Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 2, 39, 2001 DYNAMICS OF ELASTIC BODIES IN TERMS OF PLANE FRICTIONAL MOTION Gwidon Szefer Institute of Structural Mechanics, Cracow Unuversity of Technology email: szefer@limba.wil.pk.edu.pl In the paper a general approch to dynamics of flexible systems in which displacements are resolved into displacements due to deformation and displacements due to rigid bodymotion, will be applied. A contact pro- blemofbodies resting on frictional foundationandbeing inplanemotion is stated and qualitatively discussed. Key words: rigid body, dynamical equations, rough surface, frictional motion 1. Introduction The majority of contact problems formulated and considered in mecha- nics represents two, qualitatively different approaches: the first, typical for mechanics of solids (elastic, plastic etc.) is concentrated on determination of deformations, stress distributions and interaction processes in the contact zo- ne; the second one, typical for multibody dynamics, is looking for motion of the system described obviously as finite-dimensional. Contact is taken into accountmainly by forces respresenting reactions of obstacles or interactions of contacting bodies (see e.g. Bremer andPfeifer, 1992). One of the few explored problems in contact dynamics is a planar contact of deformable body mo- ving on a rough surface. Some results in this topic was given by Fischer and Rammerstorfer (1991), Fischer et al. (1991), Mogilevsky and Nikitin (1997), Nikitin et al. (1996), Stupkiewicz andMróz (1994), Stupkiewicz (1996). Ben- ding of beams resting on frictional surface, torsion of plates pressed between two rough planes etc. are examples of this type Nikitin (1998). In the present paper the mentioned problem of the plane motion of an elastic body resting on a rough rigid foundation will be considered. Contrary 396 G.Szefer to the existing formulations, the body is treated as a highly flexible system in which the location of each particle is resolved into displacements due to defor- mation and displacements due to rigid bodymotion. Such description enables to determine the mutual interaction between rigid motion and deformation. Governing equations of dynamics in presence of two-dimensional friction and some qualitative results will be given. Thepaper is organizedas follows:we startwith couplingof the rigidmotion and deformation according to the general statment given in the paper Szefer (2000). Next we pass to the two-dimensional problem of frictional motion of an elastic body in the plane state of stress. At the end some conclusions are presented. 2. Coupling of rigid body motion and deformation Consider a deformable body B, itsmotion from its reference configuration BR into the current location Bt at instant t beingmeasured with respect to a global inertial system {0xi}, i = 1,2,3. Denoting by {XK}, K = 1,2,3 thematerial coordinates of an arbitrarymaterial pointwith its position vector X(XK), one describes the motion xi = xi(XK, t) as a mapping of X onto x(XK, t) where x means the current position vector of the point at time t. Thustheconfiguration Bt of thebodycanbetreatedasa resultof deformation described by the displacement field u(XK, t) followed by a rigid bodymotion defined by a translation vector x0(t) and a rotation tensor Q(t) in the form (Fig.1) x(X, t)=x0+Q(t)[X+u(X, t)] (2.1) Remark.Weassume, that thevector x0(t) stands formotion of the center of mass 0∗, what constituites the most convenient description. Velocity and acceleration of each point yields (Szefer, 2000) ẋ=vu+vw ẍ=aw+au+ac (2.2) where vw =Qu̇ vu = ẋ0+Q[ω× (X+u)] aw =Qü au = ẍ0+Q{ω× [ω× (X+u)]+ ω̇× (X+u)} ac =2Qω× u̇ Dynamics of elastic bodies... 