.mI-m-‘n—v—rv—ArV-_ 4 , I -1 ”-7". 1:. Ik4r,-.gu . V 1"" .,-, w .‘..‘. .. , I, _ “hrizdhrruthwiné. . In \.‘.."‘ H , ., ’ . , I... I'...-““,':‘.-,,' 5‘ ul‘,,‘." ."' I ’ ‘. ‘1 | .‘v, ., , HARMONIC DISPERSION ANALYSIS OF INCREMENTAL WAVES IN UNIAXIALLY PRESTRESSED PLASTIC AND VISCO-PLASTIC BARS, PLATES AND UNBOUNDED MEDIA Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY ROY A. WILLIAMS 1967 Thlshtocettifgtlntthe thesis entitled HARMONIC DISPERSION ANALYSIS OF INCREMENTAL WAVES UNIAXIALLY PRESTRESSED PLASTIC AND VISCO-PLASTIC BARS, PLATES, AND UNBOUNDED MEDIA presented by Roy A. Williams has been accepted towards fulfillment of the requirements for _Eh.D.._degree mJechanics LIBRARY Michigan State University IN Major professor 0-169 . ABSTRACT HARMONIC DISPERSION ANALYSIS OF INCREMENTAL WAVES IN UNIAXIALLY PR ESTRESSED PLASTIC AND VISCO-PLASTIC BARS, PLATES, AND UNBOUNDED MEDIA by Roy A. Williams In this investigation the equations governing the propagation of infinitesimal incremental loading stress waves in a medium prestressed uniaxially into the plastic region are derived for a strain-rate -independent and a strain-rate -dependent constitutive equation. With the assumption that a loading wave exists in the medium and that such a loading wave could be constructed with the use of generalized Fourier analysis and synthesis, the equations derived were applied to a harmonic analysis of the following three problems: (a) Plane waves in an unbounded medium (b) Longitudinal waves in an infinitely long cylindrical rod (c) Symmetrical longitudinal waves of plane strain in an infinitely long and infinitely wide thin flat plate. For each of the above problems the frequency equation relating wave speed to the frequency of the wave is found and solved by numerical methods and the dispersion curves for the two theories are constructed and compared with the corresponding dispersion curves for the corresponding elastic problem. The results showed that the dispersion curves for the rate- dependent theory have a maximum phase velocity very close to the ROY A. WILLIAMS elastic bar-wave velocity co = W for the bar, and close to the elastic plate longitudinal wave velocity CL = cO/N/l - 1/2 for the plate, while for the rate-independent theory the maximum phase velocities were of the order of half these values. This means that if the assumptions above are valid, the leading edge of a Fourier synthesized pulse constructed according to the rate-independent theory could not travel at the elastic wave speed as observed in experiments. However, the leading edge of a pulse constructed according to the rate-dependent theory could travel with velocities very near the elastic velocities. Also the rate-dependent theory predicts that high frequency longitudinal waves in an unbounded medium travel at velocities very near the elastic dilatation wave speed cl = [(x + 2G)/p]1/Z Associated with the rate-dependent theory is a geometric damping factor which in each of the three problems appears to approach a non-zero finite value at high frequencies. The high— frequency asymptotic value was obtained exactly for the unbounded r edium. HARMONIC DISPERSION ANALYSIS OF INCREMENTAL WAVES IN UNIAXIALLY PR ESTRESSED PLASTIC AND VISCO-PLASTIC BARS, PLATES, AND UNBOUNDED MEDIA BY 4%. R oy A0 Williams A T HESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOC TOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Materials Science 1967 ACKNOWLEDGMENTS The author expresses his appreciation to Professor L. E. Malvern who suggested this problem for help in formulation and solution of this problem, to Professors R. W. Little and D. H. Y. Yen for suggestions which contributed to the solution, to Professors G. E. Mase and W. A. Bradley for serving on the Guidance Committee, and to his wife for typing the first draft from barely legible notes . ii TABLE OF CONTENTS AC KNOW LEDGMENTS LIST OF FIGURES Chapter I. II. INTRODUCTION ................ 1.1. Purpose, and Summary of Investigation Methods . . ........ 1.2. Background ............. DERIVATION OF EQUATIONS FOR THREE DIMENSIONAL ANALYSIS OF INFINITESIMAL INCREMENTAL WAVES ............ 2.1. General Principles and Assumptions . . . Z. Z. Constitutive Equations . ......... ' 2. 2. 1. Strain-Rate -Independent Formulation ............ 2. Z. 2. Strain -Rate -Dependent Formulation ............ 2. 3. Harmonic Analysis - Constitutive Equations ................ 2.. 4. Wave Speeds of Plane Waves Propagating in an Unbounded Medium. ...... . 2. 4. 1. Strain -R ate -Independent Formulation ........... 2. 4. 2. Strain-R ate -Dependent Formulation - Harmonic Analysis 2. 5. Cylindrical Bar: Longitudinal Waves in Prestressed Direction . . . ..... 2. 5. 1. Strain-Rate ~1ndependent Formulation. . . . ..... 2. 5. Z. Strain-Rate- Dependent Formulation ........... 2. 6. Frequency Equation for Cylindrical Rod 2. 7. Flat Plate. Longitudinal Waves in the Prestressed Direction ........ 2. 7. 1 . Strain -Rate -Independent Formulation ............ Z. 7. Z. Strain-Rate-Dependent Formulation . . . ......... 2. 8. Frequency Equation for Flat Plate iii Page ii p—a l4 14 16 16 20 24 27 27 31 31 34 38 41 41 43 45 III. NUMERICAL SOLUTION OF FREQUENCY EQUATIONS .................. 3.1. General Procedures and Properties of Solutions ................ 3. 2. Cylindrical Bar .............. 3.3. Flat Plate . . . .............. IV. RESULTS .................... 4.1 Dispersion Curves for Longitudinal Waves in an Unbounded Medium ...... 4. 2 Discussion of Dispersion Curves - Unbounded Medium ........ . 4. 3. Dispersion Curves for Longitudinal Waves in a Cylindrical Bar ........ 4. 4. Discussion of Dispersion Curves - Cylindrical Bar .............. 4. 5. Dispersion Curves for Symmetric Longitudinal Waves in a Flat Plate . . . . 4. 6. Discussion of Dispersion Curves - Flat Plate ......... . ....... 4. 7. Summary of Results and Conclusions . . BIBLIOGRAPHY ...................... iv Page 49 49 56 59 59 59 62. 63 75 79 80 84 LIST OF FIGURES Figure Page 1. 1 Schematic Representation of the Strain Distribution Due to Constant Velocity Impact in a Rate-Independent Material . . . . . ..... 7 1. 2 Strain Distribution (After Malvern) ....... 9 4.1 Dispersion Curves - Unbounded Medium: Phase Velocity of Longitudinal Waves for Rate- Dependent Constitutive Equations (1/ = 0. 29) . . . 60a 4. 2 Dispersion Curves - Unbounded Medium: Damping Factor for Rate-Dependent Constitutive Equations (1/ = 0. 29) .................... 60b 4. 3 Dispersion Curves - Cylindrical Rod: Phase Velocity of Extensional Waves for 1/ = 0.29 . . . 64 4. 4 Dispersion Curve - Cylindrical Rod: Damping- Factor for Rate-Dependent Constitutive Equations ([31 = 0.1, V = 0.29). . . . . ..... 65 4. 5 Dispersion Curves - Cylindrical Rod: Group Velocity of Extensional Waves for 1/ = 0.29 . . . 66 4. 6 Dispersion Curves - Cylindrical Rod: Comparison of Strain-Rate-Independent Phase Velocity Curve with Approximate Theory of Hunter and Johnson. 67 4. 7 Dispersion Curves - Cylindrical Rod: Comparison of Strain-Rate-Independent Group Velocity Curve with Approximate Theory of Hunter and Johnson . . .............. 68 4. 8 Dispersion Curves - Cylindrical Rod: Comparison of Strain-Rate-Dependent Phase Velocity Curve with Approximate Theory of Bejda and W'ie’rzbicki 69 4. 9 Dispersion Curves - Cylindrical Rod: Comparison of Strain-Rate-Dependent Group Velocity Curve with Approximate Theory of Bejda and Wierzbicki 70 Figure 4.10 Page Dispersion Curves - Flat Plate: Phase Velocity of Symmetrical Longitudinal Waves for 1/ = 0. 29 ......... . ...... 76 Dispersion Curves - Flat Plate: Damping Factor for Rate-Dependent Constitutive Equations((31=0.l,V=0.29).......... 77 Dispersion Curves - Flat Plate: Group Velocity of Symmetrical Longitudinal Waves for v = 0.29 ....... . ........ 78 vi I. INTRODUCTION 1. 1. Purpose, and Summary of Investigation Methods In the mechanics of solids, constitutive equations occupy a central position. The determination of the basic parameters and the formulation of constitutive equations which includes these parameters are essential to understand the mechanical behavior of the material. In the study of plastic-stress-wave propagation in metals, the propagation of stresses exceeding the elastic limit of the metal, neither the constitutive equations nor even the basic parameters have been determined with universal acceptability. The two most common one-dimensional plastic wave propagation theories are: (A) Strain-rate-independent theory of von Karman [ l, 2] , using the constitutive equation 0' = 0(6) (1. 