w urea/am a. :ESM hga‘i'-¢@ 5.; 31:15: Unstrmity 3 1293 003838 2:" lnnnnwummululumum MICHIGAN STATE UNNERSFW LIBRARV - 1- This is to certify that the thesis entitled SOME APPROXIMATIONS IN VIBRATIONS AND WAVE MOTION OF ELASTIC MEDIA presented by MICHEL YVON RONDEAU has been accepted towards fulfillment of the requirements for PH.D.Jegreein MECHANICAL ENG. 5 Major profgr Date W 0-7639 MICHIGAN STATE UNIVERSITY LIBRARY OVERDUE FINES ARE 25¢ PER DAY PER ITEM Return to book drop to remove this checkout from your record. d I‘IQII SOME APPROXIMATIONS IN VIBRATIONS AND WAVE MOTION OF ELASTIC MEDIA BY Michel Yvon Rondeau A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Engineering 1979 ABSTRACT SOME APPROXIMATIONS IN VIBRATIONS AND WAVE MOTION OF ELASTIC MEDIA BY Michel Yvon Rondeau This paper is a study of wave propagation phenomena in a linear, elastic, isotropic, homogeneous layer. A layer is defined as an infinite plate bounded by a pair of parallel planes at x2 = t b as referred to a rectangular coordinate system Xi(i = 1,2,3). The motion is described by the inte- gral method of Kirchhoff [ll], derived through Hamilton's principle, which serves as the starting point for the various approximations to be developed. In chapter one, a series expansion procedure coupled with a new truncation concept is used to construct two-dimensional partial differential equations for motions in the layer. Frequency spectra associated with traveling waves in a traction- free layer are studied in detail and compared with the spectra predictions of Medick [l4], Nikodem [18], and that of the three-dimensional theory [1]. These two-dimensional layer equations also enter into the developments of chapters three and four. In chapter two, wave propagation problems are approached from a different viewpoint employing a symbolic technique. Applications are made to one, two and three—dimensional problems. This Michel Yvon Rondeau technique will also be employed in chapter four where we discuss transient waves due to non-homogeneous boundary conditions. In chapter three we subdivide a plate into (n) - subdomains. In each subdomain Hamilton's principle and the truncation procedure used in chapter one will be applied. This will give us two non-homogeneous P.D.E. coupling the uniform and linear amplitudes distributions. The non- homogeneous terms are stated at the interface of each sub- domain. To further simplify the analysis, a new technique called Finite-point approximation is introduced. It gives a discrete solution in the direction of the plate thickness (i.e. x2-variable). The points are located at the tOp, bottom and interface of each subdomain. This technique will reduce the two non-homogeneous P.D.E. to a system of Differential-Difference equation. Finally, a solution in the form of a travelling wave with free traction at the top and bottom of the plate will generate an approximate Frequency Spectrum. The number of dispersion curves that we obtain through the spectrum depends on N the number of subdomains in the plate. In chapter four, impact boundary conditions are applied at the top of the plate and free stresses are applied at the bottom of the plate. The Differential-Difference equations obtained in chapter three are reduced to Difference Michel Yvon Rondeau equation using the Symbolic technique of chapter two. A solution for the Difference equation is developed for arbitrary M (the number of subdomains). Utilization of this solution in the impact boundary conditions mentioned above gives rise to a set of P.D.E. involving variables xl,t. These equations may readily be solved by transform techniques or numerical means. A DEDICATION This thesis is dedicated to my wife, Ah-ling, without whose encouragement and confidence this work would not have been completed. ii ACKNOWLEDGMENTS I wish to thank Professor Mathew A. Medick whose counsel and encouragement helped me establish a bridge between my earlier education in mathematics and the exciting and challenging problems abounding in physical phenomena. He introduced me to the creative use of intuition in problem solving. He first kindled my interest in elastic wave guide theories in his lectures, and subsequent encounters, he helped me to identify and shape the direction of my dissertation research. I am grateful for his guidance and encouragement, and most importantly, for his faith in me. iii TABLE OF CONTENTS SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . Vi CHAPTER ONE . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . 1 Series Expansion of the Displacement . . . . . . . 2 Derivation of the (N)-order Equation of Motion . . 3 Derivation of the (N)—order Strain-Displacements and Stress-Strains Relations . . . . . . . . . . . 5 Derivation of the (N)-order Strain Density and Kinetic Energy . . . . . . . . . . . . . . . . . . 7 Truncation Procedure . . . . . . . . . . . . . . . 8 The Truncation Application for [N,q] = [1,1], [2,1] and Arbitrary . . . . . . . . . . . . . . . . 10 Frequency Spectrum . . . . . . . . . . . . . . . . 14 CHAPTER TWO . . . . . . . . . . . . . . . . . . . . . . 28 Introduction . . . . . . . . . . . . . . . . . . . 28 Symbolic Method to Solve the String Equation . . . 29 Symbolic Method to Solve the S.H. in an Elastic Layer . . . . . . . . . . . . . . . . . . . . . . . 34 Symbolic Method Approximation to Solve Three- Dimensional Wave Problem . . . . . . . . . . . . . 39 CHAPTER THREE 0 O O O O O O O O O O O O O O O O O O O O 4 7 Introduction . . . . . . . . . . . . . . . . . . . 47 Transformation and Partition of a Plate in (N)-Subdomain . . . . . . . . . . . . . . . . . . . 48 Truncation and Finite-Point Method Application . . 49 iv Dispersion Relationships of Harmonic Waves Longitudinal Strain Problem . CHAPTER FOUR Introduction Plane Strain Transient Wave Due to Impact Longitudinal Strain Transient Wave Due to BIBLIOGRAPHY Impact 54 60 7O 70 70 78 83 SYMBOLS CHAPTER ONE u . 1 CI m x m U 5 III displacements strain tensor stress tensor (n)-order amplitudes of the cosine distributions (n)-order amplitudes of the sine distributions the (n)-order average thickness distributions (n)-order face traction the plate strain energy density the plate Kinetic energy density (n)-order strain energy density (n)—order Kinetic energy density vi (n) _ In) (n) = .. . e — 111’l + u3'3 ~ (n)-order cillatation Sn = n'r/Zb C = l E shear velocity T 0 CL = (X+d)/o 5 pressure velocity v = Poisson ratio 2 _ 2 2 x — CL/CT x 5 frequency k E wavenumber T CT 2 = 32'k T c1,c2 E correction coefficients ,2 _ 2 ‘2 2 . — 5X + 02 + 53 CHAPTER TWO (S.O.D.E.) E symbolic O.D.E. (S.I.C.) E symbolic initial condition 2 2 a = 3 xx 5 / X vii SECTION 2.2 {2 = c"2 d2/dt2 L r = X/Z transformation c T—‘Zt m = éL-d/dT L E = b/z SECTION 2.3 2 -2 .2 2 2 Q = (C a - a — a ) T t X1 X2 2 _ -2 i2 2 ii — (CT ot — 5X ) l “I" =Xl transformation a — xz/z e = b/z m2 = 2222 = [(-9—)2 32 + 32] C T e T SECTION 2.4 R2 = v2 - C'2 52 d d t Cm = CL and CT , for d = 1,2 viii if = rc:2 5E — a: - a: 3, ~ = 1,2 * “ 1 3 Sfl = sinh £1 b/iw E = cosh a; b , a = 1,2 O. J CHAPTER III and IV N 5 number of subdomain N E the order of truncation un(xl,x2,t) = displacement on the (n)-subdomain u n (n)-displacement and (n)—stresses at the interface