397 Fig. 1. and ω(t) is the axial vector of the skew-symmetric tensor W=Q⊤Q̇=⇒Wa=ω×a ∀a The vectors vw and vu measured in the reference system {0x i} can be interpreted as the relative and transporting velocities of the particle due to deformationwhereas the vectors aw,au,ac are the relative, transporting and Coriolis accelerations, respectively. Thus, using the principle of momentum and the principle of the angular momentum (or the equivalent virtual power principle), one obtains the system of equations of motion of any flexible system in the form (Szefer, 2000) x0 : Mẍ0+Q[ω× (ω×A(u)]+Qω̇×A(u)+2Qω×P(u̇)+ +QB(ü)=Fext+FC ω : Juω̇+ω×Juω−2ω×K0(u̇)+ ẍ0×QA(u)+L0(ü)= =M0(u)+M C 0 (u) u : DivS(1+∇⊤u)+ρRb= ρR(aw+au+ac) (2.3) 398 G.Szefer Here the following notations have been used M = ∫ VR ρR dVR A(u)= ∫ VR ρR(X+u) dVR P(u̇)= ∫ VR ρRu̇ dVR B(ü)= ∫ VR ρRü dVR FC = ∫ ΓC R tR dSR F ext = ∫ VR ρRb dVR+ ∫ SR pR dSR J u = ∫ VR ρR[(X+u)(X+u)1− (X+u)⊗ (X+u)] dVR (2.4) K0(u̇)= ∫ VR ρRu̇× (X+u) dVR L0(ü)= ∫ VR ρRü× (X+u) dVR M ext 0 (u)= ∫ VR ρR[b×Q(X+u)] dVR+ ∫ SR pR×Q(X+u) dSR MC0 = ∫ ΓC R tR×Q(X+u) dSR where ρR – mass density in BR VR – volume domain in BR SR – boundary surface loaded byprescribed external tractions pR ΓCR – contact zone with contact tractions tR b – body forces ∇ – stands for the gradient operator with respect to BR 1 – identity tensor S – second Piola-Kirchhoff stress tensor. Vectors FC and M C 0 expresses the presence of contact forces or reactions of constraints. Remark. Denoting the sum of the second to the fifth terms on the left- hand side of equation (2.3)1 by Fu =Q[ω× (ω×A(u)]+Qω̇×A(u)+2Qω×P(u̇)+QB(ü) (2.5) and similarly the coresponding sum in (2.3)2 by Mu0 =−2ω×K0(u̇)+ ẍ0×QA(u)+L0(ü) (2.6) Dynamics of elastic bodies... 399 one can write the mentioned equations in the form Mẍ0 =F ext+FC −Fu (2.7) J uω̇+ω×Juω=Mext0 (u)+M C 0 (u)−M u 0 This form coincides with the known system of equations of rigid body dy- namics but with the inertial tensor Ju (see Eqs (2.4)). System (2.7) together with equation (2.3)3 possess a clear structure and provide a simple interpre- tation for the coupled rigid motion and deformation: translation and rotation influences the motion of a continuum by additional transportation and the Coriolis acceleration, whereas deformation influences the rigid body motion by configuration-dependent force (2.5), moment (2.6) and inertial tensor Ju. System (2.3) must be completed by the constitutive equations S=F(X,E) (2.8) the kinematical equations for Green’s strain tensor E= 1 2 (∇u+∇⊤u+∇u⊤∇u) (2.9) and by the boundary and initial conditions S(1+∇⊤u)N = { pR on SR tR on ΓCR (2.10) x0(t0)= r0 ẋ0(t0)=v0 ω(t0)=ω0 u(X, t0)=u0(X) u̇(X, t0)=v0(X) X ∈BR (2.11) Here N means the unit outward vector normal to SR and ΓCR is the map- ping of the contact zone ΓC onto the reference configuration. Equations (2.3), (2.8), (2.9) constitute a coupled system with unknown functions x0(t), ω(t) and u(XK, t),K =1,2,3, which describe the complex motion of any flexible body with displacements explicitely decomposed into rigid motion and pure deformation. Such statement of any dynamical problem represents a third, and in fact, themost general approach to dynamics of deformable bodies. The system (2.3) which consist of two ordinary and one partial differential equ- ations shows evidently the mutual dependence of translation, rigid rotation and deformation. Simultaneously, the displacements due to deformations de- pend strongly on rigid motion what can be seen from (2.3)3, where dynamic body forces are supplemented by transportation and Coriolis members. The 400 G.Szefer presence of deformation shows additionally that contrary to pure rigidmotion, there is a coupling between translation and rotation. It is worth to observed that equations (2.3) (or in the form (2.7)) are va- lid for elastic bodies the external constraints of which may be nonholonomic, rheonomic, unilateral and rough. The material system may posses large di- splacements and rotations, too. Thus the impact, friction, rolling with and without sliding etc. can be taken into account. 