university of notre dame tribological investigation of the carbon- carbon composite brake system...

University of Notre Dame

Tribological Investigation of the Carbon-Carbon Composite Brake System

Ling HeAdvisor: Timothy C Ovaert

University of Notre DameOct. 19th, 2006


Objection of My Research• Aircraft Brake System Control

• Racing Automobile Brake System Design


Outline Introduction

Three-Dimensional Thermoelastic Rough Surface Contact Model for Isotropic Materials

Three-Dimensional Asperity Contact Model for Anisotropic Materials with General Boundary Condition

Future Work


Introduction


Contact Problem for Elastic Material

1 2 0, 0 ( )z i z i i i i cu u h P i

1 2 0, 0 ( )z i z i i i i cu u h P i

Governing Equation:

( )

s

ii

P P

1 2i i ih S S

1 2i

1 2

z1 z2

1 2

P : total external load = + : total approach

+ : total deflectionS = S + S : initial gapzu u u


Total area: A=∑Ai

Three-Dimensional Asperity Contact Model

Analysis Flowchart Discretize the brake surface region

Read characteristic data for each separate region

Solve 3D rough surface thermoelastic problem using

CGM technique.

Obtain subsurface stress/strain…etc. based on

contact pressureAnalysis and graph the result

Ai


Rough Surface Contact

is iu

c s

OD

node

0P

0

i

c

s

P : total external load : total approach : total deflection

s : initial gap : contact area : seperate area

Curve O : origional profileCurve D : deformed profile

iu


Why Need Analytical Solution

Numerical Method(FEM, BEM, etc)

Analytical Method

Problem Type Complex Case Simple CaseResult Transient/Steady State Steady State

Solving Time Slow Relatively FasterAccuracy Good Relatively worse

Can numerical methods handle rough contact problem? No!


Three-Dimensional Thermoelastic Rough Surface Contact Model for Isotropic Materials


Deflection ( ) of B when pressure is applied at A (elastic effect)

3D Thermoelastic Asperity Contact Model for Isotropic Materials

Normal pressure applied to circular region

)()()(

)()()(ln)(

)()()()()()(ln)(

)()()(

)()()(ln)(

)()()()()()(ln)(

1

2/122

2/122

2/122

2/122

2/122

2/122

2/122

2/122

2

axbyax

axbyaxby

axbybyaxbybyax

axbyax

axbyaxby

axbybyaxbybyax

puE z

pdxdyzu

x=(x(B)-x(A)); y=(y(B)-y(A));a=dx/2;b=dy/2;

zu

z

AB

y [m]x [m]

[m]

zu

pdxdyzu



Normal pressure applied to circular region

Deflection (uz) of B when pressure is applied at point A (thermal effect).

' ln))((ln))(( ln))((ln))((

tantan2

)(

tantan2

)(

tantan2

)(

tantan2

)()(2

43

21

112

112

112

112

Crybxarybxarybxarybxa

ybxa

ybxayb

ybxa

ybxayb

xayb

xaybxa

xayb

xaybxaABCDu

ch z

z

AB

ω

x [m]y [m]

[m]zu

zu

pdxdypdxdy

ω : rotational velocity (rad/s)r1,r2,r3,r4 : the distance from B to each corner of rectangle Ac = a (1+v) : a is thermal expansion coefficient



z

ω

P0

Smooth Surface

Contact Force

Surface Displacement

(ω = 0 rad/s )

y [m]x [m]

[m]

P [N]

[m]

zu

zu

x [m]

y [m]x [m]

y [m]


3D Thermoelastic Contact Model Verification

Pij (contact force)Pij (contact force)

uz (numerical result)uz (theory result)3D Hertz problem

P0 = 2×105 N

E = 72×109 Pa

= 0.2

Rsphere = 0.25 m

uz(0,0) = 2.44×10-4 m

uz(0,0) = 2.52×10-4 m (theory)

P(0,0) = 9.26×103 N

P(0,0) = 9.47×103 N (theory)

