Tribology model for determination of shear stress distribution along the tyre-soil interface

Journal of Terramechanics, Vol. 27, No. 2, pp. 93-114, 1990. Printed in Great Britain.

TRIBOLOGY MODEL STRESS DISTRIBUTION

0022-4898/90 $3.00 + 0.0(I Pergamon Press plc. ~) 1990 ISTVS

FOR DETERMINATION OF SHEAR ALONG THE TYRE-SOIL INTERFACE

R. N. YONG and M. A. FODA*

Summary--The principles of tribology are introduced to the general problem of tyre-soil interaction with a view of development of a better knowledge of the manner in which the tyre-soil interface behaves under rolling motion subjected to tractive force. A trial and error procedure is adopted to deduce the shear stress distributed along the tyre-soil interface. The general agreement between the constructed and measured shear stress distribution is presented.

INTRODUCTION

IN MOVING pneumatic tyres subjected to tractive applied torque, the horizontal forces developed at the tyre-supporting ground interface (soil thrust) part of this thrust, are wasted through slip losses, and the rest remains as a useful force to accelerate the vehicle, slope climbing, pull loads (tractive effort). The relationship between these two components of the soil thrust is believed to be one of the most important aspects of the overall complex problem of tyre-ground interaction. Two approaches can be used for prediction of the traction driving forces for a wheel moving on a soil: (1) dimensional analysis, (2) mathematical modelling of the traction behaviour at the tyre-soil interface. The two aforementioned approaches pay little attention to the frictional mechanism along the tyre-soil interface which in turn is responsible for the traction production process. The lack of a complete understanding of the development of the tractive force along the contact patch, together with the development of the longitudinal slippage along the same contact patch simultaneously, is clear from the simplified assumptions involved in the limit equilibrium analysis approach, where the state of failure along the contact patch is assumed to follow Mohr-Coulomb failure criteria. Moreover, the relative displacement between the tyre and the supporting ground is assumed to increase linearly from the front edge of the contact patch to the rear edge (Fig. 1). ~=iX where

i X

= = =

degree of normal slip; distance from the first contact, and tyre-soil displacement.

In an attempt to understand the mechanism of rolling and slippage motion, the previous assumption states that at any particular time while the tyre is moving with certain *Geotechnical Research Centre, McGill University, 817 Sherbrooke St.W. Montreal, PQ, Canada H3A 2K6 93

94

R.N. YONG and M. A. F()DA Direction of Travel D.O.T.

j-"" FPL

Fl(;. 1. Tyre-soil shear displacement variation.

constant horizontal velocity (V), there is relative displacement along the entire contact length. Therefore, there is slip velocity along the entire contact length. In essence, one can ask simple questions: (1) If the tyre moves horizontally as a result of the rolling part of the motion, what provides the forward advance of the tyre if there are slip velocities along the entire contact patch? (2) Can the tyre-ground interface interaction transmit the required shearing stresses to the substrate to provide the necessary traction while the relative displacements take place along the entire contact length with different slip velocities? It seems that the main motive for the previous basic assumption is to relate the shear stresses and the degree of normal slip to predict the tractive force as a function of the degree of slip. In other words, to predict the "cause" as a function of the "effect", rather than the "effect" as a function of the "cause"• The questions that can be posed now are: (1) What is traction mechanism and how is it produced? (2) How can the "soil thrust" be divided into the two components, i.e. wasted part and useful part, simultaneously? This presentation represents an analytical-experimental study designed to investigate the above-mentioned problem. In this present phase of reporting, the use of tribological principles is elucidated in regard to the tyre-soil contact phenomenon. The other phases of reporting covering experimental information and validation will follow subsequently. To be more specific, in the second part, the experimentally evaluated frictional characteristic of the tyre-soil interface will be used to evaluate its performance during rolling motion subjected to tractive forces in terms of the local variation in the slip velocity, coefficient of friction and shear stress for different interface features (buffed/treaded) and various boundary conditions (applied torque, applied angular velocity). In the third part, the applicability of the finite element method to evaluate the tyre-soil longitudinal slippage will be shown in detail.

