1 ! " # $ % " & ' %( ) * + ', +"!! & *,)+&,, $ -,+*, $$ -..
3 Dissertation for the Degree of Doctor of Philosophy in Engineering Science with Specialization in Solid Mechanics presented at Uppsala University in 2002 Abstract Widehammar, S A Method for Dispersive Split Hopkinson Pressure Bar Analysis Applied to High Strain Rate Testing of Spruce Wood. Acta Universitatis Upsaliensis. Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology pp. Uppsala. ISBN The aim was to establish a method for studying the relationbetweenstressandstraininspruce wood at high strain rate. This was achieved by adapting and somewhat further developing the split Hopkinson pressure bar (SHPB) technique. Hopkinson bars usually have a circular cross-section and a diameter much smaller than the operative wavelengths. The wave propagation in the bar is then approximately non-dispersive and a one-dimensional (1D) wave propagation model can be used. When, as in this study, it is not certain that the transverse dimensions of the bars are small in relation to the wavelengths, a solely 1D wave propagation model is insufficient and the geometry of the cross-section, which was square in this study, must be taken into account. Therefore, an approximate 3D wave propagation model for bars with arbitrary cross-section was developed using Hamilton s principle. The model provides a dispersion relation (wavenumber vs. angular frequency) and average values for displacements and stresses over the bar/specimen interfaces. A calibration procedure was also developed. Tests on spruce wood specimens were carried out at a high strain rate (about 10 3 s 1 )using the adapted SHPB technique, and for comparison at low ( s 1 ) and medium (17 s 1 ) strain rates using a servohydraulic testing machine. The moisture contents of the wood specimens corresponded to oven dry, fibre saturated and fully saturated, and the testing was performed in the radial, tangential and axial directions relative to the stem of the tree. In each case, five tests were run at room temperature. The results show the strain rate dependence of the relation between stress and strain for spruce wood under all conditions studied. Keywords: split Hopkinson, wave propagation, dispersion, Hamilton s principle, strain rate, calibration, spruce, wood Svante Widehammar, Department of Materials Science, Uppsala University, Box 534, SE Uppsala, Sweden c Svante Widehammar 2002 ISSN X ISBN Printed in Sweden by Uppsala University, Tryck och medier, Uppsala 2002
4 Akademisk avhandling för avläggande av teknologie doktorsexamen i teknisk fysik med inriktning mot hållfasthetslära vid Uppsala universitet år 2002 Sammanfattning Widehammar, S En metod för dispersiv analys av försök med delad hopkinsonstång tillämpad på provning av granved vid hög töjningshastighet. Syftet var att etablera en metod för att studera sambandet mellan spänning och töjning för granved vid hög töjningshastighet. Detta åstadkoms genom att anpassa och något vidareutveckla tekniken med delad hopkinsonstång ( Split Hopkinson Pressure Bar, SHPB). Vanligtvis har hopkinsonstavar cirkulärt tvärsnitt och en diameter som är mycket mindre än de verksamma våglängderna. Under sådana förhållanden är vågutbredningen i stängerna approximativt ickedispersiv, och en endimensionell (1D) vågutbredningsmodell kan användas. När det, som är fallet i denna studie, däremot inte kan säkerställas att stängernas tvärdimensioner är små i förhållande till våglängderna, är en helt igenom 1D vågutbredningsmodell otillräcklig, och tvärsnittets geometri, vilken var kvadratisk i denna studie, måste beaktas. Därför utvecklades med hjälp av Hamiltons princip en approximativ 3D vågutbredningsmodell för stänger med godtyckligt tvärsnitt. Modellen ger ett dispersionssamband (vågtal som funktion av vinkelfrekvens) samt medelvärden för förskjutningar och spänningar över gränsytorna mellan stänger och provstav. En kalibreringsprocedur utvecklades också. Provning av granved genomfördes vid hög töjningshastighet (omkring 10 3 s 1 ) med den anpassade SHPB-tekniken, samt för jämförelse vid låg ( s 1 ) och måttlig (17 s 1 ) töjningshastighet med en servohydraulisk provningsmaskin. Fukthalterna i veden motsvarade ugnstorr, fibermättnad och fullständig mättnad, och proven utfördes i radiell, tangentiell och axiell riktning i förhållande till trädets stam. För vart fall utfördes fem försök vid rumstemperatur. Resultaten visar töjningshastighetsberoendet för sambandet mellan spänning och töjning för granved under alla studerade förhållanden. Svante Widehammar, Institutionen för materialvetenskap, Uppsala universitet, Box 534, 75121Uppsala