P0

z

X1 [m]

Z [m]

X2 [m]

X1 [m]

Normal Displacement [m]

X2 [m]

X1 [m] X2 [m]


Normal Contact Force [N]

X2 [m] X1 [m]


X1 [m] X2 [m]



3D Hertz problem

P0 = 2×105 NE = 72×109 Pan = 0.2Rsphere = 0.25 m

Subsurface von-Mises stress (x-z plane)[Pa]

[m]1[m]

P0

z

X1 [m]

Z [m]

X2 [m]



Thermoelastic Hertzian result (smooth surface)

X1 [m]

X1 [m] X1 [m]

X1 [m] X2 [m] X2 [m]

X2 [m] X2 [m]

Normal Displacement [m] (ω =10 rad/s)






Thermoelastic Hertzian result (rough surface)

X1 [m] X1 [m]

X2 [m] X2 [m] X2 [m]

Original profile [10-3 m]

X1 [m]

Normal Displacement [10-4 m] (ω =200 rad/s)

Normal Displacement [10-4 m] (ω=10 rad/s)


3D Thermoelastic Asperity Contact Model for Isotropic Material (Ring Surface)

(Original profile) (smooth) (rough)

(Surface Displacement )

(Contact Force )

(ω = 10)

y [m]x [m]

h[m] uz[m]uz[m]

Pij[N] Pij[N]h[m]

x [m] x [m] x [m]

x [m]x [m] y [m] y [m]

y [m]y [m]


Three-Dimensional Thermoelastic Rough Surface Contact Model for Anisotropic Materials


Asperity Contact Model for Anisotropic Material

Basic Equations of Anisotropic Elasticity ksijksij C

Type of Anisotropic Material (Based on Symmetry Planes)Triclinic Materials (21)

Monoclinic Materials (13)

Orthotropic Materials (9)

Trigonal Materials (6)

Tetragonal Materials (6)

Transversely Isotropic (or Hexagonal) Materials (5)

Cubic materials (3)

Isotropic Materials (2)


3D Contact Model for Anisotropic Material

3D Green’s function for infinite anisotropic medium:

Idea: obtain the result in term of a line integral on an oblique plane in the three-dimensional space

General deformation:

Note: f will be calculated due to different conditions

11

1 2 000

1 1 [ ] [ ]( , ,0) [ ]2 sin( )

x x dr

S Lu L f

0

0

0

1 ( )

1 ( )

1 ( )

d

d

d

1

2

3

S[k] N

H[k] N

L[k] N

1

1

1

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( )

T

T

1

2

3

N T R

N T

N R T R Q

1 1 1 2 2 2, , .ik i k j s ik i k j s ik i k j sQ C n n R C n m T C m m


3D Anisotropic Contact Model Verification

Pij (Contact Force)Pij (Contact Force)

uz (numerical result)uz (theory result)3D Hertz problem

P0 = 2×105 NE = 72×109 Pa = 0.2Rsphere = 0.25 m

x y

uz(0,0) = 2.48×10-4 muz(0,0) = 2.52×10-4 m (theory)P(0,0) = 9.61×103 NP(0,0) = 9.47×103 N (theory)

P0

z

X1 [m]

Z [m]

X2 [m]

X1 [m]


X2 [m]

X1 [m] X2 [m]



X2 [m] X1 [m]


X1 [m] X2 [m]

2

2

1 2[ ] ( )2(1 )

1[ ] ((3 4 ) )4 (1 )

[ ](1 )

T T

T

T

vxv

x vv r

x vv r

S mn nm

xxH I

xxL I


3D Contact Model for Anisotropic Material

Anisotropic Application E1=144x109 pa, E2=E3=72x109 pa, v=0.2, G=E2/[2*(1+v)]

Normal pressure distribution [pa] Contour for the normal pressure distribution


Future Work

Investigate on the case that have different fiber property on the different region.

Improve the algorithm to reduce the calculation time.


Thank you

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