TRIBOLOGY MODEL FOR SHEAR STRESS DISTRIBUTION

95

The need to elaborate on the tribological model for the problem is considered important, since the analysis of laboratory information (to follow) and confirmation of the developed slip phenomenon require application of the relationships described in this presentation. From the above, it is clear that a detailed study of the behaviour of the contact patch is greatly needed, and a better understanding of the traction evaluation process lies in the tyre-ground interface behaviour during rolling motion subjected to tractive applied torque. It should be noted that the applied torque is specified as tractive to distinguish between the driving condition and the braking condition where the applied torque in the later case will be braking applied torque. APPLICATION OF THE PRINCIPLE OF TR1BOLOGY TO THE TYRE-SOIL INTERFACE MECHANISM PROBLEM

Tribology is defined as the science and technology of interacting surfaces in relative motion, having its origin in the Greek word "tribos" meaning rubbing. It is a study of the friction, lubrication, and wear of engineering surfaces with a view to understanding surface interactions in detail and then prescribing improvements in given applications. In the following section, a brief review will be given to some previous tribological analysis for On-The-Road Contact Patch Behaviour. Hailing [1] discussed the principles of stick zone and slip zone within the contact patch during rolling subjected to traction as shown in the details of Fig. 2. A part of the contact patch is subjected to slipping whilst in the remainder at the front of the contact zone no relative motion occurs. In order to explain this mechanism we must recognize three things. The degree of slip is microscopically small, the materials in contact may deform elastically, and the motion being rolling means that particles on the rim of the wheel pass through the contact zone and out the other side as motion proceeds. Thus, if we imagine particles approaching the contact zone they do not slip relative to the wheel rim because of a build-up of elastic deformation, but in due course these built-up deformations are released and microslip between the particles and the wheel rim ensues. The particles then pass out of the contact zone and all the strains are finally released. Moore [2] considered another aspect of the mechanism where he discussed the distribution of the slip velocities in the slippage zone. In the case of driven rolling, the band velocity of the tyre decreases to the "road velocity" value e0r upon entering the contact

4(

~Slip

Stick Zone

FIG. 2.

FD

Zone

Contact patch behaviour under the action of tractive applied torque (after Hailing [1]).

96

R . N . YONG and M. A. FODA

patch, and maintains this value until approximately one-half of the contact length has been traversed, as shown clearly in Fig. 3. At this point the tyre band velocity increases non-linearly towards the rear of the contact, thereby producing a variable kmgitudinal slip velocity in the opposite direction to the forward direction. The magnitude of this slip velocity increases rapidly with speed. In the case of driving, the longitudinal slip velocity has a tendency to overshoot the rearmost point in the contact patch, and this is interpreted physically as a discontinuity as the rubber tyre element suddenly adapts from longitudinal to circumferential motion during the process of rolling. C O N T A C T PATCH B E H A V I O U R U N D E R THE ACTION OF T R A C T I V E A P P L I E D T O R Q U E

The energy concept for evaluation of traction was formally introduced by Yong and Webb [3] to evaluate the motion performance of a rigid wheel on soft soil. Subsequently, the energy model was extended by Yong and Fattah [4], Yong et al. [5, 6] to predict pneumatic tyre performance on soft soils. The basic principle used in the energy model evaluates the performance of a tractive element on ground surface in terms of the well-known energy balance equation [7]; INPUT E N E R G Y (energy supplied to the tractive element)

SLIP ENERGY (wasted at the interface in forms other than work done) =

MOTION RESISTANCE (energy expended in overcoming resistance to motion)

OUTPUT ENERGY (work available, drawbar-pull).

