Asymptotic Solution of the Anti-Plane Problem for a Two-Dimensional Lattice

Transcription

1 Asympoic Solion of he Ani-Plane Problem for a Two-Dimensional Laice N.I. Aleksandrova N.A. Chinakal Insie of Mining, Siberian Branch, Rssian Academy of Sciences, Krasnyi pr. 91, Novosibirsk, 6391 Rssia, alex@mah.nsc.r Absrac. Propagaion of nseady waves nder he effec of a sep poin load on a sqare laice of spring-conneced masses is invesigaed. The problem is solved by wo mehods. Asympoic solions a large ime inervals, which describe he behavior of long-wave perrbaions, are derived analyically. The solion over he whole ime inerval for he waves of he enire specral range is derived by he finie difference mehod. These solions are compared, and heir good agreemen is shown. Keywords: Block medim, sqare laice, sep load, low freqency wave Previosly, he heory of deforming he rock massif as a homogeneos medim was widely sed in geomechanics. A serios reason o reconsider he convenional views has been given by invesigaions of recen years, which indicae he necessiy o ake ino accon he block srcre of monain rocks in mahemaical models inended for geomechanics and seismology [1, ]. In his case, he monain massif is considered as a sysem of nesed blocks of varios scale levels conneced by inerlayers consising of weaker cracked rocks. The presence of sch liable inerlayers leads o he case when he block massif is deformed mainly de o heir deformaion boh saically and dynamically. In he simples case, he dynamics of he block medim is invesigaed approximaely considering ha sch blocks are incompressible and all deformaions and displacemens occr de o compressibiliy of inerlayers [3 5]. The comping model can be a laice of masses conneced wih each oher by springs. In erms of his model, we consider he ani-plane deformaion of a wo-dimensional sqare laice consising of masses M conneced by springs wih lengh L having idenical rigidiies k in boh direcions. The nseady propagaion of perrbaions nder he effec of a sep load is invesigaed. The problem in his saemen wih he sinsoidal load having he resonan freqency is solved analyically in [6]. The eqaions of moion of masses have he following form: M& k ) Q( ), (1) m, n = ( m+ 1, n m 1, n m, n+ 1 m, n 1 m, n + where, is he displacemen of masses in he direcion orhogonal o he laice plane, m and n m n are he nmbers of masses in he direcion of axes x and y, Q( ) = Q H ( ) δ mδ n is a sep load applied in he poin wih coordinaes (, ), H () is he Heaviside fncion, and δ n is he Kronecker dela. Applying he discree Forier ransformaion over variables m and n and he Laplace ransformaion over ime, we derive he solion in images: 1

2 LFmFn LF F m n Q LF Q mfn =, Q =. () Mp + k[ cos( q L) cos( q L)] p x y Here, L denoes he Laplace ransformaion over ime wih parameer p ; F m and discree Forier ransformaions over m and n wih parameers q x and q y, respecively. Invering derived expression () over variable q y, we obain ( B B 1) n LF Q Mp n m =, B = + cos( q L). x kp B 1 k F n are he Le s invesigae he nseady behavior of long-wave perrbaions a large ime inervals. We assme ha p.this corresponds o in he space of he originals. The asympoics of images of displacemens a n = will be as follows: Q p q L x qxl B +, sin 1 sin ω kp B 1 4ω LF m, = k M. (3) Here and below, formla v( w( as z z means ha lim [ v( w( ] = z z Using formlas of inversion of he Forier and Laplace ransformaions [7 9], we derive he asympoics of long-wave perrbaions as m, Q H kπ ln : ( z 1) ln( z + z 1 ), ( 4 ω ) + γ, z = ω m =, m, m, Q & m, J m( ω), (5) km [ ] Q & m, J m( ω ) J m 1( ω) J m+ 1( ω). (6) M Here, J m is a Bessel fncion of he firs kind of order m [8] and γ = is he Eler consan. Using he approximae represenaion of he Bessel fncions hrogh he Airy fncion, which is valid a m >> 1 [8, 9], le s rewrie solions (5) and (6) in anoher form: Q Ai( κ ) m ω & m,, κ =,. (7) km ( ω) ( ω) QAi( κ )Ai ( κ ) & m,,. (8) km Here, Ai is he Airy fncion [9], and he prime means he derivaive wih respec o he argmen.. (4)