H of each subdomain "n u(o) and v(O) E uniform amplitude of displacements u(l) and V(l) 5 linear amplitudes of displacements F(O), H(O) E uniform amplitude of stresses F(l), H(l) 5 linear amplitude of stresses - 2 2 2 D11 _ 4b /“ 6xx 5 - W l — 2b/ ax - 2 2 2 A2 Dtt 4b /" CT ”tt 2 = 2b/Tr k ix A /_.. Ah I CT wavenumber frequency 2, 2 CT,CT d2/dt2 SERIES EXPANSION-METHOD WITH TRUNCATION PROCEDURES FOR APPROXIMATION THEORY 1.1 Introduction In this chapter we will use the general procedure for deducing approximate two-dimensional equations for elastic plates from the three-dimensional theory of elasticity which was introduced by Mindlin (16) based on the series expansion methods of Poisson (23) and Cauchy (3) and the integral method of Kirchhoff (11). Legendre orthogonal polynomials were first used by Mindlin-Medick (18) as a basis in their series expansion. They constructed a 2nd-order extensional plate theory that for the first time exhibited and accounted for complex branches in the frequently spectra. Later Lee-Nikodem (18) repeated the development of Mindlin-Medick using the simple thickness modes as orthogonal basis. In this chapter, representing the field variables by fourier series, we briefly sketch the conversion of the three-dimensional field equations to two-dimensional partial differential equations. These results were previously obtained by Lee-Nikodem (14). A new formulation is introduced based upon a redefinition of the nth order stress—strain relations. A new truncation procedure is then used to generate appropriate plate theories. Frequency spectra predictions of the new theories are obtained and compared with those of previous theories and with exact results where available. 1 l\) 1.2 Series Expansion of the Displacement We first need to define the coordinates system and orthogonal basis to be used. Consider an infinite plate bounded by a pair of parallel planes at x2 = i b as referred to a rectangular coordinate system xi(i = 1,2,3). The thickness of the plate will be in the direction of x2 and the displacement will be expanded using an orthogonal base ¢n(x2). Thus, the finite domain [-b,b] restricts our choice to Legendre or Fourier analysis. In this chapter we let ¢n(x2) = cos £1 (1 - 52). Hence, the 2 b displacement becomes (D x _ (n) nn ._ 2 . = ui(xj,t) - E ui (Xa’t) cos ——-2 <1 —-—-b> 1,) 1,2,3 (1.1) =0 a = 1,3 } Using the orthogonality of the base we can write the amplitudes distributions as follows: b i(n) (xa’t> = %/ ui(xj,t) cos 321- (1 - f52->dx2 . -b u (1.2) The ui(n) are the amplitudes of sinusoidal distribu- tions of displacements across the thickness of the plates. For convenience, however, they will be referred to as n-order displacements. Since the plate is isotropic in this analysis, motion symmetric (extensional) and antisymmetric (flexural), with respect to the middle plane may be considered separately. In the case of extensional motion, only those components of displacement ui(n) are retained for which i + n is odd. Similarly, i + n even give rise to flexural motion. In-plane amplitudes distributions are called compres- sional modes and the one normal to the plane are called face-shear modes. In the case of compressional modes we (0) (O) as extensional and transversal modes have ul and u2 and ul(l) and u2(l) as thickness-shear and thickness-stretch (0) modes. For face-shear modes, we have u3 as shear mode (1) as face shear modes. Further explanations of and u3 their physical meanings can be found in Medick-Mindlin [18]. 1.3 Derivation of the (n)-Order Equation of Motion The motion is described by the integral method of Kirchnorf [11] which is derived through Hamilton's principle and become: jf }{(Oij,j-pui,tt) duidvdt = 0 AD T V (1.3) N H s N s D) Recall Equations (1.1) m x ui .. - 9.1 - .2 Ti2 -f Oi2 $111 2 (1 b) dx2 -b i = 1,2,3 (n) _ _ n+1 __ = Fi — 012(bixd't) + ( 1) Oi2( bixa't) a 1,3 which are the (n)-order average thickness stresses and (n)- order components of face-traction. Thus, (Vn) and duén) # 0 Equation (1.17) gives us U"! (n) _ - (.n) _l_ (n) _ (n) _ (1 9 Tij,j Sn 12 + b Pi pui,tt ’ 0 I which represents the (nl-order stress equations of motion. 1.4 Derivation of the (n)-Order Strain-Displacements and Stress-Strains Relations The three-dimensional strain-displacements relations are _1 ..__ eij - 2 [ui,j + uj,i] , 1] - 1,2,3 (1.10) Then if we substitute the displacements Equation (1.4), Equation (1.9) becomes .- c” (n) nw _ x2 ~(n) . _ x2 Eij - Z Eij C03 ‘2— ( -b—'> + Eij Sln ( '8') n=0 where (n) _ l (n) (n)] Eij - :— [ui’j "I' 11in (1.11) —(n) 3n (n) (n) .. = . . + . . . 81.3 T (521“) 623111 > The above Equation (1.11) represents the (n)-order strain- displacement relations. To obtain compatible n-order auxiliary stress-strain relations we start with the stress-strain relations for isotropic medium in the form: 0.. = 15 1] ijekk + Zuei. . (1.12) J O\ , where 1,; are lame constants. Combining Equation (1.11) to above we obtain m (1.13) _ 2:010m) £11 _ :2 -(n) . mt _ x2 Oij -1j cos 2 < b) + Gij Sln 3—- (1 I?) n=0 where (n) _ (n) (n) Oij - Aaijekk + Zusij (1.14) l—(n) _ i-(n) -(n) Gij - Aéijekk + Zueij . This represents the (n)-order auxiliary stress-strain relations. At this point, if we substitute Equation (1.13) in the first two equations of (1.8) we get 1 T(n) __ C,(n) —(n1) ET ij '- Oij + Z Amn 0ij (1°15) m=0 1 -—(n) __ —(n) dim) ET i2 - +:::A Anm where Anm = 0 for n + m = even;Ann1= 4m/(m2-n2)n for n + m = odd. Then the (n)-order stress equations of motion can be written in terms of the (n)-order auxiliary stress equations of motion. (n) —(n) -(m) (m) Gij,j " Sn 012 +2 Anm Oij,j ‘ Sn C21 (1.16) m=0 (n) pbu (n) :1g5 Derivation of the (nI-Order Strain Density and Kinetic Energy The strain-density energy is defined as b — _ 1 U - 7/ OijEij dXZ . (1.17) -b By substituting Equations (1.4) and (1.8) in the above equation it leads to _-b m U—fz Un (1.18) n=0 where (1.19) _ (n) (n) -(n)-(.n) -(m) (n) (m)-(n) Un - Gij Eij + Oij 8ij + Z Anm[cij Eij + Uij eij ] . m=0 This represents the (n)-order strain-density energy equation. The Kinetic energy is defined as b ‘ — E K - 2./. ui,tui,t dx2 (1.20) then if we substitute Equation (1.1) in the above equation, we get 13:93er (1.21) 2 n n=0 where _ (n) (n) Kn — ui,t ui,t . (1.22) This represents the (n)-order Kinetic energy equation. At this point we can use the (n)-order strain and Kinetic energy Equations (1.19) and (1.22) to derive Equations(1.15): 00 EU (n) _ l n _ (n) 2 : —(n) Tij _ 2 asIn) — 0ij + Anm Gij (1.23) ij =0 BU m —(n) _, J; n __ —(n) (m) T12 ‘28E-n) ‘012 +§:Anm 012 ii =0 (n) _ 1 3L ui,tt ‘ 2 at Kn ° 1.6 Truncation Procedure In this section we will first define our truncation procedure and correction coefficients necessary in the approximation theory. The coefficients will be added to the (n)-order strain density and Kinetic energy equations. Definition 1.1 The [NJql-order truncation procedure is defined as follows: a) us“) = o n > q (1.24) (n) _ —(n) b) Tij — Tij 0 n > q 2 .. _ 2 (n) _ c) axixj (13 — 1,2,3), at of uj — 0, N < n S q . where q is the order of the series truncation and N is the order of higher elastics and inertia terms retained. Conditionfa) and b) have the same physical meanings and explanations used by Poisson [2], Medick-Mindlin [l8] and Lee-Nikodem [14]. The addition condition c) will reduce the system of P.D.E. in terms of the first q amplitudes distributions of both compressional and face-shear modes. Analytically it will reduce the wavernumber k (in the x1- direction) and frequency w relationships (which can be derived from the Partial Differential Equation by using straight-crested waves propagating in the xl-direction) to a polynomial of order q. Definition 1.2 Correction coefficients c1 and c2 are going to be added to the (n)-order strain-density and kinetic energy Equations (1.19) and (1.22) as follows: m (1.25) _ (2) (n) (n) -(n)-(n) (1) (m)-(n) —(m) (n) Un - Sln cij eij + Cij eij + Z Anm 61m [Uij Sij + Oij eij] =0 .. (2) (n) (n) Kn ‘ 50m ui,tt ui,tt (1°26) where l , m # 1 5(1) = (1.27) 1m cl , m = 1 l , m # a (2) _ _ 2 11‘ 5an - , p — cos (2‘) - 0,1 N'U 8 II Q Q I 10 Now, using the above Equations (1.25) and (1.26), Equation (1.23) becomes 00 TI?) = (3(2) 05’.” +2 A a”) Sim) (1.28) 1] 1n 1] nm 1m 13 m=0 ,-I.