In the present study, the energy concept is enhanced through the adoption of the principles of tribology to provide the needed link between the input energy, slip energy and the useful output energy in terms of the tyre-soil interface mechanism during rolling subjected to tractive forces. Hence, possible answers to the previously posed questions can be reached. From a tribological viewpoint we require a fundamental understanding of the frictional

I oaR .o

CL IL°ngitudinal [ _ ,slip velocity .[.if __ ±

colq I of slip I Contact Length-

FiG. 3.

R o ~ -'~\\\\\ s.p speod

X ,[

+

Longitudinal slip velocity distribution along the slip zone (after Moore [2]).

T R I B O L O G Y M O D E L F O R S H E A R STRESS D I S T R I B U T I O N

97

mechanism which develops in the tyre-soil contact region. Such an understanding is of prime importance to our ultimate goal of evaluating both the useful and wasted components of the soil thrust forces. In essence, it is appropriate to have a basic review of the mechanism of rolling motion subjected to traction. Figure 4 shows (schematically) successive views of a tyre moving with a constant horizontal velocity V on a rigid surface under the pure rolling condition, (the towed condition in which the sum of shear stresses along the interface equals zero). Our objective is to trace an element A on the tyre circumference before it enters the contact patch and follow it till it passes out from the other side. We notice that element A from the moment it touches the rigid surface remains stationary, in other words, it does not have any longitudinal advance in either direction as illustrated in the advanced distance-time plot in Fig. 5. Accordingly, the entire contact patch consists of a no-relative-displacement zone (stick zone) due to the absence of tractive applied torque. On the other hand, Fig. 6 shows successive views of a tyre while it is moving under constant applied torque and constant applied angular velocity. Additionally, it is moving with a constant translational velocity V. In this case, the contact patch consists of two zones, slip zone and stick zone. It must be noted that as the tyre is moving with constant V, it would be expected that

I t

Advanced Distance

\

[]

t

Position 1

t ÷At

Position 2

t ÷ 2At

Position 3

t +3At

Position 4

t ÷4At

Position 5

Stick Zone

Time

FIG. 4.

Successive views of the contact patch while travelling with constant V and under pure rolling condition.

98

R.N. YONG and M. A. FODA Advanced Distance ~-

Time

F~¢;. 5. The behaviour of element A while in contact with the ground surface.

V=

C

Advanced distance ~_

)A

~

t

Position 1

t+&t

Position 2

t +2'~t

Position 3

t+3At

Position 4

t+4&t

Position 5

t+5At Position6

Time

FIG. 6.

[]

slip zone

[]

stick zone

Successive views of contact patch while travelling with constant V and constant degree of normal slip and under rolling subjected to tractive constant applied torque (plan view).

b o t h the stick a n d slip z o n e s m a i n t a i n a c o n s t a n t a r e a d u r i n g the m o t i o n . In o t h e r w o r d s the d e g r e e o f n o r m a l slip is k e p t c o n s t a n t d u r i n g t h e travel. F i g u r e 7 shows the e l e v a t i o n views c o r r e s p o n d i n g to the p l a n views o f Fig. 6, the c o n t i n u a l c h a n g e in the s h e a r strain e n e r g y o f e l e m e n t A d u e to the i n c r e a s i n g a d h e s i o n a l forces i m p o s e d on

T R I B O L O G Y M O D E L F O R S H E A R STRESS D I S T R I B U T I O N

99

D.O.T.

7

Position 1

Position 2

--TJ

Lut~ r

Position 3

r Position 4

Y Position 5

"7 Position

6

T : applied torque D.O.T.: direction of travel

FIG. 7.

