Rolling contact stresses between two rigid, axial and flat cylinders

Subhankar Das BakshiDepartment of Materials Science and Metallurgy, University of Cambridge, U.K.

E-mail : sd444@cam.ac.uk/subhankar.dasbakshi@gmail.com

1 Statement of Purpose of the code

The code is written to calculate the distribution of forces and stresses during rolling contact of to rigid,parallel and flat cylinders assuming Hertzian contact between the mating surfaces under various conditionsof slip. It aims to estimate the magnitude and distribution of normal and tangential forces acting over theHertzian contact half-width and subsequently calculates the tangential, normal and shear stresses underplane strain condition. A range of stress distribution from perfect rolling to perfect sliding between matingcylinders can be calculated.

2 Distribution of force over the Hertzian contact width

The schematic of the twin-disc set up is shown in Fig. 1. The discs compressed against each other will

(a) (b)

Figure 1: Schematic of the (a) twin-disc set up, (b) stress distribution over the Hertzian contact.

initially have a straight line of contact having length equal to the thickness of the disc and the width of theone-half of the contact strip is expressed by;

b = 2

¿ÁÁÁÀ1 − ν2

π

P ( 1E1+ 1

E2)

l( 1R1+ 1

R2)

(1)

where,ν = Poisson’s ratio for the materials pressed against each other,P = applied force on the cylinders,

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E1, E2 = modulus of elasticity of the materials under compression,R1,R2 = radii of the compressed cylinders, andl = length of cylinders.

Under plain strain condition, the normal force distribution, pz(x) and the tangential force distrinution,px(x) over the Hertzian contact zone (-b ⩽ x ⩽ b) is exprressed as ;

pz(x) =⎧⎪⎪⎨⎪⎪⎩

2pnπb2

√b2 − x2, ∣x∣<b,

0, ∣x∣ ⩾ b, (2)

This gives raise to a parabolic distribution of the normal force over the Hertzian contact width which dropsdown to zero at both the boundaries.

px(x) =⎧⎪⎪⎨⎪⎪⎩

0, ∣x∣ ⩾ b,−sgn(s1)2µpnπb2

√b2 − x2, ∣x∣ ⩽ b, ∣x − b + b1∣>b1,

−sgn(s1)2µpnπb2[√b2 − x2 −

√b21 − (x − b − b1)2], ∣x∣ ⩽ b, ∣x − b + b1∣ ⩽ b1.

(3)

where,

sgn(s1) =⎧⎪⎪⎨⎪⎪⎩

+1, fors1>0,0, fors1 = 0,−1, fors1<0.

s1 is the symbolic variable, µ is the dynamic coefficient of friction, pn is the normal load per unit length(P/l), b1 is the half-width of the stick zone. The stick-slip ratio, ξ, is defined as the ratio between contacthalf-width of the stick zone to the Hertzian contact half-width ,i.e., ξ = b1/b.Now, Integrating eq.3 within the entire contact width (a,-a) results in;

b1b=⎛⎝

1 − ptµpn

⎞⎠

12

(4)

where pt is the tangential load per unit length. Substituting eq.4 in eq.3;

px(x) =⎧⎪⎪⎨⎪⎪⎩

0, ∣x∣ ⩾ b,

−sgn(s1)2µpnπb

√1 − (xb )

2, −b ⩽ x<b − 2bξ

−sgn(s1)2µpnπb

⎡⎢⎢⎢⎢⎣

√1 − (xb )

2−√ξ2 − (ξ + x

b − 1)2⎤⎥⎥⎥⎥⎦, b − 2bξ ⩽ x ⩽ b.

(5)

normal load per unit contact half-width, pn/b, can be taken as constant for the given pair of rollers andexperimental condition. The coefficient of friction, µ during stable rolling/sliding regime are to be consideredduring calculations. These set of equations are numerically solved by writting a code in C language for variousvalues of ξ and distribution of tangential force, px(x), and normal force, px(z) can be calculated.

3 Calculation of stresses in the x-z plane

The normal and shear stresses due to the distributed normal and tangential force acting on the Hertziancontact width is expressed as [1, 2];

σx(x, z) = −2z

π∫

b

−b

pz(s).(x − s)2[(x − s)2 + z2]2ds − 2

π∫

b

−b

px(s).(x − s)3[(x − s)2 + z2]2ds (6)

σz(x, z) = −2z3

π∫

b

−b

pz(s)[(x − s)2 + z2]2ds − 2z2

π∫

b

−b

px(s).(x − s)[(x − s)2 + z2]2ds (7)

2

τxz(x, z) = −2z2

π∫

b

−b

pz(s).(x − s)[(x − s)2 + z2]2ds − 2z

π∫

b

−b

px(s).(x − s)2[(x − s)2 + z2]2ds (8)

substituting pz(x), px(x) and x/b = i, z/b = j and s/b = t;

