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© Sa Majesté la Reine en droit du Canada (Ministère de la Défense nationale), 2019

CAN UNCLASSIFIED

IMPORTANT INFORMATIVE STATEMENTS

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Disclaimer: This publication was prepared by Defence Research and Development Canada an agency of the Department of National Defence. The information contained in this publication has been derived and determined through best practice and adherence to the highest standards of responsible conduct of scientific research. This information is intended for the use of the Department of National Defence, the Canadian Armed Forces (“Canada”) and Public Safety partners and, as permitted, may be shared with academia, industry, Canada’s allies, and the public (“Third Parties”). Any use by, or any reliance on or decisions made based on this publication by Third Parties, are done at their own risk and responsibility. Canada does not assume any liability for any damages or losses which may arise from any use of, or reliance on, the publication.

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DRDC-RDDC-2019-R147 i

Abstract

The dynamic mechanical properties of materials used in defense applications, under high strain rate and shock loading, are necessary in designing and optimizing their capabilities under such conditions. The Split Hopkinson Pressure Bar (SHPB) technique has been a popular test method for characterizing the properties of materials under dynamic loading at moderate to high strain rates by predicting the propagation of pressure waves in long slender bars. This Scientific Report outlines the procedure for predicting the propagation of pressure waves in an elastic split Hopkinson bar system and correcting for dispersion and radial inertia effects. The procedure is then used to characterize the high strain rate properties of aluminum alloy 6061-T6 under impact and validated against the literature.

Significance to defence and security

This Scientific Report aims to facilitate and improve the efficiency of the characterization of metals and ceramic materials under impact loading using the elastic split Hopkinson bar system at DRDC – Valcartier Research Centre by providing a clear step-by-step procedure for analyzing raw strain gage data and accurately predicting the dynamic response of the test material. The procedure is versatile in that it accommodates all necessary corrections for pressure bar distortion effects using spectral wave analysis for any isotropic elastic bar material with Poisson’s ratio between 0.1 and 0.4, and any bar radius and length.

ii DRDC-RDDC-2019-R147

Résumé

Pour concevoir des matériaux employés aux fins d’applications de défense impliquant des vitesses de déformation élevées et de puissants chocs de chargement, ainsi que pour optimiser les capacités de ces matériaux dans telles conditions, il est important de tenir compte de leurs propriétés mécaniques dynamiques. La technique de la barre de pression Hopkinson (Split Hopkinson Pressure Bar – SHPB) constitue une méthode répandue de caractérisation des propriétés de matériaux soumis à des charges dynamiques impliquant des vitesses de déformation moyennes à élevées, grâce à la prévision de la propagation d’ondes de pression dans de longues barres effilées. Le présent article donne un aperçu d’une procédure suivie pour prévoir la propagation d’ondes de pression dans un système élastique de barre Hopkinson, de même que pour corriger des effets de dispersion et d’inertie radiale. La procédure sert ensuite à caractériser les propriétés de vitesse de déformation élevée de l’alliage d’aluminium 6061-T6 lorsqu’il subit des impacts, ainsi qu’à valider ces propriétés en fonction de ce qui figure dans d’autres articles.

Importance pour la défense et la sécurité

Le présent rapport scientifique vise à faciliter et à améliorer la caractérisation de métaux et de matériaux céramiques soumis à des chocs de chargement d’après le système élastique de barre Hopkinson au RDDC – Centre de recherches Valcartier grâce à l’élaboration d’une procédure systématique claire d’analyse de données extensométriques brutes et de prévision exacte de la réaction dynamique des matériaux éprouvés. La procédure est polyvalente en ce sens qu’elle permet toutes les corrections nécessaires en ce qui concerne les effets de déformation de barre de pression, à l’aide d’une analyse d’onde spectrale ciblant toute barre élastique isotrope présentant un coefficient de Poisson de 0,1 à 0,4, ainsi que tout rayon et toute longueur de barre.

DRDC-RDDC-2019-R147 iii

Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Significance to defence and security . . . . . . . . . . . . . . . . . . . . . . . . . i

Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Importance pour la défense et la sécurité . . . . . . . . . . . . . . . . . . . . . . . ii

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 The Hopkinson Bar Technique . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 Hopkinson bar measurement techniques . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Strain-gage non-linearity correction . . . . . . . . . . . . . . . . . . . 3 2.1.2 Strain-gage static and dynamic calibration correction . . . . . . . . . . . . 4 2.1.3 Strain-gage attenuation correction based on gage length . . . . . . . . . . . 6 2.1.4 Strain-gage misalignment and transverse sensitivity corrections . . . . . . . . 6

