Normal Impact of A Rigid Cone-shaped against A Viscoelastic Plate on Viscoelastic Foundation

Nội dung text: Normal Impact of A Rigid Cone-shaped against A Viscoelastic Plate on Viscoelastic Foundation

1. VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 4 (2021) 95-101 Original Article Normal Impact of A Rigid Cone-shaped against A Viscoelastic Plate on Viscoelastic Foundation Duong Tuan Manh* VNU University of Engineering and Technology, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam Received 10 July 2021 Revised 18 July 2021; Accepted 18 July 2021 Abstract: This paper considers the problem of a low-velocity normal impact of a rigid cone-shaped upon a viscoelastic plate. The contact force is defined by the modified Hertz's contact law. Approximate solutions of the system of nonlinear integro-differential equations for the contact force and local indentation have been obtained. Keywords: Normal impact, viscoelastic plate, Kelvin-Voigt model. 1. Introduction * The problems with the analysis of the impact of two bodies have a widespread application in various fields of science and technology. Because these problems belong to dynamic contact interaction, their solutions are connected with severe mathematical and calculation difficulties. Some approaches and methods have been suggested in articles by Abrate [1], Rossikhin and Shitikova [2]. Rossikhin et al., [3] investigated the collision of two viscoelastic shells, viscoelastic features of which are described by the standard linear solid model. During the impact process, the local bearing of the materials of the colliding viscoelastic shells is taken into account, the solution in the contact domain is found via the modified Hertz contact theory involving the operator representation of viscoelastic analogs of Young’s modulus and Poisson’s ratio. In the paper, we will consider a special but very important for engineering practice case when a rigid cone-shaped impacts a viscoelastic plate. ___ * Corresponding author. E-mail address: iam.mr.manh@gmail.com https//doi.org/10.25073/2588-1124/vnumap.4661 95
2. 96 D. T. Manh / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 4 (2021) 95-101 2. Problem Formulation Let us consider the problem on a transverse impact of a rigid cone-shaped upon a viscoelastic Kirchhoff-Love plate when the viscoelastic features of the surrounding medium are described by a Kelvin-Voigt model. In this case, the equations of motion of a rigid cone-shaped of mass ms and the viscoelastic rectangular plate with the dimensions a and b and of thickness h have, respectively, the form z Cone-shaped ? ? d h Viscoelastic plate Foundation Damping coefficient Spring foundation Shear foundation Figure 1. Scheme of the normal impact of a rigid cone-shaped against a plate. mztPs ()=− ()t , (1) 2 ab Dw +−=−− x(, y thw ,)(, x  y ,)(tL w , FP )() txy 22 (2) w(,,) x y t −−+ K wxytCKwxyt( , , )( , , ) wTG t 11 =2FLww − (,) , (3) E 2 where ρ is the density of plate, w(x,y,t) is the displacements of an arbitrary point in the plate in the z- a direction, P(t) is the contact force, F is Airy function stress, is the Laplace operator,  x − , 2 b  x − is the Dirac delta function, d and z(t) are the penetration depths, θ is the angle between the 2
3. D. T. Manh / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 4 (2021) 95-101 97 plane and cone, Kw , KG , CT are the Winkler, Pasternak and Damper modulus parameters, respectively, and  EE=1+ t E Eh3 D = 12 1( − 2 ) Utilizing the KelvinVoigt model with the bulk modulus K remaining constant during the process of mechanical loading of this material [4], it is found [3] that E  =+EL1 t ( ) − L(  ) , 1− 2 E where E,, are the relaxed Young’s modulus, Poisson’s ratio and viscosity, respectively, −1  == is the creep time, and  E −+−−2121(vv)( ) (1− 2 ) t ===− ;;expex' ;Ltt( ) dt ( ) p'( ) (1−− 21 ) 2 (  ) 4 0 Equations (1), (2) and (3) are subjected to the following initial conditions: zVzP(0),(0)0,(0)0===0 (4) w(,,0)0,(,,0)0 xyw xy== (5) wy(0, tw , a )0,(=== y tw , xtw, )( x ,0, b )0,(t , , )0 (6) Myxxyy(0, tMa , )(y=== tMy , , )(0,tMb ,y )( t , , )0 (7) xa= 0, N(, x y ,)(, tp===− Nx ,)(, y tp Nx ,)0: y tat , (8) xyxy 1;2; yb= 0, where all four edges of a rectangular plate are simply supported and immovable, V0 is the initial velocity of the collision. Integrating twice Eq. (1) yields 1 t z( t )= − P ( t ')( t − t ') dt ' + V t (9) 0 ms 0 In the case of the collision of a rigid cone-shaped against a plate (Figure 1), the solution in the contact domain could be found using Hertz’ theory. Thus, the contact force is defined as [5] 2( ) Ez t 2 Pt( ) =, (10) (1− 2 ) tan