1) (B) Strain-rate-dependent theory of Malvern [ 3, 4], using the constitutive equation 66 _ 80' E05? — 5E- +g(0’,€) (1'2) where E0 is the modulus of elasticity. Experimental results on cylindrical rods [10] indicate that although the strain-rate-independent theory accounts for many of the effects encountered, there is one important exception, the situation in which a cylindrical rod is prestressed plastically and then an incremental loading stress wave is propagated. The leading edge of this incremental wave is observed to pr0pagate at the elastic wave speed of the bar co = (EC/pII/Z (1.3) and not at the lower plastic wave speed predicted by the strain- rate -independent theory, namely 1 d 1/2 cp=(;a-:— (1.4) The strain-rate -dependent constitutive equation does predict the observed result. Neither of these one-dimensional theories considers lateral inertia. In studies of elastic waves it has been observed that lateral inertia has an important influence on propagation when the wave lengths involved are of the order of magnitude of five or ten times the bar diameter or smaller. In plastic-wave propagation such wave lengths are also to be expected. Thus to improve plastic wave theory lateral inertia effects must be included. The purpose of this investigation is to study incremental plastic -wave pr0pagation and to determine which of the two most common theories gives the best account of it. This investigation is to be accomplished in the following manner: The equations describing the propagation of infinitesimal incremental loading waves in a plastically prestressed medium are derived for the most common strain-rate -independent and strain-rate -dependent constitutive relations. We use harmonic analysis to represent these loading waves by their Fourier components. The component phase velocity then depends upon the frequency of the wave. As a result of the different propagation speeds of the different components, a pulse changes its shape as it propagates; this is the phenomenon of geometric dispersion. Since the Fourier components are sinusoidal in time, the individual components involve unloading behavior to which the governing equations of the rate -independent theory do not apply. Hence some justification is needed for the use of Fourier components and superposition to study the dispersion and change of pulse shape. We assume the following: (1) There exists a loading incremental wave in the medium. (2) This incremental wave can be analyzed by generalized Fourier analysis and synthesis. If these assumptions are correct,unloading of the individual components is irrelevant. To establish these assumptions we would have to synthesize the pulse and verify that 32 Z 0 during the propagation of the pulse, where l2 is the second invariant of the deviatoric stress tensor. This synthesis and verification have not yet been done. The rate-dependent formulation can be applied to a hypothetical situation in which we have (:12)0 > 0 and (J?)0 > k‘2 in the initial prestressed state. Then a sinusoidal component of the incremental wave would not represent unloading so long as its amplitude was sufficiently small that J2 > k‘2 even though we might have :12 negative. For the rate-dependent theory we make the same two assumptions (1) and (2) listed above for the rate -independent theory but now loading means only that J2 > k2 and does not require :12 > 0 . Three problems of small-amplitude incremental waves are investigated: (1) Plane waves propagating in a uniaxially-prestressed direction in an unbounded medium. (2) Longitudinal waves propagating in an axially-pre stressed bar. (3) Longitudinal waves propagating in a uniaxially-prestressed plate. The limitation to small-amplitude incremental waves in the three- dimensional analysis permits linearization by use of the equations derived for infinitesimal incremental waves. The results of these investigations are given in Chapter 4. Briefly summarized, these results are: The dispersion curves for the rate-dependent theory have a maximum phase velocity very close to the bar-wave velocity co = (E7; for the bar, and close to the plate longitudinal wave velocity CL = co/V 1-1/2 for the plate, while for the rate- independent theory the maximum phase velocities were of the order of half these values. This means that if the assumptions (1) and (2) listed above are valid, the leading edge of a Fourier- synthesized loading pulse could not travel at a speed near co in the bar, according to the rate -independent theory. According to the rate-dependent theory, however, the leading edge would travel at a speed near cO in agreement with the experimental evidence available. For infinitesimal plane waves of uniaxial strain in an unbounded medium the rate -dependent theory predicts a limiting high-frequency phase velocity equal to c1 = [( k +2G)/p]l/2 and finite nonzero damping at high frequencies. The equations for this three -dimensional analysis are derived in Chapter 2 and the numerical solution of the frequency equations is discussed in Chapter 3. All results, discussions and conclusions are given in Chapter 4. Before these analyses and results are presented, a more detailed account of the background of the present investigation is presented in Section 1. 2. 1. 2. Background The first study of stress waves with amplitudes greater than the proportional limit of a material was made by Donnell [ 5] . By using a bilinear stress-strain curve and a superposition technique, he investigated the propagation of longitudinal waves in a long slender bar. In the early 1940's the equations describing longitudinal plastic -stress-wave propagation were developed independently by Von Karman [ l, 2] in the United States, Taylor [6] in England, and Rakhmatulin [7] in Russia. Since that time many researchers have been interested in plastic -wave pr0pagation and many publications have appeared. Survey papers by Hopkins [ 8, 9] , Abramson, Plass, and Ripperger [ 10] , and Craggs [ 11] provide a comprehensive review of publications prior to 1965, and their bibliographies constitute an extensive bibliography on the subject of anelastic wave propagation. For this reason only the papers which are pertinent to the present investigation will be cited here. The analyses made by von Karman [ l, 2] in Lagrangian coordinates and Taylor [ 6] in Eulerian coordinates, which after a suitable transformation were identical, involved a constant- velocity tensile impact on an infinitely-long, uniform bar. Both analyses made the following assumptions: The stress is uniform across the cross section; lateral inertia effects are neglected; the relation between stress and strain in dynamic loading and in static loading is invariable. The results found for a stress-strain curve concave toward the strain axis, with a maximum slope of E can be interpreted as follows (using Lagrangian coordinates): O (a) for x > cot, 6x = 0 the disturbance has not reached this region. do x 2 ' < < _.... _—_ __ (b) for c t x cot, d6 p(t ) , 61 1/2 (c) for x< c't, 6 = 6 is constant, defined by v =5 (1- Q1) d6 1 l p d6 1/2 0 where c' = 1'5- 3% depends on the impact velocity vl . 6:6 1 This solution is shown in Figure 1.1. Experiments by Duwez and Clark [ 12] and Duwez and von Karman [ 2] were performed to verify (a) the existence of a plastic- wave front of a given amplitude, (b) the relationship between the maximum strain of the plastic-wave front and the impact velocity O L_______. _.____ Figure l. 1. Schematic Representation of the Strain Distribution Due to Constant Velocity Impact in a Rate-Independent Material. and (c) the distribution of plastic strain between the elastic and plastic wave fronts. The relationship between the amplitude of the plastic -wave front at the constant-strain plateau and the impact velocity was evident but there were major differences between the calculated and observed velocity of the plastic -wave front and in the distribution of plastic strain between the elastic and plastic fronts. It was suggested [ 12] that the discrepancies might be due to strain-rate effects in the material. Some discrepancies were in fact also found between predicted and observed constant-strain plateau amplitudes, which were also thought possibly due to strain-rate effects. However, later experiments by Johnson, Wood and Clark [ l3] and many experiments by Bell [ 14] have shown good agreement between the predicted and observed plateau amplitudes, indicating that the earlier discrepancies in strain-plateau amplitude may have been due to inaccuracies in the experimental measurements. In 1950 Malvern [ 3, 4] formulated a theory of wave propagation incluuing strain-rate effects in a strain-hardening material. Sokolovsky[ 15] had earlier given a plastic-wave analysis for a non-hardening material with a similar dependence of plastic strain rate on the excess of the current stress above the initial yield stress. The problem investigated by Malvern was a constant-velcmty impact with a linear-overstress rate- dependence function. A comparison of Malvern's calculated results from this theory with the rate-independent theory is shown on Figure l. 2. Adam 1 a. .NS n .S 45332 3:3 gonzo—EBB cambm Amofioné N .N .H SSE on ma OH m o I. w + _I O .. moo . .. woo . a ncoo. N “Gopnomoflnoumm .. N o>HSO / H / unopnomopfiu 3mm .. H 95.90 xwoo . ('ui/ 'ut) utexis 10 These calculations agree qualitatively with experiments in that the smaller strains are propagated faster than the non-rate theory predicted. But these calculations did not reveal the strain plateau near the impact end predicted by the rate -independent theory and confirmed by experiments. In later calculations with Malvern's rate-dependent theory, Mercado [l6] and Efron [17] found that the strain plateau appeared if the pulse was long enough for the material to reach equilibrium, for constant stress or constant velocity imposed at X = 0 . Another important consequence of the rate-independent theory, pointed out by Rubin [ 18] , was that incremental loading- waves in plastically-prestressed rods traveled with the elastic bar-wave speed and not with the plastic-wave speed predicted by rate -independent t eory. This result had already been found experimentally by Bell [ 19] in tests made on steel bars and later by Sternglass and Stuart [20] with copper and Alter and Curtis [21] with lead. A study by Bell and Stein [22] of incremental loading waves in dynamically prestressed aluminum bars indicated that only the initial portion of the incremental pulse traveled at the elastic wave speed. The remainder of the pulse appeared to propagate at the plastic wave speed expected from the rate-independent theory. According to Bell and Stein [ 22] this behavior of the initial elastic wave is due to a trigger mechanism which is required to re-inaugurate plastic deformation after it has ceased. This has the effect of putting steps into the stress-strain diagram ll of aluminum, with the rising portions of the curve (between steps) having the