2.2 Specimen response in a Hopkinson Bar System . . . . . . . . . . . . . . . . . 7

3 Hopkinson bar distortion effects and corrections . . . . . . . . . . . . . . . . . . 12 3.1 Spectral wave analysis using the Discrete-Time Fourier Transform (DTFT) . . . . . . 12 3.2 Phase velocity dispersion correction . . . . . . . . . . . . . . . . . . . . 14 3.3 Radial inertia effects and prediction of axial bar stress and strain . . . . . . . . . . 16 3.4 Summary of procedure . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Characterization of aluminum alloy 6061-T6 . . . . . . . . . . . . . . . . . . . 20 4.1 Strain-Gage Calibration and time-domain corrections . . . . . . . . . . . . . . 20 4.2 Spectral wave analysis and frequency-domain corrections . . . . . . . . . . . . 22

4.2.1 Evaluation of dispersed phase velocities . . . . . . . . . . . . . . . . 23 4.2.2 Separation of Incident, Reflecting, and Transmitting Wave Pulses for Dispersion

Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2.3 Evaluation of specimen stress-strain response . . . . . . . . . . . . . . 28

4.3 Frequency filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.4 Validation of results . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

List of symbols/abbreviations/acronyms/initialisms . . . . . . . . . . . . . . . . . . 36

iv DRDC-RDDC-2019-R147

List of figures

Figure 1: Schematic of Split Hopkinson Pressure Bar with two strain gages. . . . . . . . . . 2

Figure 2: Wheatstone bridge configuration and linear foil gage alignment on the Hopkinson bars at DRDC – Valcartier Research Centre. . . . . . . . . . . . . . . . . . . . . 3

Figure 3: The various forms of drift between two data sets observed in static calibration of strain gages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Figure 4: Expanded view of a split Hopkinson bar showing the direction of forces, velocities, displacements and the prediction of typical strain signal pulses from the location of the strain gage to the bar-specimen interface. . . . . . . . . . . . . . . . . . . . 7

Figure 5: Free body diagram of a one dimensional bar element showing stresses acting on each face of the element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Figure 6: Resultsof a dynamic calibration of the Hopkinson bar system at 15 m/s. . . . . . . . 21

Figure 7: Strain signals measured from input and output bar strain gages in a test of AA 6061-T6 impacted at 17.6 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Figure 8: Amplitude spectrum of strain signals from a test of AA 6061-T6 impacted at 17.6 m/s. 23

Figure 9: Effect of dispersion on dilatational phase velocity of pressure waves in maraging steel pressure bars based on bar radius, for Poisson’s ratio = 0.3. . . . . . . . . . . . . 24

Figure 10: Incident, reflected, and transmitted bar strain signals for evaluating specimen response of AA 6061-T6 at 17.6 m/s. . . . . . . . . . . . . . . . . . . . . . . . . 25

Figure 11: Predicted incident, reflected, and transmitted bar strain pulses at bar-specimen interfaces for AA 6061-T6 at 17.6 m/s. . . . . . . . . . . . . . . . . . . . . 26

Figure 12: Effect of dispersion on incident pulse for AA 6061-T6 at 17.6 m/s. . . . . . . . . 26

Figure 13: Effect of dispersion on reflected pulse for AA 6061-T6 at 17.6 m/s. . . . . . . . . 27

Figure 14: Effect of dispersion on transmitted pulse for AA 6061-T6 at 17.6 m/s. . . . . . . . 27

Figure 15: Engineering stress-strain responses of all tested specimens of AA 6061-T6. . . . . . 28

Figure 16: Effect of varying the cutoff frequency on the attenuation of the input bar voltage signal of Specimen #3 using a critically-damped recursive low-pass filter with 50 passes. . . 30

Figure 17: Effect of critically-damped filtering on the attenuation of the stress-strain response of Specimen #3 showing an overestimation of the yield point as cut-off frequency is decreased. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

DRDC-RDDC-2019-R147 v

List of tables

Table 1: Experimental parameters for testing AA 6061-T6. . . . . . . . . . . . . . . . 20

Table 2: Static calibration of strain gages on split Hopkinson bar at DRDC – Valcartier Research Centre. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Table 3: Static calibration coefficients used to correct strain signal. . . . . . . . . . . . . 21

Table 4: Fitting function coefficients for evaluating phase velocities of pressure waves in maraging steel bars with a Poisson’s ratio of 0.3 as a function of wavelength, and bar radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

vi DRDC-RDDC-2019-R147

Acknowledgements

The author would like to thank Luc Légaré for his assistance in calibration, testing, and troubleshooting on the split Hopkinson pressure bar system at DRDC – Valcartier Research Centre.

DRDC-RDDC-2019-R147 1

1 Introduction

A necessary step in designing structures and components for defense applications is characterizing the behavior of the material components under dynamic loading and shock. High strength armor materials, such as metallic alloys and ceramics, are designed to withstand high rates of deformation ranging from blast loading to projectile impact. The popular Split Hopkinson Pressure Bar (SHPB) technique, developed by Kolsky [1], was created from the need to reliably quantify the dynamic response of materials under extreme loading conditions by confining all the energy of impact and the propagation of pressure waves to one-dimension through long slender elastic pressure bars. By placing a much shorter cylindrical specimen between two pressure bars, end-to-end, and impacting one end of the bars, it was possible to estimate the dynamic stress and strain response of the specimen under uni-axial impact based upon the characteristics of the pressure waves propagating through the bars. The experimental technique is limited to impact forces that do not plastically deform the bars so that the bars remain elastic throughout testing and the specimen deforms uniformly. Since its inception the Hopkinson bar technique has been modified to test other modes of deformation including tension and torsion.