σx(i, j) = −4pnbπ2

⎡⎢⎢⎢⎢⎣jIx1 + µIx2(ξ) + µIx3(ξ)

⎤⎥⎥⎥⎥⎦(9)

σz(i, j) = −4pnj

2

bπ2

⎡⎢⎢⎢⎢⎣jIz1 + µIz2(ξ) + µIz3(ξ)

⎤⎥⎥⎥⎥⎦(10)

τxz(i, j) = −4pnj

bπ2

⎡⎢⎢⎢⎢⎣jIxz1 + µIxz2(ξ) + µIxz3(ξ)

⎤⎥⎥⎥⎥⎦. (11)

where,

Ix1 = ∫1

1

√1 − t2(i − t)2

[(i − t)2 + j2]2 dt, (12)

Ix2(ξ) = ∫1−2ξ

−1

√1 − t2(i − t)3

[(i − t)2 + j2]2 dt, (13)

Ix3(ξ) = ∫1

1−2ξ

[√

1 − t2 −√ξ2 − (ξ + t − 1)2](i − t)3

[(i − t)2 + j2]2 dt, (14)

Iz1 = ∫1

−1

√1 − t2

[(i − t)2 + j2]2dt, (15)

Iz2(ξ) = ∫1−2ξ

−1

√1 − t2(i − t)

[(i − t)2 + j2]2dt, (16)

Iz3(ξ) = ∫1

1−2ξ

[√

1 − t2 −√ξ2 − (ξ + t − 1)2](i − t)

[(i − t)2 + j2]2 dt, (17)

Ixz1 = ∫1

−1

√1 − t2(i − t)

[(i − t)2 + j2]2dt, (18)

Ixz2(ξ) = ∫1−2ξ

−1

√1 − t2(i − t)2

[(i − t)2 + j2]2 dt, (19)

Ixz3(ξ) = ∫1

1−2ξ

[√

1 − t2 −√ξ2 − (ξ + t − 1)2](i − t)2

[(i − t)2 + j2]2 dt, (20)

Having calculated σx, σz and τxz, the two principle stresses σ1,xz and σ2,xz are exprressed as;

σ1,xz =σx + σz

2+

¿ÁÁÁÀ⎛

⎝σx − σz

2

⎞⎠

2

+ τ2xz (21)

σ2,xz =σx + σz

2−

¿ÁÁÁÀ⎛

⎝σx − σz

2

⎞⎠

2

+ τ2xz (22)

These set of integrations are numerically solved in a program written in C programming language.

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4 Input and output parameters

A list of input parameters to be called from an input data.txt file are listed below in Table 1. The list ofthe output parameters, generated in an output file is listed in Table 2.

Table 1: List of input parameters for the code.

Parameter, unit variable type

Poisson’s ratio doubleLoad, N double

Overlap length, mm doubleYoung’s modulus, disc 1, GPa doubleYoung’s modulus, disc 2, GPa double

Radius, disc 1, mm doubleRadius, disc 2, mm double

Coefficient of friction double1-(%Slip/100) double

Table 2: List of input parameters for the code.

Parameter, unit variable type

Distance in x-direction, mm doubleDistance in z-direction, mm doubleTractionals stress, σx, MPa double

Tractional force/Nornal load, σx.b/Pnormal doubleNormal stress, σz, MPa double

Normal force/Nornal load, σz.b/Pnormal doubleShear stress, τxz, MPa double

Shear force/Nornal load, τxz.b/Pnormal double

5 Accuracy limits

The stress values are accurate to the errors equivalent to that of the input values.

6 Key words

Hertzian contact stress, rolling contact, %slip in rolling/sliding.

7 Sample program

To be submitted in a separate ASCII text format. The code has been tested on Windows/Linux/Mac OSxplatforms and found to work perfectly. An example of the output of the code, plotted using GNUplot isshown in Fig. 2.

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Fig

ure

2:C

alcu

lati

on

of

tract

ion

alst

ressσx,

nor

mal

stre

ssσz

and

shea

rst

ressτ xz

forl=

5m

m.

(a-c

)A

ssu

min

gp

erfe

ctro

llin

g,(d

-f)

roll

-sli

de

par

amet

erξ=

0.9

5,

assu

min

gm

arg

inal

slip

,an

d(g

-i)

assu

min

gp

erfe

ctsl

idin

gof

roll

ing/

slid

ing

cylin

der

s.

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8 Notification of the use of the code

The code has been used and cited in article titled Dry rolling/sliding wear of nanostructured bainite by S.Das Bakshi, A. Leiro, B. Prakash and H. K. D. H. Bhadeshia, submitted in Wear.

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References

[1] K. L. Johnson. Contact Mechanics. Cambridge University Press, Cambridge, UK, 1985.

[2] L. WenTao, Y. Zhang, F. ZhiJing, and Z. JingShan. Effects of stick-slip on stress intensity factors forsubsurface short cracks in rolling contact. Science China Technological Sciences, 56:2413–2421, 2013.

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