Successive views of contact patch while travelling with constant V and constant degree of normal slip and under rolling subjected to tractive constant applied torque (elevated view).

it is demonstrated. In both Figs 6 and 7 as shown in position (4), element A is located directly on the boundary of the stick-slip zones where the element has its maximum shear strain energy which can be stored by the rubber material of the tyre. Finally, as shown in positions (5) and (6), relative displacement starts to cease with increasing rate till the element passes out of the contact patch with its highest slip (sliding) velocity. It should be pointed out that although a linear relationship between the shear stress and the shear strain is indicated in the schematic diagram of Fig. 7, such an assumption is not needed in the present analysis as will be shown in the following sections. Figure 8 compares the pure rolling case with rolling subjected to a tractive force. In the later case, the element moves in the direction opposite to that of the tyre travel with increasing rates from the moment it enters the slippage zone till it passes out from the rear edge of the contact patch. From the above, and keeping in mind the behaviour of element A during its journey along the contact length, it is likely that the same behaviour occurs to different elements

100

R . N . Y O N G and M. A. F O D A

Travel ( D.O.T.)

Direction of

IL

Advanced distance

Rolling subjected

to traction 7

/ Pure

rolling

Time

FIG. 8.

Behaviour of element A while in contact with the ground surface for both cases of pure rolling, and rolling subjected to tractive force.

along the entire tyre circumference. Therefore, if we imagine that a picture was taken of the contact patch while travelling with constant V, various elements are expected to be located inside the contact patch at different stages of stress history depending on their relative locations within the contact length as shown in Fig. 9 for elements Y, Z, A, B, C, D and E. FRICTIONAL CHARACTERISTICS OF THE TRAVELLED ELEMENT

Up to this stage of analysis, no mention has been made regarding the influence of the normal stresses acting on the travelled element. It is reasonable to ask for instance . . . "does normal stress play a direct or indirect role in the determination of the boundary between the stick-slip zones?" To address this question, it is necessary to consider both the local normal stress and the local shear stress in the analysis of the travelled element through the contact length. Accordingly, the analytical treatment should consider the local coefficient of friction and the interfacial stresses simultaneously. In the field of tribology, several molecular kinetic and mechanical model theories of adhesion exist to describe the frictional behaviour of elastometric materials [8]. Whilst it is beyond the scope of this study to investigate the various orders of logic associated with the different theories, it is nevertheless necessary to consider the general shape of the dynamic coefficient of friction-sliding velocity relationship. Such a relationship takes the form described in Fig. 10 where an adhesional peak value occurs at a certain sliding velocity followed by a decreasing trend in /~d as the sliding velocity increases. In the meantime, the variation in the coefficient of friction with time is shown schematically in

TRIBOLOGY MODEL FOR SHEAR STRESS DISTRIBUTION

101

v

~ E

"(

D

"/

C

Y

Y

B

q"

A

"Y

v

"/

Z

Loading Unloading

FIG. 9.

.

.

.

.

.

.

.

.

.

An instant view of the contact patch under the action of constant tractive applied torque.

Fd

vS

FIG. 10.

Dynamic coefficient of friction-sliding velocity relationship for elastomeric material (after Moore

[81).

Fig. 11 where/z S is the value at rest which increases slightly due to junction growth. The junctions break down, and slip and the dynamic component/x d persist [9]. It is important to visualize the history of the coefficient of friction from the moment the tangential force

102

R . N . YONG and M. A, F O D A

Time

FIG. 11.

~

Variation in coefficient of friction with time (after Sarkar [9[).

#

VS~

FIG. 12.

An attempt to combine the history of the coefficient of friction on the g-Y~ plot.