The SHPB was originally designed to operate at impact velocities producing low to moderate strain rates of deformation (102–103/s) so that wave propagation through the pressure bars can be assumed one-dimensional and errors due to dispersion effects neglected. However, at higher strain rates (103–104/s), often measured during blast loading or high speed projectile impact, high frequency vibrations are produced within the bars. The phase velocities of these frequency components do not travel at the fundamental speed of sound in the bar material, but decrease with increasing frequency of vibration creating phase lag as the pressure waves propagate along the bar. This phase lag is often referred to as dilatational (longitudinal) phase velocity dispersion and is a function of the bar Poisson’s ratio, diameter, distance traveled, and wavelength or frequency of vibration. Additionally, as the wavelengths of the higher frequency components approach the bar lateral dimension, radial inertia effects produce a variation in the stress and strain response through the cross-section of the bars, even for a uniformly applied impact load. Increasing the force of impact typically increases dispersion in the bars while increasing the bar diameter reduces dispersion, by reducing vibration, but at the cost of losing one-dimensional propagation and producing radial (distortional) dispersion. It therefore becomes necessary to correct for these distortion effects when predicting the wave form at any location in the bar and is an integral step towards reliably characterizing the dynamic response of tested specimens using the Hopkinson bar technique.

This paper reviews the procedure for predicting the propagation of pressure waves in elastic pressure bars using spectral wave analysis while correcting for dispersion and radial inertia effects. The procedure is then implemented to characterize the dynamic response of specimens of aluminum alloy 6061-T6 using the Split Hopkinson Pressure Bar system at DRDC – Valcartier Research Centre. Validation of the procedure is carried out by comparison with results from the literature.

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6 DRDC-RDDC-2019-R147

2.1.3 Strain-gage attenuation correction based on gage length

In the development of the Hopkinson bar system at DRDC – Valcartier Research Centre, Bolduc and Arsenault [3] and Kaiser [4] showed that there exists an error induced by strain gage length whereby the frequency resolution is lost for wavelengths near that of the gage length. Also, the output from the gage is zero when the gage length is equal to the wavelength or when the gage length is an integer multiple of the wavelength. Shorter gages offer greater frequency response. Kaiser analyzed the effect of strain gage length on the strain signal and evaluated the associated errors related to attenuation, or loss in signal amplitude. Based on this analysis, Bolduc and Arsenault developed a correction factor for the strain signal to compensate for this error, expressed in the frequency-domain as

where ε(ω) is the measured strain signal in the frequency-domain, transformed using Fourier analysis; ‘ω = γω0 =2πγ/T’ is the angular frequency of each frequency component of the strain signal, where ‘γ’ is the wave number, and ‘T’ is the total time of the signal; ‘C0’ is the bar velocity or sound speed in the bar, and ‘Lg’ is the length of the gage. It will be shown in later sections, that when bar dispersion effects are considered, the attenuation correction is no longer a function of the bar velocity, C0, but of the phase velocity, Cω, which diminishes with increasing frequency.

2.1.4 Strain-gage misalignment and transverse sensitivity corrections

Transverse sensitivity in a strain gage refers to its response to strains perpendicular to the primary sensing direction of the gage [5]. In Hopkinson bar testing, there exist some lateral vibrations that can affect the measurements of strain. It is important to correct the strain measurements for these effects. Additionally, any misalignment in the gages needs to also be corrected. Bolduc and Arsenault [3] showed that the strain signal can be corrected for misalignment and transverse sensitivity using Equation (10), expressed as

where ‘ν’ is Poisson’s ratio, ts is the transverse sensitivity factor supplied with the strain gages, ε(t) is the uncorrected strain measurement, ‘α’ is the angle between the gage’s axial direction, and the bar’s axial direction, and sin and cosine terms are the mean of the absolute of the sine and cosine of the offset angle, respectively.