slope of the elastic modulus, and causing a small pulse to propagate at the elastic wave speed. This idea has been further explored by Dillon [ 23] and Kenig and Dillon [ 24] . Some additional theoretical studies of the inclusion of rate dependence have investigated more realistic strain-rate functions than the linear overstress theory. Plass [25] compared the linear strain-rate law of Malvern with an exponential-law and found no essential difference in the effects. Efron [17] made a study with a power-law strain-rate function and compared results with an experimental study. The first theoretical study concerned with the prepagation of one -dimensional, incremental, longitudinal waves in a pre- stressed rod was made by Rubin [ l8] , assuming a rate-dependent material similar to that of Malvern. J. W. Craggs [11, 26] studied the propagation of infinitesimal plane waves in an unbounded medium of a non-strain-rate-ser.sit1ve material and found that waves of two types may exist, each in general involving both dilatation and shear strain; he gave an equation describing the velocity of each wave. Hunter and Johnson I 27] investigated approximately the effect of geometric dispersmn of axisymmetric, longitudinal, infinitesimal incremental waves in cylindrical rods by expanding the radial and axial displacements as power series in the radial coordinate and keeping only two terms of each series. Assuming that that the pulse can be synthesized in a generalized Fourier sense with an eigenfunction solution this analysis showed that 12 individual harmonic components of the incremental pulses may travel at phase and group velocities of the order of half the elastic velocity even when the plastic wave velocity vanished (231% = 0). This analysis employed the Levy-Mises incremental pla stic ity equation (16?. =dYUI. (1.5) 1.3 U with isotrOpic hardening and a universal constitutive relation of the form E : HI) dz?) (1.6) where F: (3/2 aijciij/Z, dc‘p = (2/3 defj defj)1/2 and erg). are the components of the deviatoric stress tensor. Hunter and Johnson noted that the eigenfunction solutions are of physical significance only in the context of a generalized Fourier synthesis for a loading pulse, since unloading is totally elastic. Bejda and Wierzbicki [ 28] made a similar approximate analysis using rate-sensitive constitutive equations of Perzyna [29] . For the case they considered, without hardening, the constitutive equation is the same as had been given earlier by Hohenemser and Prager [30] generalizing the one-dimensional formulation of Bingham [ 31] similar to that of Sokolovsky and Malvern. Their constitutive equations were; op _ 8F > sij - yet“ as,” for J12 k 13 (1.7) :0 fo k2. (c) Since the variation from the initial prestressed state is small, derivatives with respect to state variables in the neighborhood of the prestressed state are considered constant. This is consistent with the linearization of the incremental response. (d) Linearization of the constitutive equations may be accomplished by expanding certain functions in power series about the initial prestressed state and retaining only linear terms. Throughout this three-dimensional study two assumptions are essential. They are as stated in Chapter 1: (1) There exists an incremental loading wave in the material. l6 (2) This wave can be analyzed by generalized Fourier analysis and synthesis. (See Section 1. 4) Under these assumptions the leading-edge signal velocity for the incremental pulse would not be greater than the maximum phase velocity (or possibly group velocity) of the harmonic components of the pulse. The linearized constitutive equations for incremental loading are presented in Section 2. 2 and the harmonic analysis in Section 2.3. 2. 2. Constitutive Equations 2. 2.1. Strain-R ate-Independent Formulation P The equations used for the incremental plastic strain deij in the non-rate-dependent-theory are the Levy-Mises equations def]. = dioi'j (2.6) which imply that d€§j are deviatoric. With isotropic hardening and a universal stress-strain relation of the form 6? = 11(5 (1??) (2.7) see Hill[34] p. 30, where - 1/2 —p p p 1/2 .. l l _ 0' - (3/2 0'13. O'ij) , d6 — (2/3 deij deij) . The multiplier d: in equation (2. 6) is given by _ 21.2.3 _ . d _ 3/2 dt 9 de a, do'km (2 8) dt — - _ —2 - km dt ° 0' 4 0' do This formulation is equivalent to the plastic work formulation of l7 Craggs [ ll, 26] . For uniaxial prestress 0'11 and small incremental stresses Aaij we have 0.. = 0'11 6.. + AU If we neglect Acrij 11 11 i1' ' in comparison to 0'll whenever added, then _ (2. 9) 0' = I 0' Also if 0'11 )4 0 and Erij (1)5 j) are not an order of magnitude greater than 0'll we may neglect 2(A0'12 0'12 + A013 013 + A023 0'23) 1n 0 .' compar1son to 0'11 011 . Then 0" 0' = 0‘ 0" (2.10) _ _ 5-! Let (3 _ 3/2 ‘16 Then 9— = 313—11— (2.11) — dt 2 0' d0 11 where the superposed dot denotes a time derivative. For infinitesimal incremental waves, 0 will be considered constant. The constitutive relations (2.12) are obtained by substituting equations (2. 4) and (2. 6) into (2. 3) and using equations (2. 8) through (2.11). The resulting linear differential equations in component form are: 1 - 1 - _ l __ l — E11 ‘ F3"1+2c. “11+9K Gkk . — E '1 _'1 .1... 622 '2 “11+2G“22+9K “kk e = -1: .1 .14.. +3.. (“2’ 33 2 11 2c; 33 9x kk 18 Let the initial prestressed state be 6% , a"; , u: . The incremental stresses and strains E. and E... are defined by 3'... = 0.. - 0'9. , 1J 1J 1J 1J 1J — _ o _ 8 _ o . . eij - eij - €ij — _8x. (ui ui) . Equations (2.12) may be integrated J explicitly (for constant [3) to give incremental strain in terms of incremental stress. The assumed constancy of B restricts the results to small increments. Then ql ;q| e :13; +751—kk L 12G l I 11 1 + 11 - _ 11—. 1—. 1- 6 - - 0' +—0'2 +—-0'k 22 2 11 2G 9K (2.13) _ - “Q —1 _L_I +._1__: 633‘ 2011+2GG33 +K9 “k — _ i— . . eij — ZGUij (1)53). Equations (2.13) may be inverted to give incremental stress in terms of incremental strain. Addition of the first three equations (2.13) yields O'kk : 3KA (2.14) where Z is the incremental dilatation. The first equation (2.13) then yields (B +2153) “’11 - E11 - 3; (2.15) whence, since 0'11 6:11 - KZ , _ _ 1 _ 1 _ “11 " (K '3{;3+(1/2c;)]1)A + [3+(1/ZG) €11 Let _ 13 - é 19 where x is the Lamé elastic constant,to obtain the first equation (2. 17) for 0'11. is eliminated by using (2.15). The next two equations are obtained similarly -1 after 0'11 C) II (x + 26¢)Z + (26 - 66¢) E11 11 I? = (x -G¢)Z +20? +3G¢€ 22 22 11 (2’17) 0'3,3 = (X - G¢)A +2G€33 +3G¢ €11 lj = ZGEij (i/éJ) The dependence of (b on the prestress state can be seen by writing it as 4, z 1/3 _ (2.18) 1+2§1+v) [I do] 3 E - 0 d6 where v is Poisson's ratio, E0 is Young's modulus. For moderate plastic prestrain dF/de—p is small compared to E0, and therefore d> - 1/3 very quickly as the prestrain increases. On the other hand equations (2.17) reduce to the elastic equations for ¢ = O . If the stress-strain curve has a continuous tangent at yield, B is zero at yield, and Ch varies from zero at yield to approach the value 1/3. Before considering the harmonic analysis with these equations, the incremental constitutive equations for the rate -dependent theory are derived in Section 2. 2. 2. We will see that in the rate-dependent case the inversion analogous to that leading to equation (2.17) cannot be performed in general. But in the harmonic analysis an operator 20 form of the inverted equations is obtained, equation (2. 35) of Section 2. 3, which is formally the same for both the rate-dependent and the rate -independent theories. 2. 2. 2. Strain-Rate-Dependent Formulation For the strain-rate -dependent theory the equations we employ to represent the plastic strains are the Hohenemser-Prager [30] equations . 13 ° aoij (2.19) where k is a parameter determined by the amount of pre-strain hardening, _ ___Z_ _ 11 F _ k -1 - k -1 (2.20) W (é-crij 0".)1/2 and Y0 is a function of the material. Since 1 (7.. F3: 2 4%{1-_k_} (2.21) ij 2k ~sz 65 is deviatoric. Note that for this viscoplastic type of rate dependence, the material is assumed to be governed by the plasticity equation (2.19), even when 312 is negative, provided that Nsz is greater than the static equilibrium value of k corresponding to the current condition of the material. In the linearization we will treat k as a constant, equal to its value in the initial prestressed state. If the initial prestressed state is not chosen to be the state of first yield, this value of k will reflect 21 the hardening produced by the prestrain, which we assume to depend only on the prestrain and not on the rate of prestrain. The initial state may be one of plastic loading slow enough to avoid inertia effects, but rapid enough that (V212) 0* exceeds k by a finite amount. Bejda and Wierzbicki [ 28] pr0posed to analyze such an initial loading state in order to justify their harmonic analysis of the linearized incremental wave. An incremental sinusoidal plastic wave would then not imply unloading during part of the cycle, provided the current value of \[JZ remained greater than k . Differentiating equations (2. 3) and (2. 4) with respect to time and using equations (2.19), (2. 20), and (2. 21), we obtain . ' v eij = 215&ij+3ééij+—92_ O'i'j{1- L} , JJ2>1< (2.22) 2k ~sz We linearize equation (2. 22) by expanding F in a power series about the initial prestress 22L.= 0 o _ . . 011 (Uij - 0 14241) and also take 80 (F) This is the same linearization process as was ij ij used by Bejda and Wierzbicki [ 28] . «f «f 0 .3: >3 NJZ }° 0 0 Z 2 2 J2 —( J2) +1:1 _1 {3:17— (O'ij - O'ij) + 0(0'ij - O'ij) ( . 3) NJ ' 0 2 11 0 ll 0' Z 0 ( ) s (Kl—J) : — s 0' — O- 9 30'1J Z'foCZ) 2 3 ll 3 ll 00':o~°'=-1—e° o°'=01£' (224) 22 33 3 11 ij J ° Superscript zero denotes the initial prestressed state of the material. 