A Solution of Rigid Plastic Cylindrical Indentation in Plane Strain Prof. Robert L. Jackson Director, Tribology Minor Multi-Scale Tribology Laboratory Department of Mechanical Engineering Auburn University
[email protected] (334)844-3340
Problem Definition • We are considering indentation of a rigid cylinder into a plastically deforming flat in plain strain.
Applications • • • • • • • •
Road Roller/ Wheel Contacts Rounded Edge Contacts (Fillet) Cutting Edges Metal Rolling and Working Wire Crimping Highly Anisotropic Rough Surface Contact Artificial Surface Features Indentation Testing
Spherical Indentation • Ishlinkii (1947) performed a slip line analysis of a spherical contact and found that the contact pressure for a perfectly plastic contact without hardening (i.e. Brinell hardness) was between 2.61 and 2.84. • However, it was shown that the average pressure decreases with indentation depth (Yu and Blanchard (1996), Kogut-Komvopoulos (2004), Alcalá and Esqué-de los Ojos (2010), Jackson, Ghaednia and Pope (2015)).
Initiation of Yielding • Green (2005) provided CS y 2 E * 1 equations to predict the c R * 2 ln E CS y initiation of yielding in 2 elastic-plastic cylindrical CS y Fc R contact: L E* 2
• For plane stress, C=1.
1 C 1 4(v 1)v C 1.164 2.975v 2.906v 2
Ac 4 LR
CS y E*
Slip-Line Theory Methodology • The slip line theory and derivation is not thoroughly described here, but additional details can be found in the book by Tabor (1951). • For the theory to be applicable the following assumptions are made, • Plane strain • The loading is quasi-static • There are no body forces • The material yields according to the von Mises criteria as a rigidperfectly plastic material. • No elasticity is considered. • Here no friction is accounted for, although this effect is often important. • The effects of pile-up and sink-in due to elasticity and volume conservation are also neglected.
Slip Line Examples
β slip line
α slip line
Slip-line Equations • Slip lines are defined as a curved line which is tangential along its length to directions of maximum shear stress. • Slip lines in a material are made up of two curvilinear and orthogonal lines (α and β) described by the following equations. • h 2k C1 along the α line S Y For Von k Mises 3 Criteria • h 2k C2 along the β line • h is the hydrostatic stress and k is the shear yield strength
Schematic
Slip-line Relation • On the labeled free surface there is no normal or shear traction. Therefore the shear stress tangential to the surface is zero there. • On the indenter surface there is an applied pressure, p, but also no shear stress tangential to the surface, due to the surface being frictionless. • From Tabor (1951), this gives:
p 2k 2k
Derivation • Since the contact pressure on the indenter is always normal to it, we must consider that only the vertical portion contributes to F a
F 2L p sin dx
pv p sin
0
x R cos
1
2k 2k sin 2 d /2
F LR
2 2 1 a 1 a cos 2 cos 1 p F 1 a 4 R R 2 S y 2bLS y 2 3 R a a a cos 1 1 sin 2 cos 1 R R R
Limits • Theoretically, the upper limit to the pressure appears to be a factor of approximately 2.97 times the yield strength. • However, as indentation is increased and the contact effectively changes shape, the pressure reduces. The lower limit is approximately 1.91 times the yield strength. 2 2 1 a 1 a cos 2 cos 1 p F 1 a 4 R R 2 S y 2bLS y 2 3 R a a a cos 1 1 sin 2 cos 1 R R R
Finite Element Analysis • In order to verify the derived equation, a finite element model was implemented. • The cylinder is rigid, but the surface is elasticplastic and in plain strain. • Typical material properties have been assumed (E=200 GPa, v=0.3) and the yield strength (Sy) is varied between 200 MPa and 800 MPa. • The current results have focused on cases mainly in the fully plastic regime (approximately >100c).
Finite Element Mesh • Mesh Convergence confirmed (67,288 elements). Mesh refined near contact region.
Stress Contours
Contact Pressure Results 3.2 Slip-line Eq. 20 FEM (Sy=200 MPa) FEM (Sy=400 MPa) FEM (Sy=800 MPa) Pileup Adj., Eq. 20 & 21
3
p/Sy
2.8
2.6
2.4
2.2
2 0
0.2
0.4
0.6 b/R
0.8
1
Error 0.14 FEM (Sy=200 MPa) FEM (Sy=400 MPa) FEM (Sy=800 MPa)
0.12
Abs. Error
0.10 0.08
0.06 0.04
0.02 0.00 0.E+00
2.E+04
4.E+04
6.E+04
/c
Above the region where elasticity influences the problem, the error is less than 1% for most cases, and less than 5% until a/R reaches approximately 0.77.
• Elastic deformation appears to become negligible at approximately 400c (or at least that is when the model becomes applicable) 0.14 FEM (Sy=200 MPa) FEM (Sy=400 MPa) FEM (Sy=800 MPa)
0.12
Abs. Error
0.10 0.08 0.06 0.04 0.02 0.00 0.E+00
2.E+03
4.E+03 6.E+03 /c
8.E+03
1.E+04
Correcting for Pile-up • All cases appear to fall on same normalized curve. • An equation is fit to correct for this (for a/R>0.5):
p pileup Sy
2 3 a a 1 p 1 5 R R 4 S y
Modified Model 3.2 Slip-line Eq. 20 FEM (Sy=200 MPa) FEM (Sy=400 MPa) FEM (Sy=800 MPa) Pileup Adj., Eq. 20 & 21
3
p/Sy
2.8
2.6
2.4
2.2
2 0
0.2
0.4
0.6 b/R
0.8
1
Conclusions • First this work derives analytically, using slip-line theory, a prediction of the average pressure occurring in the indentation of a rigid frictionless cylindrical punch into a perfectly plastic flat surface. • The FEM and slip-line predictions are in surprisingly good agreement. • An adjustment for pileup is also made to the model for a/R>0.5. • The resulting model has very good agreement for all the fully plastic FEM cases.
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