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where ‘f’ and ‘g’ are arbitrary functions describing the shape of the propagating wave. The function ‘f’ corresponds to a forward traveling wave in the positive x direction; while function ‘g’ corresponds to a wave traveling backward in the negative x direction, and can be thought of as a reflected wave. The strain in the bar can be derived as a function of the forward and backward traveling strain pulses by taking the partial derivative of Equation (14) with respect to x, as

The particle velocity, or the velocity at which a single point within the bar moves as the wave passes through it, can be derived from Equation (13), and Hooke’s laws as,

Particle velocities depend on the applied loading while the wave velocities, or phase velocities, are an intrinsic material property. From Equation (11) and (13), the particle forces in bar can be expressed as

In a Hopkinson bar system using elastic pressure bars, the instantaneous forces at each bar-end in contact with the specimen can be expressed using Equation (18) as

(14)

(15)

(16)

(17)

(18)

, 0 0

0 0

2

2 02

2

2 02

2

2

, , ,

, 0 0 0 0 , ,

,2

2

2

2

2

2 , ,


where Abar is the cross sectional area of the bar, and ‘Ebar’ is the Young’s modulus. Each bar contains forward and backward traveling waves denoted as incident (forward) and reflected (backward) in the input bar and transmitted (forward) in the output bar. The average (engineering) stress response of the specimen can be evaluated by taking the average of the forces at the two bar-ends in contact with the specimen on the face of the specimen, expressed in Equation (20), as

where Aspec is the cross sectional area of the specimen. The engineering strain rate response in the specimen can be evaluated as a function of the difference between the particle velocities determined at each bar-specimen interface, shown in Equation (21), as

and the engineering strain can be evaluated by taking the integral of the strain rate with respect to time, expressed in Equation (22), as

The true strain and strain rate in the material can be determined using high speed imaging or digital image correlation techniques to measure the instantaneous change length and cross-sectional area of the specimen during deformation. The true strain response of a specimen can be evaluated using Equation (23), expressed as

(19)

(20)

(21)

(22)

2⁄

2

2

⁄ 0

⁄ 0

0


where ‘Lf(t)’ is the instantaneous length and ‘Li’ is the initial length of the specimen. The true strain rate is evaluated by taking the derivative of Equation (23) with respect to time and the true stress can be evaluated using Equation (24), expressed as

(23)

(24)

ln

exp

Under certain conditions, bar distortion effects become significant. The degree of distortion depends on a number of factors including the bar diameter, the energy of impact, and the bar material. A large diameter bar will have more distortion than a small diameter due to an increase in lateral vibrations. Similarly, a higher impact force will produce higher vibrational frequencies which can distort the signals. When distortion effects are considered, wave propagation is no longer one-dimensional. The measured strain signals in the bars vary axially and radially. To properly predict the signals at the bar-specimen interaces, the measured signals must be corrected for bar distortion.

Follansbee and Frantz [6] and Zhao et al. [7] argued that in typical Hopkinson bar tests, using either elastic or viscoelastic bars, errors induced by dispersion effects are negligible and the stress state is almost one-dimensional through the cross section. On the other hand, Tyas and Watson [8] showed that a pressure wave signal in a Hopkinson bar will propagate one dimensionally at the bar velocity,

⁄ , only if all the energy of the impact load is contained within frequency components of low wavelength, below which a/Λ < 0.05-0.10, where ‘a’ is the bar radius and ‘Λ’ is the wavelength of a frequency component of the signal. For a steel bar this is approximately 250/a to 500/a kHz where ‘a’ is measured in mm. The magnitude of dispersion also depends on the Poisson’s ratio of the material. If the wave signal contains higher frequency ultrasonic signal components, as is often the case in the measurement of impact or blast pressures, the phase velocities of the frequency components of the signal are no longer equal to the bar velocity but are a function of the frequency of vibration, typically decreasing with increasing frequency. This effect is called dilatational phase velocity dispersion. Additionally, as the wavelengths of the higher frequency components approach the bar lateral dimension, radial inertia effects become important, which produce a variation in the stress and strain generated across the cross-section of the bars, even for uniformly applied loading. The quality of results from a Hopkinson bar analysis is improved if corrections are made for dispersion and radial inertia. The steps involved in this process are outlined in the next section.


3 Hopkinson bar distortion effects and corrections

Two major sources of distortion in Hopkinson bars are longitudinal (or dilatational) phase velocity dispersion and radial inertia effects whereby the signals disperse axially and radially and wave propagation is no longer one-dimensional. In Hopkinson bar analysis, dispersion refers to the lateral “widening” of a signal or wave as it propagates axially. At the same time, the signal may attenuate where its amplitude or the amplitude of its frequency components diminishes. Attenuation is especially significant in viscoelastic bar materials, which act as acoustic dampeners. The phase velocities of the frequency components of a pressure wave signal affected by dispersion are no longer equal to the bar velocity but diminish with increasing frequency, the degree of which is dependent on the bar diameter, Poisson’s ratio, and wavelength of the frequency components. Additionally, at some higher frequencies, radial inertia produces a variation in the stress and strain generated across the cross-section of the bars, even for a uniformly applied load, such that any strain measurements at the bar surface are not the same as along the axis. Thus, it becomes necessary to correct a strain signal both for longitudinal dispersion and radial inertia if it is to be predicted correctly from the strain gage reading to the specimen-bar interface. Both phase velocity dispersion and radial inertia effect corrections are applied to the measured signals in the frequency domain using discrete Fourier analysis.