22 From equations (2. 23) and (2. 24) we get, for the current state, the linearization 3 JJ =-—1- {20~ -0' -0' }=’\/-:'—O" , (2.25) 2 243 ll 22 33 2 11 hence from equation (2. 20) we get «[3 0" 20' .. 0' - o' F: 71:14-1: 11 22 33-1. (2.26) 2k~f3 Since 8F : 1 8F : 8F = _ l 8"11 J3 k 80'22 8"33 2J3 k and 1, BF _ > 817.. — O 1’4] 1.] the linearized equations relating loading stresses and strains in the current state are for ~sz > k ' v 20' -O' -0' é :_1__&,+é+ 11 22 33_1} 11 26 11 3 \[3k 2k~f3 é : _1_ 6' +9 _ Yo 2611'022'033 -1} 27‘ ZG 22 3 2J3k 2k~f3 (2.27) :5 :_1_U, +é_ Yo 2C’11"’22"°’33 _1} 33 ZG 33’ 3 2J3k 2k\(3 1 eij ‘ 2‘6 “ij ”£3 23 Similar equations result for the initial state '60 _ -1_ ll — 2G '60 - _1_.. 22 - 2G ‘0 _ _1_ E33 ‘ 26 where F0 and AC .0 Y O. . l efl +39 + #— ii. -1} «[3k «f3k '0' AC Yo 11 “22+? ‘ {— ”1} 2J3k «Bk 0 A0 y cr “33%“ 0 {—11'1} 2~f3k J3 0’0 z 11 _1 «f3k = éo (the initial dilatation rate) (2. 28) By subtracting equations (2. 28) from (2. 27) and defining incremental variables (2.29) we obtain the linearized equations relating incremental loading stres s and strain €11 t22 33 ii — _L..—l g 3.9—1 “26‘11+3+k"11 : _1__;l +§_:2.;l 2c; 22 3 2k 11 — ._l__.—'I i; :9 _1 “2c; 33+3'2k “1 — _L.— - 2G 1j 1%] (2. 30) 24 where 0.0 = yO/Zk The equations (2. 30) cannot in general be inverted to give stress explicitly as a function of strain. However for this investigation we have assumed the possibility of harmonic analysis; hence we invert these equations using a linear operator method (section 2. 3) and apply these inverted equations to study plane waves prepagating in an unbounded medium (section 2. 4), longitudinal waves in cylindrical rod (section 2. 5), and longitudinal waves in flat plates (section 2. 7). 2. 3. Harmonic Analysis - Constitutive Equations To invert the rate -dependent constitutive equations using harmonic analysis we let x1 be the direction of the initial prestress and let C'ij = Tij exp [i(pt - yx1)] 61,- = eij exp[i5, 0 o 733‘{K'3—G' 3k }A'+ZG 633+ k e11 Tij 2 2G eij 1,143 2 , (1060 Since A = K -3— G (Lame Constant), we let <1) = 3k in(2.34) to obtain in operator form the inverted infinitesimal incremental constitutive equations for the strain-rate-dependent theory. 26 711:0. +2G<1>}A*+{2G--6G<)>}e11 722 = {1 -G¢}A*+ZGeZZ+3G¢ e11 (2.35) 733 = {)1 -G<)>}A*+ZG e33+3G¢ ell Tij 2 2G eij ifiJ where ch = 1113 k = 1(3 2 . (2.36) 1+2G0 l+-1—P—k— o GYO Note that the operator form of the constitutive equations, equations (2. 35), is also applicable for the rate -independent formulation and that in fact these equations have the same form as the actual incremental stress-strain relations of the rate- independent formulation equations (2.17). The, only difference between the operator forms of the constitutive relations in the two formulations is in the definition of the parameter (P appearing in them. For the rate-dependent case (I) is given by equation (2. 36); note that it is a complex number, which depends on the circular frequency p as well as on the physical constants G and v and on the state of hardening in the initial state through the value of k. For the rate-independent case Cb is real, given by equation (2.18), and depends only on the initial state through dF/de—p and on the shear modulus 2(l+I/)/E0 . The harmonic solutions for the two cases will also differ in that the wave number Y of equation (2. 31) is considered to be a complex number y = Y1 - iyz in the rate-dependent formulation. 27 Its imaginary part leads to a damping factor of the form exp [- Y2 x1] in all of the harmonic solutions for the rate- dependent case. It turns out that no harmonic analysis is required to determine the wave speed for infinitesimal incremental plane waves in a uniaxially prestressed unbounded medium, according to the rate -independent theory (section 2. 4.1). But for all the other wave speed equations obtained in sections 2. 4. 2 through 2. 8 harmonic analysis is used. 2. 4. Wave Speeds of Plane Waves Propagating in an Unbounded Medium 2. 4. 1. Strain-Rate -Independent Formulation We consider only the case of an incremental plane wave propagating in the x-direction in a medium under uniform uniaxial prestress 6:3: . With all stress and displacement components independent of y and z, and Z = ——X (2.37) equations (2.17) yield _ aux O"XX = (h + 2G - 4G¢) 532-" _ _ 35X 0FYY : 0—zz : (K + 26¢) 8x (2. 38) _ all _ _ 63 0' = G —-X , 0' = 0, 0' = G 2 xy 8x yz xz 8x The three rectangular Cartesian component equations of motion (2.1) then yield the following three uncoupled wave equations for the displac ement c omponents . 28 azu azu (X+2G-4G<)>) ,z.=p__..£ a 2 8t2 X (2. azu azu €3qu azuz G —X = 9...}. , G = 9 8x2 at2 8x2 8t2 39) Hence, infinitesimal incremental plane waves of transverse displacement prepagate at the elastic shear wave speed c2 given by p c? = G (2. while longitudinal incremental waves propagate at CL given by 2 ch=k+2G-4G¢. (2. When the prestress is just equal to the initial yield stress, then, if the stress-strain curve has a continuous tangent at yield, d'é’P/dF = o, whence ¢ = o by equations (2.11) and (2.16, and this reduces to the case of irrotational plane waves in an elastic solid 2 p cl = X + 2G . (2. On the other hand, for dE/dgp < < C , Gd) approaches G/3 very quickly and equation (2. 41) becomes very nearly pCiZX‘tg-GIK (Z. for moderate plastic prestrain. The results of this section can also be obtained by specializing the results of Craggs [ ll, 26] to the case of uniaxial prestress. 40) 41) 42) 43) Z9 2. 4. 2. Strain-Rate Dependent Formulation - Harmonic Analysis For an infinitesimal incremental plane wave propagating in the x-direction in a strain-rate-dependent unbounded medium with uniaxial prestress 0:}: , we employ the harmonic forms of the variables. Again,with all stress and displacement components independent of y and z and A>i< : - 1y Ux , where 51: Ui exp[i(pt -yx)], and Z = A95 exp[i(pt -Yx)] , equations (2. 35) yield 'TXX = -1“). +2c3-4<3<)>)Ux 7w = ’Tzz = -1y(>. +2G¢)Ux (2.45) 'r =-inU,T =0,'T =-iYGU . xy y yz xz z We substitute (2. 45) into the three rectangular Cartesian component equations of motion to obtain the following equations (2. 46) 2 [(1 +2G-4GC)>)—pB-2~] UX=O v 2 (G - p 2:) Uy=o (2.46) v ei (G‘P Y2) Ut:0 Hence we get for the transverse waves 2 2 p /Y = G/p and for the longitudinal wave (2. 47) 132/312: 35 {.1 +ZG-4G¢}, 30 but in this case we observe 2 L -.- {———£L——}Z where c = —P- . (2- 48) Y2 Y1 - 1Y2 Y1 From the second equation (2. 47) we get 1’1 ”2 7- 1 V'T} 3C2_4C2{ 1 } (2'49) 1 3 21+ip¢ where 2_x+zo Cz_;_3 andqp k 1_ p ' Z-p ’ -ZGao The phase velocity and damping factor are found from equation (2. 49) by solving for the real and the imaginary part of equation (2. 49). The solution of these algebraic equations yields 4 22 22 {-3—}2: 21(1-3'372) +P LP 1- . (:1 4 2 4 2 2 16 4 2 {/2 (1 '3'1724'PZLPZ) +[(1 -§n2+PZ¢Z) +7772!) 4‘2] where the plus sign is taken to ensure real phase velocities and (2. 50) 2222 '3-nzPLP (1-§n§+p¢ where D N 772: O g—a A plot of the phase velocity and the damping factor against frequency is. given in Figures 4. 1 and 4. 2. For the uniaxially prestressed unbounded medium explicit results were easy to obtain. But when the geometric dispersion in 31 the bar or plate is considered a transcendental frequency equation is obtained, requiring a numerical solution. 2. 5. Cylindrical Bar: Longitudinal Waves in Prestressed Direction 2. 5.1. Strain-Rate -Independent Formulation For the bar wave we take the prestressed xl direction of equations (2.17) as the axial z direction of a circular bar of radius a, while 622 and 633 denote err and 699 , physmal components in cylindrical coordinates. For a radially-symmetric wave all quantities are independent of 9 , and the transverse displacement component 36 vanishes. Then _ aiIr _ Er _ afiz _ air 332 err: 8r ’699 z-r—’€ zaz ’ 26rzzaz +8r (2°51) give the relevant incremental strains, where 111. and uz are radial and axial incremental displacements, respectively. Also __ Hz _ 1 air aIIz (rur)+az and we :2— {E— -fi—} (2.52) 09 3:1— 1‘ Q30.) *1 where {:39 is the non-vanishing rotation. The two surviving equations of motion in cylindrical coordinates are a; a; e -2; 62: rr + rz + rr 99 : p r Br 82 r atZ (2.53) _ 2— 80' a u 22 + l 5’— (r? ) - z 82 r 8r rz - 9 8t2 Equations (2.17), (2. 51), (2. 52) are substituted into (2. 53) to yield the displacement equations of motion 32 2- 2— 63 3:59 a uz a nr ()\+ZG-G<)>)-a-;-+2Gaz +3G¢arez=pat2 (2. 54) _ 2_ .. ”mew; 4.2.. -, .e.iz.‘:z._ a “z az‘rarrwe" 2‘92 Bz at The boundary conditions are ; z; :0 at r=a (2-55) rr I‘Z The classical treatment of elastic waves by Pochhammer (1876) employed the same system of equations with 4) : O, combining these equations in two different ways to give one equation for Z and one for we , with solutions of the form 3: GJo(h'r) , we 2 T-IJl(k'r) where G and I? are functions of z and t and . 2 2 2 h'2 =[pp2/(1 + 26>]— v . k' = (p p /G> - v2. and 106:). mm are Bessel functions of order zero and one, respectively; see, for example, Kolsky [36] , pp. 54-65. For harmonic solutions"< of (2. 54) for a symmetrical longitudinal wave we assume incremental displacements, Er : U exp [1(pt - YZF] _. _ (2. 56) uz '2 i W exp[i(pt - yz}j where U, W are functions of r only, Y is the wave number, ZTT/Y is the wave length, p/Zn is frequency, c = p/Y is phase Recall that an individual Sinusoidal plastic wave has no meaning, because a different set of equations governs during the unloading part of the cycle. This type of analysis would require for its complete justification a demonstration that a loading pulse can be synthesized from the harmonic components. Then the fact that the individual components involve unloading would be irrelevant. Such a demonstration has not yet been made. 33 velocity. When equations (2. 