4. 98 D. T. Manh / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 4 (2021) 95-101 Replacing Eq. (1) into Eq. (10) yields 2() Ezt 2 mzt()0+= (11) s (1− 2 ) tan The approximate solution of the equation (3) satisfying the boundary conditions (6), (7) and (8) can be written as mxny w(,,)sinsinxytf = (12) ab E  a222 mxb 2 nypypx 2 2 Ff=1+ 2 cos + cos +12 + , (13) t 22 32 E b a a b 2 2 where f is function of time t, m, n are odd numbers. Replacing Eq. (13) into Eq. (2) yields Emh3 xn ym nmnm xn y 2 2 44 44 4 ffsinsin2sinsin ++ + h 2 2 244 12(1− ) aba babab 22 mm  xn yEn y 2 2 2− 2 2 −−ff napsinsin4cos 1+ ++ 2 t 1 aabEb 32 − (14) 22 nm  xn yEm x 22 2 2 − 2 − ff mbsinsi4 no−1+ c s + p 2 t 2 babEa 32 ab m xn y m xn y = P() txyKC −−−−  wTffsinsinsinsin 2 2 abab mn22m xymn xn y −K f 2 ()sinsinsinsin + G a22aba b b With due account for orthogonality of sines on the segments 0 ≤ x ≤ a; 0 ≤ y≤ b, we obtain: Eh3 m x n y m2 n 2 4 m 4 4 n 4 4 f sin sin 2 ++ 2 2 2 4 4 12(1− ) a b a b a b m x n y + hf sin sin ab 22 mm x n y E 2 2 2− 2 2 n y fsin sin1+ + f 4 n a cos − p ab 2 t 1 a a b 32 Eb m x n y − sin sin dydx = 0 22  00 n m x n y E 2 2 2− 2 2 m x ab fsin sin1+ − f 4m b cos p 2 t 2 b a b 32 E a ab m x n y m x n y −−P() t x y−+ + KwTffsin sin C sin sin 22 a b a b 22 2 mnm x n y m x n y +Kf ( sinsin+ sin si n ) G 22 a abb ab
5. D. T. Manh / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 4 (2021) 95-101 99 The result of which could be written in the form of 32 P()tALLfALLfAfA=−+−+++1236 (  ) ( ) ( ) ( ) f + A4 5 fA f f , (15) where 2 mnab224 h3 AE1 =+ 22 ab 48 2 mnabab224 h3 A2 =++ 22 CT ab 484 mnabababmn222222 App312=+++ 2222 KKwG+ abab 444 ab Ah= 4 4 m2 nbm 2 42 na 2 4 AEE=+ 5 6464ab33 m2 nbm 2 42 n 2 4a A = + 6 64ab3364 2 A7 = ; tan (16) The system of nonlinear integro-differential equations describing the dynamic behavior of two bodies during their collision takes the form of P()t = −m z() t s E m z( tz )(+= t )0A 2 (17) s 7 2 (1 − ) 32 ms z( t )0++AL12356 ( ) −++LfA( ) LLfAfAA ( ) −+( ) fA4 + f f f = 3. Approximate Calculations First we substitute the approximate solution of the equation (11) to form 2 Hz()()()() t=+N z tM z t z t , (18) where AE NLL= −7 ( ) − ( ) m s A7 MLL= − ( ) − ( ) ms
6. 100 D. T. Manh / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 4 (2021) 95-101 z t( a ) ( t )=  k (19) k=0 Replacing Eq. (19) into Eqs. (18) yields t ttt''''' atV( )(( teeAteeAtdtdtdt=− ''')(− '''))''''''A −− tttt'''''''''' kkk 0 8 kkk===000 0 000 , (20) t ttt''''' −A ((eeBteeBtdt−− ''')( tttt'''''''''' '''))'''''' − dtdt 9 kk 0 00 0 kk==00 where AE7  A8 = ms A7 A9 = ms Note that ik t k k! (−−11) ( ) eex−−− tx dxtke kk it =−! (21)  ik++11 0 i=0 (ki− )! The first two of them have the form a0 ( t) = Vo t 2 V0 AA889 1 1143 112 A 1 a10() tV= −−+−−− tt 22 12   33 11 1 1 11 222 (22) −−+−VtVt00 A898 3 32AA 24 4 +−6    −−−  tt tt− 222 1 1 e ee e −−+44AAA989VtV000 3 355 6 −− V 44−    22 1 11 1 −−+−64A89V00 55 A V 44   We obtain the solution
7. D. T. Manh / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 4 (2021) 95-101 101 2 V0 AAA889 111111 432 z()ttt −−+−−−Vo tV 0 22 1233    222 1111 11 −−+−VtV00 AAA898 3322 +−6 44t    (23) −−−−  t ttt 222 11 eee e −−+−−−464AA989Vt0 3435 AV0 5 V0 4    22 1111 −−+−64AA89VV0 55 0 44   In the experiments of the collision problems, the collision time is about 10-3s, so the series (19) converges very quickly. Here, we only need the first two terms to approximate the value of the solution. Another method such as balancing the coefficients of the series can be found in [3]. The obtained equation by replacing Eq. (23) into Eq. (15) can be used for nonlinear dynamical analysis of the collision. 4. Conclusion In the present paper, the problem on the normal impact of a rigid cone-shaped upon a viscoelastic plate has been studied using the damping features of the impactor modeled by the linear Kelvin-Voigt model. An approximate analytical solution has also been found. Acknowledgments This work was supported by VNU University of Engineering and Technology under Project CN.19.05. References [1] S. Abrate, Impact on Composite Structures, Cambridge University Press, Cambridge, 1998. [2] Y. A. Rossikhin, M. V. Shitikova, Analysis of Two Colliding Fractionally Damped Spherical Shells in Modelling Blunt Human Head Impacts, Central European Journal of Physics, Vol. 11, 2013, pp. 760-778, [3] Y. A. Rossikhin, M. V. Shitikova, D. T. Manh, Modelling of the Collision of Two Viscoelastic Spherical Shells, Mechanics of Time-Dependent Materials, Vol. 20, 2016, pp. 481-509, [4] Y. N. Rabotnov, Creep of Structure Elements, Naka, Moscow, 1966. [5] I. N. Sneddon, The Relation Between Load and Penetration in The Axisymmetric Boussinesq Problem for A Punch of Arbitrary Profile, International Journal of Engineering Science, Vol. 3, 1965, pp. 47-57,