This section provides the theory and procedure for correctly predicting the stress and strain signals at any location in the bars, while correcting bar distortion effects.

3.1 Spectral wave analysis using the Discrete-Time Fourier Transform (DTFT)

Spectral wave analysis is the analysis of waves in terms of their spectrum of frequencies. When used to analyze wave propagation in pressure bars, it provides a way to shift a signal in time or spacial location while accounting for dispersion and radial inertia effects by altering the phase angles and amplitudes of the frequency components of the signal. A strain signal measured as a function of time, can be expressed as the sum of harmonic waves using a Discrete-Time Fourier Transformation (DTFT) formula by converting it to the the frequency domain, shown in Equation (25) as

(25)

where ‘ω=γω0’ is the angular frequency of each frequency component of the wave signal, ‘γ’ is the wave number in the frequency domain, ‘ω0=2π/T’ is the fundamental angular frequency, ‘T’ is the total time of the signal, ‘N’ is the total number of time increments or strain data readings, and ‘Δt=T/N’ is the time increment. A sampling rate of ‘N/2T’ is chosen to limit a distortion typically associated with this transformation called aliasing, which results in differences between the original signal and the reconstructed one at higher frequencies. This is called the Nyquist sampling frequency or Nyquist criterion. As a result the reliable frequency-domain signal is limited to N/2 frequencies. The phase angle and amplitude of the frequency-domain strain signals are

0

∙ 0 ∆ ∙ 0 ∆ ∙ 0 ∆0


(26)

and

(27)

The frequency domain strain signal can also be expressed using its amplitude and phase angle as

(28)

Any corrections for dispersion and radial inertia effects are made to the amplitude and the phase angle of each frequency component prior to converting back to the time-domain using the Inverse Discrete-Time Fourier Transform (IDTFT), expressed in Equation (29), as

(29)

Typically, only the real portion of the bar strain signal is useful in characterizing the test material and imaginary portions are ignored. The real portion of Equation (29) is expressed as

(30)

The values determined by Equation (30) represent the distortion corrected strain signal at the bar-specimen interface. It will be shown that dispersion corrections are made by shifting the phase angle of each frequency component, φ, by an amount dependent on the the phase velocity and distance between the strain gages and the bar-specimen interfaces. Radial inertia effect corrections are made to the amplitude of the signals |ε(ω)|.

PhaseAngle arg atan2

Amplitude/2

2 2

/2

cos sin

2

∙ 0 ∆

2⁄

0

2 cos sin ∙ 0 ∆

2⁄

0

2 cos sin ∙ cos 0 ∆ sin 0 ∆

2⁄

0

2 cos 0 ∆ sin 0 ∆

/2

0

2 cos cos 0 ∆ sin sin 0 ∆

/2

0


3.2 Phase velocity dispersion correction

The effect of distortion on pressure waves in long slender bars has been studied and analyzed over many years. Several algorithms to correct for distortion effects have been developed and continue to be improved upon for several bar materials in order to expand the capabilities of the Hopkinson bar technique. For elastic bars, these analyses are based upon the closed form solution of the Pochhammer-Chree equation of motion for infinitely long bars, despite it not being strictly valid for analysis of transient pulses in finite bars but far more computationally efficient than numerical solutions. Pochhammer (1876) [9] and Chree (1889) [10] independently developed solutions of equations of motion for idealized cases of infinitely long bars subjected to forcing functions of infinite duration. Love (1934) [11] was the first to develop a frequency equation by transforming the three-dimensional Pochhammer-Chree equation of motion of a solid circular rod into cylindrical coordinates and applying traction-freesurface boundary conditions. Bancroft (1941) [12] solved Love’s frequency equation numerically for thefirst root differing from zero, referring to the first mode of vibration (longitudinal wave propagation). Hissolution relates dilatational phase velocity to bar radius and wavelength of frequency components forPoisson’s ratios ranging between 0.1 and 0.4. The second and third roots of the Pochhammer-Chreefrequency equation correspond to the second mode of vibration (distortional) and third mode of vibration(Rayleigh surface wave propagation), respectively. From Bancroft [12], the frequency equation, derivedby Love [11] takes the form,

(31)

where

(32)

and ‘µ’ is the shear modulus or modulus of rigidity of the bar material, ‘E’ is Young’s modulus, ‘v’ is Poisson’s ratio, ‘λ’ is Lame’s first parameter, ‘ρ’ is the density of the bar material, ‘a’ is the bar radius, ‘Cω’ and ‘Ct’ are the dilatational and shear (transverse) phase velocities of the frequency components in an infinite bar, ‘C0’ is the bar velocity or fundamental sound speed in the bar, Λ=(2π/γ) is the wavelength of the frequency components, and ‘Jn’ defines a Bessel function of the first kind of order ‘n’. Bancroft simplified Love’s equation to

