Applied Mamatics and Mechanics (English Edition, Vol. 12,.No. 2, Feb. 1991) Published by SUT, Shanghai, China THE EXACT INTEGRAL EQUATION OF HERTZ'S CONTACT PROBLEM* Yun Tian-quan (Z-:~) (South China University Technology. Guangzhou) (Received March 5, 1990) This "paper presents exact integral equation Hertz's contact problem, which is obtahwd by taking hlto account horizontal displacement pohlts in contacted surfaces due to pressure. The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution Key words only when elasticity, polytropic contact problem, index intej:al detonation equations products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock I. Introduction behavior reflection shock in explosive products, and applying small parameter purterbation Hertz's method, contact an problem analytic, is first-order a well-kr, approximate own classical solution problem is obtained elasticity, for and problem it is introduced flying by plate many driven text books, by various e.g. [1]. high In explosives 188 I, H. Hertz with first polytropic solved indices problem or than pressure but nearly distribution equal to between three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus two elastic spherical bodies in contact. Then, he used similar method for more general case two elastic bodies in contact. The assumptions in Hertz's method (or it is called Hertz's ory) index) for estimation velocity flying plate is established. are: contacting solids are isotropic and linear elastic; representative dimensions contact area are very small compared to various radii curvature undeformed bodies, in vicinity contact interface; contact is perfectly smooth, i.e. only normal stress exists in contact Explosive interface. driven According flying-plate to se technique assumptions, ffmds its by important use use Boussinesq's in study solution, behavior i.e. materials solution under a concentrated intense impulsive force perpendicularly loading, shock applied synsis to diamonds, surface and a half explosive space, welding Hertz derived and cladding integral metals. equation The method contact estimation problem flyor from velocity and geometric way condition raising it are questions vertical displacement common interest. and got solfition by assumption method. Hertz's works has interested a lot mechanics Under researchers assumptions and mamaticians. one-dimensional During plane detonation last hundred and years, rigid flying Hertz's plate, ory normal has been approach developed in solving manyaspects problem espectially motion in following flyor is to aspects: solve 1, following more contact system problems equations with governing flow field detonation products behind flyor (Fig. I): various types contact interfaces or geometrical shapes bodies have been solved ; 2, contact problems with more complex form stress between contact interfaces, e.g. shear stress is added to normal stress (friction contact), have been developed; ap +u_~_xp + au 3, various kinds materials instead isotropic linear elastic material have been studied, e.g. case a rigid solid (rigid punch) contacted to an elastic solid, or for grandlar media y etc. : 4, =0, powerful mamatical methods closely concerned with above problems (including two-dimensional cases), e.g. complex variable as as method, integral transform, singular integral equations etc., have been developed. Or development such as combination elastic contact ory with hydrodynamics has also been dcveloped. The summary development above 1-3 items can be found in chapter 42 where [2] (1962) p, p, for S, u details: are pressure, development density, specific entropy above and 4-th particle item and velocity advance detonation elastic products contact respectively, problems before with 1980 trajectory can be found R in reflected [3]. Recently, shock detonation study wave elastic D as contact a boundary problem and is still trajectory F flyor as anor boundary. Both are unknown; position R and state parameters *Dedicated on it are governed to Tenth by Anniversary flow field and I One central Hundred rarefaction Numbers wave behind AMM (11) detonation wave D and by Project initial supported stage motion by National flyor Natural also; Science position Foundation F and China state parameters products 181