5 Preface This work was initiated by the former Sunds Defibrator Industries AB (now Metso Paper AB) and the Mid Sweden University. The work was carried out at the Fibre Science and Communication Network at Mid Sweden University in Sundsvall, in co-operation with the Division of Solid Mechanics at Uppsala University. It was supported economically by the following external agencies: Foundation for Knowledge and Competence Development, the National Graduate School of Scientific ComputingandMetsoPaperAB. I would like to thank Professor Per Gradin, Mid Sweden University, and Professor Bengt Lundberg, Uppsala University, for their guidance and support during the course of this work. I would also like to thank all my colleagues in Sundsvall and Uppsala for providing a pleasant and creative atmosphere, in particular Staffan Nyström and Max Lundström for their assistance in the laboratory, including the superb coffee. Special thanks also to Göran Widehammar and Örjan Bäcklund who used their free time to find and cut a suitable tree and to manufacture the wood specimens. Finally, I would like to thank my children Ronja and Hugo for being there, even though they could not see the meaning of storing small cubes of wood in water and writing funny signs and curves on the computer, and my beloved wife, Anna-Karin, who, after all, managed to cope with me during these years. Sundsvall in December, 2002 Svante Widehammar
6 List of papers This thesis comprises a survey of the following four papers: I. Widehammar S, Gradin P.A. and Lundberg B. (2001). Approximate determination of dispersion relations and displacement fields associated with elastic waves in bars. Method based on matrix formulation of Hamilton s principle. Journal of Sound and Vibration 246(5): II. Widehammar S. Estimation of 3D field quantities and energy flux associated with elastic waves in a bar. Journal of Sound and Vibration (Accepted) III. Widehammar S. Split Hopkinson pressure bar testing: procedures for evaluation and calibration. Experimental Mechanics (Submitted) IV. Widehammar S. Stress-strain relationships for spruce wood: influences of strain rate, moisture content and loading direction. Experimental Mechanics (In progress)
7 Contents 1. Introduction 9 2. The refining process Fundamentals of SHPB testing D formulation Displacement, particle velocity, strain and stress in a SHPB Dispersionrelation EvaluatingSHPBtests Two-pulsemethod Determinationofamplitudes Discussion D formulation Displacement, particle velocity, strain and stress in a SHPB Dispersionrelation EvaluatingSHPBtests Two-pulsemethod Determinationofamplitudes Discussion Calibration Dispersionrelation Poisson sratio Preparatory tests Experimentalset-up EstimationofPoisson sratio EstimationofYoung smodulus Evaluation of redundant measurements Comparisonofevaluationoptions Discussion... 34
8 8. Testing of spruce wood Experimentalconditions SHPBtests Resultsanddiscussion Conclusions 38 References 39
10 DISPERSIVE SHPB ANALYSIS OF WOOD 9 1. Introduction The aim of this work was to establish and use a method for determining the relation between stress and strain in spruce wood under conditions which are similar to those which prevail in a refiner in the thermomechanical pulping process. Characteristic of these conditions are high temperature, a moisture content above fibre saturation and a high strain rate. The last condition excludes the use of a traditional servohydraulic testing machine for determining the stress-strain relations. In order to achieve a high strain rate, the split Hopkinson pressure bar (SHPB) technique has proven suitable. A general presentation of the technique is given by Al-Mousawi et al. . Hopkinson bars usually have a circular cross-section and a diameter which is much smaller than the wavelengths associated with the measured pulses. Under such conditions, the wave propagation in the bars is approximately non-dispersive, and a one-dimensional (1D) wave propagation model can be used with good accuracy. This is the traditional method, described by, e.g., Graff . Preliminary tests showed that dispersion could not be neglected in this work. Dispersion occurs when the transverse dimensions of the bars are of the same order of magnitude as the operative wavelengths. Compensation for dispersion is a standard procedure nowadays, and it has been used by many researchers, e.g., Follansbee and Frantz , Gong et al. , Zhao and Gary  and Li and Lambros . Common for all these investigations is that a dispersion compensation procedure is applied to the standard 1D model of the bar, in which warping of the bar cross-section is neglected and the geometry of the cross-section is assumed to be insignificant. The dispersion relation (wavenumber vs. angular frequency) must then be known, and the exact solution for a bar with a circular cross-section, which was originally presented by Pochammer  and Chree  and further developed