Dynamic model of introducing rigid smooth sphere into homogeneous elastoplastic hardenable solid body has been considered. On the basis of the model numerical-analytical dependencies describing the behaviour of solid body in elastoplastic region of contact interaction were suggested. The numerical-analytical dependencies allow us to take into consideration additional approach of contacting bodies owing to dynamic loading

Koteneva, N. V.

Maximenko, А. А.

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mode of pneumatic percussion device operation. Sharpchange in air consumption of МО9П hammer at pressing force less than 260 Н is explained by relative shift ofhelve, springloaded valve and case during operation.Decreasing pressing force helve and valve withdraw fromhammer intermediate link, because of this efficientsquare of admission port reduces.ConclusionThe regularities of changes in energy parameters andcompressed air consumption of hand hammers at changing air parameters in the system and pressing force have been experimentally stated.When increasing the temperature of compressed air,the energy characteristics, maximum and perexhaustair pressure in the back chamber grow. The lower thepressing degree and maximum air pressure in the backchamber, the higher the growth of stroke energy andpower. Among the types of hammers under study themost significant growth of energy power is observed inhammers of M types having relatively large volume ofclearance space and low degree of air compression inthe back chamber.At heating the compressed air its consumption decreases sufficiently. The degree of saving in compressedair is not the same for different types of hammers, themore the payload volume and the less the stroke frequency, the higher the degree of saving in compressed air.In production and operation of pneumatic percussion mechanisms of hammers and drills the influence oftemperature of compressed air on its consumption anddevice operation should be taken into account.Using hot compressed air or its special heating ischaracterized by high energy efficiency, for hammers ofM type in particular.Increase in pressing force on hammer helve is accompanied by the growth of compressed air consumption to the definite level. This level which corresponds to beginning of quasistationary operation mode is mainly determined by thevalue of shaking force in hammer. In hammers having valveintake in the helve, air consumption changes sharply and significantly at changes in pressing force up to limit value.Bulletin of the Томsк Pоlytеchnic University. 2007. V. 310. № 260REFERENCES1. Karabin А.I. Compressed air. – Moscow: Machinostroyeniye, 1964.– 343 p.2. Hand pneumatic hammers / Edited by V.F. Gorbunov [et all]. –Moscow: Machinostroyeniye, 1967. – 184 p.Received on 31.05.2006Demands of modern technology for constructiondesign possessing mechanical reliability at low materialconsumption lead to development of optimal design instrength taking into account contact deformation. Interms of improvement of physical and mechanical concept on destruction mechanism of solid bodies the section of deformable body mechanics called fracturemechanics has been formed within the last three decades.Creation of fracture mechanics made possible to interpret the processes of contact interaction from absolutely new point of view. Geometric localisation (in the contact region) of all forms of deformation (elastic and plastic) and destruction is known to be characteristics forcontact interaction of solid bodies. Being at first local,destruction can then progress catastrophically and resultin break of a sample or a detail. At present it is generallyaccepted that more than 80 % of failures in mechanisms,machines and devices are conditioned by the processes taking place in the contact zone of solid bodies. Investigations in this field as well as in strength tests are concernedwith study of behaviour of material surface layers at impression of one body (indenter) into another. The strengthtests provide information on mechanical properties ofmaterial and in study of this phenomenon significant success has been achieved. Investigations of fracture mechanics is wrongly paid little attention, which is obviously explained by complexity of formation and growth processesof surface fractures and absence of corresponding theoretical methods of analysis. However, study of contact destruction is of great scientific and practical importance.UDC 621.891(048):539.178(048)DYNAMIC MODEL OF ELASTOPLASTIC CONTACT INTERACTION OF SMOOTH BODIESА.А. Maximenko, N.V. KоtenevaAltai State Technical University, BarnaulЕmail: sle@agtu.secna.ruDynamic model of introducing rigid smooth sphere into homogeneous elastoplastic hardenable solid body has been considered. On thebasis of the model numericalanalytical dependencies describing the behaviour of solid body in elastoplastic region of contact interaction were suggested. The numericalanalytical dependencies allow us to take into consideration additional approach of contacting bodies owing to dynamic loading.The case of contact of spherical indenter of R radiuswith elastic halfspace under the action of force P, thesocalled Hertzian contact or Hertzian loading (Fig. 1)is the best studied and widely used one in practice.Fig. 1. Introduction of spherical indenter into elastic halfspace(the Hertzian contact)According to the Hertzian theory the body approachδ0 is the power function of contact load P [1], i. е.where is the elastic constant, elasticity normal module and the Poisson’s ratio respectively.In increasing contact load P there appears plastic deformation for the first time, which then extends graduallyboth in depth and to the surface of a rider. At some valueof load the Hertzian dependence is disturbed on the surface of rider. After removing the load elastic recovery ofthe rider material takes place and general approach decreased by the δy value of rider becomes equal to δпл (Fig. 2).Fig. 2. Scheme of introduction of rigid ball into flat boundary ofelastoplastic riderWith the advent of residual impression on the contact surface elastic deformations are known to continuefollowing the dependencies of the elasticity theory,however, it is obvious that in this case conventional «elastic» formulas have to be corrected in accordance withnew conditions of contact. Fundamental difference ofthese conditions from those of purely elastic contact ofbodies consists in the fact that in the presence of residual impression the circle of R radius contacts not in thepoint, but with the surface of residual impression, thecurvature radius of which is equal to R1 (Fig. 2).To determine the magnitude R1 let us take the following assumptions:1) after removing load spherical form of imprint doesnot change;2) form of impression under load and after unloadingin the planes of main curvatures are outlined by thecircle of R1 radius;3) surface of rider is not deformed without contact.Based on the proceeding correction to the Hertzianformula, including influence of rider plastic deformation in the contact region on the value of elastic approach, can be calculated by the formula [2]:(1)where δy is the elastic constituent of complete approach;δпл is the plastic constituent of complete approach.Finally, we haveor taking into account the formula (1)(2)Approach, corresponding to appearance and presence of plastic deformation, can be calculated by theformula [3]:(3)Complete approach in elastoplastic contact of circlewith rider consists of two summands, residual approachδпл, equal to the depth of residual imprint, and elasticapproach δy, disappearing after removing the load dueto elastic recovery of rider. Thus,(4)Substituting the values of elastic and residual approaches from the formulas (2) and (3) into the equation(4) respectively, we obtain:(5)whereThe formulas presented above permit the calculation of complete approach in elastoplastic contact atstatic loading. In some cases contact of solid bodies occurs at dynamic, in particular shock loading. Dynamicsof loading is characterised by either shock velocity ordeformation rate, or stress rate, which depends on rateof load application. At elastoplastic contact of materialsunder the condition of dynamic loading additional approach takes place that can result in changes of mechanical properties of the surface. Elastic contact occurs rarely at the initial collision of solid bodies, especially materials. Analysis of stress state of material at dynamicelastoplastic introduction of rigid indenters into it represents a very complex problem; the investigations inthis field are being continued. The absence of commonmethods of construction in solution of nonlinear pro3/ 21 ; ; .2 k RRα β γ ππ= = = ΗΗ1/ 2( ) ,P Pδ α β γ= +.ó ïëδ δ δ= +,2ïëPRδ π= Η3/ 2 1/ 2( ) ,ó k P Rδ π= Η0 ,óδδ = Ω1/31 ,ïëóδδ⎛ ⎞Ω = +⎜ ⎟⎜ ⎟⎝ ⎠2(1 )2 , ,k EEμ μ−=2/30 ,kPδ =Mathematics and mechanics. Physics61blems leads to necessity of developing efficient approximate numericalanalytical methods.In many mechanic systems motion is described bynonlinear differential equations. The model of contactinteraction considered in the paper is nonlinear; therefore, differential equation of motion under the condition of free oscillations has the view:where x=δ, but P(x) is expressed from the formula (5).Taking into account the mentioned above the differential motion equation of rigid plain sphere along elastoplastic halfspace at shock will have the view:whereConsidering that at the initial moment of collision after the first integration one can find the velocity of approach in the form(6)The most value of the approach magnitude is achieved at the moment when Solving the equations(6) one can calculate the value of dynamic approach,maximum shock force and maximum pressure in thecentre of contact. To calculate the shock duration division of variables in the equation (6) is performed, andthen integration from the beginning of shock to the moment of maximum approach is made(7)Solving the given dependence and approximatingthe obtained solution one can plot the dependencies x(t)and P(x) at any moment of collision.If one takes the value of initial velocity equal to zero(v0=0), then one can calculate static elastoplastic valueof approach. Solution has the form:(8)Values calculated by the formula (8) are similar tothose calculated by the formula (5).In conclusion it should be noted that the given numericalanalytical method makes possible to calculate the parametersof contact hardness and strength both in the conditions of static and dynamic loading. Application of this algorithm allowsestimation in influence of elastoplastic deformation on contact strength for the case of collision of rigid plain ball withplain elastoplastic halfspace. The proposed numericalanalytical method of calculation can also be used for design ofelastoplastic contact interaction of rough bodies. For this purpose it is necessary to take into account the microgeometryparameters of contacting bodies in the formula (5).22122 sin 1 .3 22t aax a mπ⎛ ⎞⎡ ⎤⎛ ⎞⎜ ⎟= − +⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠max3/ 2 20 2 1 20.43xdxta x a xv m m= ±⎛ ⎞− +⎜ ⎟⎝ ⎠∫0.dxdt =3/ 2 22 1 204 .3a x a xdx vdt m m⎛ ⎞= − +⎜ ⎟⎝ ⎠0 ,dx vdt =1/ 21 23/ 21; .a aβ γ αα= − =1 2 0,mx a x a x+ + =( ) 0,mx P x+ =Bulletin of the Томsк Pоlytеchnic University. 2007. V. 310. № 262REFERENCES1. Тimoshenko S.P., Gurier J. Elasticity theory. – Moscow: Nauka,1975. – 567 p.2. Drozd М.S., Matlin М.М., Sidyakhin Yu.I. Engineering calculations of elastoplastic contact deformation. – Moscow: Machinostroyeniye, 1986. – 220 p.3. Koteneva N.V. Elastoplastic contact of smooth sphere with flat surface ay dynamic loading // Bulletin of Tomsk Polytechnic University. – 2005. – V. 308. – № 2. – P. 114–116.Received on 31.10.2006

Dynamic model of elastoplastic contact interaction of smooth bodies

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