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Unformatted text preview: ElasticPlastic Spherical
Elastic
Contact
Robert L. Jackson
Robert
Mechanical Engineering Department
Auburn University Hertz Contact Solution (1882)
Hertz
Closedform expressions to the deformations
Closed
and stresses of two spheres in a purely elastic
contact (Theory of Elasticity).
The Hertz solution assumes that the interference
The
is small enough such that the geometry does not
change significantly.
The solution also approximates the sphere
The
surface as a parabolic curve with an equivalent
radius of curvature at its tip.
It is also assumed that the contact surfaces are
It
frictionless. Hertz Solution Results
Hertz 2
1 1 − ν 12 1 − ν 2
=
+
E′
E1
E2 AE = πRω 1
1
1
=
+
R R1 R2 4
3/ 2
PE = E ′ R (ω )
3 Critical Interference
2 ⎞ −1 ⎛ ⎛z⎞
σ 1 = − p o ⎜1 + ⎜ ⎟ ⎟
⎜ ⎝a⎠ ⎟
⎝
⎠
σ 2,3 −1
⎧⎡ ⎛
⎫
2 ⎞⎤
⎡z
⎪
⎛z⎞
⎛ a ⎞⎤ ⎪
= po ⎨⎢2⎜1 + ⎜ ⎟ ⎟⎥ − (1 + ν )⎢1 − tan −1 ⎜ ⎟⎥ ⎬
⎜
⎟
⎝ z ⎠⎦ ⎪
⎣a
⎪⎢ ⎝ ⎝ a ⎠ ⎠⎥
⎣
⎦
⎩
⎭ Substitute into the von Mises Yield Criterion: Sy = [ 1
(σ1 − σ 2 )2 + (σ 2 − σ 3 )2 + (σ 3 − σ1 )2
2 Critical Interference
Sy
poc 2 ⎞ −1 3⎛ ⎛ z ⎞ ⎟
= ⎜1 + ⎜ ⎟
2⎜ ⎝a⎠ ⎟
⎝
⎠ ⎡z
−1 ⎛ a ⎞⎤
− (1 + ν )⎢1 − tan ⎜ ⎟⎥
⎝ z ⎠⎦
⎣a Then taking derivative and setting to zero, the point of initial yielding is
numerically located: d ⎛ Sy ⎞
2
2
2
2 2⎡
−1 ⎛ a ⎞ ⎤
⎡
⎤
⎜
⎟ = −az ⎣ a (4 +ν ) + (1 +ν ) z ⎦ + (1 +ν ) ( a + z ) ⎢ tan ⎜ ⎟ ⎥ = 0
dz ⎝ poc ⎠
⎝ z ⎠⎦
⎣ p oc
= C = 1.295 exp(0.736ν )
Sy ⎛ π ⋅ po ⎞ ω =⎜ 2 ⎟R
⎝ 2E ′ ⎠ Critical Interference
Critical
(Initial Yielding)
Using the von Mises Yield Criteria and the
Using
Hertz Contact solution the following
numerically fit solution is obtained. ⎛π ⋅C ⋅ Sy
ωc = ⎜
⎜ 2E ′
⎝ 2 ⎞
⎟R
⎟
⎠ C = 1.295 exp(0.736ν ) Critical Contact Area and Force
Critical
⎛ CS y R ⎞
3
⎟
Ac = π ⎜
⎜ 2E ′ ⎟
⎝
⎠
2 2 4⎛ R ⎞ ⎛C
⎞
Pc = ⎜ ⎟ ⎜ π ⋅ S y ⎟
3 ⎝ E′ ⎠ ⎝ 2
⎠ 3 Normalization
Normalization
ω = ω / ωc
* P = P / Pc A =ω
*
E * * A = A / Ac
* () P =ω
*
E * 3/ 2 Evolution of Deformation
Evolution
As long as the deformations are purely elastic, i.e.,
As
below the critical interferences, the entire hemisphere
will abide by the 3D Hooke’s law.