3. Elastic plate undergoing frictional motion Consider a thin elastic plate resting on a rough rigid foundation loaded by prescribed tangential boundary tractions pt and compressed by normal forces with density pn (Fig.2). Fig. 2. When the body starts to move due to the external boundary load pt, friction occurs. Thus the body forces b=−µpn(X, t) vT |vT | (3.1) Dynamics of elastic bodies... 401 where vT means the sliding velocity, arise at all points of the plate areawhere pn(X, t) 6=0. The intensity of the friction forces |b|=µpn is known whereas their direction results from the Coulomb law (3.1). They have the body force character since they act on the internal points of theplanebody.For sticking it will be vT =0; otherwise the body is sliding. Let thematerial reference frame be a cartesian coordinate system (0∗,X,Y,Z) with its origin in the center of mass and let the global inertial system will be denoted by (0,x,y,z). Thus the kinematics of the body yields ω= [0,0,ω= α̇] u= [ux,uy] x0 = [x0,y0] Q=    cosα −sinα 0 sinα cosα 0 0 0 1    Q̇=−α̇    sinα cosα 0 −cosα sinα 0 0 0 0    x ⊤ = [ x y ] = [ x0+Acosα−B sinα y0+Asinα+Bcosα ] v⊤ = [ vx vy ] = [ ẋ0−ω[Asinα+Bcosα]+ u̇xcosα− u̇y sinα ẏ0−ω[−Acosα+Bsinα]+ u̇x sinα+ u̇y cosα ] (3.2) a⊤w = [ awx awy ] = [ üxcosα− üy sinα üx sinα+ üy cosα ] a ⊤ u = [ aux auy ] = [ ẍ0−ω 2[Acosα+B sinα]− ω̇[Asinα+Bcosα] ÿ0−ω 2[Asinα−Bcosα]− ω̇[−Acosα+B sinα] ] a⊤c = [ acx acy ] = [ −2ω(u̇x sinα+ u̇y cosα) −2ω(−u̇xcosα+ u̇y sinα) ] where A=X+ux, B=Y +uy. Taking into account the fact that in a plane motion it is Jω‖ω, K0‖ω, calculating next all the integrals (2.4) Ax(t)= ∫ VR ρRux dVR Ay(t)= ∫ VR ρRuy dVR Px(t)= ∫ VR ρRu̇x dVR Py(t)= ∫ VR ρRu̇y dVR Bx(t)= ∫ VR ρRüx dVR By(t)= ∫ VR ρRüy dVR (3.3) 402 G.Szefer L0z(t)= ∫ VR ρR(Büx−Aüy) dVR Jzz(t)= ∫ VR ρR(A 2+B2) dVR and introducing for clarity the unknownmatrix q⊤ = [x0,y0,α], one obtains the system of equations (2.3) for the two-dimensional case as follows     M 0 −Ax sinα+Ay cosα 0 M Axcosα−Ay sinα Ax sinα+Ay cosα −Axcosα+Ay sinα Jzz     q̈− −2     0 0 Px sinα+Py cosα 0 0 Pxcosα−Py sinα 0 0 0     q̇+     0 0 −Axcosα−Ay sinα 0 0 −Ax sinα+Ay cosα 0 0 0     q̇2+ +     Bxcosα−By sinα Bx sinα+By cosα L0z     =     Fextx +FCx Fexty +FCy Mext0z +M C 0z     (3.4) Div [ Sxx(1+ux,x)+Sxyuy,y Syx(1+ux,x)+Syyuy,y Sxxuy,x+Sxy(1+uy,y) Syxuy,x+Syy(1+uy,y) ] − −µpn      vx √ v2x+v 2 y vy √ v2x+v 2 y      = ρR [ awx+aux+acx awy+auy +acy ] The quantity q̇2 means multiplication of matrices q̇⊤q̇. The external resultant force Fext andmoment Mext0∗ have the components Fextx =−µ ∫ VR pnvx √ v2x+v 2 y dV + ∫ SR ptx dS Fexty =−µ ∫ VR pnvy √ v2x+v 2 y dV + ∫ SR pty dS (3.5) Mext0z = ∫ SR (ptxry−ptyrx) dS−µ ∫ VR pn vxry−vyrx √ v2x+v 2 y dV rx =Acosα−B sinα ry =Asinα−Bcosα Dynamics of elastic bodies... 