-(n) —(n) mzA 6(1) (m) 12 2m 1m 012 u(n) = 6(2)u (n) ui,tt on ui, tt ' Next, if we substitute Equation (1.28) in the (n)-order equation of motion Equation (1.16), we get m (1.29) (2) (.n) -(n) (1) -(m) (m) (n) _ (2)u (n) [aln CIij,j-$nO 012] +2 Anmélm [Gij,j . snOiZ] + F1 T 0b sonu 1, ti =0 .. 211 Sn ‘ 2b ' which represents the (n)-order auxiliary equation of motion with correction coefficients. 1.7 The Truncation Application for [qu] = [1,1], [2,1] and Arbitrary In this section we will truncate for N = 1,2 and q = 1 which will be needed for future discussions. An arbitrary truncation will also be generalized at the end of this section. Case I (N = 1, q = 1) Using definition 1.1 for [1,1] then Equation (1.29) gives us ll (0) (1) (0) _ p_ (0) Oij,j + A01c1 Oij,j + Fi — pbc2 ui,tt (1.30) (1) _ —(1) (1) _ (1) Cgoij,j s1 012 + F1 ‘ Db ui,tt From Equations (1.11) and (1.14), the (n)-order strain- displacements and stress-strain equations, Equation (1.15) becomes, in the case of the flexural motion (1 + n = EVEN, a = 1,3) 2 (0) _2_1_J_ (1) A (0) _9_ (0) uV u2 + l: cle + b F2 c2 u2,tt (1.31) 2 (1) (l)_ _T_T_ 2 (l) _ 23 (0) 3; (l) _ (l) “V ua + (1 + u)ea u(2b) ua k) clus,a + b Fa " pua,tt and for the extensional motion (1 + n = ODD 1 a = 1,3) 2 (0) (0) 21 (l) l (O) __ (0) uV ua + (1 + u)ea + IT clu2,a + 5 Fa - pua,tt (1.32) 2 (.1) _ 21 (0) l (1) _ (l) 2 2 2b (n) __ (n) (n) where e — ul’1 + u3l3. These represent the zero and first orders displacement equations of motion. 1) Case II (N = 2, q Using definition 1.1 for [2.1] then Equation (1.29) gives us 12 0(0) -v(l) (O) _ (0) 0ij, j + A0101 Oij,j + F1 T pbczu i,tt (1'33) p (l) _ -(l) (2) _ (2) (l) (l) Czoij,j$1012 4” A12l:31j, j 51012] + F1= pbu1, tt O(2) _ -—(2) (1) (l) (2) __ Oij j 52012 + A21|:Gij j - 52012] + 1:"i — 0 . From Equations (1.11) and (1.14) the fluxural motion (1 + n = EVEN, a = 1.3) becomes 2 (0) 2n (1) 1. (0) _. o (0) LIV 112 + ‘5— C18 4" 13' F2 — a: uz'tt (1.34) 2 (1) (l) 2 (1) 2n u(0) 2(41+u) (2) uV ua + (1+u)ea T u(Zlb>u ua '- TI'C l 2, + TTT§E_T L12, 1 (l) __ (l) + 5 Fa T on a, tt 2 + (1 + zu)(%) uéz) + Zlégggl e(l) - % F§ZI = o and for the extensional motion (1 + n - ODD, a = 1,3) we get 2 (0) (O) 21 (l) 1. (0) __ (0) (1.35) uV ua + (1+u)ea + I; c1112,a + 3 Fa - pu a tt 2 HZUVZ £1) - (1+2U)(§%) u2(1) _ 21 a e(0) + 2(1+4u) e(2) 1 3b 1 (l) _ (l) 3 F2 ‘ pu2,tt 2 + u<%> u(2) + 2(1+4u) u<1) 1 F(2) a 3b 2,a b a T 0 13 These represent the zero, first and second orders displacement equations of motion. Case III (N E EVEN, q = 1) Using definition 1.1 for [EVEN, 1] Equation (1.29) then gives us (1.36) or(0) (l) -(2m+l) (0) _ (0) orij,j +:23AO(2m+1)6l(2m+1) Uij,j + F1 pbc2u i, tt N p (1) _ -(1) —(2m) _ (2m) (1) C2"1j.j 31012 +2 A1(2m)[cij,j S1012 ] + F1 m=0 (1) Tobui, tt ' and for l < n i N E EVEN (1.37) (2) (2) N-l (1) (2 1) (2 1) n — n - m+ m+ 013,3' ’ 3211012 +2 A(2m+1) (2n)51(2m+1)[°1j,j ' 5211012 I =0 + F(2n) 0 l N U(2n+1) _ -(,2n+1) —(2m) _ (2m) 013', j 32114-1012 *2 A(2m) (2n+1)[cij,j s211+1‘712 ] m=0 + F(2n+l) 1 Similarly for N E ODD, the displacements equation of motion can be derived using Equations (1.11) and (1.14) in the same fashion as the first and second orders. In the above section we did not assign any numerical values for the correction coefficients c1 and C2. According to Lee-Nikoden [14] it can be shown that in order to make 14 the slope of the lowest flexural branch in the first order theory coincide with that from the three-dimensional theory when both the frequency and wave number approach zero, the value of cl must be taken as c1 = n/4. In order to make the phase velocities of the lowest extensional branches approach that of the Rayleigh surface waves [1] as both the values of the frequency and wave number get large, c2 must be set equal to the real root of 3 2 2 2 _ c2 - 8c2 + 8(3-2/x )c2 - 16 ( 7 - - B = x - ——— 4x - 7 - —— D + 1 - D (1.40) 12 ‘ 9Tr2 x2) 11 tt r - _ 4 ( _ )2 2 _ — B44 - 1 T9? 4X '7 Dll + X Dtt - _ _4_ - A13 ’ c1 w D1 _ 4 - A22 ‘ C1 F D1 A - 6 - J; B 23 T 11 c2 tt 16 31 34‘°1%51 41 44‘C2 and (1.41) 0| I Q) 11 T 22 xx 0| n I Q) 1:th The quadratic amplitude becomes .12) I I can b 2 1 :7: (4 - x)“1()1 (1.42) .3) I I cum which can be solved later. The approximate frequency Spectrum for flexural motion can be derived by assuming a solution as follows: 17 nil) — uf1)ei[kxlTw€] uéo) = Ué0)ei[kxl-wt]. Substituting the above in Equation (1.39) leads to __ I "(1fI B12 A13 U1 = 0 - — (0) IA22 A23) U2 3 I. - where le=- x2--4-§(4x2-7)<4-J§> 22-11-102 9w x _ _ , 2 A13 - 1 cl n z - _ 4 A22 ' 1 c1 E z — _ _ 2 1. 2 A23- Z+-c—2-Q and z = 32-k n _ 2b (n 0 — 77 C; . Thus, the determinant of Equation (1.44) becomes PI B12 A23 ‘ 13 A22 = 0 (1.43) (1.44) (1.45) (1.46) (1.47) 18 It represents the approximate frequency spectrum for the flexural motion. The dispersion curves in Equation (1.47) are computed and compared with those of Lee-Nikodem and Rayleigh-Lamb [1] frequency equation as shown in Figures (1.1)- [gj31-Order Truncation (First Approximation) Using Equations (1.29), (1.14) and (1.11) for N = 3 and q = 2 the displacement equations for extensional motion become: P j - (0)1 R11 R12 R13 U1 (1-48) (1) “ R21 R22 R23 U2 ‘ ° ° (2) 1R31 R32 R33.. L01 - where ._ ‘_ 16 ( 2 _ > __ 2 2 — _ - (1.49 R11 - g? X 2 (.1 3) 'I' X 01]. Dtt ) .. 1(2- )- R12 ‘ C1 n X 2 D1 8 < 2 > 2 .