In plasticity, however, volume is conserved. As the
In
deformation increases, yielded material flows plastically
and it will not compress. FEM Elastoplastic Model
FEM
Kogut and Etsion (2002) performed a FEM analysis
Kogut
of the same case of an elasticperfectly plastic
sphere in contact with a rigid flat.
In this analysis, the value of H is set to be fixed at
In
2.8⋅Sy.
Very similar to current model, although the finite
Very
element mesh used is much more course than the
current mesh. Spherical Contact Model
Spherical Just Before Contact Mostly Elastic
Deformation Mostly Plastic
Deformation Finite Element Model
Finite
Perfectly plastic material yields according to the von Mises
Perfectly
yield criterion.
100 Contact Elements are used to model the contact at the
100
interface between the sphere and the rigid flat.
Iterative scheme used to relax problem to convergence.
Iterative
Mesh convergence was satisfied.
Mesh Finite Element Mesh
Finite ω*=171 von Mises Equivalent stress at
ω*=0.571 ω*=1.43 ω*=2.14 ω*=3.57 ω*=4.29 ω*=5.72 ω*=8.57 ω*=20.0 ω*=31.4 ω*=40.0 ω*=62.9 ω*=80.0 ω*=100 ω*=114 ω*=171 Material Properties
Material
Sphere properties: Radius = 1 μm, E=200 GPa,
Sphere
ν=0.32, Sy=0.210 GPa, 0.5608 GPa, 0.9115
Sy
GPa, 1.2653 GPa, and 1.619 GPa (Results also
confirmed for other materials).
These properties represent the typical material
These
properties of steel alloys used in engineering.
Normalized the results by the Hertz solution.
Normalized Empirical Formulation
Empirical
HG /Sy ≈3
a/R=0+ 3> HG /Sy >1
. HG/Sy ≈1 0<a/R<1 a/R=1 Diagram of progression of change in hardness with geometry. HG
⎛
⎛ a ⎞⎞
= 2.84 − 0.92⎜1 − cos⎜ π ⎟ ⎟
⎜
Sy
R ⎠⎟
⎝
⎝
⎠ Empirical Formulation
Empirical
⎛ω ⎞
π (CS y R ) ω ⎜ * ⎟
*
⎜ω ⎟
Ac ⋅ AF
⎝ t⎠
=
2
4π (RE ′)
πR
* 3 a
=
R 2 * B
B
πCe y ⎡ * ⎛ ω * ⎞ ⎤
⎢ω ⎜ * ⎟ ⎥
=
2 ⎢ ⎜ ωt ⎟ ⎥
⎝⎠ ⎣ This results in a formulation for HG as a function of the material
properties, E, Sy, and ν (not just upon Sy as suggested by Tabor
(1951)). ⎦ 1
2 Empirical Formulation
Empirical
0≤ω ≤ω
* *
t () P =ω
*
F ω ≤ω
*
t * 3/ 2 * ⎡ ⎛1*
*
PF = ⎢exp⎜ − ω
⎜4
⎣⎝ () 5
12 ⎞⎤ * 3 / 2 4 H G ⎡
⎛1*
⎟⎥ (ω ) +
⎢1 − exp⎜ −
⎟
⎜ 25 ω
CS y ⎣
⎠⎦
⎝ Statistically this formulation differs from the FEM data for all five
materials by an average error of 0.94% and a maximum of 3.5%. () 5
9 ⎞⎤ *
⎟⎥ω
⎟
⎠⎦ Empirical Formulation
Empirical
0≤ω ≤ω
* *
t A =ω
*
F ey =
* Sy
E′ ω t* = 1.9 ω t* ≤ ω * ⎛ω ⎞
A =ω ⎜ * ⎟
⎜ω ⎟
⎝ t⎠
* *
F B * B = 0.14 exp(23 ⋅ e y ) Equation differs from the FEM
data for all five materials by
an average of 1.3% and a
maximum of 4.3%. Empirical Formulation
Empirical
Since these equations are normalized by their critical values, the
resulting formulation for the average pressure is: P
2 PF*
=C*
A ⋅ S y 3 AF Large Deformation Motivation
Importance in areas of asperity contact, anisotropic conductive films,
Importance
metal forming, compression of wear particles, etc.