by, e.g., Kolsky , Davis  and Bancroft , is usually used. For practical reasons, a square cross-section was chosen for the spruce specimens as well as for the bars in this work. No exact solution seems to be available for such bars. For arbitrary cross-sections, finite element methods have been used by, e.g., Hladky-Hennion , Volovoi et al.  andtaweel et al. . InPaperI, an approximate 3D wave propagation model for bars with arbitrary (and in particular square) cross-section was developed using Hamilton s principle. The model uses Fourier series and leads to an eigenvalue problem which has to be solved for each angular frequency associated with the Fourier components. The eigenvalue problem provides a dispersion relation, and average values for displacements and stresses over the bar/specimen interfaces can be calculated, as is shown in Paper II. This means that warping can also be included in the evaluation of SHPB tests. The effects of warping have been studied for circular bars by Tyas and Watson [15 17], who used
11 10 SVANTE WIDEHAMMAR the Pochammer-Chree equations. They showed that the effects of warping may not always be negligible. The formulations with Fourier series used in the wave propagation model are suitable for redundant measurements, which may increase the accuracy in the results. This is evaluated in Paper II. The technique with redundant measurements was previously used by Hillström et al.  to estimate the complex modulus of the material of a bar. A calibration procedure for the experimental set-up using an ordinary SHPB projectile and an optimizing procedure has also been developed from the wave propagation model. The calibration procedure is developed in Papers II and III. The fully 3D model, including redundant measurements and parts of the calibration procedure, is validated on a bar with a free end in Paper II. A mixed evaluation procedure, using the dispersion relation from the 3D model and the 1D formulations for variables such as displacements and stresses in the bars, was applied to SHPB tests and validated in Paper III. Finally, high strain rate testing of spruce wood was performed in the work described in Paper IV. Low and medium strain rate testing was performed in a servohydraulic testing machine for comparison with the high strain rate SHPB tests. Oven dry, fibresaturatedandfullysaturatedwoodweretestedintheradial,tangential and axial directions with reference to the stem of the tree. 2. The refining process The thermomechanical pulping (TMP) process produces pulp for papermaking out of wood chips. The central part of the process takes place in a refiner, which consists of two concentric circular discs. Either one disc rotates and one is fixed, or both rotate in opposite directions. Preheated wood chips and water are fed into the centre of the refiner and are transported radially outwards between the discs due to inertia. Through contact with radial bars on the discs, the chips are defibrated into fibres. The discs have a coarse bar pattern near the centre, the breaker bar zone, and a much finer bar pattern at the periphery. The gap between the two discs decreases from the centre to the periphery, where it is of the order of less than one millimetre. The discs are about 1.5 meters in diameter and normally operate at 1500 or 1800 revolutions per minute. All aspects of mechanical pulping are treated in a book by Sundholm . The TMP process is very energy-consuming and much effort has been spent on reducing the energy consumption without any loss of pulp quality. A better understanding of the refining process requires a better knowledge of the behaviour of wood
12 DISPERSIVE SHPB ANALYSIS OF WOOD 11 chips under realistic refining conditions. The initial breaking of the wood chips is performed at high temperature (about 125 C),atamoisturecontentabovefibre saturation, and at very high strain rate (about 10 3 s 1 ). It is generally recognized that species of the spruce family are the most favourable raw materials for mechanical pulping, and spruce is also the most widely used species for that purpose in Europe and Canada. Experiments aimed at quantifying the behaviour of wood under such conditions have been carried out, e.g., by Uhmeier and Salmén  and by Renaud et al. [21, 22]. Uhmeier and Salmén used servohydraulic testing equipment and reached 25 s 1 at 98 C for spruce wood, while Renaud et al. used the SHPB technique in order to achieve a high strain rate at room temperature. The latter authors used three different kinds of hardwood which are all of minor importance for the Swedish pulping industry. Other tests on wood at high strain rates have been carried out, e.g., by Bragov and Lomunov  and by Reid and Peng . The latter investigators performed their experiments on dry