— R = 9x - l4 1 - D 13 351T2 < 3') 11 _ 4 2 - — R21 ‘ c1 E (I 2) D1 R = x2 + c D - D 22 2 11 tt - _ JL_ 2 ‘ R23 - 3w (x + 2) D1 8 < 2 > 2 — R = 9x - l4 1 - D 31 35Tr2 ( ;2) 11 _ - i (2 > — R32 — 3” x + 2 D1 R33= - 162 <9x2-14)< -1; +x2 511+4-Dtt. 225w x The extensional third amplitude becomes (3) 8b 2 (0) 1 l4 (2) (1 50) u = - -——- - —— u + — —1—— u - 2 9“"2 < x2) 1,1 5 ( x2) 1,1 . which can be solved later. The approximate frequency spectrum for extensional motion can be found by assuming a solution as follows: ' kx -wt (0) _ (0) 1[ 1 ] (1.51) 111 - Ul e ' kx -wt (1) _ (1) 1[ 1 ‘] u2 — U2 e ' kx -mt (2) _ (2) 1[ 1 I ul - U1 e . Then Equation (1.48) becomes '- - - " (0)“ (1.52) R11 R12 R13I U1 — — — (1) R21 R22 R23 U2 — - - (2) -R31 R32 R331 LUl 20 The coefficients Rij are obtained by substituting the followings in Equation (1.49): Dll = - z (1.53) D1 = 12 - _ _ 2 Dtt - 0 Thus, the determinant of Equation (1.52) becomes + R11R22R33 ' R11R23R32 ' R12R21R33 R12R23R31 (1.54) R13R21R32 ‘ R13R22R31 T and represents the approximate frequency spectrum for extensional motion. The dispersion curves of Equation (1.54) are computed and compared with those of Medick and Rayleigh-Lamb [1] frequency equation as shown in Figures (1.2). We can conclude that the first approxi- mation technique improves the approximate frequency spectrum. We can also find the cubic amplitude distribution by using Equation (1.50) which will improve the solution for the displacements. [2,4l-Order Truncation (Second Approximation) Using Equation (1.29), (1.14) and (1.11) for N = 4 and q = 2 the displacement equations for extensional motion become: P '1 I T (0) R11 R12 R13 U1 (1) _ R21 R22 R23 U2 - 0 (1.55) (2) R31 R32 R33 U1 1 J _ .1 where _ 2 _ 19 2 _ _ 2 Rll — [x 9w2 (x 2)(l X2)Dll ott] (1.56) _ 16 2 _ R12 "F‘ C1‘X 3) D1 - - 12 2 - - 11 R13 ’ [451“X X)(9 X2) D11J _ _ i 2 - R21 ‘ n Cl (X 2) D1 _ _ 4 2 2 _ — _ 2 R22 - [c2 2 (x + 14) D11 Dtt x ] 9001 _ __1 2 — R23 ‘ 3n (X + 2) D1 16 2 14 R = I (x - 2) (9 - ——) D 1 31 451T2 X2 11 _ __i 2 — R32 ‘ 31 (x + 2) D1 _ 2 _ 16 2 _ _ 14 — _ — _ R33 - [(x 2 (4x 14) (9 2))Dll Dtt 2] 255w x The extensional third and fourth amplitudes are (3) _ -4 _ 2__ (0) 1 _ 1_4_ (2) Uz ‘ 9 2 (l 2) U1 , 1 + 5(9 2) U1 , 1 11' X X (1.57) (4) _ _ 1 2 (1) U1 — 60w (x + 14) U2 I 1 which can be solved later. The approximate frequency spectrum for extensional motion can be found by assuming a solution as follows: 111(0) = Ulm) elIkxlthI (1.58) (1) (1) i[kx -o)t:| U2 — U2 e 1 U12) = U(2) ei[kxl-wt] Then Equation (1.55) becomes P— - — 1 - (0)~ R11 R12 R13 U1 (1-59) - — r (l) _ R21 R22 R23 U2 ‘ ° — — — (2) -R31 R32 R33. L91 ] The coefficients R15 are obtained by substituting the following in Equation (1.55): — _ _ 2 D11 - z (1.60) D]. = 12 — _ _ 2 Dtt - 9 Thus, the determinant of Equation (1.39) becomes R11R22R33 T R11R23R32 T R12R21R33 + R13R23R31 (1°61) R13R21R32 T R13R22R31 T 23 and represents the approximate frequency spectrum for extensional motion. The dispersion curves of Equation (1.61) are computed and compared with those of Medick, of the first and second approximation and Rayleigh-Lamb [1] frequency equation as shown in Figure (1.3). We can conclude that the second approximation improves the approximate frequency spectrum. 9 - Frequency 6.0 - 5.0 - 4.0 - 9," O l 5’ O I 2.0 4:. 1’ =25 .. _ -— - MEDICK - NIKODEM APPROXIMATE 1 l l 1 1 3.0 4.0 5.0 6.0 70 RE Z - Wave Number Figure 1.] OU‘NCT Q'FKE 2.0 1.4L .. -1. J. -- q, l I ". I l l A l OI -'1 —| d. I I 1 -I TI (2)... . ‘25 4.x 0 - 3.1 C L ‘ u a II. (T v" 30 .~ a CIRCLExVETICZ L T‘IHHSLE=TTTTT QPPPGKIMQTTTN 1 590334137 WT APPROKZNQTTCN , I .;. C. -fi-f . . 1 - , fl FIGURE (1.2) + + A + o A r O A A- n A r o A .U A o \— D A I; o A Mm c A MK 0 A + 0 I + 3. O c 0&- Q 05' C «3-43:. . . 4 . 4‘ J J o.m r 1 .H. T . N 04 7.. WI .2. .U 0.. .L M,. T: T; “H VA NH. O Q... r 0.. wk. D 5!. 3 PM 01 D. - D- T: n2..- w.” .5. T] .P... 3.. n... . .2. T. “.1. . . n... f- H... -.. C. : P. MI 7.. _.._ : 1. C... E ”.0 : n. L uni. ..... - ass PI» .d- 1..., . R T. .{u + 2 .i. «H mm. U r... T C A. c . K o. o f .u + r A «. r T- . A a 141 J. .r i 4. 4414. 1 f u A t. A .u A A.— L L .H u an A“. .d & u w C c r... O V» O .U p C h. J U H C m . . # kuzusommg- a o.— Jl. .dll ulfllllfllul..- -IJIIIJ «ItulIJ IJ-‘;_I lfi‘l..;ul..£l...—!| J!‘ . all. . o.o _‘~‘§ \q -~3v_‘..-:r\ I inf. .. Hr 7. (1.3} FIGURE CHAPTER II SYMBOLIC METHOD IN WAVE PROPAGATION PROBLEMS 2.1 Introduction In this chapter we are going to generate an approxima- tion theory using symbolic technique [13] . The P.D.E. and B.C. are reduced to an infinite differential equation. This is done by letting all partial derivatives behave as constants, except for one of them. Thus, this will reduce the P.D.E. to a symbolic ordinary differential equation (S.O.D.E.) with symbolic initial condition (S.I.C.). Then, to generate the infinite differential equation, we substitute the sym- bolic solution (8.8.) into the B.C. First we will examine the string equation because the simplicity of the Operator permits us to solve this problem exactly. Then we non-dimensionalize the equation and use the parameter to find an approximation solution. Similarly a two-dimensional problem will be presented. At the end of~ this chapter the three-dimensional elastic plates problem with free stresses at the tOp and bottom of plates will be considered. The approximation will be compared to other techniques used in the preceding chapter. 28 2.2 Symbolic Method to Solve the String Equation The partial differential equation for the string problem is (2.1) uXX - c3211tt = o (P.D.E.) with B.C. (2.2a) ux(_._»_ b,t) = o (B.C.) and I.C. (2.210) u(x,0) = o (I.C) ut(x,O) = f(x) 2 Let 5t behaves as a constant then equation (2.1) can be written as follows: II 2 2 d2 (2.3) u - £.u = O (S.O.D.E) where £ = 2 2 c dt 2 u(O,t) = H(t) (2.4) (S.I.C.) u’(o.t) = £F(t) The solution for the (S.O.D.E) is u(x,t) = a(t) cosh £5<+'b(t) sinh £§< and by using the (S.I.C.), the above equation becomes (2.5) u(x,t) = H(t) cosh £0: + F(t) sinh £>< Next, we substitute the above in the B.C. equation (2.2a) which will give us an infinite O.D.E. 30 (2.6 H(t) sinh £J3 i F(t) cosh £1: = O or [sinh £b]H(t) = O (2.7) II 0 [cosh £b] F(t) The above infinite operator on H(t) and F(t) can be solved exactly by assuming a solution in the form of an expotential such as (2.8) Expanding cosh {b and sinh ib in terms of power series and Operating equation (2.8) term by term yields [sinh %EJH(t) = O (2.9) L [cosh gE]F(t) = O . L If A and B are arbitraries, then the above equa— tions are satisfied for w = i n—TT- EVE (2.10) INT JJ — 1 EC]; r 1’1 ODD. Therefore, using the above result, the solution (2.8) becomes . n=EVEN n L (2.11) 2°. . m F(t) = x. 13m Sin 33 th. n= ODD Next, if we substitute equation (2.11) in equation (2.5), operate and rearrange terms, we obtain a solution for our problem as follows: (2.12) u(x,t) = 2: Ancosgchtcosg-g-x + Z anin-g-gCLtsin%%x n=EVEN n=ODD with the time variable I.C. equation (2.2b). u(x,O) ll 0 (2.13) f(x) . ut(x,O) Note that equation (2.12) is the exact solution of the P.D.E. with boundaries. We are now going to non- dimensionalize equation (2.14) given below. II 0 [sinh £b] H(t) (2.14) [cosh £b] F(t) II 0 Using these transformations given below n = %- , 2 = max-wavelength (2.15) T =‘% t, c: is the velocity then, __9_._ where (2.36) 2 = (c' The solution for the (S.O.D.E) is (2.37) u(xl,x2,t) = a(xl,t) cosh {x2 +b(xl,t)sinh ix 2 and by using the (S.I.C) equation (2.35) the above equation becomes (2.38) u(x1,x2,t) = H(xl,t)cosh £x2 + F(Xl,t)Slnh £x2 . Next, we substitute the above in the B.C. equation (2.32) which gives us an infinite P.D.E., (2.39) H(xl,t) sinh £b ;: F(xl,t) cosh £b = O or sinh £1) H(Xl't) I O X (2.40) 1 cosh it) F(xl,t) = 0. These infinite operators can be solved exactly by assuming an exponential solution i[kx t) = Aee -5312] H(Xl' l (2.41) i[kx t) = iIBe -It] F(Xl' 1 If we substitute (2.41) in (2.40), then for A and B arbitrarieS. we are going to get T2 2 nF 2-k =§EI nEVEN CT (2.42) 2 m 2 _ El CT Equation (2.42) are the SH-symmetrical and SH- antisymmetrical modes of the Frequency Spectrum. which are derived in Achenback [11]. Hence, if we substitute equations (7.42) and (2.41) in equation (2.38) we get the exact solution as follows: 21(x2) (2.43) u(xl,x2,t) = exp 1(kxl-ut) zII(X2) where p.) \ J n? n7 2 m2 9 zI(x2) = A cos 2b'x2 , n E EVEN, (as) ==E;3-- k“ (2.44) 2 '2 zII(x2) = B sin %% x2, n E ODD, (%%- ==E:E-- k We are now going to non-dimensionalize equation (2.40) which. is sinh £b H(x1,t) = 0 (2.45) cosh £b F(xl,t) = O . Using these transformations given below x U ='j% E =:% x 2 c 2 2 = — = _ A (2.46) e z then at (2 ST _.S 2 _ .1 2.2 T — It 5X — (fl) 06 2 then 2 (2.47) m2 = .8212 = [(3) 52 - 5‘3] C T t. T and equation (2.45) becomes sinh Em h(n,T) — o H(hn, 3%) = h(n,T) (2.48) where cosh Em f(n,T) = o F(hn, gm) = f(mT) and equation (2.38) becomes (2.49) G(n,e,T) = h(n,T)cosh Ema + f(n,T)sinh Ema, where (2.50) 501.24) = u(bmz . 