182 Yun Tian-quan appearing. For example. [4] (1987) gave solution method for frictionless elastic contact problem with a curved rigid punch arbitrary s,lape. Most above developments are devoted to use or to completing Hertz's ory to solve more contact problems. There seems to be no paper concerned with properties or derivation integral equation Hertz's contact problem. The authorts) showed from mamatical sense solution that re could be many solutions for homogeneous part Hertz's integral equation.'this paper studies derivation integral equation Hertz's contact problem. Under same assumptions as Hertz's, thispaper derives a new integral equation different from that Hertz by taking into account all components (vertical and horizontal) displacement points in contacted interfaces due to pressure. However, Hertz's integral equation is derived by only taking into account vertical displacement points in contacted interfaces due to pressure, and its solution does not satisfy integral equation this paper. Hertz's integral equation is not an exact integral equation for frictionless contact problem since influence horizontal displacement on equation has been neglected. Therefore, it becomes a question that wher Hertz's solution is a true exact solution frictiontess contact problem or not. It is necessary The one-dimensional to seek a solution problem satisfying motion exact integral a rigid equation flying plate this under paper. explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, II. The a numerical Exact Integral analysis Equation is required. In The this paper, Problem however, Two by utilizing Spherical "weak" Bodies shock in behavior Contact reflection shock in explosive products, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate Using driven by various same assumptions high explosives and with similar polytropic method indices Hertz's or, but than considering but nearly equal horizontal to three. displacement Final velocities points flying in plate centact obtained interfaces agree very we well can with n numerical derive a new results exact by integral computers. equation Thus an differed analytic from formula Hertz's. with two parameters high explosive (i.e. detonation velocity and polytropic index) Let for R. estimation and R, be radii velocity two flying undeformed plate is established. spheres contacted at O respectively, R be radius contact base (i.e. projection contact interfaces on plane tangent at O). According to assumption that 1. representative Introduction dimensions contact area are very small compared to R D and R~, we can use results obtained by half space to represent resu- Explosive driven flying-plate technique ffmds its important use in study behavior lts a local deformed sphere with large curvature, and use quadratic form materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding metals. The method z~ =r~ estimation /( 2Rl ), flyor velocity z2=r2, and /( 2R~ way ) raising it are questions (2.1) to represent equation points M~ and M,, on a meridian section undeformed spheres R~ Under assumptions one-dimensional plane detonation and rigid flying plate, normal and approach R_, at very solving small distances problem r~ and r: motion from =~-axis, flyor is =~-axis to solve respectively following with system sufficient equations accuracy. governing When we flow treat field frictionless detonation contact products problem behind by flyor above (Fig. assumptions, I): we see from solution concentrated force perpendicular to surface half space ((215) p. 402 [l]) that both vertical and horizontal components displacement a point in surface half space belong to same order, i.e. O(r -~tz ), ap that +u_~_xp is, + influence au horizontal displacement can not be neglected. Now, we take terms au due to au horizontal 1 y displacement =0, into account to derive integral equation. In Fig. 1, R represents base radius. as A as point M in base with radius r represents identical po,int deformed from M~ on sphere I and M_, on sphere 2. Let r~ and r, be radii undeformed point 3,4~ and M~ respectively, U t and U~ be horizontal displacements M t and M_, respectively. From geometric condition horizontal displacement, we get r =rl + UI =rz + U~ (2.2) respectively, with trajectory R reflected shock detonation wave D as a boundary and From trajectory geometric F flyor condition as anor boundary. vertical displacement, Both are unknown; we have position R and state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D and by initial stage motion flyor also; position F and state parameters products