2202 2

J0 2 1

2 0 2 22

1

0,

1 ,

2 1 ,

1 2 1⁄ ,

0

2

1 ,

2 ⁄ ⁄ ,

⁄ ,

0 ⁄ ,

0.5 1⁄ ,


(33)

where

(34)

The first root of Equation (33), differing from zero and solved explicitly for ‘x’ corresponds to the first mode vibration for the propagation of longitudinal waves. Bancroft computed the first roots for values 0 < γa < ∞ and for Poisson’s ratios 0.1 < ν 0.4 in increments of 0.05 and tabulated his results in [12]. Later Gong et al. [13] developed a fitting function, shown in Equation (35), that accurately predicts the phase velocity as a function of wavelength and bar radius and fits well the dispersion data computed by Bancroft. It is expressed by

(35)

where Cω/C0 is the normalized phase velocity, Λγ is the wavelength of the frequency component ‘γ’, ‘a’ is the bar radius, and A, B, C, D, E, and F are the coefficients derived for each Poisson’s ratio. These coefficients can be solved using regression analysis techniques.

From these analyses, Gorham (1983) [14] and Follansbee and Franz (1983) [6] were the first to to develop an algorithm to correctly predict dispersion of a traveling wave based on the first mode of the Pochhammer-Chree dispersion relation using a Hopkinson bar to test for ductile metallic materials. They showed that a properly dispersed waveform could be predicted at any location along the bar by applying a phase shift to each phase angle of the frequency components of the strain signal in the frequency domain, which is dependent on the phase velocity of each frequency component as well as the axial distance to shift the signal. The accuracy of this method was limited to moderately dispersed signals [8] [15], losing accuracy for highly dispersed signals due to the possibility that some of the energy of the signal carried by higher frequency components travels at higher modes. Li and Lambros [16] modified this method to better visualize the shift in the pressure pulse in both distance and time. This phase shift is expressed by Equation (36), as

(36)

where Δx is the axial distance—a positive value referring to a forward shift and a negative value referring to a backward shift. The phase shift represents the amount of dispersion produced in a frequency component of a signal traveling an axial distance of Δx along the bar. In Hopkinson bar analyses, Δx would be the distance between the strain gage and the bar-specimen interface. The dispersion corrected phase angle would then be calculated as

(37)

, , 1 2 1 0,

0

1

0Λ

4

Λ

3

Λ

2

Λ

1.5

1,

0∆

,

arg ,


where arg(εγ(ω)) is the phase angle of the strain signal in the frequency domain.

3.3 Radial inertia effects and prediction of axial bar stress and strain

In his extensive analysis of Hopkinson bars, Davies (1948) [17] discovered that at some higher frequencies, the lateral inertia in Hopkinson bars causes the magnitude of the stress to vary across the radial ordinate and that the ratio of the axial stress to axial strain (Young’s modulus) is not constant across the bar cross-section and varies with both frequency and radial ordinate. Davies developed equations to calculate the average stress and particle displacement over the bar cross section which were then analyzed by Tyas and Watson [8] who developed correction factors that transformed the surface strains to the average axial strain and then from the axial strain to the average axial stress. According to Davies, for an infinitely long rod subjected to a plane sinusoidal axial disturbance of infinite duration, at any time ‘t’ and at a location (x,r) where ‘x’ refers to the axial direction and ‘r’ refers to the radial direction, the axial stress, σxx, and particle displacement, ux, can be expressed as

(38)

and

(39)

where ‘A’ is a constant defining the magnitude of oscillation, ‘i=√ 1’, and

(40)

and

(41)

Davies pointed out that the values of f1 and f2 cannot be complex meaning that the particle velocities and displacements must be in the same phase or in antiphase at any instant at all points on the cross-section of the bar. Davies showed that the average values of particle velocity and axial stress could be derived using the average value of f1 and f2 over the bar cross-section as

(42)

and

2 2 11 ,

1 ,

1 1 0

1

11

0

1,

0

1

11

0

1.

1

22 1

02 1 2 1

1

1,

1

0


(43)

From Davies’ analysis, Tyas and Watson [8] derived two correction factors, M1(ω) , and M2(ω) , which are multiplied with the amplitude of each frequency component of the strain signal to convert the surface strain measured by the strain gages to the average axial strain and then to the average axial stress. The first correction factor, M1(ω), is essentially the ratio of the average axial strain over the bar cross section to the axial strain at the bar surface and the second correction factor, M2(ω), is the ratio of the average axial stress to the average axial strain for a given wavelength (equal to the dynamic elastic modulus of the bar). These factors are defined in Equations (44) and (45), as

(44)

and

(45)

The magnitudes of the average axial strain and stress are then calculated using Equations (46) and (47), expressed as

(46)

and

(47)

These magnitudes can then be used to construct the frequency domain average axial bar strain, shown in Equation (48), and average axial bar stress, shown in Equation (49), using the dispersion corrected phase angles from Equation (37), as

(48)

and

2

22 2

02 2 2 1

1

1.