wood. There is an evident lack in the literature of test data on wet spruce wood at strain rates similar to those which occur in a refiner. The SHPB technique is well suited to obtain such data. 3. Fundamentals of SHPB testing For compression testing with the SHPB technique, a specimen of the material to be tested is inserted between the ends of two bars with colinear axes, as shown in Figure 1. The material of the bars, which has known properties, is commonly linearly elastic and isotropic, and small deformations can be assumed in the bars. A projectile is fired onto the free end of one of the bars, the sender bar. This generates an incident pulse, which propagates towards the specimen where the pulse is partially reflected and partially transmitted into the second bar, the receiver bar. Measurement and analysis of the incident, reflected and transmitted pulses enables the mechanical Incident pulse Reflected pulse Transmitted pulse Projectile Sender bar Specimen Receiver bar Figure 1: Principle set-up of SHPB experiments.
13 12 SVANTE WIDEHAMMAR behaviour of the material of the specimen to be estimated. The stress in the specimen at time is approximated by the mean value of the forces from the two bar ends in contact with the specimen, divided by the specimen cross-sectional area, i.e., () =[ b A (0)+ b B (0)] 2 (1) where b is the cross-sectional area of the bars, A ( ) and B ( ) are the axial stresses in the sender and receiver bars, respectively, and is the axial coordinate. The two bars have individual coordinate systems, where =0is the axial position of the bar/specimen interface in each system. An overbar denotes the mean value over the bar cross-section. The strain in the specimen is approximated by the difference between the averages of the axial displacements of the two bar ends divided by the initial length of the specimen, i.e., () =[ B (0) A (0)] (2) where A ( ) and B () are the axial displacements in the two bars. Differentiation with respect to time gives the strain rate () =[ B (0) A (0)] (3) where a dot denotes time derivative. From relations (1) and (2), a stress-strain curve can be established with time as a parameter. The strain rate (3) generally varies with time, but it is often approximately constant during a test. In order to be able to evaluate the conditions at the two sides of the specimen, axial strains in the bars are measured at some distances from the bar/specimen interfaces. A procedure will be formulated in which the specimen behaviour can be calculated from the measured strains. 4. 1D formulation 4.1. Displacement, particle velocity, strain and stress inashpb Using a discrete Fourier transform (DFT), a pulse can be expressed as a sum of harmonic waves. If the wavelength of such a harmonic wave is large compared to the cross-sectional dimensions of the bar, it can be assumed that transverse displacements and stresses are small compared to the axial displacements and stresses.
14 DISPERSIVE SHPB ANALYSIS OF WOOD 13 If the transverse quantities are neglected, axial displacement, axial particle velocity and axial strain and stress in the SHPB:s depend only on and, and not on the cross-sectional coordinates and. The mean value over the cross-section is then trivial, so the overbar is left out in this section. Under this assumption, the axial displacement in the sender bar can be approximated by 2 X A ( ) = 0A + 0A + 0A +Re h³ i i e i + r e e ii (4) where the first two terms on the right-hand side represent rigid body motion with constant velocity 0A,and 0A is a constant axial strain. The sum represents harmonic waves travelling in positive and negative -directions. These waves are obtained from the DFT:s of the incident (index i) andreflected (index r) pulses, respectively. The constants i and r are complex-valued amplitudes and and are the wavenumber and the angular frequency, respectively, associated with Fourier component number. The constant is recognized as the length of a measuring record, and Re denotes the real part. The axial particle velocity in the bar is obtained by differentiating equation (4) with respect to time, i.e., 2 X A ( ) = 0A +Re h³ i i e i + r e i e ii (5) and the axial strain is obtained by differentiating equation (4) with respect to, i.e., 2 X A ( ) = 0A +Re h³ i i e i + r e i e ii (6) With the assumptions that the bar material is linearly elastic and the stress is uniaxial, the axial stress is obtained by multiplying the strain by Young s modulus, i.e., 2 X A ( ) = 0A + Re h³ i i e i + r e i e ii (7) For the receiver bar, correspondingly, the axial displacement is 2 X B ( ) = 0B + 0B + 0B +Re h i t e i e i (8) where index t denotes the transmitted pulse. The axial particle velocity, strain and stress are obtained in the same way as for the sender bar.