is applied with increasing rate till the coefficient reaches the static value--which is by definition the value required to initiate the motion. It then drops almost instantaneously and the dynamic c o m p o n e n t takes over and starts to build up at an increasing rate till it reaches its p e a k value. After that, the effect of sliding velocity begins to influence its value when a continual fracturing process of the adhesive bonds occurs at different sliding velocities. Figure 12 attempts to combine the entire history on t h e / z - vS plot. The effect of the sliding velocity on the dynamic coefficient of friction can be evaluated experimentally in the soil bin by performing a set of stationary spinning tests as shown in Fig. 13. Unfortunately, in this study, the entire range of interest on the p. - vs plot cannot be obtained due to equipment limitations. In Fig. 14, the idealization of/z - v~ is demonstrated where the static value is taken as the extrapolated value corresponding to zero sliding velocity. In this Figure, the possible peak occurrence range is shown between vs = 0.0 and v s = lowest possible value obtained experimentally. It is now appropriate to re-evaluate the behaviour of the travelled element in terms of its local coefficient of friction. Figure 15 shows an instantaneous view of the contact patch with different elements through different stages of behaviour depending on their local positions within the contact length. As shown in the figure, the elements are subjected to local normal and tangential forces in connection with tyre load and applied torque respectively. As the static coefficient of friction is the value required to just initiate motion, it is reasonable to expect that the value of/x s is located at the boundary between the stick-slip zones, as shown in Fig. 15 on the/z-contact length plot. On the same plot, the discontinuity at the boundary is illustrated when a drop in the value of element B from/~s to zero occurs. The remaining elements are identified on the plot according to

TRIBOLOGY MODEL FOR SHEAR STRESS DISTRIBUTION V

= Zero

V

=

Zero

V

=

103

Zero

#d

Ve

VS =

o3 x R

T /Zd

-

WxR

Where :

Flo. 13.

vs =

Slip velocity

o3 =

Applied angular velocity

W

Tyre load

=

,

R

,

=

T

Rolling radius =

Applied torque

Determination of dynamic coefficient of friction-sliding velocity relationship (schematic illustration).

their local/x. In the meantime, the projections of the element values from the/x-contact length plot to t h e / x - v s plot are shown. DEDUCTION OF THE SHEAR STRESS DISTRIBUTION ALONG THE CONTACT LENGTH

Generally speaking and based on supporting studies [10, 11], a flexible tyre moving on soft ground may be postulated to yield a relatively uniform normal stress distribution while a rigid tyre produces a parabolic normal stress distribution. In this study, the normal stress distribution will be approximated by either uniform or parabolic distribution patterns at all slip rates. However, in the following schematic diagrams we will limit ourselves to the parabolic distribution. In this section, a trial and error procedure is adopted to construct the shear stress distribution that not only agrees with the analysis of the behaviour of the travelled element but also satisfies, together with the normal stress distribution, the frictional characteristics of the tyre-soil interface as obtained experimentally.

104

R.N. YONG and M. A. FODA

Fitted relationship

#s

between

,u,d -

vs

Lowest possible sliding velocity obtained experimentally

_~ Range of oossible= peak occurrence

F~(~. 14.

vs

Idealization of coefficient of friction-slip velocity relationship.

Moore [2] and Clark [12] show that due to the linear stretching of the rubber element from the moment it enters the contact patch, the adhesional forces can take a linear distribution. Accordingly, it seems appropriate to initially approximate the shear diagram by a triangular shape and then apply the trial and error procedure to adjust its shape in the slippage zone to agree with the analysis of the contact patch under rolling subjected to applied tractive torque. The following steps summarize the trial and error method, while Fig. 16 illustrates it schematically. (1) In order to locate the boundary between the stick-slip zones, the values of the local coefficient of friction are checked from the front edge until the static value is obtained, which determines the edge of the slippage zone. (2) Having determined the length of the slippage zone, and following the general nonlinear distribution of the slip velocity as shown in Fig. 3, an initially assumed slip velocity distribution can be considered. This distribution has a value equal to zero at the boundary and a value equal to (co r-V) at the rear point. It must be noted that the value (co r-V) represents the tyre global slip velocity where the travelled element is about to leave the contact patch. (3) In the range of a possible peak occurrence as shown in Fig. 14, a value of sliding velocity equal to 1.0 cm/s is chosen to represent the peak value. This is in accordance with the usual observation that afinite velocity of sliding is required (usually of the order of 1 cm/s or less) to attain maximum adhesion [2]. (4) From the sliding (slip) velocity distribution as shown in Fig. 16, the local slip velocity can be determined for different local elements along the slip zone as A , B , C , D , E , and F. In essence, the local dynamic coefficient of friction can be obtained by using the Ix - vs diagram as shown in the figure. (5) Knowing the local Ix and the local normal stress, the local "r (adhesional) can be obtained.

