Loyola Marymount University Department of Mechanical Engineering MECH 570 – Tribology of Mechanical Systems May 4 th , 2011 HERTZIAN LINE CONTACT ANALYSIS WITH MATLAB Eric Fleming Graduate Student of Mechanical Engineering LOS ANGELES, CALIFORNIA, USA ABSTRACT This report inspects the details of the development of two MATLAB programs (normalized and non-normalized results) which models the Hertzian Contact of cylinders. Material properties may be user defined and the material is assumed to be of elastic-perfectly plastic nature for analysis. Results of the subsurface stresses are plotted and verified with reference [6], special cases are discussed. MATHEMATICAL NOTATIONS E* Composite Modulous R e Composite Radius υ Poisson’s Ratio a Radius of Contact (Half-Width) δ Elastic Displacement P 0 ,P h Hertzian Pressure (P is also used for distributions) P,W Forcing Load σ Stress Component τ Shear Component (Unless noted, all units throughout this report are in SI format) 1.0 INTRODUCTION Contact between surface asperities can me modeled in the simplest form with point or line contact among asperities. Point contact involves two spherical/elliptical surfaces with relative radii of curvature. Line contact involves two cylindrical surfaces with relative radii of curvature as point contact involves two spherical surfaces. [2] The model involves a Silicon Carbide cylinder on an AISI 52100 Steel plate. Properties are as follows: SiC Clyinder: E=550 GPa υ=0.3 R=0.001m AISI 52100 Steel: E=200 GPa υ=0.3 R=Infinite (Flat Plate) For calculation of subsurface stress fields, several critical parameters must be calculated first and foremost. This is required for equivalent moduli and radii of the two materials, the contact half- width, and (peak) Hertzian pressure. (eq1) (eq2) The subscripted properties above are those of the two different materials. It is important to note that, for concave surfaces, the relative radius is of negative value. [2] 1
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Loyola Marymount UniversityDepartment of Mechanical Engineering
MECH 570 – Tribology of Mechanical SystemsMay 4th, 2011
HERTZIAN LINE CONTACT ANALYSIS WITH MATLAB
Eric FlemingGraduate Student of Mechanical Engineering
LOS ANGELES, CALIFORNIA, USA
ABSTRACT This report inspects the details of the development of two MATLAB programs (normalized and non-normalized results) which models the Hertzian Contact of cylinders. Material properties may be user defined and the material is assumed to be of elastic-perfectly plastic nature for analysis. Results of the subsurface stresses are plotted and verified with reference [6], special cases are discussed.
MATHEMATICAL NOTATIONSE* Composite ModulousRe Composite Radiusυ Poisson’s Ratioa Radius of Contact (Half-Width)δ Elastic DisplacementP0,Ph Hertzian Pressure (P is also used for distributions)P,W Forcing Loadσ Stress Componentτ Shear Component(Unless noted, all units throughout this report are in SI format)
1.0 INTRODUCTIONContact between surface asperities can me modeled in the
simplest form with point or line contact among asperities. Point contact involves two spherical/elliptical surfaces with relative radii of curvature. Line contact involves two cylindrical surfaces with relative radii of curvature as point contact involves two spherical surfaces. [2] The model involves a Silicon Carbide cylinder on an AISI 52100 Steel plate. Properties are as follows:SiC Clyinder:
E=550 GPa υ=0.3 R=0.001m
AISI 52100 Steel:E=200 GPaυ=0.3
R=Infinite (Flat Plate)For calculation of subsurface stress fields, several critical
parameters must be calculated first and foremost. This is required for equivalent moduli and radii of the two materials, the contact half-width, and (peak) Hertzian pressure.