56) are substituted into (2. 54) and (2. 55) the result is: Z dU ldU U dW dW (X+ZG-G¢){——Z-+;E;-—Z+YE'} “€1le U+Ydr} dr r dW 2 +3G¢YEr—--ppU (2.57) -(1 +2G+2u¢){Y dU+—)+y W} +G{Y(g— U+rH)} dZW l 2 +G{—-Z-—+——}+6Gct>y2W=-p pW r dr [(X-G¢)E-‘:-{ U+y-+ YYWW}+2G(—i-[-J+3G¢ W] =0 dr r=a dW _ [YU+dr]r:a_ We let r=aR and -2: 2_9_ 2 1+2G ”2‘61 ' Cz‘ p ' C1 p c -_} A>-< +2G err+3G¢ ezz (2.61) Tee : {x -G¢}A* +ZG eee +3G¢' e 22 'r =2Ge rz rz Note that these operator forms apply either to the rate -dependent or the rate -independent formulation. The only difference is in the definition of the parameter (b , given by equation (2. 36) for the rate -dependent case and by equation (2.18) for the rate- independent case. We substitute equations (2. 60) and (2. 61) into 36 the equations of motion in cylindrical coordinates (2. 53) to obtain the equation of motion (2. 62) Z dU ldU U dW‘ 2 dW {1 +2G-G<)>}{—z-dr +?_dr -—z-r +Y—dr } -G{Y U+Y—dr} dW 2 +3G¢¥F='P PU - {1 + 26+ 26¢}{y(§—§- +1.3) + YZW1+G{y(§§-+§-)} +G{%217-W+%g:—V}+6G¢YZW=-pzpw (2.62) and the boundary conditions [(1 - 66) {g—rI-J- +rH +y w} + 26-3":l + 3G¢YW]r=a = o [- Y U + ‘3‘}, r-a 0 Note that these have exactly the same form as equations (2. 57) for the rate-independent case. We again let r = aR and equations (2. 62) assume the non-dimensional form 22 22 d2U+ld_Ll+{(xno-Zn2)_l}U+ dRz fidR 1.4”): E7 2 2' 1+2¢’72"72 dW z{ 2 a—-.—o l-tn R 2 22 2 2 dZW 1dW {Xno-Z(l-4¢n2) d—R‘z"fi3§+ 2 V“ ’72 nz-Ztni-l dU U ”2 [(1- ¢n§>§§+ [1 - n§(2+¢)] -,‘%+ z [1 - 2n§<1~¢1W11R=1 = 0 37 where X, Z, 172, no are defined in equation (2. 58) of the preceding section. We note again that, except for the form of Cb equations (2. 63) and the strain-rate-independent equations (2. 59) are the same. Also, since in the strain-rate-dependent formulation ¢ is complex Y is taken as the complex wave number Y = Yl - iYZ and Y2 may be interpreted as a damping factor (see Section 2. 2. 2). The phase velocity c for the rate-dependent formulation is c = —L - 0 (2.64) In the rate-dependent theory <1) assumes the form 1/3 iX(1+V) (2'65) 381 .9. 1+ where v is Poisson's ratio and [31 is a function of the material parameters and the bar radius a Yo E0 ‘31 z 2 6 c k 0 Bejda and Wierzbicki [ 28] suggest that representative values of material constants for a mild steel bar of radius a = 1. 0 cm are: TABLE I E p k V a (31 6 3 2.1x10 7.8x10- 2x10 .3 148 0.1 .kg cm."2 kg cm-‘Itsec‘2 k g cm-2 sec 38 Since the differential equations and boundary conditions are the same in form as for the rate -independent theory we can consider their frequency equations by a single derivation in Section 2. 6. 2. 6. Frequency Equations for Cylindrical Rod The solution of the governing differential equations for the cylindrical rod subject to the boundary condition given is formally the same for the rate-independent case as for the rate-dependent case. The system of equations to solve is; 2 Sl—I-‘ZI+-}fi£1d—U—-+{C..LZ}U+D%¥{if = 0 dR R 2 (2.66) dW ldW dU U _ 7+§E§+EW+FlEfii+§}”O’ dR and the boundary conditions at R = 1 are dU U _ la—fi +m§+qW—O (2.67) dW_ -ZU+?§— O , where XZnZ-Z‘ZnZ 1+2¢nZ-n 2 2 2 C: , D=Z 1 ¢ 2 1 ¢ 2 ‘ ”2 ' "2 2.68) 2 2 2 2 2 ( Xn-Z(1-4¢n) n-2¢n -1 o 2 2 2 E: 9 F:Z{ 2 2 ”2 "2 1:1-¢n§, m:1-n§(2+¢), q=Z{1’Zn:(l'¢)} X, Z, 172 and no are defined in equation (2. 55), while equation (2.18) gives for the rate-independent theory 39 ¢ = 1/3 1 + 2(1+u) 1 d? 3 E -p 0 d6 and for the rate-dependent theory, equation (2. 65) gives 4) __ 1/3 — 1+iX(1+1/) 361 Guided by the Pochhammer-Chree elastic wave solution, we try as solution a linear combination of two solution vectors, of the form U hJ1(hR) k J1(kR) = C1 + C2 . (2.69) W B1J0(hR) BZ JO(kR) By substituting the first prOposed solution vector (U = h Jl(hR), W = B1J0(hR)) into the differential equation system we determine that this will be a solution provided that B1 = D = 7—— (2.70) where h is a root of the biquadratic equation 4 2 h -(C+E-DF)h +CE=O, while the second proposed solution vector is likewise a solution - k2 F k2 provided that B2 = C D = .2 _ where k is a root of the k - E biquadratic equation 4 2 k -(C+E-DF)k +CE=O. (2.71) Since h and k each satisfy the same biquadratic we arbitrarily 40 take h2 = C+EZ‘DF -%—J(C+E-DF)Z-4EC e (2.72) k2 = C+E'DF+1§flc+F-DF)Z-4Fc and for h and k we take the principal square roots of the right sides of equations (2. 72). Although the four roots of equation (2. 70) are d: h and :t k, we note that since the solution vectors are even functions of h and k the roots -h and - k add nothing to the solution. For distinct values of the principal square roots h, k, the two solution vectors are linearly independent and the complete solution of equations (2. 66) is given by equation (2. 68) as U=C hJ(hR)+C kJ (RR) 1 l 2 1 (2.73) W: C1B1J0(hR)+ C2 B2 JO(kR) . We substitute equations (2. 73) into the boundary conditions (2. 67) to obtain a set of linear homogeneous algebraic equations (2. 74). clumz + qu>Jo

exey—p atz 42 where vl:§2—+?_Z2" 511:???" Fuzz—:31 3x 8y Y 85 85 (2'78) zé'lzz—y2E +5321, Z=EII+EZZ all other — =0 Substitution of the assumed displacements of equations (2. 76) into equations (2. 77) gives the first two equations (2. 80) below. The last two follow from the boundary conditions 0' 26-" =0 at =:i:a 2.79 W xv Y ( ) {pap - (1+ 26 - 4G¢)Y2} U + GU" - (1+ 6 + 2G¢)YV' = 0 (1+ 6 + 26¢)YU' + (p‘2 p - GYZ)V + (1+ 26 - 6¢)V" = o (2.80) [(1 +26¢)YU + (1+ 26 - C:¢)V']y==l=a = 0 I _ [U -vv] Foo—0. where primes denote derivatives with respect to y. We non- dimensionalize these equations (2. 80) by letting y = a Y to obtain 2 2 2 Z (x no -2 22(1-4:<1>n))U+n 51— U -Z(1+n (2<1>-1))‘-i—Y = o 2 2 dYZ Z dY (2.81) 2 2 U 2 Z Z d V -Z<1+n(2¢-1))9— -(Xno -Zzzn)V-(l-¢n) = o 2 dY Z 2 2 dY with boundary conditions at y =:I: l [Z{1+Zn22(¢-1)}U+(1-n¢) = o 2 Z diY Y==tl dU _ [dY - z V] - 0 'Yzil 43 where c -22 - 2 - __£ X—C ,Z—aY,CO—E/P, nz—C o l 2 G 2 1+ 26 Co (2'82) c : -— , : , :— 2 p l p 0 c1 and ¢ = 1/3 where V is Poisson's ratio. The solution of this system of equations yields the incremental displacements U and V as functions of Y. containing the parameters Y and c . After considering in Section 2. 7. 2 the comparable formulation for the rate-dependent material of Section 2. 2. 2 we obtain the frequency equation relating c and Y for the lowest mode in Section 2. 8. 2. 7. 2 Strain-R ate -Dependent Formulation For a flat plate prestressed in the x-direction, with strain- rate -dependent constitutive equation, the condition of plane strain is again assumed and the rectangular-Cartesian incremental displace- ments are equations (2. 83). E = 0 Z Ex = i U exp[i(pt - Yx)] (2.83) {I = V exp[i(pt - Yx)] Y 44 Assuming incremental stresses of the form Fij = Tij exp[i(pt - Yx)] the non-vanishing equations of motion take the linear operator form 87 _. ___x_y : _. 2 1 Y Txx + 3y i p p U 87' 2 (2.84) -iYTXy+—flay =-ppV. We substitute equations (2. 35) and (2. 83) into (2. 84) to obtain equation (2.85) 2 (pzp - (X+ 2G -4G¢)Y2} U+ Gg—g-(X+G+2G¢)Y% = 0 dv dU 2. 2 dZV (X+G+ZG¢)YE+(p p -GY )V+(X+2G- (NU-ET =0 (2.85) [(1+ 2G¢)YU +(1+ZG - G¢)d—Y dY yzia. dU _ [dy - vvlyio — 0 which have the same form as equations (2. 80) of the rate -independent formulation. Again let y = aY to obtain equations (2. 86) 2 {inz - 22(1-4¢n2)} U + n‘2 9-3-1 - z{1 +nZ(Z¢-1)} 5'1 = o o 2 2 2 2 dY dY 2 2 dU 22 22 2 d -z{1+n <2¢-1>}— -(x n -z n )V-(1-¢n )—V =0 2 dY o 2 2 dY2 (2.86) with boundary conditions at y : :1: l [z{1+2n§(<1>-1)}U+(1-n§¢)cog—oily:i1 = o dU _ dY " Z V]y=41‘ 0 where now 45 while the other variables are the same as for the rate-independent theory, except that Z = aY is considered to be a complex number in the rate -dependent case. Since the equations and boundary conditions are the same in form as for the rate-independent theory we can consider their frequency equations by a single derivation in Section 2. 8. 2. 8. Frequency Egpation for Flat Plate For the two types of constitutive relations given, the investigation of incremental longitudinal waves in an axially prestressed flat plate necessitates the solution of a system of two linear, second-order,homogeneous, constant-coefficient differential equations of the form 2 aU+bg—I—J +ed—V 0 (”2 dY (2.87) 2 Cg—g+dV+ed¥ = 0 dY subject to boundary conditions at Y = i 1 dV _ £U+md—'Y — 0 (2.88) dU _ a. -ZV — 0 where _ 2 2 2 2 a- x no-Z(1-4¢n2) 2 b — 172 c = -z[1+n:(2¢-1)] (2.89) 2 2 2 2 d " '(XT) “2772) 2 e - -(1-¢le) 46 h. H z[1 +2n§(¢-1>] 2 (2. 89) B We write equations (2. 87) in linear-differentia1-operator form as follows. Let L1: a+bD2 L2 = cD 3 = c1+en2 where then equations (2. 86) take the form LU+LV 1 2 0 (2.90) LZU+L3V H C) Since the differential operators are commutative, eliminate V by applying L3 to the first equation and L2 to the second and subtracting to obtain the first equation (2. 91) below. The second equation is obtained by eliminating U . {1.31. -LZL2} U = o 1 (2.91) {L213Z 4.11.3} v = 0 Since both U and V satisfy the same equation we let 4; represent either U or V to obtain {(eDz + d)(bD2 + a.) - c2132} 4, = 0 or (2.92 {(be 134+(ee +db -cz)Dz +ad}¢ = o 47 Hence U and V each have the form 11Y 12Y 13y 14Y ¢=C1e +C2e +C3e +C4e (2.93) where the X 's satisfy the biquadratic equation be X4+(ae +db-c2))\2+ad = 0 2 _j (2.94) 2 _1. c ..ae -db 'l‘./ 2 2 . 11,3_§{ be 4106 (c -ae-db) -4abde} We take 12 : - )‘l and X4 = - )13 and give the solutions in terms of hyperbolic functions as U = C1 Cosh le + C2 Sinh XIY + C3 Cosh X3Y + C4 SinhX3Y (Z 95) V = D1 Cosh )le + DZ Sinh XIY + D3 Cosh X3Y + D4 Sinh X3Y For waves symmetrical with respect to the mid plane, U is an even function of Y, and V is odd. Hence CZ=C4=D1=D3 = 0 (2.96) and F— T — U — l U Cosh XlY Cosh X3Y 7 LV B181nh )xl Y B3 Sinh X3Y Substituting each of the solution vectors (2. 