TRIBOLOGY MODEL FOR SHEAR STRESS DISTRIBUTION

105

V--C w

E

E

D

D

C c

=( Vs

VsD~C E

C

B

A

B a A"

Z

Y

A

Z C

Y .......

-~yZ~B IB ~y ;ontactlengtE~

slipzone

T FIG. 15.

An instant view of the contact patch under the action of tyre load and constant tractive applied torque.

(6) Along the slippage zone, the previously calculated local -r (adhesional) is superimposed on the local "r (elastic rebound) to obtain the local -r (total) along the slippage zone. (7) Combine the local linear elastic "r along the stick zone with the local -r (total) along the slippage zone, the distribution of local -r (total) can be obtained for the entire contact length. (8) To be consistent with the previous assumption of peak sliding value equal to 1 cm/s, the location of maximum adhesional shear stress is determined and its corresponding

106

R . N . Y O N G and M. A. F O D A

Initially assumed triangular distribution

local normal stress /Xs

-~.

Be

DE F

Vs

E De B

t

t Constructed shear diagram

/:J

~_. Final shear diagram

FJ(;. 16.

Schematic representation of trial and error procedure to construct the shear stress distribution along the slip zone.

sliding (slip) velocity is checked. If the two velocities do not match, a value of l cm/s is taken at the location of maximum adhesional stress and the slip velocity distribution is adjusted accordingly. In this case, the nonlinear curve is fitted through three points, zero at boundary, 1 at the location of maximum T and (~o r-V) at the rear point. The trial procedure is repeated from step (4) till the location of maximum adhesional shear stress coincides with a value of sliding velocity equal to 1 cm/s. As shown in Fig. 16, curves 1, 2 and 3 represent three different trials. The flowchart in Fig. 17 illustrates the different tyre-soil system parameters involved in the trial and error procedure and the role of each. In addition, the flowchart shows the sequence of the various steps followed in the trial and error to reach the final local distribution of the shear stresses along the entire contact length.

TRIBOLOGY MODEL FOR SHEAR STRESS DISTRIBUTION

107

FPL oJ : applied angular velocity

T : applied torque FPL : foot print length

V " constant horizontal velocity #d - V s : dynamic co. of friction-sllding

R : rolling radius

velocity relationship

/~s " static coefficient of friction

W : tyre load

© I

global slip velocity

steak& slip zones

I

vs

I

I

local /~d-- slip zone

[

I I

local 7"

local ~'elast ic

elast ic rebo-

I

local 7" adhesion

-und siP zone

steak zone

slip zone [ superposition

[ ]

1 local

Ttotal

I

TtOtal FIG. 17.

--

slip zone

[

1 --

F

Flow chart to illustrate the different parameters involved in the trial and error procedure.

GENERAL DISCUSSION

In view of the proposed mechanism of the tyre-soil interface during rolling motion subjected to tractive applied torque, it is important to discuss the following points which directly and/or indirectly support the general ideas of the present analysis. (1) In regard to the constructed shear stress distribution as outlined in the previous section, it may be appropriate to review the measured shear stress distribution reported by Krick [13], and compare between the general trend of the measured and constructed

108

R . N . Y O N G and M. A. F O D A

L s,.0 o o . , . / j ) % / / / F V - - k

90 °

~ o ;N

/

"'

"

•

90" Fl{;. 18.