* Better understanding of the deforming geometry of sphere
Better
Defining a single model to closely predict contact area, pressure force
Defining
for elastic, elasto plastic and fully plastic range of deformations
Comparisons with existing experimental results
Comparisons
Studying the relationship between the yield strength and hardness of
Studying
materials 41 * http://www.theshortspan.com/photo/friction/contactarea.png ; http://www.flipchips.com/images/postbond.gif ; http://advanceindustriesgroup.com/images/metal6.jpg FEM Analysis
Axisymmetric problem
Axisymmetric
Sphere modeled initially as elastic
Sphere
perfectly plastic behavior
Models built based on boundary
Models
conditions across the base
Modeled using contact elements.
Modeled
Number of elements is 1493
Elements used
Elements 2D 8node element for sphere
2D 3 node element for contact Lagrange and penalty method contact
Lagrange
algorithm
Results verified for 3 different material
Results
properties.
Sy/E = 0.001,0.0025 and 0.005
42 Mesh Convergence
0.2365 Mesh convergence for
Mesh
maximum displacement
of nodes and reaction
forces
Selection criterion less
Selection
than 0.1% difference
Mesh size of 1493
Mesh
elements converges Max. displacement 0.236 0.2355 0.235 0.2345 0.234 0.2335 0.233
Max. Displacement 86 321 706 1493 1926 0.234 0.235 0.236 0.236 0.236 No. of Elements Reaction forces 86 3.821E+09 321 3.790E+09 0.817942 706 3.780E+09 0.26455 1493 3.758E+09 0.585418 1926 43 No. of elements % Error 3.756E+09 0.053248 FEM Boundary Conditions
Rigid Base Deformable Base 44 Deformable base case
Base allowed to deform in xdirection
Base Volume of hemisphere equated to volume of a hemisphere truncated by
Volume
interference to predict deformed radius, R1,
π
2
3
2πR 3
V2 = πR1 (R − ω ) − (R − ω )
V1 =
3
3 Using Volume conservation principle and solving for radius R1,
Using
R1 = 2R 3
(R − ω )2
+
3( R − ω )
3 ⎡
⎛ a ⎞⎤
P
= 2.84 − 0.92⎢1 − cos⎜ π ⎟⎥
⎜ R⎟
Sy
1 ⎠⎦
⎝
⎣ This is then used to make a better prediction of a/R1,
45 Deformable Base results –
De
Base
average pressure
3 2.5 Pm /Sy 2
Eq. (6)
JG
JMG
FEM mtl 1
FEM mtl 2
FEM mtl 3 1.5 1 0.5
0.2 0.3 0.4 0.5 0.6 a/R 0.7 0.8 Average error between FEM and new model predictions is 4.78%
46 0.9 1.0 Contact Radius Calculations
Original equations only valid till a/R ≤ 0.412
Original
a/R
Current study aims to modify them such that it is
Current
valid till a/R =1
a/R
For deformable base case,
For ⎛ω
⎛a⎞
⎛a⎞
⎜ ⎟ = ⎜ ⎟ + A1 ⎜
⎜ω
R ⎠ new ⎝ R ⎠ JG
⎝
⎝c
⎛ Sy
A1 = 0.0826 ⋅ ⎜
⎜E
⎝ 47 ⎞
⎟
⎟
⎠ 3.148 2 ⎞
⎛ω
⎟ − A2 ⎜
⎟
⎜ω
⎠
⎝c ⎛ Sy
A2 = 0.3805 ⋅ ⎜
⎜E
⎝ ⎞
⎟
⎟
⎠ ⎞
⎟