403 The system (3.4) is strongly nonlinear and can be solved numerically only. It is seen from (3.4) that the rigid part of planemotion depends on defor- mation through functions (3.3) and (3.5) only. This propertymakes it possible to solve the system (3.4)1 (with suitable initial conditions) formally indepen- dently on (3.4)2 (e.g. bymeans of the Runge-Kutta method). On the other hand, the nonlinearity of (3.4)2 causes that the incremental approach is obviously used. The incremental form of (3.4)2 is then as follows Div[S∆H+∆S(1+H)]+∆b= ρR(∆aw+∆au+∆ac) (3.6) where H=∇u. Thus the system (3.4)1 must be solved iteratively for any increment ∆u. Leaving the numerical details and analysis for separate discussion, one can however, in particular cases, come to some general qualitative conclusions A. Constant body force If the density of the external body force is constant, then Fext = ∫ VR ρRb dV =Mb (3.7) and we obtain from (2.7)1 ẍ0 = b+ 1 M (FC −Fu) (3.8) Substituting this expression into (2.3)3 we obtain DivS(1+∆u⊤)+ρRb= ρRQü+ρR [ b+ 1 M (FC −Fu)+a ω u +ac ] (3.9) where aωu means this part of au which results from rotation (see (2.2)). One can see from the above equation that the term ρRb vanishes and it reads finally DivS(1+∆u⊤)= ρR(Qü+a ω u +ac)+ ρR M (FC −Fu) (3.10) This resultmeans that, in thecaseof constantbody force, puredeformation does not depend on b; the constant body force density influences translation only. This fact is invisible if displacements are not presented in the form (2.1). 404 G.Szefer B. Sliding without rotation under symmetricmonotonic load and uniform pressure The result obtained above can be applied to a plate being in translatory sliding motion (Fig.3a). Thus ω= 0. Assume that the lateral velocity vy is small (e.g. if the plate have dimensions of a rod). Then b=−µpn[1,0] (since the direction of velocity v= u̇+ẋ0 for all points of the body is the same and known) and the property of case A holds true. Fig. 3. So, friction disappears in equations of motion (3.4)2. If stick-slip process occurs (and this takes place when v have to vary under nonmonotonic or nonsymmetrical loads pt) then friction influences the deformations. On the other hand, when the lateral velocity vy cannot be neglected and the plate will be clamped on one side (Fig.3b), rigid rotation vanishes ω=0, Q=1 and pure deformation results now from the equation DivS(1+∆u⊤)−µpneT = ρR(ü+ ẍ0) (3.11) Dynamics of elastic bodies... 405 where eT = [ vx √ v2x+v 2 y , vy √ v2x+v 2 y ] vx = u̇x+ ẋ0 vy = u̇y whereas the system (2.7) (and hence ;(3.4)1) yields Mẍ0 =F ext x +FCx−Bx (3.12) Remark. In case of symmetry all the terms in (2.7)2 vanishes. Simultane- ously it is y0 =0. Hence the above result. C. Dynamic bending of a beam Consider an elastic uniform slender beam using the standard Bernoulli- Eulermodel of small deformation butwith large rigid rotations. Let the beam of length L, cross-sectional area A, inertiamoment J andYoungmodulus E rest on the plane {0xy} (Fig.3c). Thematerial coordinate system {0∗XYZ} rotateswith thebeam.Thecentroidal axis is assumedtobe inextensible.Under the action of prescribed load pt(X,t), the beam moves and bends laterally with the deflection w(X,t). Thus the functions (3.3) take the values ux ≡ 0 =⇒ Ax =Px =Bx =0 (3.13) Ay = ρA L/2 ∫ −L/2 w(X,t) dX Py = ρA L/2 ∫ −L/2 ẇ(X,t) dX By = ρA L/2 ∫ −L/2 ẅ(X,t) dX L0z =−ρA L/2 ∫ −L/2 ẅ(X,t)X dX For the loading terms one obtains the components Fx = L/2 ∫ −L/2 ptx dX−µb L/2 ∫ −L/2 pn vx √ v2x+v 2 y dX Fy = L/2 ∫ −L/2 pty dX−µb L/2 ∫ −L/2 pn vy √ v2x+v 2 y dX (3.14) M0z = L/2 ∫ −L/2 pt(X,t)X dX−µb L/2 ∫ −L/2 pn vxX sinα−vyX cosα √ v2x+v 2 y dX 406 G.Szefer where b is the width of the beam. From (3.4) result the equations of plane rigid motion of the beam Mẍ0+Ay(t)(α̈cosα− α̇ 2 sinα)−2Py(t)α̇cosα−By(t)sinα=Fx(t) Mÿ0+Ay(t)(α̈sinα− α̇ 2cosα)+2Py(t)α̇sinα+By(t)cosα=Fy(t) (3.15) ρ bL3 12 α̈+Ay(t)(ẍ0 cosα+ ÿ0 sinα)+L0z =M0z(t) To obtain the most convenient form of bending, the local coordinate system {0∗XY} which is moving together with the beam will