Integral Equation Hertz's Contact Problem 183 a--(wl+w2)=zl+ z2 (2.3) where wl and Wz represent vertical displacement M~ and M: respectively, a represents approach any two points spheres on axes z~ and z. at large distances from O. Letting unknown stress distribution between contacted interfaces be q. using Solution concentrated force perpendicular to surface half space, we can write expressions for displacement U~. U.. wl, w2 due to total stress distribution q U,=-(l--2v')(1+vl)2n:E, J" Iq'e~176 " (2.4) (l-v1 z) The formulas for U,. w2 are similar, which are obtained by replacing subscript 1 by 2. where El, E2, vt, v2 denote Young's modulus and Poisson ratio spheres 1 and 2. respectively. The s. $ in (2.4) are polar coordinates with origin ai M as shown in Fig. 2. The integral limit in The one-dimensional problem motion a rigid flying plate under explosive attack has (2.4) is base with radius R (for convenience and as it also agrees with [I]. integral limit is an analytic solution only when polytropic index detonation products equals to three. In general, omitted). a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock in explosive products, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by various R= high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate r, ~L,r_r~ obtained agree very well with numerical results by computers. Thus index) for estimation velocity flying plate is established. 121 Explosive driven flying-plate technique ffmds its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding metals. The method estimation flyor velocity and way raising it are questions Fig. 1 Fig. 2 Under assumptions one-dimensional plane detonation and rigid flying plate, normal approach solving problem motion flyor is to solve following system equations Substituting (2.4), (2. I), and (2.2) into (2.3), we have governing flow field detonation products behind flyor (Fig. I): where ~Cl. --/3r"- p. 4.. )2 ap +u_~_xp + 1 --v] as 1--v5 as -, 1! k I = -7~ 1, k2 =.,fez ' /) =~1 + 2Rz (2.5) (1--2,'1)(~+,',) (1--2,'2)(! :-vz) (2.6) ~_--- 2.'t P-;'I R~ 4 2:cE.:R2-- respectively, with trajectory R reflected shock detonation wave D as a boundary and,~ l - >, )( 1 + z', ) ( 1-2,'2 )( J. +,,., ) 1 ~ trajectory F flyor as anor boundary. Both are unknown; position R and state parameters on it are governed by flow field I central rarefaction wave behind detonation f wave D and (2.5) by initial is exact stage integral motion equation flyor for also; frictionless position spherical F and solids state contact parameters problem. products This is a nonlinear integral equation with small parameters. Furr study for solvability and au
184 Yun Tian-quan solution method this integral equation is needed. If nonlinear term is neglected, n (2.5) becomes I q. d,e( k, + ( 2.7 ) Here (2.7) is called approximate integral equation frictionless spherical solid contact problem. In (2.7), order integration has been changed. If term due to horizontal displacement is furr neglected, n we get Hertz's integral equation IIq.dsd~=( a -flrz)/(kl + kz) ( 9.. 8 ) Since solutions (2.5) and (2.7) have.not yet been found, degree accuracy solution q, Hertz's integral equation (2.8) q, =( qo/ R),,/ R~-r 2 (2.9) (where The q0 is one-dimensional a constant) to problem exact integral motion equation a (2.5) rigid or flying to plate approximate under explosive integral attack equation has an (2.7) analytic can not solution be judged. only when polytropic index detonation products equals to three. In general, It is a worthy numerical while analysis to point out is required. that in an In integral this paper, equation however, with small by utilizing parameter, if "weak" term shock with behavior small parameter reflection is omitted, shock obtained explosive solution products, could have and a applying large error. small parameter purterbation It is method, known that an analytic, q, does not first-order satisfy approximate (2.5) or (2.7). solution is obtained for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final III. velocities The Exact Integral flying plate Equation obtained agree for General very well Case with numerical Two Elastic results by Solids computers. in Contact Thus index) According for estimation to Hertz's ory, velocity surfaces flying plate bodies is established. with general shape near point contact can be represented approximately by equation with quadratic terms (see p.414 [1]) zl=alx] +AzxlYl+A3y~, zz=blx~ +B,x2yz+B3y] (3.1) where Explosive