3 .37.3 v Fig. 1. Oscillograms of he displacemens Fig.. Oscillograms of he velociy of displacemens The advanage of solions (7) and (8) compared wih (5) and (6) is in he fac ha hese solions explicily describe he degree of decay of long-wave perrbaions near he qasi-fron ml = c, where c = Lω is he velociy of infiniely long waves in he laice. The analysis of solions (7) and (8) shows ha he maximal amplides of he velociy of displacemen drop as 3 and hose of acceleraions drop as 1 as. The qasi-fron zone, where perrbaions vary from zero o maximm, is exended as Here, as. Eqaion (1) was also solved by he finie difference mehod sing he explici scheme r r r r r ( Q δ δ ), r r+ 1 r r 1 m, n m, n + m, n = m+ 1, n m 1, n m, n+ 1 m, n 1 m, n m n τ. (9) = rτ, where τ is he ime sep of he difference mesh, and r is he layer nmber over ime. Figs. 1 3 show oscillograms of displacemens = m, (Fig. 1), velociies of displacemens v = & m, (Fig. ), and acceleraions w = & m, (Fig. 3) nder he sep load. The parameers of he problem are as follows: m =, k = 1, L = 1, M = 1 and Q 1. Thin crves correspond o he finie = difference solion by scheme (9) ( τ =. 7 ), and hick crves correspond o analyical solions (4), (7), and (8). The comparison of nmerical and asympoic solions (4), (7), and (8) shows ha he long-wave asympoics accraely describes he solion near he qasi-fron m = c. Saring from momen of ime = m c ~ 44, where c = π is he phase velociy of shor-wave perrbaions, perrbaions appear in oscillograms of nmerical solions of v and w, which correspond o oscillaions of shor waves ( q q x = y = π ); he laer are absen in asympoic solions (7) and (8). We also compared he finie-difference solions for velociies of displacemens v and 3

4 .3 w Fig. 3. Oscillograms of he acceleraions acceleraions w wih analyical solions (5) and (6). As shold be expeced, heir coincidence is even beer han wih solions (7) and (8). Ths, nder he sep impac on he sqare laice of spring-conneced masses, he following effecs are observed in he laice as : (i) he amplides of displacemens grow proporionally o ln( z + z 1) z = ω m, m ;, where (ii) he maximal amplides of velociies of displacemens of long-wave disrbances drop wih ime as 3, and hose of acceleraions as 1 ; (iii) he qasi-fron zone is exended as. This comparison of he nmerical and analyical solions shows ha he asympoic solion derived as accraely describes long-wave perrbaions. The agreemen of hese solions appears a a finie impac ime or a a finie disance from he place of applying he load. REFERENCES 1. M. A. Sadovskiy, Naral block size of rock and crysal nis, Dokl. Earh Sci. Sec., 47, 4 (1979).. M. V. Krlenya, V. N. Oparin, and V. I. Vosrikov, Originaion of elasic wave packes in block srcred medim nder implse inp. Pendlm-ype waves U µ, Dokl. Akad. Nak 333 (4), 3 13 (1993). 4

5 3. N. I. Aleksandrova, Elasic wave propagaion in block medim nder implse loading, J. Mining Sci. 39 (6), (3). 4. N. I. Aleksandrova, M. V. Aizenberg-Sepanenko, and E. N. Sher, Modelling he elasic wave propagaion in a block medim nder he implse load, J. Mining Sci. 45 (5), (9). 5. N. I. Aleksandrova and E. N. Sher, Wave propagaion in he D periodical model of a block srcred medim. Par I: Сharacerisics of waves nder implsive impac, J. Mining Sci. 46 (6), (1). 6. M. Ayzenberg-Sepanenko and L. Slepyan, Resonan-freqency primiive waveforms and sar waves in laices J. Sond Vibr. 313, (8). 7. A. Erdélyi, W. Magns, F. Oberheinger, and F. G. Tricomi (eds.), Tables of Inegral Transforms, Vol. 1, New York: McGraw-Hill Book Co, I. S. Gradsheyn and I. M. Ryzhik, Table of Inegrals, Series, and Prodcs, 7h ed., Elsevier, 7 (Translaed from he Rssian). 9. M. Abramowiz and I.A. Segn (eds.), Handbook of Mahemaical Fncions wih Formlas, Graphs and Mahemaical Tables, U.S. Deparmen of Commerce, Washingon, Please cie his aricle in press as: N.I. Aleksandrova, Asympoic solion of he ani-plane problem for a wo-dimensional laice, Doklady Physics. 59(3) (14), pp hp://dx.doi.org/1.1134/s x. 5