The contct stress prolem for pieceisely defined punch indenting n elstic hlf spce Jcques Woirgrd 5 Rue du Châteu de l Arceu 8633 Notre Dme d Or FRANC Astrct A solution for the contct stress prolem of n indenter ith geometry defined in severl regions is given. Both the lod-penetrtion reltionship nd the stress distriution re given. The stresses in the different regions re otined y mens of n esily clculle numericl integrtion. Introduction The prolem of determining the distriution of contct stresses eteen rigid indenter nd n elstic hlf spce hs een solved in the cse of indenters ith simple shpes, tht cn e descried y single smooth functions, such s flt punches, cones, spheres or profiles descried y poer ls ith ritrry eponents [,]. Hoever the prolem susists ith rel indenters hose profiles cnnot e descried ith single epression, ut must e defined in different regions. It is, for emple, the cse of tips ith flt or rounded etremities. In the folloing, the contct stresses for such indenters, s ell s the lod-penetrtion reltionship, ill e presented. Theoreticl results In this ork Sneddon s results [,3] ill e used. In cylindricl coordintes nd using Hnkel s trnsforms, the ihrmonic equilirium eqution cn e epressed y the folloing set of integrl equtions:
G (r,z J ( r d z u r G (r,z ( z u z GJ ( r d G rz ( r,z ( G J( r d z 3 G G zz ( r,z ( (3 4 J ( r d 3 z z z nd r re cylindricl coordintes respectively norml nd prllel to the surfce, u r (r,z nd u z (r,z the rdil nd norml displcements, τ rz (r,z the sher stress nd σ zz the norml stress. J (ξ,r nd J (ξ,r re Bessel functions of the first kind. G(,r C(,r zd(,re z is function vnishing fr from the surfce nd stisfying the oundry conditions. For frictionless punch τ rz =. Sustituting ρ=r/ nd t=ξ/, eing the contct rdius, the integrl equtions cn e ritten on the surfce: ( ( ( ( u( g(tj ( tdt tg(tj ( tdt g(tj ( tdt Applying the inverse Hnkel s trnsform nd since σ(ρ= outside the contct re (ρ>: g(t ( J ( t d ( λ nd µ eing Lme s coefficients. Replcing g(t y this vlue leds to:
( (z ( u(z ( ( d ( d J ( tj (ztdt J ( tj (ztdt Using elementry properties of Bessel functions: J ( tj J ( tj (ztdt min(z,,z (ztdt,z z ( (z The norml displcement cn e ritten: (z ( z z ( d The contct stress distriution is otined y inverting the Ael s integrls: nd introducing the uiliry function F(: d z(zdz F( z 4( d ( ( d F( Let us consider rigid frictionless isymmetric indenter, hose profile is defined in pieceise mnner nd introducing the dimensionless prmeters i, limiting the different regions of the profile:......... n h f (z h f (z h f n (z n z z z n
The uiliry function F( tkes the form: F( d ( / z (zdz z / / z (zdz z... n / zn(zdz z With f (=, n = nd h the indenttion depth, this function cn ritten: n i / f ' (zdz F ( h i [] i i / z the shpe of the indenter eing descried y continuous function, not necessrily differentile, i fi ( fi i ( then, introducing the Young modulus nd the Poisson s rtio ν: n i / d F i( ( [] ( d i i / To void numericl differentition σ(ρ is etter epress s: ( ( ( i / df i( n i i / i / dfi ( i [3] i We hve lso: leding to: P ( d [4] n i / z i(z P dz [5] i i / z
When the punch is smooth t the contct circle, the stress hs finite vlue for r=, nd the penetrtion h is the solution of the eqution []: F( [6] qutions (5 nd (6 led to generliztion of Sneddon s formul (, more convenient to derive h nd P: h n i i / i P / (zdz n ' i i i z / i / z (zdz ' i z [7] This s lredy proposed y Gunghui fu [4]. The fundmentl reltionship: dp dh [8] Cn e directly verified differentiting P nd h reltive to or, in simpler y using the reciprocl theorem [5]. Punches ith specil shpes In the folloing, different shpes of punches ill e emined. Truncted cone: The first cse to e consider, is the cse of truncted cone terminted y flt end of rdius, used for emple in Shore tests.
