimensional elastic foundation. This MDR model
gives the correct solution to the three-dimensional problem only if the conditions for
sticking and sliding are fulfilled also for in-plane stresses in relation with normal stresses
in the initial (truly three-dimensional) problem. We thus begin our analysis by checking
the fulfillment of the sticking conditions and go later to an additional validation by
comparison with results of direct 3D simulation given in [10].
According to the MDR rules, the distribution of normal pressure p in the three-
dimensional problem may be calculated using the following integral transformation [11]:
22
()
1
( ) d
z
r
qx
p r x
xr
, (16)
where qz(x)=ΔFz(x)/Δx is a linear density of the normal force. A similar transformation is
valid for the tangential stress:
22
( )d
1
() x
x
r
q x x
rxr
, (17)
where qx(x)=ΔFx(x)/Δx is a linear density of the tangential reaction force, respectively.
The proof for these rules can be found in Appendix D of Ref. [5]. This proof can be easily
generalized to an arbitrary two in-plane dimensions and shows that the transformation (17)
can be applied separately to each component of tangential stress, so we can obtain
tangential stresses in y-direction similar to Eq. (17):
22
( )d
1
() y
y
r
q x x
rxr
. (18)
Thus, for calculating the stress component we have to determine first the linear force
densities qx(x)=ΔFx(x)/Δx and qy(x)=ΔFy(x)/Δx.
Let us denote the coordinates of a spring tip as (ux,tip, uy,tip) and the coordinates of the
upper point of the spring as (ux, uy). Assume that in an iteration step the coordinates of the
spring ux and uy, are changed by δux and δuy, so that
x x x
y y y
u u u
u u u
.
(19)
If new coordinates
x
u
and/or
y
u
now lie outside a circle having a central point (ux,tip, uy,tip)
and a radius l0(x):
0( ) ( )/
zx
l x F x k
, (20)
then the spring tip will start to slide in the direction of the tangential reaction force (see
Fig. 4) until it reaches the point
, ip ,tip
( , )
x t y
uu
:
22
, , 0
( ) ( ) ( )
x x tip y y tip
u u u u l x
.
(21)
Simulation of Frictional Dissipation under Biaxial Tangential Loading... 301
In other words, the new equilibrium point lays on the straight line connecting the points
(ux,tip, uy,tip) and
( , )
xy
uu
, at distance l0(x) from
( , )
xy
uu
(see Fig. 4).
Fig. 4 The slip displacement of a single spring in XY plane under lateral motion
The components of the tangential reaction force of the spring can be found as follows:
,
,
( ) ( )
( ) ( )
x x x x tip
y y y y tip
F x k u u
F x k u u
. (22)
We have studied the frictional energy dissipation for the parabolic indenter with the
following fictive parameters: r0=1 m, E*=1 GPa, d=0.001 m, ν=0.28, μ=0.3. The indenter
was initially moved to the point (Ux0, 0) and then subjected to an in-plane harmonic
displacement
0
0
( ) cos( )
( ) sin( )
xx
yy
U t U t
U t U t
. (23)
Controlling the tangential reaction forces in Ox and Oy directions, it is possible to
introduce the force-dependent governing parameters, following the paper of Ciavarella [10]:
/z
QF
and
/
M x y
R Q Q
, (24)
where
22
max ( ) , max ( ) ,
x x y y x y
Q F t Q F t Q Q Q
. (25)
Note that the value of Q, defined in Eq. (25), does not correspond to any real
tangential force acting on the indenter, but it serves only as a governing parameter in the
parametric study of the problem under consideration.
With Eq. (22) we determine linear force densities qx(x)=ΔFx(x)/Δx and qy(x)=ΔFy(x)/Δx.
We then calculate the tangential stress components given by Eqs. (17) and (18) and finally
the absolute value of the tangential stress:
22
( ) ( ) ( )
MDR x y
r r r
(26)
302 A.V. DIMAKI, R. POHRT, V.L. POPOV
The corresponding dependencies are presented in Fig. 5 together with the normal stress
distribution multiplied with the coefficient of friction and the formal Mindlin solution
with the same radius of stick region (dashed lines in Fig. 5). One can see that the obtained
stress distributions do not exactly fulfill the conditions for stick and slip. In most ranges
of radii smaller than the stick radius, the tangential stress is smaller than the normal stress
multiplied with the coefficient of friction; there is only a small region inside the stick radius
with too high. Thus the stick condition is fulfilled not exactly but in good approximation.
However, for radii moderately larger the stick radius, the tangential stress is higher than the
normal stress times the coefficient of friction, which means that the sliding condition is not
fulfilled. At even larger radii, the condition that in the sliding region the tangential stress
must be equal to normal stress times the coefficient of friction is fulfilled with good accuracy.
Thus, the tangential stress distribution has a qualitatively correct shape but it does not exactly
match the stick und slip conditions.
The mentioned discrepancy is observed only in a relatively narrow interval of radii.
Thus, the integral influence of this error may be moderate. This situation can be compared
with the solution by Cattaneo and Mindlin which also has a local error, but the global error in
the force-displacement relations is moderate and is generally tolerated.