Measured normal and tangential stresses for rigid wheel for the same test conditions (after Krick [13]).

diagrams. Figure 18 shows both the radial and tangential stress distribution for a rigid wheel at 40% slip, and Fig. 19 shows the same information for pneumatic tyres at slip rates 10% and 40%. Several observations arise from a study of the figures: (a) (b)

The initial increase in the shear stress at the front edge of the contact patch is more or less linear for both rigid and flexible cases. For tyres at low slip rate (10%), the tangential stress takes a triangular distribution for most of the contact length and then decreases linearly towards the rear edge. At a higher slip rate (40%) the triangular shape tends to dominate the front third of the contact length. In the meantime, the middle third shows an almost uniform distribution, while the rear third again shows a triangular distribution which decreases linearly towards the rear edge.

The parabolic normal stress distribution seems reasonable in the case of a tyre under low slip rate (10%). As the slip rate increases to 40%, the normal stress distribution appears to adjust itself to become more or less uniform in the major region of the contact length.

TRIBOLOGY MODEL FOR SHEAR STRESS DISTRIBUTION

90"

FIG. 19.

Measured normal and tangential stresses for pneumatic tyre (after Krick, 1969 [13]).

109

1 l(l

R . N . Y O N G and M. A. F O D A

Torque

C B

f

~

~

D i

I I I i

Slip

100%

D

r :Shear stress distribution C B

A Direction

Fl(;. 20.

of Travel

Behaviour of contact patch at different locations in the torque-slip relationship,

B

A

Applied Torque

Normal Slip%

Fro. 21.

i

Effect of elastic shear rebound on the t o r q u c - n o r m a l slip relationship.

T R I B O L O G Y M O D E L F O R S H E A R STRESS D I S T R I B U T I O N

I II

From the above observations, the agreement between the general characteristics of the constructed shear distribution and the measured shear distribution is evident. (2) It must be noted that since the applied torque is the integration of the shear stress distribution multiplied by the rolling radius, and since the constructed shear stress depends on the characteristics of the shear strength of the interface, it is not surprising to find that the applied torque has a limiting value. This limiting value cannot be exceeded unless the shear strength of the interface is modified by, for example, introducing the soil cutting shear strength component through the proper tyre tread pattern. Moreover, in agreement with the usual observation of the applied torque that shows a slight decrease in its value after reaching its limiting value at high slip rate [7], the trend of the proposed analysis shows a similar behaviour at relatively high slip rates as illustrated schematically in Fig. 20. In this figure, at high slip rates, the portion of the contact patch subjected

V = Const.

IR

B'

Action

A : wasted

component

useful

component

B:

Reaction

FIG. 22.

B

B'

Possible view of the tyre-soil contact patch with action and reaction.

112

R.N. YONG and M. A. FODA

to slippage dominates almost the entire contact length, and the slip velocity is expected to be very high. Hence, the local dynamic coefficient of friction along the slip zone will accordingly have a low value. In essence, this results in low adhesional shear stress, and after integration and multiplication with the rolling radius, a slightly lesser value of torque than its limiting value will be obtained. (3) The question whether or not the travelled element starts to release the stored shear deformation energy from the moment the junctions break down, corresponding to the transition from static friction to dynamic friction, remains unanswered. However, in this study it is assumed that the elastic shear rebound starts at the stick-slip boundary and increases linearly towards the rear edge of the contact length. To investigate the significance of the elastic rebound approximation, Fig. 21 compares two cases: case (A) where the elastic rebound takes place and case (B) where the element is assumed to sustain its stored shear deformation energy during travel through the slippage zone. As the adhesional component of the local T is not affected by either case, it is not unusual to discover that both cases have little effect on the produced longitudinal slippage (as will be demonstrated in detail in forthcoming reports). Nevertheless, the overall shear diagram is obviously affected, and thus the torque is expected to differ with a slightly increasing rate as the slip rate increases, as shown in the figure. However, from the

DBP

B

~D

Slip

100%

D C

B A

O Direction

of Travel

C

q

FPL

-~

FIG. 23. The general shape between the tribology model and the shape of drawhar pull-slip relationship.