5T) 0) The parameter 9 can be used as a means to truncate equation (2.48). The approximation will be of order ~ (N = 0,1,2-°-). Now, let us proceed by considering two cases (N = 0,1)- 0 CaseI (N=o. 6 ED This reduces equation (2.48) to (2.51) h(T).“I‘) o m g(n,T) 0 where g(n,T) = m f(n,T). And equation (2.49) leads to a - - _ (2-52) '33'u(n,e,T) - O or (2.53) u(xl,x2,t) = constant. Case II (N = l ,E ) This reduces equation (2.48) to (2 +m262) h(n.T) (2.54) where 9(fi.t) = m f(T) (6+ m262) 9(n.T) H 0 II C G(xl,t)==£ F(xl,t) and equation (2.49) gives us 222 22-2 (2.55) G(mT) = h(n,T) (1+m—i—) +9(T‘.,T) (“LIB—L) t5. 2 (T\ By using equation (2.46), equation (2.54) and (2.55) become a (2.56) (2+b2£) H(x ,t) = o l where J - c2 - c-282-82 22 ‘” ’ T— T t x (6+b.—£) G(x1,t) =0 2 bzx2 (2.57) u(xl,x2,t)==H(xl,t)+-G(xl,t)bx2+-£1H(xl,t) 2 3 3 2 b x + i G(Xl,t)T—- . Thus, the above equation gives us the approximation of order (N==1, 62). 2.4 gymbolic Method Approximation to Solve Three-Dimensional Wave Problem Consider the free stresses condition at x2: ib , then we have Navier equation and B.C. as follows: 2 —- — — 2 - - - I“! _ (2.58) CL v(v°u) “CT Vx(qu) - u - O (2.58a) 02j(x1, :_b,x3,t) = O and for solutions, Fredholm vector decomposition is used in e—l <1! 0 E- ll 0 (2.59) E = Em + 5 x where u(xj,t), j = 1,2,3. From equation (2.59) the divergence and curl of the displacement vector are <1| o (2.60) E = V 3, scalar potential ’H A C II I <1 (:1— s vector potential and by substituting the above equation in (2.58) we get 4O 2 - - Rl Ul - 0 and 7 x Ul — 0 (2.61) 2 ' _ : o _ _ R2 U2 - O and 2 U2 — 0 where RZ—vz 4.52 c-c .6... —12 d _ ’ ' C2 T—7" a 7 L a T or a ‘ ot a , v2 = a: + 52 + 82 (2.62) 1 X2 X3 U1=".o U2 = V x) The displacement vector (2.59) becomes 2.63 Ex.,t =6 x.,t +6 x.,t , '=1,2,3. ( ) (j ) l( 3 ) 2( j > 3 2 2 2 . Let a , 5 , a behave as constants, then equation t X1 X3 (2.61) can be written as follows: -11 2 - _ 2 _ 1- .2 2 .2 1 Ua - £3 Ua — o (S.O.D.E), 2a - [7 cat-5X -6X ..' c l 3 a Ua(x1,0,x3,t) = Fa(x1,x3,t) (2.65) (S.I.C.) , d=1,2 -1 _ Ua(X1'O'X3't) - £6 Ga(x1.x3.t) The solution for the (S.O.D.E.) with (S.I.C.) is (2.66) U = F. d = 1,2 1(d )cosh £53< +~G sin {fix 1(a) 2 2(C0 v2 2 ' where U(m) = (U of vector U(a), a = 1,2. are the components 41 We also have two additional conditions which are the diver— gence of U and the curl of U to be satisfied. Hence, 2 1 we let (2.67) T 0 U2 = Ul(2),l + U2(2)'2 + U3(2)'3 = [F1(2),1 + F3(2),3 + £2 G2(2))”Sh =£2 X2 + [31(2),1 + G3”),3 + £2 F2(2)]51nh £2x2 = 0. Thus, (V x2) the above equation gives us F1(2).1 + F3(2).3 + £2 G2(2) = O (2.68) G1(2).1 + G3<2>.3 + £2 F2(2) ' 0 Similarly, the curl of 61 is 269 " ‘ -( A — ( . ) V x Ul — -eijk Uj(1),k]ei - 0. So for (Vi) we must have (2.70) eijk Uj(1),k = O 0]: (2°71) F2(1).a = *1 Gs<1> I 3:10.173 62(1),a = .5 65(1) F3(1),1 = F1(1),3 G3(1),1 =G1(1),3 42 Next let us consider the stresses equation at the plane (x2 = i.b) (2'72) :23 = azjkru1.1‘*“2.2‘*u3,3) + ZHEUZ, *‘u ,2) or .321 = ufuz'l + 111,2] (2.73) 022 = l[ul'l+-u3'3] + (14-2u)u2'2 O23 = “[u3,2*'u2,3] and substitute equations (2.63) and (2.66) in the above equation and use the B.C. (2.58a) (at x = :_b) we then 2 have 2 - (2'74) 021 = GE; [”(F2(a).1 + H1(a)’ C6 + ”(H2(d),1 + :5 Fl(a)) :31) = o 3—1 - - F (2'75) 023 7 3:1 LMF2(6),3 + H3(a)) Ca £2 F ) 5 - o + Ll(H2301)r3+ on 3(a) - a] - 2 F - (2'76) 022 = aEiL[X(Fl(O),1 + F3(a).3)*’(44'2“)H203)]Ca 2 + WHi<8. Shear waves. Ther for , ‘ - E ', , = = e e if we let U1 0 (i e Fa(l) G2(l) 0). equation (2.79) can be written as ua(xi,t) = F cosh £2x2 , a(2) (2.82) I _ . . _ u2(xi,t) — 142(2) Sinh £2x2, l — 1,2,3 and the infinite P.D.E. (2.80 ) are reduced to , 2 (2.83) 921 = [112(2),1 + 22 51(2)] 32 o J - (H + £2 F 2) S - 0 23 7 ~ 2(2),3 2 3( 1 2 7 C22 = [4(F1(2),1*'F3(2),3)*'(kl'2“)H2(2)]C2 7 O or r 2 49(X+-29) 2 -2 1 - - . 4 ' - . (2 8 ) Latt p(l+2u) (axll'dx3). Slcl Ua(X1'X3't) O 7 = 1,3 For the theory of order 0 (Sl==Cl==1) equation (2.84) is identical to the results of Poisson-Cauchy [3] and Hegemier-Bache [6 L 45 Case II Flexural motion In the case of flexural motion,the following condi- tions must hold: (2.85) a) ulru3 ODD WITH X2-VARIABLE b) u2 EVEN WITH XZ-VARIABLE d) 62(1) = G2(2) = 0. Then the displacement (2.66) becomes (2.86) u2(xi,t) = F2(l) cosh {iXZ + F2(2) cosh £2x2 ua(xi,t) = Gd(l) Slnh {ixz + Ga(2) Sinh £2x2 a = 1,3 and I — (2.87) ua(xi,t) — Ha(l) cosh iaXZ + Ha(2) cosh £2x2 a = , If we let U1 5 0, then F2(l) = Ga(1) = O (d==l,3) and equation (2.86) becomes ’ — ua(xi,t) — Ha(2) cosh £1x2, a = 1,3 (2.88) u2(xi,t) = F2(2) cosh £2x2 and the stresses (2.74-76) take the forms G21 7 [F2(2),1 + Hl(2)] E2 7 0 023 7 [F2(2),3 + H3(2)] C2 7 0 022 = [A(Fl(2)'l + F3(2)’3) + (A + 2u)H2(2)] C2 = o. The above equation can be reduced to equation (2.84). CHAPTER III REDUCTION OF SERIES-EXPANSION TO A FINITE-POINTS APPROXIMATION 3.1 Introduction In this chapter an elastic plate with thickness 2bN is being analyzed. N—subdomain thickness 2b with local coordinates xj(j = 1,2,3) will be related to a global coordinates xj(j = 1,2,3) by a transformation. Hamilton's principle and the truncation procedure used in Chapter one will be applied in each subdomain. Continuity of the stresses and displacements is required at each interface dividing subdomains. Hamilton's principle gives us two non-homogeneous P.D.E. coupling the zero and first amplitudes distributions. The non-homogeneous terms are stated at the interface of each subdomain. The finite points approximation gives a discrete solution in the direc- tion of the plate thickness (i.e. xz-variable). The points are located at the tOp, bottom and interface of each sub- domain. This technique will reduce the two non-homogeneous P.D.E. to a system of Differential-Difference equation. Dispersion Relationships of Harmonic Wave will be studies for various values of N, the number of subdomain. 47 4.. 0‘) 3.2 Transformation and Partition of a Plate in (n)- Subdomain Recall Hamilton's principle and displacements obtained in the first chapter: (5 (3.1) i (Gij,j-pui,tt) 5ukdvdt=0, i,j,k=l,2,3. T V co X (3.2) u.(x.,t) = Z uFmH; ,t) cos 5‘11 (1———2-), (1:1,3. l j m=O l a 2 b We define a transformation XI II X (3.3) 1 1 2 = x + P , where P = 2nb 2 2 n x37x3 which relate the global coordinate to the local description. The figure below will show the plate is partitioned in (n)-subdomain. TOP OF PLATE ‘// 2b ‘7 BH2/; 2b x / "' " Un+2 ------- }‘ 1' 7" / / r X3 /l?,....