15 14 SVANTE WIDEHAMMAR 4.2. Dispersion relation The angular frequencies are obtained from the definition of the DFT, but a dispersion relation = () is lacking in the previous expressions. Hamilton s principle can be utilized to find the dispersion relation. In the absence of external forces, Hamilton s principle states that µz 2 ( )d =0 (9) 1 where is the kinetic energy, is the elastic strain energy, and 1 and 2 are two arbitrary instants of time The kinetic energy is here given by ZZZ µ 1 = 2 2 d (10) where is the mass density and is the volume of the body. The elastic strain energy stored in the body is ZZZ µ 1 = 2 d (11) Here, the axial strain is = 0, where the prime denotes partial differentiation with respect to, and the axial stress is = 0. Inserting these relations and equations (10) and (11) into Hamilton s principle (9), gives µz 2 ZZZ µ ( ) d d =0 (12) By performing the variational operation, an integration by parts with respect to time of the first term and an integration by parts with respect to of the second term, it is obtained that + 00 =0 (13) together with initial and axial boundary conditions, which will not be treated here. Inserting a harmonic wave similar to those in equation (4), i.e., n =Re e i( )o (14) gives which leads to where the phase velocity is constant. 2 e i( ) 2 e i( ) =0 (15) = 0 (16) 0 = p (17)
16 DISPERSIVE SHPB ANALYSIS OF WOOD Evaluating SHPB tests Inserting expressions (4), (5) and (7) evaluated at =0, and similar expressions for the receiver bar, into relations (1)-(3) gives () = b 2 Re X ( i + r t )i e i (18) and () = Re X (t i r )e i (19) () = Re X (t i r )i e i (20) where the constants 0, 0 and 0 are most easily determined directly from the conditions that () = () = () =0before the incident pulse reaches the specimen. This is done by plotting the results in the time domain using only the Fourier series in equations (18) and (19). In these plots, the initial parts of the curves are generally straight sloping lines. The constants are then determined so as to make these lines coincide with the horizontal axis. This evaluation method is called the three-pulse method in Paper III, as all three pulses, incident, reflected and transmitted, are used in the evaluation Two-pulse method If the SHPB test is well designed, it can be assumed that the stress is constant throughout the specimen, i.e., A (0)= B (0). Expression(7)andthecorresponding expression for the receiver bar then give i = r + t. Using this equality in equations (18)-(20) gives () = 0 2 b Re X t i e i (21) and () = Re X r e i (22) () = Re X r i e i (23)
17 16 SVANTE WIDEHAMMAR Here, only the reflected and transmitted pulses are used in the evaluation Determination of amplitudes To be able to determine the amplitudes associated with harmonic waves, axial strains are measured at axial positions =1 2 on each bar. After a DFT of each strain record has been performed, the axial strain at = can be approximated by 2 X ( )= 1 +Re e i (24) where are complex-valued amplitudes obtained from the DFT. Comparison of this expression with the axial strain in the sender bar given by equation (6) yields = ³ i i e i + r e i (25) for =2 32. For each Fourier component, this provides a system of equations e i1 e i1 1 (i ) i.. = e i e i. (26) r (i ) for the two unknown amplitudes i and r. Thus, for 2 this system of equations is overdetermined and an appropriate approximation of i and r can be obtained by the least-squares method. An overdetermined system of equations contains redundant information, and as a result, it may give more accurate results for i and r than a system which does not contain redundancies. If a measuring section is placed so far from the bar/specimen interface that the incident and reflected pulses are separated in time at this particular section, the measuring record can be split into two parts, viz., one part containing the incident pulse and one part containing the reflected pulse. The two split records should be padded with zeros at the end or at the beginning, respectively, to maintain the original record length (), and to let each of the two pulses appear at the same times in the split records as they did in the original record. This simplifies the analysis. Performing a DFT on each of the two split measuring records gives the constants i and r for the incident and reflected pulses, respectively. The equation corresponding to row in the system of equations (26) can then be separated into two new equations, viz., e i 0 i 0 e i r = i (i ) r (i ) (27)