97) into (2. 87) we determine 1) (a+b1§ Bl=-——;')q—— . B =-—';7\3——— (2.98) (a + bkz We substitute equations (2. 97) and (2. 98) into the boundary conditions (2. 88) to obtain a system of two linear homogeneous algebraic 48 equations (2. 99) m 2, m 2 . _ [I --C—(a+b11)] 665111161 +[1 -—C—(a+b13)] Cosh 1363 - o (2.99) [1 +l—(a+b12)] Sinh1 c +[1 +——Z—(a+b12)]smh1 = o 1 e1 1 1 1 3 c1 3 3C3 l 3 The condition for a non-trivial solution of this system of equations is found by setting the determinant of the coefficients of Cl and (3:3 equal to zero. This yields the frequency equation (2.100) relating the these velocity c = p/Y to the circular frequency p Tanh )x [ cl - m(a+b)\2)] [ck2 + Z(a+b)\2)] X 3 3 1 1 3 m = 2 2 2 '1— (2-100) 1 [cl - m(a+b11)] [e13 + Z(a+b13)] 1 where a, b,c, d, e, I, m are given in equations (2. 89) and )11, 13 are given in equations (2. 94). Solution of this frequency equation is considered in chapter 3. III. NUMERICAL SOLUTION OF FREQUENCY EQUATIONS 3.1. General Procedures and Properties of Solutions The solutions of the frequency equations for the cylindrical bar and flat plate (2. 75, 2.100) relating the phase velocity c to the circular frequency p for the lowest mode were accomplished numerically on a CDC 3600 digital computer. The method used to solve these nonlinear, transcendental equations was the Newton- Raphson iterative method (see Ralston [35] p. 373), which proceeds as follows: Let a complex function F (Z) = 0 be defined and analytic for Z in some neighborhood N (Z0) . Let F'(Z) )6 0 for Z in N(Zo) and let F(Z) = 0 have the solution S with S in N(ZO). We approximate F(Z) with a power series of quadratic accuracy about an estimate of the root (S '2 ZO) , 2 F(Z) — 0 — F(ZO) + (Z - 20) F (ZO) + 0(Z - 20) For Zofi S we iterate on Z to find S zi+1 = zi _ F(Zi)/F'(Zi) . (3.1) This procedure will converge for ZO sufficiently close to S . The computational procedure used to solve the frequency equations for Z as a function of X is: (a) Expressions for the first derivatives of the frequency equations with respect to Z were calculated. (b) The values of Z for X = 0 were determined exactly. 49 50 (c) Using the value Z from (b) as an initial estimate for equation (3.1) X is increased by an increment of 0. 01 and the Newton- Raphson iterative process is initiated. (d) This procedure is continued withthe solution Z of the previous X used as an initial estimate for the new X. (e) The convergence criterion used in this iterative process was 1+1 1, 7 lz -z <10' The only external computer program required for the above procedure is a subroutine to calculate Bessel functions of zero and first order for complex arguments. The subroutine used in this investigation was obtained from Professor S. B. Childs [38] through private communication. Also to facilitate the use of complex library functions available on the computer the hyperbolic tangent functions were written in the form Tanh (u) = - i Sin (ia)/Cos (16) Neither the cylindrical-rod equation nor the flat-plate equation were conveniently solved in the form given (2.75, 2.100), and both were reduced to more suitable forms. Even in the reduced form some problems were encountered. These problems were of two types: (a) In both the cylindrical rod and flat plate, for all wave lengths or frequencies, the frequency equation is satisfied for the phase velocity equal to the elastic shear wave velocity c2 2_G_2 22_22 c- —C2 or XnO-an This solution was also found and rejected as unwanted for the 51 Pochhammer-Chree elastic longitudinal wave solution; see Kolsky[36] p. 58. In the numerical solution when the variable phase velocity c neared c2 there was a tendency for the iteration process to be unstable. (b) For a few discrete values X* of the independent variable X in the frequency equation, the solution showed X1 = A3 (for the plate) or h = k (for the rod). This means that at these values of X* the two solution vectors are identical, and we do not have two independent solutions. In the neighborhood of these values X* the numerical solution was also unstable. The influence of the above rejected situations was reduced in the cylindrical-bar problem by factoring [(in: - Zzniflhz - k2)] from the frequency equation. But this was not done in the plate equation; instead the continuation of the plate dispersion curve was made in the following manner. When a value of X* was encountered in the calculation, the independent variable X was incremented past X* and the process worked backwards toward X* using as starting ZO values the results of calculations immediately before X’l< . A similar procedure was used to get past the points where c2 = G/p . The results of this process indicated that the dispersion curves were smooth. These values of X* will be labeled on the plate dispersion curves in Chapter 4. It should be noted here that no problems were encountered in the solution of the rate -dependent frequency equation for the plate. It has been suggested by Hunter and Johnson [27] and by Bejda and Wierzbicki [28] that the measurable quantity in experiments 52 on wave propagation is group velocity cg rather than the phase velocity. The group velocity is defined as the velocity with which a packet of waves is propagated. For the group velocity of the wave-packet produced by superposing several infinitely-long trains of sinusoidal waves of wave lengths not differing much from their mean wave -length A , Kolsky [36] p. 52 gives the group velocity as _ es Cg — c - A dA or (3.2) _ 21y. vg - v -A dA where A is the wave length and v = c/cO . But in a single loading pulse of the type usually considered in incremental plastic -wave propagation the wave-length variation required to represent the pulse in terms of its harmonic components, e. g. , by a Fourier integral, is not small. The significance of the group velocity associated by equation (3. 2) with each wave-length A is then not so clear. If the complete pulse solution could be synthesized by a generalized Fourier integral, then all wave-lengths would be involved and it seems reasonable to expect that the fastest experimentally- observed wave speeds would be not greater than those of the highest phase velocity of any of the components. R. B. Lindsay [37], p. 111, points out that in defining group velocity of a pulse it is necessary to assume that the pulse can be represented by a group of waves whose wave-number Y is limited to a small interval AY on either side of the mean wave-number Y . Hunter and Johnson and Bejda and 53 Wierzbicki have not claimed a group velocity for a pulse, but only for the harmonic waves whose wave -length is near each component. With some reservations about their physical significance, group-velocity curves have been obtained from the present analysis and compared with the approximate analysis of Hunter and Johnson and of Bejda and Wierzbicki. To find the group velocity according to equation (3. 2) as a function of the variable used in this investi- gation we note that A A v-f—-—B—- X (3.3) O O and substitute (3. 3) in to the second of (3. 2) to obtain (L22) _ ZTra ZTra vg — v(1 - X dA ) (3.4) which reduces to _ 22 d_V -1 vg — v[1- v dX] (3.5) In the following two sections the phase velocity and the group velocity, for a cylindrical rod and a flat plate were calculated and the results plotted in Chapter 4 as a function of X = ap/cO for strain-rate-independent theory and strain-rate-dependent theory. A comparison of the results of each theory for the two geometric shapes is made. With this is also a comparison with the dispersion curves for the elastic case and with the results of the approximate analyses of Hunter and Johnson and of Bejda and Wierzbicki. 54 3. 2. Cylindrical Bar The dispersion curve for the cylindrical bar is obtained by numerically solving the frequency equation (2. 75). To avoid the difficulties discussed in section 3.1 we factor from this equation (2. 75) the term {(ini - Zzngflhz - k2)} and reduce it to a more suitable form for solving with the computer. The resulting equation becomes: hJO(h)J1(k) - kJO(k)J1(h) kJ0(h)J1(k)- hJo(k)Jl(h)} ASG{ }+ (BS -ATG){ hk(h2 — k2) h2 - k2 3. 3 k J (h)J (k) -h J (k)J (h) J (h)J (k) -BT{ ° 2' 2 ° 1 }-RTG 'hk' =0 (3.6) (h -k) where A = 17:0— ¢n§) B =1-2n§(1-¢) 2 G = (ino - 22(1 - 44> n§))/A (3.7) 2 2 2 R = 2n2(1+2¢nZ-n2) 2 T‘l'anZ 2 2 2 22 S—Z(l+2¢nZ-2n2)+X no and where 4) _ 1/3 ‘ 1+iX(l+1/) 381 for the rate -dependent formulation and 55 Co 2 (2“) -1 ¢ = P C 2 (3.8) 2(1+v)+3((;9-) -1) p for the rate -independent formulation. This form of ([9 for the rate -independent formulation is found by noting for a simple tension test. d; _ Si dEE — 51.6. _ _1. (3 9) - ‘ d0 " d0‘ ‘ dcr E ' ds 0 Hence (1; I20 E _ = _____E : ————O (3.10) d6 p __o_ _ 1 c0 2 £510; (3“) '1 d6 p We note again that in the rate-dependent case Z = Z1 - i Z2 is complex where Z.2 is considered a damping factor and the phase velocity c is given as the ratio of the circular frequency p to the real part of the wave number Y or in terms of the variables used in this calculation since Z is real. The dispersion curvesof c/cO : X/Z versus X are plotted in Figure 4. 3 for a mild steel cylindrical bar (U = . 29). Plots of the group velocity against frequency computed from equation (3. 5) 56 for the phase velocity dispersion curves given in Figure 4. 3 are shown in Figure 4. 5. To reduce the complexity of Figure 4. 5 the group velocity curve corresponding to Co/Cp = 7 for the rate- independent case is omitted and presented in Figure 4. 7. This group velocity is found in the following manner. We substitute v = X Re(Z) into equation (3. 5) to yield v— {1_§E1_Y_}-1- {1_§ l - X d(Re(Z)) -1 g ‘ V v dX " V v Re(Z) {Re(z)}2 dx Hence dX : —— 3. 