(eq1)
(eq2)
The subscripted properties above are those of the two different materials. It is important to note that, for concave surfaces, the relative radius is of negative value. [2]
1.1 POINT CONTACT FORMULAESince line and point (cylindrical and spherical) contact
behave differently, there are separate representations for the same parameters. The contact radius and others for point contact are as follows:
(eq3)
Hertzian Pressure:
(eq4)Displacement:
(eq5)
1.2 LINE CONTACT FORMULAEEqn’s 3-5 are modeled in line contact mode in the
following formulas. The forcing load is in units of force per unit length, unlike point loading where the load is solely in units of force. [6]
Contact Radius:
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(eq6)
Hertzian Pressure:
(eq7)
Displacement:
(eq8)
1.3 LINE CONTACT SUBSURFACE STRESSES FORMULAE
McEwen (1949) represents stress in terms of arbitrary constants m and n: [6]
(eq9)
(eq10)
Where equations 9 and 10 are the signs of the z-axis and x-axis respectively. [2] The constants are then substituted into the following formulas:
(eq11)
(eq12)
(eq13)
These formulas are used to find stress components σx, σz, and shear in the ‘xz’ plane,. For computation of the principal shear stress, the following formula is used. [4] This is also known as Tresca Stress.
(eq14)
The y-component may also be calculated if desired:
(eq15)
It is also beneficial to creat a pressure distribution function which is based of the peak pressure, or Hertzian pressure.
(eq16)
2.0 ANALYSIS AND PROCEDUREMATLAB code was utilized to study the effects of the
stress distribution across the contact area. The basis of the code was studied from reference [5]; the logical “for” loops were altered for the formulation of the stress and shear contour plots while encoding the formulas from references [4] and [6]. Code was developed for cylindrical cases, verified, and then studied.
2.1 STRESS FUNCTIONS ALONG AXIS-OF-SYMMETRY
Functions along the z-axis (x=0) were observed by extracting the data from the center (line of symmetry) column of the stress matrices. This column in the data happens to be column 68.
3.0 VERIFICATION OF PROGRAM INTEGRITYIn order to confirm the integrity of the formulated code
found in Appendix B, results for normalized stress were compared with those in the appendix of reference [4]. The weights for normalized stress in the x direction (σx) from the aforementioned reference are as follows:
From observation, the obtained results are very similar to the published results. There are slight discrepancies due to
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how relative positions were discretized in the program. Further verifications for σz and τxz may be found in Appendix C.
Further verification is proved through the maximum value and location of normalized principal shear along the z-axis. It was found to be 0.302@z=0.78a. [3]
4.0 RESULTS AND DISCUSSIONAfter the program was verified, the case for normalized
results was run several times to observe the change in contours as initial values were significantly altered. The contours of principal shear stress (eq14) was plotted.
(Fig1, Normalized Principal Shear)
Poisson’s Ratio, load, radii, and elastic moduli were modified for both materials and the normalized contour plots were unaffected, except σy which has no influence on other plots, and is directly influenced by Poisson’s Ratio. Some parameters such as Hertzian Pressure, composite modulus, and contact radius were however affected. This will in part affect results for simulations which are not normalized. It is imperative to recognize the fact that modification of user-defined variables does not have an effect on normalized results.
Several plots for the normalized case can be found in Appendix A, which include σx, σy, σz, τxz, pressure distribution, and stress values across the axis of symmetry.
4.1 THE NON-NORMALIZED CASEIn order to grasp a better idea of what is happening in the
subsurface, it is important to study results which are not normalized to the Hertzian pressure. By keeping results un-normalized, the contours shift and change value accordingly to input. Below is figure 1 without normalization, forcing load is set to 1000 Newtons.
(Fig2, Principal Shear, 1 kN case)
Now the load is changed to be 50 Newtons, obseve the changes in the principal shear stress plot:
(Fig3, Principal Shear, 50N case)
As it can be observed, the contours have indeed shifted in magnitude (much smaller contours) and location. The loading force is now reverted back to the original value of 1000N, and the Poisson’s Ratio for both materials is changed to 0.9.
(Fig4, Principal Shear, υ=0.9 case)
From here it can be noted that Poisson’s Ratio has a significant effect on the principal contours. The maximum location is now effectively one order of magnitude greater and its location is significantly deeper into the subsurface. Material properties are again reverted back to their original
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state, and the radius is changes from a millimeter to a micrometer.
(Fig4, Principal Shear, R=1µm case)
Now the principal shear contour has much more density in magnitude, the maximum value is much greater than the standard case, and it is condensed into a much smaller area. A complete overview of the graphical results can be observed in Appendix A.