// 2b / — - Un+1 ---------- V — Fr’. 2Nb / _ .. un ___________ / 72 BOWTONICF'PLATE We shall proceed to analyze a plate of thickness 2bN. In each subdomain we have: a. Hamilton's principle (3.1) describing the motion. b. The truncation procedure of Chapter I applied to Hamilton's principle. (m) c. The truncation displacements u = O, m > 1. Therefore equation (3.2) becomes 11(1), _ (0) .. ~ .11. (3.4) ui(xj,t,n)-u (xa,t) + cosri. xa,t)31n x 2b 2 where the transformation (3.3) is used and n describes the displacements in the n -subdomain. d. The displacements at the interface which are con- continuous (i.e.) (3.5) ui(xa,b,t,n) = ui(xa, -b,t,n+ 1), i 3 1,2,3 ($6) uwx,bfign+1)=1L(x,4mtfir+m, q: L3. 1 d i a Similar results can be obtained for the stresses. 3.3 Truncation and Finite-Point Method Application From definition 1.1 in Chapter I, the general trunca- tion procedure for the [N,l]-order iS (3.7) a) (13)“) = o n > N b) 3):?) =E)r.‘)=o n>N 1] 13 . . _ 2 (n) _ c) a ixj(i,j — 1,2,3), at of uj -O SO Plain Strain will also be added to further reducing the complexity of our problem, (i.e.) u3 E 0 and 8X3( ) E O. A new notation is now introduced, .3... . = .{o> , F(o> = F{o> u(l) = u1(1) ' F(1) = F1(1) V(O) = u£0) ' H(O) ___ F£0) V(1) ___, uél) ' H(l) ___, Fél) for the displacements and stress components. Hence, equation (3.4) becomes _ (O) (l) . .1; un(xl,x2,t) - u (x1,t) +u (xl,t)cos nrr Sin 2b x2 (3.9) .. (0) (l) . _7r__ vn(xl,x2,t) — v (x1,t)-+v (xl,t)cosrnr Sln 2b x2 and un+1(x1,x2,t) =u(O) (x1,t) -u(l) (xl,t)cos nrr sin {13- x2 (3.10) _ (0) (l) . .I. Vn+l(X1'X2’t) —v (xl,t) -v (xl,t)cos nrr Sln 2b x2 The Finite-point method at (x2 = b) reduces the above equation to u = u(0) +_ u(l) n _ (O) (l) un+1 _ u '+ u (3.11) V = V(0) + V(l) n V _ V(0) + v(l) n+1 _ 51 which represents points in the plates. The approximation depends on the number of subdomains. If N increases, then better results can be obtained from the above theory. Next, the inversion of equation (3.11) yields [un+1 + un:I :3 ll (0| l--' C. II NHA [un+1 n (3.12) [vn+l + VnJ th <1 II ml H [vn+l n which are the uniform and linear terms of the series expansion. Similarly, the stresses can be obtained as follows: (0) _ _ F - On+1 0n (l) _ F _ On+1 + 0n (3.13) (0) _ _ H - Tn+l In (1) _ H - Tn+l + In Next, we are going to consider a truncation for N = 1,2 and q = l (which was derived in section 1.7, Chapter I) for each subdomain then the finite-point method defined above will give rise to a set of Differential-Difference equation: Case 1 (N=1,q+l)/the zero and first order displacement equations of motion for [1,1]-order truncation are given in equations 1.31 and 1.32. Thus, if we substitute equation 3.12 in these equations, we are generating a Differential- Difference operation as follows: (3.14) where (3.15) and 12 13 22 31 41 23 _ 34 - 44 ‘ 13 n+1 A23 0 i(un+l 0 A34 %(vn+l 0 A44“) 5Vn+1 O O . {(Un+l O O :(¢n+l p3 O (Gn+l O p4dlw(Tn-+l x2 D + l — D 11 tt 7%‘C1 E51 '5 C1 131 B11 7 C21 fitt X2 IS11 7 I31:1: %'(X2"2)Cl Dl ;§(x2.-2)c1 Dl C D + x2 - D 2 11 tt U I L») pi = 8132 I i 7" 1,2,3,4 um 5 _ 4b2 52 11 7'.T2 xx (3.16) 7 — 22 A Dl - 7T 7x - 4b2 2 Dtt 7 2C2 att 7T CASE II (N=2,q=1) T Similarly, the zero and first order displacement equations for N==2 were given in equations (1.34) and (1.35)- Thus, if we substitute equation (3.12) in the first two equations, we get _ .1 F A 0 B12 A13 0 (un+l + uh)! I o A A o (u - u ) (3 17) 22 23 n+1 n I + A31 0 0 A14 (Vn+1 + Vn) L_A4l O 0 B44 (Vn+l 7 Vn) -pl 0 O 0 .- _(Gn+1 + On)- ( 0 p2 O O (n+1 - Tn) = O O 0 p3 O (On+1." on) -0 O 0 p4—J _(Un+1 + Tn)- where _ 2 .41. 2 _7_ - _ '- BlZ—[x -972 (4x -7)(4—X2))Dll+1 Dtt (3.18) ' _ 4 2 2. - 2 - 1344—[1 - 2 (4x -7) J1311+x -Dtt. 54 The Aij can be computed from equation (3.15) and - -212. 1 L - UT x (3.19) p4 - ——§-[1 + 3 (4X -7) Dlj uv The last equation of each of the equations (1.34) and (1.35) are used in the decoupling of higher modes contained in the first two equations of each of the equations (1.34) and (1.35). These modes can be written as (2) _ 22___1____ 2(4x+p> (1) _1._ <0) u2 ' (7r) (“2(1) r‘ 3b u1,1"bFz ] (3.20) 2 (2) _ 2 _l_ _ 2().+4u) (l) L (0) U1 ‘ (w) u [ 3b u2,1 + b F1 ] where (2) _ (0) F2 — F2 (3.21) (2) _ (O) F1 — P1 3.4 Dispersion Relationships of Harmonic Waves Now, we investigate the dispersion relations of harmonic waves in a plate divided in § subdomains and governed by the approximate equation of motion in last section. Equations €3.14)and (3.17) correspond to a truncation [1,1] and [2,1] respectively and can be written in the form: _. , ’ 3 A12 A12 A13 A13 i; un 'A22 A22 A23 A23 !un+l (3.22) f ‘ A31 A31 'A34 A34 ;§ Vn ; 3 A41 A41 'A44 A44 § IVn+1f L. .4 1.. _J . O 0 -p2 p2 n+1; + § = 0 —p3 p3 O O ( Tn ( ; 3 g 0 0 p4 p4 , i n+1} '—. __J L. .3 and [“312 B12 A13 A13 un ’ I "A22 A22 A23 A23 n+1 (3.23) 1 A31 A31 ‘A34 A34' E Vn A41 A41 ’344 B44i ' vn+1 51 pl 0 O on O O -p p z + , 2 2 n+1 ; = 0 : l 1 ( —p3 p3 O O Tn i ' O 0 p4 p4 n+1; First, let us consider the case where fi = 1 (one domain). Then for harmonic waves prOpagating along the x axis, we assume 1 ikal-u J (3.24) (un,vn,3 (Un,Vn,Ln,Tn)e Substituting this into the approximate equations of motions (3.22) we obtain a set of algebraic difference equations with constant coefficients in terms of Un'vn’zn' and Tn where n takes the values between zero and g for a plate made of E subdomain. Boundary conditions for the dis- persion relationships require both surfaces of the plate be traction free. For the case of fi = l (3.25) :0 = To = o 4.0 = T1 = O and the Differential—Difference equation (3.22) becomes r Z Z Z Z 2 ' U 3 o ‘ ‘ 12' 12' 13' 13 o F ‘ -5: p i p Z , E U 0 22 22 23 23 1 (3.26) _ _ _ _ = A31' A31' ‘A34' A34 V0 0 A41' A41' ‘A44' A44 V1 0 J L .1 .. J The determinant represents the approximate frequency spectrum for fi = 1 and [1,1] (the general truncation procedure); the coefficients in the above matrix are functions of :ij(k’w)’ the wavenumber and frequency. Now let us consider the cut-off frequency modes which can be shown as lim E..(k,m). k4O 13 57 It reduces the coefficients of equation (3.15) to 3’ II ll 3’ II II 'J>'|| I PI! | O 13 22 34 ' ‘41 ‘ :23 = :31 = 52 :12 = 02 - 1 A44 = 32 - x2 Therefore, we can write equation (3.26) in the form: (- m 02-1 0 o o r U 2 o o o c2 202 U1 (3.27) = o. ,2 2 Q 20 o 0 v0 0 O QZ-XZ 0 Vl \— _J L .4 The determinant for the above matrix leads to 2 Q4[02-1][G2-x] = 0 which is an equation of order eight. Thus the roots will correspond to the lowest cut-off modes of uniform and linear amplitudes distributions. These are 01 2 = o a u(O) SYM a EXTENSIONAL Q = o a v(0) AN-SYM a TRANSVERSAL 3,4 (3.28) 05 6 = :_1 a u(l) AN-SYM s THICK-SHEAR 2 = :.x a v(l) SYM s THICK—STRETCH. Next using equation (3. 