18 DISPERSIVE SHPB ANALYSIS OF WOOD 17 The splitting of one equation into two equations adds information to the original system and therefore gives a greater accuracy in the determined amplitudes. In the receiver bar, only waves propagating in the positive -direction are considered. The least-squares method can be used together with redundant data also here, even though there is only one amplitude to determine for each Fourier component Discussion It is shown in Paper III that equations (21)-(23) together with equation (16) can be formulated as () = b t (0)= µ b t t + t (28) 0 () = 2 Z 0 r (0)d = 2 Z µ 0 r r + r d (29) and () = 2 0 r (0)= 2 µ 0 r r + r (30) 0 where t ( t ) and r ( r ) are the measured axial strains in the measuring sections = t and r for the transmitted and reflected pulses, respectively, and t = t and r = r are the distances between the measuring sections and the bar/specimen interfaces. These expressions are traditionally often used for evaluation of SHPB experiments, see, e.g., Al-Mousawi et al.  and Graff . It can be seen that if the distances t and r are equal, there is no need for the time shift. The method is then very easy to use and also very robust. Expressions (28)-(30) may indicate that the formulations with Fourier series seem unnecessarily complicated here. Anyway, although not really consistent, correction for dispersion is often used in connection with the 1D formulation, see, e.g., [3 6]. That technique is also used in Paper III and Paper IV. The frequent use of a 1D formulation mixed with a dispersion correction justifies the use of Fourier series in the 1D case. Also, the obvious similarities with the 3D formulation which follow will make comparisons easier. 5. 3D formulation 5.1. Displacement, particle velocity, strain and stress inashpb If the wavelengths of the operative harmonic waves in the Fourier series in equations (4)-(8) are comparable to the cross-sectional dimensions of the bar, lateral displace-
19 18 SVANTE WIDEHAMMAR ments cannot be neglected. An expression for a general displacement vector in the sender bar corresponding to equation (4) in the 1D formulation, is then u A ( )=( 0A + 0A )[0 0 1] T + 0A [ ] T 2 X + Φ ( )Re h³ i i d e i + r d e e ii (31) where is Poisson s ratio, Φ is a 3 matrix containing given coordinate functions of the cross-sectional coordinates and, and d are normalized vectors of complex-valued constants. The amplitudes i and r have the physical dimension of length, while Φ and d are dimensionless. The coordinate functions in Φ allow for variations in the displacement vector over the cross-section, warping, and the constants in d are the weights for each coordinate function. The vectors d contain the weights for waves travelling in the negative -direction, i.e., connected with the reflected pulse. It will later be shown that d are the complex conjugates of the vectors d. Differentiation of equation (31) with respect to time gives the particle velocity u A ( )= 0A [0 0 1] T 2 X + Φ ( )Re h³ ei i d e i + r d i e ii (32) The vector of strain components ε =[ ] T and the displacement vector u are related through the kinematic relation ε = u (33) where contains the partial differential operators and. For a study of wave propagation in a bar with its axis oriented in the -direction, it is advantageous to split the operator matrix into two parts, i.e., = + (34) where = = (35)