1

0

1

2

2

2 1 11

0

1

11

0

1

2

2 2 11

2 2 12

0

2

.

| | 1 ω ,

| | 1 ω 2 ω .

1 1 cos 1 sin


(49)

From Equation (29), the average axial strain signal in the time domain, obtained by performing an Inverse Discrete-Time Fourier Transformation (IDTFT), is expressed as

(50)

And the stress signal is expressed as

(51)

where the real portions are expressed as

(52)

and

(53)

The average bar strain and stress values evaluated using Equations (52) and (53) represent the predicted and dispersion-corrected axial strain and stress response of the bar at the two bar-specimen interfaces, which can be used to evaluate the dynamic properties of the specimen using equations in Section 2.2.

1 2 1 2 cos 1 2 sin

2∙ 0 ∆

2⁄

0

21 ω cos sin cos 0 ∆ sin 0 ∆

2⁄

0

2∙ 0 ∆

2⁄

0

21 ω 2 ω cos sin cos 0 ∆ sin 0 ∆

2⁄

0

21 cos cos 0 Δ sin cos 0 Δ

/2

0

21 2 cos cos 0 Δ sin cos 0 Δ

/2

0


3.4 Summary of procedure

The following steps outline the procedure for analyzing experimental strain gage data, applying corrections, and evaluating the dynamic response of a specimen using spectral wave analysis —noting that the calibration and strain gage correction procedures specifically follow [3] and apply primarily to the elastic Hopkinson bar system at DRDC – Valcartier Research Centre:

1) Install and calibrate the strain gages: apply corrections as per in Section 2.1.

2) Perform impact tests and acquire strain gage data. Convert voltage to strain using Equation (1).

3) Apply time-domain strain-gage corrections: strain gage misalignment and transverse sensitivity corrections using Equation (10).

4) Apply frequency-domain strain gage corrections: Covert the time-domain bar strain signals, ε(t), to the frequency-domain, ε(ω), using the discrete Fourier transformation in Equation (25). Apply the attenuation correction for gage length to to ε(ω), using Equation (9). Then evaluate the amplitude and phase angle of each frequency component of corrected signal using Equations (26) and (27).

5) Determine the phase velocities, Cω, for each frequency component using the fitting function, in Equation (35), on Bancroft’s tables from [12] for the Poisson’s ratio of the bar material. Evaluate the fitting function’s coefficients using methods such as regression analysis.

6) Correct the phase angles for dispersion by evaluating the phase shifts for each frequency component using Equation (36) using a linear distance Δx equal to the distance between the strain gage and the bar-surface interface. Correct the phase angles, evaluated in step 4, by applying the phase shift to each frequency component using Equation (37).

7) Evaluate the average axial bar strain and stress to correct for radial inertia effects. Determine the amplitude correction factors M1(ω) and M2(ω) for each frequency component of the bar strain signals using Equations (44) and (45) and apply them to the surface strain signal amplitudes using Equations (48) and (49). Transform to the time domain and evaluate the real portion using Equations (52) and (53).

8) Evaluate the dispersion-corrected average stress response in the specimen using the calculated average axial bar stress for each pulse in Equation (20).

9) Evaluate the dispersion-corrected strain rate response in the specimen using the calculated average axial bar strain for each pulse in Equation (21).

10) Evaluate the dispersion-corrected strain response in the specimen using Equation (22).

11) Evaluate the true stress and strain values if instantaneous length and diameter measurements are available using Equations (23) and (24).


4 Characterization of aluminum alloy 6061-T6

A total of four impact tests were performed at various impact velocities on specimens of aluminum alloy 6061-T6 at room temperature using the split Hopkinson bar system at DRDC – Valcartier Research Centre. This alloy was chosen because of the extensive availability of impact data and is relatively strain rate insensitive up to approximately 1000/s [18]. The split Hopkinson bar system at DRDC – Valcartier Research Centre consists of two 800 mm long, 14.5 mm diameter maraging steel cylindrical bars; and a striker bar with the same diameter and a length of 15 mm. Each of the input and output bars are instrumented with two linear foil gages in the Wheatstone bridge configuration shown in Figure 2 as described in Section 2.1, and located midway (400 mm) across each bar. Cylindrical specimens were machined to a length of 5.00+0.02 mm and a diameter of 10.50+0.02 mm. The strain rate of deformation of each specimen was controlled by setting the air pressure of the system that propels the striker. Impact pressures were selected to achieve strain rates starting from approximately 1000/s and increasing in order to evaluate the strain rate sensitivity of the aluminum alloy. The strain rates were found to be very sensitive to the selected impact pressures. Impact velocities were recorded using an Enhanced Laser Velocity Sensor (ELVS) that reads the displacement of the striker during the last 25 mm of its travel, ending at the point of impact [3]. The experimental parameters are presented in Table 1 along with the calculated strain rates.

Table 1: Experimental parameters for testing AA 6061-T6.