1 V2 dm3 Ecwpdfiucod Ho. 3333/ madam 55:52 popcsoncb .. mo>u90 cowmumammfl .H .v ouswrm a ooo .ow ooo .00 000 .ov coo .ON cg n .7 1 m o>HDU A: u a a N 9550 A: n a 1 H 9550 1 w.o 8N . n 3 .mcofimsvm o>fififiumaoo unopGoQoQ1ofimm HOW Houbmm wcagmfl 55%02 popgonnb 1 mo>udU Gowmuommfifl 9 COO .ow 000 .00 COO gov COO .ON 0 v a if 60b 0 H II 1 m o>HDU a can“? 1 N®>HSU xv 1. H o>adU 0 1—1 II .N .v ouswfi LTV 0O .fim.o 61 where 2 2_C2 ”2"? C1 We see that as p -’ 0 (low frequency) Ll: CIYZ -> 0 (no damping) and the phase velocity c 4 Z {'c—l} 1-'3‘ 7’12 01‘ pcii ~x+§—GEK as p-’O which is the same phase velocity as was found in section 1. 4.1. for the rate -independent theory with d0' de—P < < C (moderate prestrain) . For waves with very high frequency the phase velocity approaches the elastic dilatation-wave speed c1 pci~pcf=X+ZG as p-rco Since elastic waves are undamped and the parameter <13 for this rate -dependent the ory ¢:__1_[_3___ 1+1pq1 approaches zero for very high frequencies, so that the constitutive equations (2. 35) take the form of the isotropic Hooke's law, we might expect the damping factor V to approach zero as p -’ 0°. But from equation (2. 50) we find that Y2 approaches a non-zero asymptotic value as p - 00 62 N 2 C 2 _11-21/ Ll":le 3:7} ‘ 3 {1-11} 1 We note here that in this uniaxial-strain case there should be no difficulty in representing a loading pulse by means of a Fourier integral. It is only in the following sections on the bar and the plate that the question of the possibility of a generalized Fourier synthesis is still unanswe red. 4. 3. Dispersion Curves for Longitudinal Waves in a Cflndrical Bar The phase-velocity dispersion curves of c/c0 = X/Re(Z) versus X = ap/co are plotted in Figure 4.3 for symmetrical longitudinal infinitesimal incremental plastic waves in a mild-steel cylindrical bar (V = . 29). For the rate-independent formulation three curves are plotted, for differential initial prestress values; the degree of initial prestress is specified by the ratio Co/Cp of the “elastic wave speed" to the "plastic wave speed". The curves presented here are for cO/cp = 3, 7, 00 . On this same figure the Pochhammer-Chree elastic dispersion curve is shown, and for the rate-dependent formulation one curve ([31: 0.1) is given. Figure 4. 4 shows the plot of the damping coefficient versus the frequency for the rate-dependent case ([51: 0.1). A plot of the group velocity, computed from equation (3.12), against the frequency for four of the five phase velocity curves given in Figure 4. 3 is shown in Figure 4. 5. In section 4. 5 the equivalent curves for the flat plate are presented. To avoid confusion in this figure the group velocity curve corresponding to CO/Cp = 7 for 63 the rate-independent theory is omitted. Figures 4. 6 and 4. 7 show a comparison of phase velocity dispersion curves obtained in this investigation with approximate theories of Hunter and Johnson [27] and Bejda and Wierzbicki [28] respectively. Comparison of group velocity curves obtained in this investigation with those of the two approximate theories are presented in Figures 4. 8 and 4. 9. 4. 4. Discussion of Dispersion Curves - Cylindrical Rod As seen in Figure 4.3, for all values of CO/Cp the phase velocity curves corresponding to the rate-independent theory (curves 3, 4, 5) are quite different from the elastic curve (1), different not only in numerical values but also in the general shape of the curves. In the elastic case, low-frequency waves are propagated with wave speeds very near c0 2 WT)- while higher frequency waves are propagated with slower phase velocities which approach the Rayleigh surface-wave speed for very-high-frequency waves. But the very-low frequency waves in the rate-independent theory propagate with the plastic wave speed cp = (l/p d0'/d€)1/2 , while as the frequency increases the phase velocity essentially increases in a manner somewhat similar to the phase velocities of elastic flexural waves in cylindrical bars (see Kolsky [36] p. 71). For all values of initial prestress except CO/Cp = 00 there is initially a decrease in phase velocity with increasing frequency (see curve 3). In most cases this decrease is too small to be noticeable in the figure. 6N .o n A new mo>m3 Ecowmcouxm mo >fioo~o> omdznw "pom 33.3510 1 mo>u90 codmuommmn— .m .v onswrm x or. o.m o.~ o; o J 4 d l1 d d d 4 64 58 u &o\oov «concoaochoudm m o>n30 :1 n mo\ouv «Gopcoamchmu—mm 1 v o>u50 7m n ao\oov acopcomoccToumm 1 m o>u50 3 .o u _m: Bandwagon—139m 1 N o>usU a 983 0393M 1 g o>u50 65 04V 3.0 n Ha .omé n 3 . mnoflmswm o>33u3mnou unopcomoflnoumm no.“ Hobwh wGaEwQ pom Hmomnpfigo 1 053330 nowmnoamwm b X o.N ~11- .v .v 0.3th II ~.o N.o m6 v.0 m.o 66 .omd n A H8 mo>m>> Hanowmcouxm mo >fioo~o> adoyO pom Hmowhpfigo mezzo ”51.309me X .m .v owfiwfim .. 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In the elastic case the limiting-wave speed, the Rayleigh surface wave speed, is approach asymptotically from above in a monotonically decreasing manner. For the higher frequencies investigated here the phase velocities c/c0 of the three rate-independent curves appear to fluctuate near the values of (c/c0)* in Table 2. TABLE 2 >l< CO/Cp (c/co) . 41 7 . 37 0° . 36 This fluctuation is oscillatory in nature and does not appear to be machine error. In the approximate theory given by Hunter and Johnson (see Figure 4.6) a limiting phase velocity of 5+2 +{C°}2 c 2 co 2 V _ {:7} = {7} 1 1C can 1 ° P (Lama-9} -(1-2v) P was found to be approached in a monotonically increasing manner, although the approximate theory ceases to be a good approximation for high frequencies; for large values of CO/Cp this limiting value is very large and seemingly unreasonable. The curves of the group velocity (Figure 4. 5) for the two rate-independent curves (3, 4) again 72 show no similarity to the elastic curve, (curve 1), and the maximum group velocity is on the order of one half the elastic wave speed cO . The effect of the small fluctuation of the phase velocity is magnified in the group velocity curves, whose calculations involve derivatives. These large fluctuations were not caused by errors in the numerical differentiation, since some of the values were verified using actual differentiation of the functions appearing in the frequency equations. For reasons given in section 3.1 concerning the applicability of group velocities we will not attempt to justify these velocities on a physical basis. Although a limiting phase velocity was not found for the rate- independent curves the character of these curves seems to imply that a phase velocity of cO cannot be attained. Hence the construction by generalized Fourier synthesis of an incremental pulse which could propagate with a leading-edge velocity of c0 does not seem feasible with this form of constitutive equation. The phase-velocity curve corresponding to the rate-dependent theory (curve 2, Figure 4. 3) is very much different in character from the rate-independent curves and for frequencies above a relatively low frequency the curve is nearly indistinguishable from the elastic dispersion curve. But the rate -dependent theory curve indicates that very-low-frequency waves travel with phase velocity very close to zero. As the frequencies increase, the phase velocities increase very quickly to approach the phase velocities corresponding to the elastic case. The maximum phase velocity is very near cO , which the one-dimensional theory predicts to be 73 the speed of the leading edge of an incremental pulse. The reason for this rapid approach to the elastic speeds can be seen by examining the function 43 which governs the effect of plasticity in the rate- dependent theory. (For 4) = O the incremental constitutive equations (2.35) reduce to the isotr0pic Hooke's law.) 4) _ 1/3 _ 1 + i X “+112 351 For X = O, (P = 313—, the 49 value of the rate-independent case co/cp = 00 (curve 5) for which c -’ O as X -’ O . As X increases 4) decreases and approaches zero which is the value of Cb for the elastic case. Since (1) also depends on [31 calculations were made to determine the effect of [31 . It was found for (31 = O. 1 (curve 2) the phase velocity was within a few percent of the elastic curve at X = 0. 5, while for Bl = O. 2 the frequency of maximum c was near X = O. 8 and for 51 = O. 05 it was near X = 0.3. For all values of $1 used in the calculations it was found that the maximum phase velocity reached to within a few percent of co . It should be noted that a large value of Bl 2 ayo EO/o c0 k2 corresponds to a large value of Y0 or to a material with little strain-rate-dependence, if the other parameters in [31 are held constant. Figure 4. 4 shows the variation of the damping factor ZZ = ay‘Z with X. This damping factor is zero for X = 0, reaches a maximum of 0. 24 for X : l. 7 and decreases slowly as X -' 0° . It is not possible to conclude rigorously from the numerical results that it is approaching a finite asymptotic value, but this appears to be the case. Since it appears that Y2 approaches 74 a finite limit as X -’ 00 the value of Z2 calculated for X = 5. 04, the last value of X used in the calculation, where variation in Y2 with X was very small, is compared with the finite limiting value for the unbounded medium. Letting 4’1 : (1+V)/3 C051 we obtain a dimensionless group chJl Y2 whose value is cO 411 Y2 = 0.238 for X = 5.04. For the unbounded medium we find _1 l-Zv _ _. CIQJYZ— §{_T“.—u} _o.197 as p on when U = . 29 according to the result of Section 4. Z. We note that LIJl is inversely proportional to the bar radius a, because BI is proportional to a, if the material parameters are held constant. Hence the damping factor V2 at X = 5. 04 depends on the radius. The group-velocity curve for the rate-dependent theory (curve 2, Figure 4. 