Fig. 5 The distributions of normal pressure and the absolute value of tangential stress. P0 is the
pressure under the axis of the indenter. The dashed line indicates the formal Mindlin
solution with the same radius of stick region. RM = 1. a) Q/μFz=0.5; b) Q/μFz=0.9
In order to estimate the possible global error, let us determine the integral discrepancy
between the obtained stress distribution and the Cattaneo-Mindlin distribution [3] (which
fulfils the stick and slip conditions):
MDR CM CM
0 0 0
100% 2 ( )rd 2 ( )rd / 2 ( )rd
a a a
r r r r r r
, (27)
where τCM(r) corresponds to the Cattaneo-Mindlin solution. This discrepancy is shown in
Fig. 6. The integral difference between tangential stresses, predicted by the theory of
Cattaneo-Mindlin, and the MDR results, is about two percent for low values of Q/μFz and
RM. This means that the above MDR theory has a good accuracy at least for oscillations
with small amplitude comparable to the full slip displacement.
Simulation of Frictional Dissipation under Biaxial Tangential Loading... 303
Further, let us compare the results of MDR simulation with the full three-dimensional
calculations. The tangential stresses are calculated using MDR as described above for the
following set of parameters: RM = 1, Q = 0.9, which correspond to the same values as used
in Ref. [10]. Comparison of the MDR results with results of full three-dimensional
simulations is presented in Fig. 7.
In Fig. 7, on the left hand side, the stress-field simulated by the MDR is presented and
so is, on the right hand side, the stress field from the three-dimensional simulation [10].
While both results are in a good qualitative agreement, one can also see some differences.
Firstly, the stick radius in the MDR results does not decrease after the start of the in-plane
rotation, which can be connected to the application of the tangential displacement instead
of tangential forces in 3D simulation. Secondly, the tangential stresses in the stick area in
the MDR solution are higher than those in the full 3D calculation. However, the
mentioned discrepancies between the MDR results and full 3D calculations are moderate.
We can conclude that the MDR can be also used with “engineering accuracy” for contact
problems with bi-axial in-plane loading.
Fig. 6 The integral difference (27) between tangential stresses,
predicted by the Cattaneo-Mindlin theory, and the MDR results
5. NUMERICAL SIMULATION OF DISSIPATION UNDER NON-CIRCULAR MOTION
In this paragraph we apply the MDR within its range of accuracy for studying energy
dissipation in a contact subject to biaxial tangential loading with different oscillation
amplitudes in two perpendicular directions. In order to normalize values of dissipated energy
we use the solution of Mindlin [3] for friction energy dissipation during one cycle of a
uniaxial tangential loading:
252
33
*
2
0
925
1 1 1 1
10 6
M
z
C
z z z
R
FQ Q Q
WW a F F F
G
. (28)
304 A.V. DIMAKI, R. POHRT, V.L. POPOV
In the performed calculations we have varied the governing parameters (24) in a wide
range of values. The results of simulation, accompanied with the corresponding results of the
full 3D simulations, given in Ref. [10], are shown in Fig. 8a. It can be seen that the MDR
results are in a good agreement with the results of the full 3D simulations, except for the curve
for RM = 1 which also is in a qualitative agreement but shows distinctive quantitative
differences.
Fig.7 The distributions of tangential stresses in the contact area - the results of the MDR
simulation in the left column, the results of the full 3D simulation, from Ref. [10],
in the right column: a) after the initial displacement; b) after one revolution;
c) after two revolutions; RM=1, Q=0.9.The inner circle indicates the stick area
Simulation of Frictional Dissipation under Biaxial Tangential Loading... 305
Fig. 8 a) The normalized dependencies of the energy, dissipated during one cycle of the
circular motion, compared with the data from Ref. [10] (indicated by the crosses);
b) The normalized dependencies of the energy, dissipated during one cycle of the
circular motion, reduced into the universal curve. The normalizing factor W1,C is given
by Eq. (29)
We have found that for various values of RM, the dependencies of W on Q/μFz can be
reduced to a universal curve (Fig. 8b). The results are normalized to the value of the
energy W1,C dissipated during one cycle of uniaxial loading with Q/μFz=1:
2
1, *
2
0,
3
10
Mz
R Q F z
C
F
WW aG
. (29)
The universality of the given curve holds for relatively small amplitudes of oscillations.
When the amplitude of oscillations becomes comparable with the value of amplitude needed
for gross slip transition, the deviations from the universal curve appear (see Fig. 8b). We
suggest a power-law approximation of the data shown in Fig. 8b as follows:
3.5
1,
0.45 ( / )
cz
W W Q F
, (30)
which fits the results of numerical simulations well for Q/μFz < 0.7. Note that in Fig. 8b
the curve for RM = 0 coincides with the results of Cattaneo and Mindlin.
6. CONCLUSIONS
By analyzing the stick and slip conditions and comparing with three-dimensional
calculations we have explored the question whether the MDR is applicable for the
simulation of bi-axial in-plane loadings. We have found that the corresponding mapping is
not exact (there are local violations of stick and slip conditions) but has an acceptable
accuracy comparable with the accuracy of the Cattaneo and Mindlin solution for tangential
contact. Comparison with three-dimensional simulations shows a good qualitative
agreement but some quantitative deviations. We have found that the dependencies of the
dissipated energy on the amplitude of loading, obtained for various values of RM, fit into
one universal curve. This curve may be approximated by a power law in the range of small
values of Q/μFz < 0.7. The obtained results may be helpful for a better understanding of the
mechanics of tangential contacts under bi-axial loading.
306 A.V. DIMAKI, R. POHRT, V.L. POPOV
Acknowledgements: A.V. Dimaki is thankful for the funding of the Fundamental Research
Program of the State Academies of Sciences for 2013–2020. R. Pohrt and V.L. Popov thank V.
Aleshin for helpful discussions.
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