⎟
⎠ 1.545 Deformable base contact radius
0.412 < a/R ≤ 1 a/R≤ 0.412
1.4
1.7 0.35
0.35
0.3
0.3 Mtl.1a FEM mtl
FEM mtl 1 1 FEM mtl 1
FEM mtl
FEM mtl 2 1
FEM mtl
FEM mtl 3 2
FEM mtl 3
JG
JG
Eq. (10)
Eq. (9)
MM 1.5
1.2 Mtl.1
a a /R
a/R 0.2
0.2 1.3 FEM mtl 2 2
FEM mtl 1 Mtl.2a FEM mtl 3 3
FEM mtl Mtl.2a JG JG 0.155
0.1
0.1
0.1 1.1 0.8 a/R
a/R 0.25
0.25 0.9 0.6 Eq. (10) (9)
Eq. MM 0.7 Mtl.3a MM MM Mtl. 3a 0.4
0.5 Mtl.1b
0.2
0.3 0.05
0.05 Mtl.2b Mtl.3b Mtl. 1b
Mtl. 2b 0 0.1
0.1 Mtl. 3b 0
0.1 0 11 1010 ω /ω cc
ωω 10000
1 1000
1
000 100
100 1000
1000 ω //ω c
ω ωc 10000
10000 100000
100000 For small deformations (a/R≤ 0.412), the Jackson and Green model (Eq. (2)) shows the best agreement with FEM
However as deformations get larger (0.412 < a/R ≤ 1), the new model (Eq. (9)) predictions seem to be more accurate
Average error between FEM and new model is 2.7%
Results have been verified for 3 material properties
48 Contact Force calculations
Jackson and Green model does not incorporate the Jackson and Green model does not incorporate the
increasing radius of the sphere during heavy loading
Current work aims to use new results to modify eq. (3)
Current
from JG resulting in more accurate predictions of contact
force
The new modified Equation for contact force is,
The
⎡⎛
F ⎢ ⎜ 1⎛ ω
= ⎢exp⎜ − ⎜
⎜
Fc
⎜ 4 ⎝ ωc
⎢⎝
⎣ 49 ⎞
⎟
⎟
⎠ 5
12 ⎞⎤
⎟⎥⎛ ω
⎟⎥⎜
⎜
⎟⎥⎝ ω c
⎠⎦ ⎡
⎛
⎜ 1⎛ω
⎞
P
⎛a⎞ ⎢
⎟ + πR 2 ⎜ ⎟
1 − exp⎜ − ⎜
⎟
⎜
Fc
⎝ R ⎠ new ⎢
⎠
⎜ 25 ⎝ ω c
⎢
⎝
⎣
3
2 2 ⎞
⎟
⎟
⎠ 5
9 ⎞⎤
⎟⎥
⎟⎥
⎟⎥
⎠⎦ (11) References
References
1. Jackson, R. L., Green, I., 2005, A Finite Element Study of Elastoplastic
Hemispherical Contact, J. of Tribol., Trans. ASME, 127, 2, pp. 343354.
J.
2. Jackson, R. L., Chusoipin, I., Green, I., A Finite Element Study of the
Residual Stress and Strain Formation in Spherical Contacts, 2005, J. of
Tribol., Trans. ASME, 127, 3, pp. 484493. 3. Quicksall, J., Jackson, R. L., Green, I., 2004, Elastoplastic Hemispherical
Contact for Varying Mechanical Properties, IMechE J. of Eng. Trib. 218,
pp.313322.
4. Jackson, R. L., Kogut, L., A Comparison of Flattening and Indentation
Kogut
Approaches for Contact Mechanics Modeling of Single Asperity Contacts,
2006, J. of Tribol., Trans. ASME, 128, 1, pp. 209212.
5. Wadwalker, S. S., Jackson, R. L., A Spherical Contact Model Including
Severe Plastic Deformation, ASME/STLE International Joint Tribology
Conference, Memphis, TN, October 1921 , 2009. ...
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 Spring '08
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 Deformation, Stress

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