subscripts driven l and flying-plate 2 coordinates technique x. y ffmds,z denote its important corresponding use in elastic study solid behavior which materials quantity belongs under intense to, respectively. impulsive loading, At, A 2, shock A 3, B, synsis, B 2, B 3 are diamonds, constants. and explosive welding and cladding Let a point metals. M(x. The y) method in base estimation be identical flyor point velocity and deformed way points raising Mj it and are M questions v where M, common and M, are interest. on surfaces elastic solids 1 and 2 with undeformed coordinates Mj (x,, Yl, z,) and M~. Under (x,.. Y2, assumptions :,_) respectively. one-dimensional Let u,, v 1 and plane u 2, vz detonation be displacements and rigid flying alongx, plate, y direction normal approach points M, and solving M 2 respectively. problem Then, motion from flyor geometrical is to solve condition following horizontal system displacement, equations governing we get flow field detonation products behind flyor (Fig. I): X=Xl+Ul=Xt +u*, Y=Ya +vl=yz+vz (3.2) ap +u_~_xp + au From geometric condition vertical displacement, we have wl +w2=a-( zl + z2) (3.3) where sense a is as in above as section, as wl, w2 denote vertical displacements M~ andm_, respectively. Letting unknown stress distribution between contacted interfaces be q, using solution concentrated force perpendicular to surface half space, we can write expressions for displacements III, 1.31~. W 1 and u 2, v2, W2, due to total stress distribution q respectively, with trajectory R reflected shock detonation wave D as a boundary and trajectory F flyor as anor boundary. Both are unknown; position R and state parameters on it are governed by flow field I central rarefaction wave behind detonation wave ul (1--2vi)(1 +vl)iiq'( xt) 1 D and by initial stage motion = -- flyor 2ore also; I position xr~ F and dxtdy' state parameters products
Integral Equation Hertz's Contact Problem 185,,,=_ - 2~rEl j j r l (3.4) arel where r~--.= (x'-- x) 2 + (y' - y)z. x', yt are coordinates loaded point. The integration runs over all pointsin base. The formulas for,~_, vz, wz are similar, which are 6btained by replacing subscript 1 by 2. Substituting.(3.4), (3.2). (3.1) into (3.3). we get exact integral equation for general case frictionless elasiic solid contact problem (k, +k,)jj-q': A ~-a-(a,+bt)x'-(as+bs)y: The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock in explosive products, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus index) for estimation velocity flying plate is established. where sense Jilt kl is same 1. as Introduction (2.6), (l-2v~)(i+vt) (i- ~.v~)(i+ vo Explosive driven flying-plate technique., ffmds its important use in da = study dx' dy' behavior materials under at intense = impulsive 2~rE1 loading, shock at-- synsis 2~rE, diamonds, ' and explosive welding and cladding metals. The method estimation flyor velocity and way raising it are questions A furr study for solvability and tlae solution method this equauon is needed. Similarly, Under solution assumptions corresponding one-dimensional Hertz's plane integral detonation equation and rigid flying plate, normal approach solving problem motion flyor is to solve following system equations governing flow field detonation ( h, products + k,) II.q':'d behind -~a-.dxz- flyor (Fig. By I): z (3.6) does not satisfy (3.5). ap +u_~_xp + au References [ 1 ] Timoshenko, S. P. and J. 2q. Goodier, Theory Elasticity, 3rd ed., McGraw-Hill Book Co., Inc., New York (1970), 409-416. as as [ 2 ] Flugge, W., Handbook Engineering Mechanics. McGraw-Hill Book Co., Inc., New York (1962). [ 3 ] Gladwell, G. M. L., Contact Problem in Classical Theory Elasticity, Sijthf& Noordhf where Alpben p, p, S, aan u are den pressure, Rijn, The density, Nerlands specific (1980). entropy and particle velocity detonation products respectively, [ 4 ] Fabrikant, with V. I., trajectory Frictionless R elastic reflected contact shock problem detonation for a curved wave D rigid as a punch boundary arbitrary and trajectory F flyor as anor boundary. Both are unknown; position R and state parashape, Acta Mechanica,. 67 (1987), 1-25. meters on it are governed by flow field I central rarefaction wave behind detonation wave D [ 5 and ] Yun by Tian-quan, initial stage On motion un-uniqueness flyor also; position solution F and Hertz's state contact parameters problem, Journal products Applied Mechanics. 7, 3 (1990). (in Chinese)