23) as the approximate equation of motion, then equation (3.24) for ’ = 1 leads to P -= = = = _1 r 1 312' 312' A13' A13 “'3 , i p A; p K (3.29) 22 22 23 23 A31' A31' ‘A34' A34 A41' A41' ‘B44' B44 L. a L J and represents the approximate Frequency Spectrum for N==l and [2,1] (the general truncation procedure). The cut-off frequency modes (wavenumber -—> O) which can be shown as (3.30) lim :.. and lim § and E . k+0 1] kao 12 44 But we have ii?) rB12 = A12] (3.31) 1im [IE = Z 1 I k-+O 44 44 therefore equations (3.26) and (3.29) become identical and the same cut-off modes can be generated. Consequently, for a general truncation of order'[N,1] and N = l, the exact cut-off frequency can also be derived. If is increased, then higher Zl (the number of subdomain) modes will be generated by this process. Next, we consider N = 2 (two subdomain), then ecuation (3.23) becomes 12 12 13 13 0 pl 0 i i uo ‘ "A22 A22 0 A23 A23 0 0 P2, u1 A31 A31 0 “A34 A34 0 93 O 3 u2 I - A41 A41 0 'B44 B44 0 O ID4 .. V0 (3.32) g _ ; =C). O ‘312 B12 0 A13 A13 Pl 0 L V1 0 "A22 A22 0 A23 A23 0 ‘92 V2 ‘ 0 A31 A31 0 'A34 A34 ‘93 0 C51 1 0 A41 A41 0 ‘B44 B44 O 1?’4 T1 Using a harmonic wave propagating along the x1 axis (equation 3.24) and considering the cut-off frequency modes (k 4 0), then the determinant of equation (3.32) can be written in the form 92—1 0 o o o 0 pl 0 ! 2 r i o o o O" 20 o 0 pl; * (32 292 o o o 0 pl 0 ' 2 o o o (I -x o o 0 p1 . (3.33) 2 i =0. 0 Q -1 o o o 0 p1 o : o o o o 02 202 0 -p1 . O (12 202 O O 0 -pl 0 60 This generates six cut-off frequency modes. Hence, a plate divided into two subdomains will give rise to two addi- tional modes corresponding to extensional and flexural motions respectively. Finally, we are going to study the Frequency Spectrum derived by the technique of this chapter. The approximate Frequency Spectrum for N = l is obtained by taking the determinant of equation (3.26) which is 41A12A23A34 41A13 22 34 + 44 31 12 23 - A {I’ll (3.34) det.E Z ‘ A44A31A13A22 = 0 ' The above coefficients are defined in equation (3.15). The dispersion curves of equation (3.34) are computed and shown in Figures (3.1). Hence, for N = 1, we generate the first and second dispersion curves of extensional and flexural motion. 3.5 Longitudinal Strain Problem Due to the symmetry in the xl,x3 coordinates, the displacements u1 = u3 = O and 112 = u (x2,t). Thus, 2 the stresses become 022 (X + 21301.12!2 G = G = 111 ll 33 2,2 and Hamilton's principle in section (3.2) becomes b ,1 a (3.35) l H T‘b )511n+1 oX(-b.t)] which represents the (n)-order displacements equation of motion. The transformation of coordinates (3.3) takes the form (3.39) x2 = x2 + Pn , Pn = 2n13 GRAPH TOP F P l n n > TYPICAL DOMAIN é BOTRJM Again, we shall proceed to analyze a plate of thick— ness ZbN with spacial variable x2 and time. In each subdomain we have a) Hamilton's principle (3.35) describing the motion. b) The truncation procedure of Chapter I as (n) v (t) = O, n > 1 where vén)(t) = v(n)(t). Setting the condition v(n) = O n > 1 in equation (3.37) and equation (3.38), we get (3.40) 6“” (t) + F(O) (t) = o G(l)(t) + cgaiv(l)(t) + F(l)(t) = o where F“) (t) %[ox(b.t) - oX(-b.t>] F”) (t) %[0X(b,t) + UX(—b,t)]. Similarly, equation (3.36) becomes __ (O) n (l) . L (3.41) vn(x,t) — v (t) + (-1) v (t) 8111 2b x where the transformation in equation (3.39) is used and n describes the displacements in the (n)-subdomain. The finite-point method at x2 = b reduces equation (3.41) to (3.42) v = v(O) + v v = v + v n+1 and the inversion of above equation is (O) _.L r (3.43) v — 2 Lvn+1 + vn] (1) _ 1 Similarly, the stress can be derived as follows: (3.44) F(l) (t) ll Uhd Substituting equations (3.43) and (3.44) in zero and first order displacement equations of motion (3.40) yields a set of differential-difference equations: 2 2 (3.45) £ [vn+l+vn] + b' [Gn+l-cn] - 0 2 7 .. 1 :-r ._ =- where 2 2 d v (3.46) i =-——— , c =‘—— c L2 = £2 + c2 Next, we investigate this dispersion relationship of harmonic waves governed by the Differential—Difference equation (3.45) by assuming a harmonic wave in time. (3.47) rv ,- j = :Vn’T 3e “ , then (3.45) becomes a Difference equation as follows: 2‘- 4 2 0 J1. i ' - _ r - .1 = (3 48) ‘ n+1 + an + ( l) b ’Tn+l Tn” O .2 1 n.2 _ _ Q [Vn+l an + (-1) b [Tn+l Tn] _ 0 where 02 = (12-C2) The Boundary conditions for the dispersion relationship require that both surfaces of the plate be traction free, i.e., 7 = (3.49) To 0 T_=O. N Three cases are going to be considered, a) N = l which corresponds to one region. b) N = 2 and c) N = 3. Case 1 (N = l) The Boundary condition (3.49) is reduced to (3.50) T = O T = 0 "FREE VIBRATION" and substituting the above in the Difference equation (3.46) yields (3 31) V1 + V0] = O 2 2 2 2 o .. 1 = = - .. [v1 VOJ o , O (1 c ) , c—Lc _ 2b L or fi 5 3:2 (1)2 V0 0 (3.52) = 422 (‘2 v1 0 The determinant of the above matrix is 2 (3.53) det.= m (m2-c2) = O. The roots will correspond to the first and second modes of vibrations. These are (3.54) '1) = O t. ll H- O 2 {3‘ (D '1 (D O I N U‘ 0 Case II (N = 2 The Boundary condition (3.49) is reduced to (3.55) T = O and substituting the equation (3.47) in the Difference equation (3.45) yields O\ \l mzr * _ (3.56) b- (v1 + v0) - 2 T1 — O .2 . _ b1 [V2 + V1) + 2 T1 — O bOZFV - v ] - 2 T = 0 L l O l bazrv - v 1 - 2 T = 0 '~ 2 1-‘ 1 or F" r- 12 12 o -17 bV 2 o O m2 m2 1 IJVl (3.57) = O . 412 :22 o -1 bv2 o 4:2 <12 -1 221 - d b .4 The determinant of the above matrix is det. = (1)2(1132-C2)(2’1‘-2-C2) = o. The roots: (3.58) w 0 12 U"34 =ic correspond to the first and second modes of vibrations and . = (E (3.59) $56 :_2 is a damping frequency due to the non-homogeneous term of equation (3.48). 68 Case III (N = 3) The B.C. (3.49) is reduced to (3.60) T = O and substituting equation (3.56) in equation (3.46) gives F 7 («.2 .2 o o -1 o 7 b v o o (112 1:2 o 1 —1 b v1 0 0 <1)2 3:2 O —1 b v2 (3.61) = —C.2 (12 O O -l O b v3 0 -92 0.2 o -1 —1 2 T1 0 o .42 <22 0 -1 2 T2 L. 4 L. .J The determinant of the above is a polynomial of order eight and its roots represent the naturals and damping frequency of our system. Hence, the determinant is det. E w2(4w2-c2)(4w2-3c2)(m2-c2) = O. The lowest and highest roots are (3.61.1) w12 — m = + c c ='—— 34—' and correspond to the first and second modes of vibra- tions. The remaining roots are (3.61.2) 156 = i c/2 u / .1678 i'v EC/Z o They are the damping frequency modes. CHAPTER IV TRANSIENT WAVES DUE TO IMPACT In this chapter impact boundary conditions are applied at the tOp of a plate. Free stresses are applied at the bottom of the plate. Similar conditions had been used by Kim-Moon [10] but in composite plates. The Differential- Difference equations obtained in Chapter III are reduced to difference equation using symbolic technique. A solution for this difference equation for various N (the number of subdomain) will be develOped under plain strain and longi- tudinal strain conditions. Then, application of this solution to the impact boundary conditions mentioned above all give rise to a set of P.D.E. involving the x1 and t variables. Various numerical techniques can be employed to solve these(P.D.E. such as transform techniques and computer application. 