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epcontactpresentation2 - Elastic-Plastic Spherical Elastic...

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Unformatted text preview: Elastic-Plastic Spherical Elastic Contact Robert L. Jackson Robert Mechanical Engineering Department Auburn University Hertz Contact Solution (1882) Hertz Closed-form 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 Elasto-plastic Model FEM Kogut and Etsion (2002) performed a FEM analysis Kogut of the same case of an elastic-perfectly 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 2-D 8-node element for sphere 2-D 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 x-direction 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 Elasto-plastic Hemispherical Contact, J. of Tribol., Trans. ASME, 127, 2, pp. 343-354. 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. 484-493. 3. Quicksall, J., Jackson, R. L., Green, I., 2004, Elasto-plastic Hemispherical Contact for Varying Mechanical Properties, IMechE J. of Eng. Trib. 218, pp.313-322. 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. 209-212. 5. Wadwalker, S. S., Jackson, R. L., A Spherical Contact Model Including Severe Plastic Deformation, ASME/STLE International Joint Tribology Conference, Memphis, TN, October 19-21 , 2009. ...
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