Figure The shpe of this indenter is defined y the functions nd : h h (z cot g From equtions [7],, the contct rdius, the penetrtion h nd the lod P correspond to: h P Arccos tg tg Arccos The derivtives df / nd df /, of the different prts of the F function deduced of eqution ( re: df ( df ( ( tg And the stress σ(ρ is given y: Arccos ( ( ( ( / df ( df (
The corresponding stress is plotted Figure, for different vlues of the rdius...5.5.5.8..5....4.6.8. Figure It my e note tht, due to the discontinuity of the derivtive of the shpe function of the punch, the stress is goes to infinity for r=. Cone terminted y sphericl cp: Most rel indenters present rounded tip tht cn e modeled y sphere of rdius R or, to first pproimtion, y proloid ith the sme rdius of curvture. R Figure 3
The shpe is defined y: z h R h (z cot g R The penetrtion depth nd the lod re then epressed s: h P R nd for the stress: ( 3 ( Arccos tg 3R tg Arccos ( ( ( ( / df ( df ( / df ( ith: df ( R df ( R R tg Arccos tg
..5.5.5.8. R/=.5.5....4.6.8. Figure 4 We cn see tht the stress goes to ininity t the edge eteen the sphere nd the cone. 3 Cone terminted y tngentil sphere: It is cse of prcticl interest, since it corresponds to most rel pyrmidl punches. The etremity of the tip cn e modeled y: R tg [9] The sphere eing tngent to the cone, the punch is smooth everyhere.
R Figure 5 We hve: z h cot g h (z cot g nd for the penetrtion nd the lod: h P 4 3 tg (4 Arccos 6 tg 3 Arccos The derivtives of the uiliry functions tke the form: df ( tg df ( ( Arccos tg The corresponding stress distriution is plotted Figure 6, for different vlues of.
..5..5.5.5.8....4.6.8. Figure 6 It cn e verified tht for this smooth indenter, the stress is continuous everyhere. 4 Smooth truncted sphere: It corresponds to the cse of n equivlent punch [6,7], indenting prtilly conforming imprint in plsticlly deformed smple. The lunt edges of the equivlent punch cn e modeled y sphere of rdius R, lrge compred ith the penetrtion depth. R The shpe of this equivlent indenter is defined y the folloing functions:
h (z h R Applying gin eqution (7, e get the folloing epressions for the penetrtion nd the lod: h P (4 Arccos R 6R 3 Arccos nd for the derivtives of the uiliry functions nd for the stress σ(ρ: df df R ( ( ( ( / R df ( df ( Arccos The stresses re plotted Figure 8 for different vlues of the rdius of the conforming prt of the imprint hith rdius of the sphere R=.
..5.5..5 R/=.5.8....4.6.8. Figure 8 5 Truncted cone ith rounded edges: It is the cse of rel truncted indenters for hich the edges hve een slightly orn out. For the punch to e smooth, e must hve: R ( tg R Figure 9 The shpe is defined on three regions y the functions:
3 h (z cot g h ( h ( cot g (z cot g The penetrtion nd the lod tke the form: h ( P (4 (4 Arc ( tg cos Arc cos 3 6 ( tg 3 ( Arc cos Arc cos And the uiliry functions, in the regions delimited y the rdii, nd : df df Arccos cot g Arccos Arccos df3 ( cot g The integrls giving the stress distriution re then computed in these domins: ( ( / / df ( / df ( 3 ( ( / df ( / df ( 3 ( ( df ( 3 The stress distriution is plotted in Figure, for /=.5 nd different vlues of /.
.5..5 /=.5..5.....4.6.8. Figure It cn e seen tht the stress in the neighorhood of the cone edges increses hen the rdius decreses, tht is to sy hen ecomes close to. Conclusion The prolem of the determintion of the contct stress distriution during indenttion of n elstic hlf spce y n isymmetric pieceisely defined indenter, hs een determined. The corresponding penetrtion depth nd the lod re lso given s functions of the contct re, lloing prmetric representtion of the lod s function of the penetrtion. Indenters ith perfectly shrp or rounded edges hve een emined. Of prcticl interest re the cses of lunt cone terminted y tngentil sphere, or the cse of truncted one ith rounded edges. It must e noted tht, in these cses, the stress gets only finite vlues, hich is not the cse for unrelistic indenters ith perfectly shrp edges. References [] In N. Sneddon, Int. J. ngng Sci.,3,47(965 [] C. M. Segedin, Mthemtik, 4, 56, (957 [3] In N. Sneddon, Fourier Trnsforms, McGr-Hill, Ne York (95 [4] Gunghui Fu, Tiesheng Co, Mt. Sci. nd ngng.,a 53-54, 76, (9 [5] R. T. Shield, Z. AgneMth. Phys., 8, 68, (967
[6] L. Solomon, lsticité Linéire, Msson, Pris, (968 [7] G. M. Phrr, Mechnicl Properties of Films, Cotings nd Interfcil Mterils, Il Chiocco, Itly, (999