5) is also very similar to that of the elastic case except for very low frequencies where group velocities slightly greater than one were found. Again for reasons given in section 3. 1 concerning the applicability of group velocities we will not attempt to justify these velocities on a physical basis. The comparison of these phase-velocity and group-velocity curves with the approximate curves of Bejda and Wierzbicki are seen in 75 Figures 4. 8 and 4. 9 respectively, with a difference between the phase velocity prediction between the approximate and exact theories of less than 2% up to about X = l. 0 with maximum deviation of 5% at X = l. 5. Because of the phase velocity values attained for the rate- dependent case the conclusion is reached that if the assumptions made in section 1.1 are valid, one could construct by Fourier synthesis an incremental pulse whose leading edge would travel with a velocity of approximately cO . In the following two sections the corresponding curves for the flat plate are presented and discussed and a similar conclusion will be reached. 4. 5. Dispersion Curves for Symmetrical Longitudinal Waves in a Flat Plate The phase velocity curves of C/CL = 'J l - V2 {X/Re(Z)} and the group velocity curves of cg/cL = N} l - V2 dX/d(Re(Z)) plotted against frequency X are presented in Figures 4.10 and 4.12 respectively for cases analogous to the five curves given for the cylindrical bar (CO/Cp = 3, 7, 0° for rate-independent theory (3, 4, 5) (31 = 0.1 for rate-dependent theory (2), Rayleigh-Lamb elastic plate curve (1)). Again to avoid confusion in the figure the group velocity curve for cO/cp = 7 is omitted. The general characteristic of this curve is the same as the other rate- independent curves. Figure 4. 11 shows the damping factor Z2 = ayz plotted against X. .mm .o n A H8 nerd? Hmfipdfimnod Hmownuoagm mo >fioo~m> ommanm "33m ugh 1 mo>n50 Gowmnommmfl .oH .w. @54th 76 X o .m o .N o; o n w . . . 4 0 830.30 93 modfimcr "EC .. N .o .. 9, .. w .o I, m w. m .. o .o w .o m 0 A8 u 0\ 3 9:35 “GopnomopnT mudm m 95.90 C. u mo\oov 9rd? unopsomoUnT 3mm v o>u50 / Am u AHo\03 055m? “noncomoch Sam M 95.90 r 0 .fi :6 N am: 9335 unopnmmofl1mumm N o>n50 H 95.? oaummnm o>udU qo/o 77 2 .o u E .35 u 3 58335 3333800 «GopGoQoQ1oumm HOW Hon—ooh MGEEMQ ”3.3% «are 1 “3.?de Gowmuommafl N .11. {1- .2 .w £st 41- To ~.o m.o 78 HowuuoEEAm Ho 5329» QDOHU .oN .o n A H3 mo>m3 WEE—£934 "3.3m «arm 1 mo>HsU commuoamwfl .NH .v 0.”:th a X o .m o .N o H q q q d 1 1 ‘ m a o “a u 0\ UV Pam? «Gopcoaochoudm 1 ¢ o>u50 3 n o\oUv 933 acopcoaochoumm 1 m o>u30 S .o u rm: 933 “connoaoa1oumm 1 N 9550 l4 953 33de A o>u90 ed w.o 79 4.6. Discussion of Dispersion Curves - Flat Plates The general character of the phase-velocity curves and group-velocity curves for the symmetrical longitudinal infinitesimal incremental plastic waves in a flat plate is the same as that of the cylindrical bar. The results also follow the same pattern in that the rate-dependent curve coincided within a few percent with the elastic curve for X > 0. 5 whereas the rate -independent curves are completely different. Again, as in the cylindrical bar, there are small fluctuations in these curves at high frequencies. The values of X* for the phase velocity and group velocity curves where calculation problems were encountered (see section 3. 1) are labeled on the curves. One interesting difference from the bar results for the rate- independent phase velocity curves is that the very low frequency waves for the highly prestressed case (Co/Cp = 0°) propagate with a non-zero phase velocity near 1 }1/2 C : C:o{5-4V (Approximately half the elastic bar wave speed for v = 0. 29) which is the phase velocity for the low frequency rate-dependent wave. As in the case of the unbounded medium and the cylindrical bar, the damping factor approaches zero as X approaches zero and seems to approach a finite non-zero value as X increases. For the plate the value at the highest frequency calculated was CLquV = 0.18 at X25.O 80 which may be near the asymptotic limit, although this conclusion cannot rigorously be drawn from the numerical results. This value is slightly below the value _1_1-2V _ CquY—3{l-V}_O°197’ when U = 0. 29 , the asymptotic limit for uniaxial strain waves in an unbounded medium (section 4. Z). In the plate problem, as in the cylindrical bar, we conclude that if the assumptions of section 1.1 are valid it would be possible to construct an infinitesimal incremental-loading pulse with leading- edge traveling at a velocity very close to co. using the rate-dependent equations, but it would not be possible using the rate-independent equations . 4. 7. Summary of Results and Conclusions The primary objective of this investigation - to determine which of the most common strain-rate -independent and strain-rate- dependent constitutive relations best govern the propagation of infinitesimal incremental plastic waves in a prestressed medium - was accomplished providing the assumptions of section 1. l are valid. For the three problems examined the results as found in the previous sections may be summarized as follows: 1. Unbounded Medium With the rate -independent constitutive equations transverse waves travel with the shear wave velocity of elastic wave propagation c = N} G7p whereas the velocity of longitudinal waves is 81 c = [ (X + 2C1 - 4C1¢)/p]1/2 . For large prestress, ¢ approaches one -third very quickly and c becomes very nearly c = [ ()1 + 5— G)/ p] 1/2 = W. With the rate -dependent constitutive equations, the shear wave again is predicted to travel with the elastic shear wave speed but the velocity of the longitudinal waves dependsupon the frequency of the wave. Very low frequency waves travel with a velocity very near c = W. As the frequency increases, the wave speed very rapidly approaches the elastic speed c = cl = [(K + 2G)/p] 1/2 . The rate-dependent theory predicts incremental waves to be damped with a damping factor which has the value zero for very low frequency and approaches a finite non-zero value depending on the amount of rate -dependence, as the frequency approaches infinity. . Cylindrical R od The rate-dependent theory predicts that the phase velocity for very low frequency waves is very close to zero but as the frequency increases the corresponding phase velocity rapidly increases to a maximum within a few percent of c0 = «FE-7})— which is the experimentally observed velocity. For larger frequencies the dispersion curve of phase velocity is approximately the same as that of the elastic curve. The group velocity dispersion curve has the same general characteristics as the phase-velocity curve. The damping factor associated with this rate-dependent theory again is zero for low frequency waves; it increases to a maximum, decreases and appears to approach a finite limit as the frequency increases. The rate- 82 independent theory on the other hand predicts a maximum phase velocity and group velocity on the order of one -half the elastic wave speed co, and the characteristics of the dispersion curves were very different from that of the elastic rod. The approximations of Hunter and Johns on (rate-independent- case) and Bejda and Wierzbicki (rate-dependent case) were found to be excellent for low-frequency waves. Flat Plate The dispersion curves for the flat plate showed characteristically the same results as that of the cylindrical bar. The rate-dependent curves had maximum phase velocity values very near the elastic plate longitudinal wave velocity cL = co/Vl—T—JZ while for the rate -independent theory the maximum phase velocity was on the order of one-half this value. The group velocity showed similar results. Again as in the unbounded medium, the damping factor associated with the rate-dependent constitutive relation appeared to approach a limit as the frequency approached infinity. The conclusion reached from this investigation is that if the assumptions of section 1.1 (that a loading incremental wave exists, and that this incremental wave can be analyzed by generalized Fourier analysis and synthesis), are valid, then the rate-dependent theory would predict that the leading edge of the loading pulse would travel at a speed near the elastic wave speed in agreement with experimental bar-wave observations reported in the literature; while any wave synthesized according to rate-independent theory would not predict this leading-edge speed. If therefore appears (1) (Z) (3) (4) 83 that the rate-dependent theory is better-suited to describe incremental rloading waves. Suggested extensions of this research include: Theoretical verifications of the assumption of section 1. 1 by performing the synthesis. Such analysis and synthesis has been performed in the elastic case by double transform methods. The transform solutions have not been completely inverted, but near-field and far-field asymptotic inversions have been obtained (see Miklowitz [ 39] ), numerical solutions of the governing equations of infinitesimal incremental waves without using harmonic analysis, experimental investigation to determine whether or not a damping factor of the type predicted by the rate-dependent theory exists, and investigation of other rate-dependent theories. 10. ll. 12. BIBLIOGRAPHY von Karman, T. , "On The Propagation of Plastic Deformation in Solids, " NDRC Report NO. A-Z9 (OSRD NO. 365), 1942. von Karman, T. and Duwez, P. , "The Propagation of Plastic Deformations in Solids," J. Appl. Physics, Vol. 21, 1950, pp. 987-994. Malvern, L. E. , ”The PrOpagation of Longitudinal Waves of Plastic Deformation in a Bar of Material Exhibiting a Strain-Rate Effect, " J. Appl. Mech., Vol. 18, 1951, pp. 203 -208. Malvern, L. 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