20 DISPERSIVE SHPB ANALYSIS OF WOOD 19 The parameter is a reference length, such as the radius of a bar with circular crosssection or half the side of a square cross-section. It is introduced in order to obtain the same physical dimension of and. From equations (31) and (33)-(35), the vector of strain components in the sender bar is ε A ( ) = 0A [ ] T 2 X + Φ ( )Re h³ ei i d e i + r d e ii 2 X + Φ ( )Re h³ i i d e i + r d e i e ii (36) For a linearly elastic and isotropic material, the vector of stress components τ = [ ] T is given by the generalized Hooke s law τ =Cε (37) where is Young s modulus and C = (38) is a symmetric matrix containing dimensionless elastic constants 1 = 1 (1 + )(1 2) 2 = (1 + )(1 2) 3 = 1 2(1+) (39) From equations (36)-(39), the vector of stress components in the sender bar is obtained as τ A ( ) = 0A [ ] T 2 X + C Φ ( )Re h³ i d e i + r d i e i e i 2 X +C Φ ( )Re h³ i d e i + r d i ei i e i (40)
21 20 SVANTE WIDEHAMMAR 5.2. Dispersion relation As in the 1D case, Hamilton s principle will be used to obtain the dispersion relation. The displacement vector to be used here is similar to the harmonic waves in equation (31), viz., n u =Re Φ ( ) de i( )o (41) The kinetic energy is here given by ZZZ = and the elastic strain energy by ZZZ = µ 1 2 ut u d (42) µ 1 2 τ T ε d (43) It is shown in Paper I that equations (33), (34), (37) and (41)-(43) together with Hamilton s principle (9) lead to the eigenvalue problem " K 0 +i µ 2 µ 2 ³K 1 K T 1 + K 2 M# d = 0 (44) where 0 = 1 are reference quantities, and ZZ K 0 = and 0 = 1 s (45) ( Φ) T C Φ d (46) ZZ K 1 = ( Φ) T C Φ d (47) ZZ K 2 = ( Φ) T C Φ d (48) M = 1 2 ZZ Φ T Φ d (49) are real and dimensionless matrices. The derivation in Paper I is more general since neither a cartesian coordinate system nor an isotropic material are assumed. The eigenvalue to solve for in the eigenvalue problem (44) is one of the quantities 0 or 0 with the other known. For a given angular frequency, equation
22 DISPERSIVE SHPB ANALYSIS OF WOOD 21 (44) is a quadratic eigenvalue problem with 2 solutions. The eigenvalue 0 is generally complex-valued, but for modes corresponding to propagating waves, 0 is real. Non-real wavenumbers correspond to end modes which decay with increasing distance from the bar end. See, e.g., Taweel et al. . Suppose that a solution ( d) with positive and real is found. From equation (41), it is evident that this corresponds to a wave travelling in the positive -direction. Now, by forming the complex conjugate of equation (44), it can be shown that ( d ) where the star denotes a complex conjugate, is also a solution of the eigenvalue problem. This second solution corresponds to a wave of the same mode which travels in the negative -direction, which has already been used in equation (31) and in the expressions derived from it. The 2 solutions of (44) approximate in general several modes of wave propagation such as flexural, torsional and longitudinal. In a SHPB, only the first longitudinal mode is of interest. The coordinate functions can then be constructed to include the symmetries related to this mode. Here, these coordinate functions are polynomials in the lateral coordinates and, given in detail in Papers I and II. For longitudinal modes, it is shown in Paper II that the matrix Φ can be partitioned according to Φ = φ T 0 φ T 0 0 φ T (50) The partitioning of Φ into subvectors also motivates a partition of d, accordingto d d = (51) It is also shown in Paper II that all the components of d have the same phase angle, and that this angle differs by 2 from the phase angle of all the components in d. If the vector d is an eigenvector, so is de i, which makes it possible to change the phase angle of the eigenvector d. If the phase angle is chosen so that d is real, the subvector d is purely imaginary. This implies that d d = (52) d The convergence rate of the method can be demonstrated by solving the eigenvalue problem (44) with different orders of the polynomials in Φ, seefigure2. In Paper I, solutions of the eigenvalue problem (44) are compared to other approximate solutions from the literature for bars with a square cross-section, and with the exact solution for bars with a circular cross-section. In Paper II, it is shown that the d