5.0 SUMMARY AND CONCLUSIONSA finite element MATLAB program for the simulation of
subsurface stresses is successfully developed for the Hertzian solution in a line contact mode. It is a program that is verified with published results for a normalized case.
Generating normalized stress contours for this case of Hertzian line contact will yield constant results, no matter what values of material properties you may define. This normalized case can be considered universal, no matter how the code is formulated (as long as it follows physical law), and what material properties are chosen, all normalized contours will be exactly the same. This applies to every normalized contour except for σy which is affected by Poisson’s Ratio, yet has no influence on principal shear stress.
In order to observe actual changes in the contours, the results shall remain in form without normalization. This enables the user of the program to see the changes in the subsurface stress fields as material parameters are altered.
It can be inferred with much certainty that these stress fields are greatly affected by material properties, contact radius, and forcing load. Being the fact that the fields would conform to the ranges of inputs specified, as the fields would shift in location and magnitude.
It is also important to note that this analysis is for the most ideal of contact cases. Foremost, the solutions of the simulation rely on the un-realistic fact that the materials are assumed to be elastic-perfectly plastic. Meaning that if the material is forced into the plastic zone of stress, there will be no work hardening and the material will not benefit from an increase in strength. The material is infinitely elastic. [1] This model does not incorporate layers which is very beneficial to see how stresses
are distributed in a hardened layer and transferred onto a softer substrate. This model also assumes a frictionless environment, which is far from realistic.
6.0 FUTURE RECCOMENDATIONSIf a forthcoming student decides to build from these
results he/she must understand that modeling of friction and substrates/layers requires a much different approach. One which involves calculation of numerous constants, Fourier transforms, and complicated program scripting possibly with ANSYS. For a much simpler challenge, it is recommended that a program be formulated for spherical point contact. It is more commonly modeled using polar/spherical coordinates; however, Cartesian coordinates may be used.
A custom forcing function was attempted to be implemented, but was unsuccessful. It disobeyed the mathematical representations in the Introduction since contours would halt at the contact radius and not extrude beyond. A non-Hertzian pressure follows different formulae. Therefore, it is also recommended that a non-Hertzian contact solution be encoded.
Numerous articles were obtained for this report and its code development, and all of them did not give their underlying equations for analysis. Future work should look into numerous cross-references of these reports which were unobtainable at Loyola Marymount University. The reports are referenced in a section named UNOBTAINABLE REFERENCES.
REFERENCES[1] B. Bushan, Introduction to Tribology, 2002[2] M. Siniawski, Tribology of Mechanical Systems, Lecture Notes of Contact Mechanics, MECH 570 – Spring 2011[3] Vizintin, J., Kalin, M., Dohda, K. and Jahanmir. , 2004, Tribology of Mechanical Systems, ASME[4] “Contact Mechanics”, Wikipedia, Modified 6 January 2011, Accessed 8 February 2011[5] M. Read, “Matlab Tool for Analyzing Stresses, Principle Stresses, and Surface Displacement on any Loading of an Elastic Half Space and for analyzing Hertz 2D contact stresses” 5/12/2004. Massachusetts Institute of Technology[6] K.L. Johnson, CONTACT MECHANICS, 1985, Cambridge University Press[7] A. Elsharkawy. (1999) “Effect of friction on subsurface stresses in sliding line contact of unilayered elastic solids” International Journal of Solids and Structures, Volume 36, Issue 26
UNOBTAINABLE REFERENCES(FUTURE STUDENTS)- H. Djabella & R.D. Arnell. (1993) “Finite element