4.2 Plane Strain Transient Wave Due to Impact In section 1.7 of Chapter 1, we found the six P.D.E. needed for [1,1] and [2,1]. In Chapter III we reduced these six P.D.E. to a Differential—Difference equation. These two cases will be studied under impact conditions. In the case of [1,1] the Differential- Difference equations (3.14) are: 70 31 41 where (4.2) and (4.3) D 11 12 13 22 23 31 34 41 44 (O) (1) (O) (1) A13 Q 10(“n+1 A23 0 ; b(un+l 0 A34:. b X0 or t > to which is a transversal loading. The other stresses are (4.20) O- = TO = T- = O. N+l N+1 q (X11t) GRAPH: TOP 31 2b TYPIC A. SUBDOMAIN (F. 41) <% INFWCT LOADWKS BOTKNM y is Using solution (4.16) which is _ inr (4.21) [un,vn.on,¢n] _ [AlBlchl] + [A232c2D21e + LA3B3C3D3]e + [A4B4C4D4Je and apply the B.C. equation (4.19) to it, we get + 0 + O I — C:‘(X1.t) l 2 3 4 D]. + D2 + D3 + D4 =0 (4.22) _ 'F- 1 ' - '. _' ‘. C + C e1~N+1Jrr + C e1[N+l]b + C e 1[N+l]y = O l 2 3 4 .r- 1 cr- _- - ‘ D1 + DzeiLN+lJV + D381(N+1]w + D4e 1[N+l]) = O . Therefore, from equations (4.17) and (4.22) we have eight P.D.E. of infinite order with 8 unknown Ci(x1,t), Di(xl't) where i==l,2,3,4 which can be solved numerically. The coefficients of the displacements Ai(xl't)'Bi(xl't) where i==l,2,3,4 can be found by using equation (4.17). Next, in the case of N==2, the Differential Difference equations (3.17) are 0 B12 A13 0 ] (un+1+un) 0 A22 A23 0 ' (”n+1 ” un) A31 0 0 A14 (Vn+l + Vn) A41 0 0 B44 (Vn+1 ‘ Vn) (4.23) - —- .— - pl’ 0 O O F(l) (o) + o 1/‘b o o H = o o o 1/b o F (O) (1) o o 0 p4 H where C1 + C2 + C3 + C4 = q(xl,t) D1 + D2 + D3 + D4 = O (4.22) _ .r_ . '- '.| -. ‘. C + C e1_N+l]rr + C e1[N+l]) + C e 1[N+l]y = O l 2 3 4 .r— 1 .r- -. - .' D1 + D2e1LN+lJV + D3e1LN+l]w + D4e 1[N+l]) = O . Therefore, from equations (4.17) and (4.22) we have eight P.D.E. of infinite order with 8 unknown Ci(x1,t), Di(x1,t) where '==l,2,3,4 which can be solved numerically. The coefficients of the displacements Ai(xl,t),Bi(xl,t) where i==l,2,3,4 can be found by using equation (4.17). Next, in the case of N==2, the Differential Difference equations (3.17) are 0 B12 A13 0 ‘1 (un+l+un) 0 A22 A23 0 ’ (”n+1 " un) A31 0 0 A14 (Vn+l + Vn) A41 0 0 B44 (Vn+l ' V ) ._ _ L - (4. 23) __ _ __ pl 0 O O F(l) (o) + o l/b o o H ___ o o o 1/b o F (O) (l) o o 0 p4 H where 2 2- _ r9v(1+ 211) -4(4L+ u) g L 2 B12 " f_ 9().+ 2u)rr‘ _D11+“(2b) ’PDtt (4.24) r 2 2 2 _ pficou 4(4414' ll) 1 , I. .. B44 ‘L ' 91m? _,D11'”+2L‘)(2b p Dtt and the A. can be computed from = _2; (“(+11) p1 Vb + 3 (1+21mr D1 (4.25) l/b+£.§§.fi-L)D p4 3 pr 1 If we replace A44 and A12 by B44 and B12 in equations (4.8) and (4.9), the solution (4.16) can be applied to the above problem. Therefore, Case I (N==l) an Case II (N==2) are identical except for the coefficients B44 and B12. 4.3 Longitudinal Strain Transient Wave due to Impact In Chapter 3, section 3.5 we set the symmetrical con- ditions with variables xl,x3 which reduced the three- dimensional problem to one spacial variable x2 and t, the time variable. The Differential—Difference equations (3.46) are £;[u owe o :3 + I..—I s n+l"un] + (4.26) 2 L[fimfl-un]+ Who where £2-43 - dt (4.27) 2 _ 2 2 _ ;L. L — £. + C , C 2b CL Let at behaves as a constant(Symbolic technique), then equation (4.26) becomes a Difference equation. We assume a solution in the form of i2n6 (4.28) [un,on] = [A,B]e , then equation (4.26) becomes F q ._ - £2 (1+-ei26) , -(l-e)i28) bA (4°29) -L2 (1-4129) , (l+e)i29) 213 =0. For A and B arbitraries the determinant must be set to zero, which yields (4.30) det E (e126 - eIBHe129 - e-ia) = 0 where 2 (4.31) cos a = ("'12‘ 33+ 1). c dt The roots of equation (4.30) are 9 = 8/2 -B/2. CD ll Therefore, equation (4.28) has two independent solutions and by superposition, we get __ in6 1 “i116 (4.32) [un.0n] — [A1,Bl]e + [A2.lee . 80 Now, we let the Boundary conditions be 3' -p(t) t 2 0 (4.33) c = 0 t < 0 The figure below will have a better representation of the B.C. which give rise to a transient wave. FIGURE: pCt) TOP 1) I) l 1 ‘WF’ CF42) Pn TYPICAL SUBDOMAI {in BOTRNM Recall solution (4.32) which is (4 34) [u o ] = [A B leinB + (A B ]e‘inB ' n' n l' 14 - 2’ 2 1 d2 where cos B = (E—- -§-+ 1). Thus, if we apply the B.C. 2 (it (4.33) to the above equation yields Bl + B2 = -p(t) (4.35) _ i'fi )3 -i[N 15 _ Ble + Bze — 0 or B1 = “[32+P(t)] (4.36) [1-cos 2(fi )B]Bz(t) = -p(t) The above equation gives us two infinite 0.D E. depend- ing on N, the number of subdomain. Next, we will consider two cases N: 1,2. Case I (N: 1) Using this condition, equation (4.36) becomes B]. = -[Bz +p(t)] (4.37) (l-cos 25)Bz(t) = -p(t) . Thus, substituting equation (4.31) in the above result, yields Elm = - [32w +p(t)] (4.38) 2 2 d l. d 1 ----(l + ----) B (t) =-p(t) dtz C2 dt 2 2 which is the O.D.E. needed to solve 82(t). From 82(t) we get Bl(t). Hence, the stress becomes (4.39) o = Bl(t)einB inB n + B2(t)e L0 N and the displacements can be obtained using equation (4.29). Case II (N = 1) Using this conditiond equation(4w36) becomes 81 = ~[BZ4-p(t)) (4.40) (1- 2 cos23)132(t) = — -12- p(t) and by substituting equation (4.31) in the above result, yields 4 2 [26‘4 137+ 462 37+ 1132(t) = p(t) dt dt (4.41) B]. = '[Bz+p(t)] which is the O.D.E. needed to solve B2(t). From B2(t) we get Bl(t)' Hence, the stress and displacement can be found using equations (4.29) and (4.36). BIBLIOGRAPHY 10. BIBLIOGRAPHY Achenbach, J.D., "Wave Propagation in Elastic Solids," North—Holland Publishing Company, Amsterdam— Oxford (1975). pp. 226-236. Bluman, Q.W. and Cole, J.D., "Similarity Methods for Differential Equations," Applied Mathematical Sciences, Vol. 13, Springer-Verlag, New York - Heidelberg - Berlin (1974), p. 15-26. Cauchy, A.L., "Sur l'Equilibre et le Mouvement d'une Plaque Solide," Exercises g§_Mathematique, Vol. 3 (1828). PP- 328-355. 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Kirchhoff, G., "Uber das gleichgewicht und die Bennegung einer Elastichen Scheibe," Crelles Journal, Vol. 40 (1850), pp. 51-88. Love, A.Z.H., "A Treatise on the Mathematical Theory of Elasticity," Dover Publications, New York (1944) p. 286-309. Lure'e, A.I., "Three-Dimensional Problems of the Theory of Elasticity," Interscience Publishers, New York - London - Sydney (1964), pp. 148-199. Lee, P.Y.C. and Nikodem, Z., "An Approximate Theory for High Frequency Vibrations of Elastic Plates," International Journal of Solids and Structures, Vol. 2 (1966). PP. 581-612. Meirovitch, C., "Analytical Methods in Vibrations," The Macmillan Company, New York (1967). Mindlin, R.D., "An Introduction to the Mathematical Theory of Vibrations of Elastic Plates," U.S. Army Signal Corps Engineering Laboratories, Fort Monmouth, New Jersey (1955), p. 304. Mindlin, R.D., "Influence of Rotary Inertia and Shear on Flexural Vibrations of IsotrOpic Elastic Plates," Journal oprpplied Mechanics, Vol. 18 (1951) pp. 31-38. Mindlin, R.D. and Medick, M.A., "Extensional Vibra- tions of Elastic Plates," Journal of Applied Mechanics, Vol. 26 (1959), pp. 561-569. Mindlin, R.D., "High Frequency Vibrations of Crystal Plates," anrterly Applied Mathematics, Vol. 19 (1961) pp. 51-61. Moon, F.C., "One-Dimensional Transient Waves in AnisotroPic Plates," Journal of Applied Mechanics, Vol. 40 (June 1973), pp. 485-490. Moon, F.C., "Wave Surfaces Due to Impact on Aniso- trOpic Plates," Journal of Composite Materials, Vol. 6 (1972). Pp. 62-79. Morse, P.M., "Vibration and Sound," McGraw-Hill Book Company, Inc., New York (1948). 23. 24. 25. 26. A.- ) "\ U a Poisson, S.D., "Memoires’sur 1'Equilibre et 1e Meuvement des corps elastiques," Mémoires de l'Academie des Sciences, Series 2, Vol. 81 Parie (1829) pp. 357-370. 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