T R I B O L O G Y M O D E L F O R S H E A R STRESS DISTRIBUTION

113

measured shear stress distribution shown in Figs 18 and 19, it is unrealistic to consider a shear distribution that, although decreasing towards the rear point, ends with a certain value rather than zero as in case (B). (4) The stick-slip zones used in the present analysis should not be confused with the so-called stick-slip phenomenon. The phenomenon is particularly prevalent with metal surfaces in contact. (5) Finally and most importantly, Fig. 22 shows a possible view of the tyre-soil contact patch under the action of the applied soil thrust, divided into the wasted component along the slip zone and the useful component along the stick zone. Also shown is the possible reaction imposed on the interface along the stick zone, where there is a complete adhesion between the tyre rubber and the soil material. Figure 23 illustrates the agreement between the present proposed model and the general shape of the drawbar-pull-normal slip relationship. It is noted that although the area of the stick zone decreases with increasing degree of normal slip, the magnitude of the component of the soil thrust force applied along the stick zone (i.e. the reaction forces) increases to a limiting value and then decreases as shown in the figure by the three triangles A,B and C. Figure 24 demonstrates the possible view of the stick zone for a tyre moving with constant V.

V

= Constant

IP

Advanced

Distance

p

Time

FIG. 24.

A possible view to the reaction acting on the moving contact patch with constant V.

Acknowledgements--The study was conducted under contract arrangement with DRES Canada. The input and cooperation of Drs G.J. Irwin and P. Boonsinsuk are greatly appreciated.

l l4

R.N. YONG and M. A. FODA REFERENCES

[1] J. HALLING, Introduction to Tribology. Wykeham, London (1976). 12] D. F. MooRE, Principles and Applications of Tribology. Pergamon Press, Oxford (1975). [31 R. N. YONG and G. L. WEBB, Energy dissipation and drawbar pull prediction in soil-wheel interaction. Proc. 3rd Int. Conf. 1STVS, Essen, Vol. 1, pp. 93-142 (1969). [41 R. N. YONG and E. A FATTAH, Prediction of wheel-soil interaction and performance using the finite element method. J. Terrarnechanics 13(4), 227-24/I (1976). [5] R. N. YONG, E. A. FATrAH and P. BOONSINSUK, Analysis and prediction of tyre-soil interaction and performance using finite element, J. Terramechanics 15(1), 43-63 (1978). [61 R. N. YONG, M. A. FODA and P. BOONSINSUK, Analysis and prediction of tyre-soil longitudinal slippage using the infinite element method. 3rd European Conf ISTVS, Warsaw, Poland (1986). [7] R. N. YONG, E. A. FAIqAII and N. SKIADAS, Vehicle Traction Mechanics. Elsevier Scientifie, Amsterdam (1984). [8] D. F. MOORE The Friction and Lubrication of Elastrometrics. Pergamon Press, Oxford (1972). [9] A. D. SARKAR, Friction and Wear. Academic Press, New York (1980). [1(1] R. N. YON(;,, P. BOONSINSUK, and E. A. FATIAH, Prediction of tyre performance on soft soils relative to carcass stiffness ,and contact areas. Proc. 6th Int. Conf. ISTVS, Vienna, Vol. II, pp. 643-675 (1978). [11] R. N. YONG, P. BOONSINSUK and E. A. FATTAH, Tyre flexibility and mobility on soft soils, J. Terramechanics 17(1), 43-58 (19801. [12] S. K. CLARK, Mechanics of Pneumatic 7"ires. The University of Michigan, Ann Arbor, Michigan (19711. [13] G. KRrCK, Radial and shear stress distribution under rigid wheels and pneumatic tires operating on yielding soils with consideration of tire deformation. J. Terramechanics 6(3), 73-98 (19691.