analysis of the contact stresses in elastic coating/substrate under normal and tangential load” Thin Solid Films, Volume 223, Issue 1
- K. Komvopoulos. (1988) “Finite Element Analysis of a Layered Elastic Solid in Normal Contact with a Rigid
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Surface” Journal of Tribology. Volume 110. Issue 3- H. Djabella & R.D. Arnell. (1994) “Finite element analysis
of elastic stresses in multilayered systems” Thin Solid Films, Volume 245, Issues 1-2
- H. Djabella & R.D. Arnell. (1992) “Finite element analysis of the contact stresses in elastic coating on an elastic substrate” Thin Solid Films, Volume 213, Issue 2
- D.M. Bailey & R.S. Sayles. (1997) “Effect of Sliding Friction on Contact Stresses for Multi-Layered Elastic Bodies with Rough Surfaces” Journal of Tribology, Volume 113, Issue 4
- K. Mao, T. Bell, & Y. Sun. (1997) “Effect of Sliding Friction on Contact Stresses for Multi-Layered Elastic Bodies with Rough Surfaces” Journal of Tribology, Volume 119, Issue 3
- E.R. Kral & K. Komvopoulos. (1997) “Three-Dimensional Finite Element Analysis of Subsurface Stress and Strain Fields Due to Sliding Contact on an Elastic-Plastic Layered Medium” Journal of Tribology. Volume 119, Issue 2
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Appendix A – Graphical Results
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Appendix A – Graphical Results
Normalized Contours of Line Contact
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Appendix A – Graphical Results
1kN, υ=0.3, R=1mm Case
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Appendix A – Graphical Results
1kN, υ=0.3, R=1mm Case
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Appendix A – Graphical Results
50N, υ=0.3, R=1mm Case
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Appendix A – Graphical Results
50N, υ=0.3, R=1mm Case
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Appendix A – Graphical Results
1kN, υ=0.9, R=1mm Case
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Appendix A – Graphical Results
1kN, υ=0.9, R=1mm Case
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Appendix A – Graphical Results
1kN, υ=0.3, R=1µm Case
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Appendix A – Graphical Results
1kN, υ=0.3, R=1µm Case
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Appendix B – MATLAB Code
clear %USER-DEFINED MATERIAL PROPERTIES (SI UNITS) R1=inf; %Asperity Radius (inf for flat surface) R2=0.1; E1=200*10^9; %Elastic Modulous E2=550*10^9; v1=0.3; %Poisson's Ratio v2=0.3; p=1000; %Forcing Load %Output Mode (Raw or Normalized Results) %Input choice=1 for normalized results %Choice=2 for dimensionalized results choice=1; %This section computes parameters necessary for subsurface calulations R=1/(1/R1+1/R2); %Composite Radius E=1/((1-v1^2)/E1+(1-v2^2)/E2); %Composite Modulous a=(0.75*p*R/E)^(1/3); %Contact Half-Width x=[-2*a:.01*3*a:2*a]; %Discrete values for the x-axis z=[0:.005*3*a:2*a]; %Discrete values for the z-axis Ph=(p.*E./(pi.*R)).^(1/2); %Hertzian Pressure for Line Contact %This loop discretizes pressure distribution. for i=1:length(x) P(i)=Ph*[1-(x(i)/a)^2]^.5; end %For contour plots, a square matrix is required for position arguments. %Therefore, this loops creates a constant value for x-axis positions %vertically (column constant) and constant values for the z-axis %horizontally (row constant). for i=1:length(z); for j=1:length(x); xx(i, j)=x(j); zz(i, j)=z(i); end end %Below is the loop which computes the subsurface stress matrices. for i=1:length(x); for j=1:length(z); m(j, i)=(0.5*(((a^2-xx(j, i)^2+zz(j, i)^2).^2+4.*xx(j, i)^2.*zz(j, i)^2).^0.5+(a.^2.-xx(j, i)^2+zz(j, i)^2))).^0.5; n(j, i)=(0.5*(((a^2-xx(j, i)^2+zz(j, i)^2).^2+4.*xx(j, i)^2.*zz(j, i)^2).^0.5-(a.^2.-xx(j, i)^2+zz(j, i)^2))).^0.5; if x(i) < 0 n(j, i)=-n(j, i); end sx(j, i)=(-Ph/a)*(m(j, i)*((1+((zz(j, i)^2+n(j, i)^2)/(m(j, i)^2+n(j, i)^2))))-2.*zz(j, i)); sz(j, i)=(-Ph/a)*m(j, i)*((1-((zz(j, i)^2+n(j, i)^2)/(m(j, i)^2+n(j, i)^2)))); sy(j, i)=v1*(sx(j, i)+sz(j, i)); txz(j, i)=(-Ph/a).*n(j, i)*((m(j, i)^2-zz(j, i)^2)./(m(j, i)^2+n(j, i)^2));