be used (see Fig.3c). We then get x=x0+(X+w)= [x0,y0]+ [X,w] v= [ẋ0−ωw,ẏ0+ ẇ+ωX] (3.16) a= [ẍ0− ω̇w−ω 2X−2ωẇ, ÿ0+ ẅ+ ω̇X−ω 2w] Using the lateral components of velocity and acceleration, one obtains the dynamical equation of the beam EJ ∂4w ∂X4 = pt(X,t)−µpn(X,t)sgn(ẏ0+ ẇ+ωX)−ρA(ÿ0+ ẅ+ ω̇X−ω 2w) (3.17) This equation generalizes the static case discovered by Nikitin (1992) and Stupkiewicz (1996). If the beammove translational one get EJ ∂4w ∂X4 = pt−µpn sgn(ẏ0+ ẇ)−ρA(ÿ0+ ẅ) (3.18) Finally, if only pure deformation (bending) is taken into account, one obtains the standard dynamic equation in terms of frictional contact EJ ∂4w ∂X4 = pt−µpn sgnẇ−ρAẅ (3.19) 4. Concluding remarks The presented approach to dynamics based on formula (2.1) differs from the standard procedure where elastic strains and stresses result from the pre- scribed rigid motion (obviously used in multibody dynamics of elastic sys- tems). No restrictions on displacements, velocities and deformation gradients Dynamics of elastic bodies... 407 were introduced. Thus the systems with high flexibility and large rigid mo- tion can be analyzed. Plane friction constitutes still a challenge in contact dynamics. Few numerical results of plane slidingmotion are known up to now (some of themwerementioned in the Introduction). The equations derived in the paper give the possibility to analyze themutual interaction between rigid motion and deformation which is of great interest today. Some simple qualitative examples of sliding were presented only. More complex cases of coupling of rigid motion and large deformation in terms of contact will be discussed separately. References 1. Bremer H., Pfeifer F., 1992,Elastische Mehrkörpersysteme, B.G., Teubner Stuttgart, p.283 2. Fischer F., Rammerstorfer F., 1991, The Thermally Loaded Heavy Be- am on a Rough Surface, Trends in Appl. of Mathem. to Mech., Edit. by W.Schneider, H.Troger, F.Ziegler, LongmanHigher Educ. BurntMill, 10-21 3. Fischer F., Hinteregger E., Rammerstorfer F., 1991, A Computatio- nal Study of the Residual Stress Distribution in Thermally Loaded Beams of ArbitraryCross Section onFrictional Support,Nonlinear Comp. Mech. – State of the Art, Edit. by P.Wrigers,W.Wagner, Springer Beriln, 737-750 4. Mogilevsky R., Nikitin L., 1997, In-PlaneBending of a BeamResting on a Rigid Rough Foundation, Ing. Arch., 67, 535-542 5. Nikitin L., 1992,Bending of aBeamonaRoughSurface,Dokl. Russ.Ac. Sci., 322, 6, 1057-1061 (in Russian) 6. Nikitin L., 1998, Statics and Dynamics of Solds with Dry Friction, (in Rus- sian), Moskovskǐı Licei, p.272 7. Nikitin L., Fischer F., Oberaigner E., Rammerstorfer F., Sietzber- gerM.,MogilevskyR., 1996,On theFrictionalBehaviour of theThermally Loaded Beams Resting on a Plane, Int. J. Mech. Sci., 38, 11, 1219-1229 8. Stupkiewicz S., 1996, Modeling of Sliding and Damage Growth in Contact Zone of Elasto-Plastic Bodies, PhD-Diss., (in Polish), IPPT,Warsaw, p.171 9. Stupkiewicz S., Mróz Z., 1994, Elastic Beam on aRigid Frictional Founda- tionunderMonotonicandCyclicLoading, Int. J. Sol. Struct.,31, 24, 3419-3442 10. Szefer G., 2000, Dynamics of Elastic Bodies Undergoing LargeMotions and Unilateral Contact, J. Tech. Phys., 41, 4, 303-319 408 G.Szefer Dynamika ciał sprężystych w warunkach płaskiego ruchu szorstkiego Streszczenie W pracy zastosowano ogólny opis dynamiki układów odkształcalnych, w których przemieszczenia są dekompozycją części wynikającej z deformacji oraz części wywoła- nej ruchemsztywnym.Sformułowano i przedyskutowanojakościowoproblemkontaktu ciała leżącego na chropowatym podłożu i będącego w ruchu płaskim. Manuscript received January 5, 2001; accepted for print Frebruary 17, 2001