Specimen # Input Pressure [psi]

Impact Velocity [m/s]

Average Strain Rate [/s]

1 13.5 16.2 821 2 13.6 16.9 1052 3 13.6 17.6 1311 4 13.8 19.1 1701

A detailed description of the Hopkinson bar system at DRDC Valcartier Research Centre and its components can be found in [3].

4.1 Strain-Gage Calibration and time-domain corrections

For all five tests, a static calibration was carried on the two bars with a load cell, each instrumented with the same wheatstone bridge configuration—the results of which are shown in Table 2. Voltage readings from the oscilloscope were converted to strain using Equation (1) carrying the 5% error from the gage factor and then non-linearity corrections were applied using Equation (2). Theoretical values of strain were calculated using Hooke’s law. The modulus of elasticity of the maraging steel bars is 190 GPa.

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There is a negligible change in the fitted stress-strain response for the two lowest strain rates (821/s and 1052/s) compared to that of the next highest response at 1311 /s for a similar jump in strain rate. This confirms the findings of [18] showing that aluminum alloy 6061-T6 is fairly strain rate insensitive up to approximately 1000/s.

4.3 Frequency filtering

To reduce noise and minimize the effects of lateral vibrations, it is possible to artificially or digitally filter out the high frequency components that cause noise. According to Tyas and Watson [8], complete removal of noise is possible by filtering out all high frequency components greater than 35 kHz or an angular frequency of 220 kHz.

The Butterworth digital filter is a second order two-pass filter that is commonly used to filter out noise from impulse signals. It is a sophisticated moving average type filter that can separate out different frequency components. However, it introduces some phase distortion that causes the smoothed data to be phase shifted forward in time. However, this shift is canceled out by also filtering the time component, which increases the filter order from second order to fourth order giving a sharper roll off. The second pass further smooths the data increasing the effective cutoff frequency by about 25%. Robertson and Dowling [19] introduced a critically damped digital filter by modifying the Butterworth signal to become zero-lag and increasing the number of achievable passes. For voltage signal processing this filtering function can be expressed as

′ ′ ′ (58)

where ‘n’ is the data point increment, ‘V’ is the unfiltered or previous pass strain, ‘V ′ ’ is the filtered or next pass, and

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Decreasing the cutoff frequency decreases noise but also increases attenuation and data loss. The results of filtering the voltage signal of Specimen #3 using 50 passes and various cutoff frequencies are shown in Figure 16.

Figure 16: Effect of varying the cutoff frequency on the attenuation of the input bar voltage signal of Specimen #3 using a critically-damped recursive low-pass filter with 50 passes.

The increase in attenuation associated with a decrease in cutoff frequency also results in a slight dampening of the strain rate response of the specimen. For Specimen #3 this dampening effect introduces error in the calculation of average strain rate by a decrease of 3.6% from 1311/s unfiltered to 1264/s at 40 kHz. For the specimens tested, decreasing the cutoff frequency below 40 kHz results in significant data loss. Despite the attenuation of the voltage signal for decreasing cutoff frequencies, there was no significant data loss in the stress-strain responses of the specimes. Low-pass filtering significantly reduced the scattering of data, which fit well with the power-law fitting function derived for the unfiltered data, as shown in Figure 17 for Specimen #3. This indicates that the fitting function provides a good estimation of the data. However, filtering produced an overestimation of the yield point (intercept of dynamic response and elastic curve) observed to be relatively consistent for all selected cutoff frequencies indicating that this point may lie on the moving average of the stress-strain signal around which the data is reduced through filtering. When using low pass filtering techniques it is important to carefully select the cutoff frequency that minimizes data scattering but also minimizes data loss.

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and A, B, C, m, and n are coefficients, the strain rate sensitivity parameter ‘C’ for aluminum alloy 6061-T6 has been found to range between 0.002 < C < 0.02 according to [22, 23]. Given a quasistatic strain rate of 0.0001/s, and assuming that all tests are perfomed at room temperature, this sensitivity range would increase the yield strength for a strain rate of 1311/s (Specimen #3) to between 247MPa and 319MPa and for a strain rate of 1730/s (Specimen #4) to between 248MPa to 320MPa. The calculated yield strengths for specimens #3 (275 MPa) and #4 (325 MPa) fall approximately within these ranges validating the analysis to within acceptable error.


5 Conclusion

Spectral wave analysis is a reliable technique that allows for both the filtering and correction of errors induced by vibrations and distortions typically found in elastic Hopkinson bar systems and provides a good approximation to the dynamic response of metallic materials under impact. This report outlined a step-by-step procedure that can be followed to analyze strain gage data and obtain meaningful results necessary in characterizing the impact properties of materials tested in an elastic Hopkinson bar system. The procedure and analysis of aluminum alloy 6061-T6 was found to produce fairly accurate results, validated by the literature.

Spectral wave analysis is also an integral method for analyzing experimental data obtained from viscoelastic Hopkinson pressure bar systems in testing soft materials such as rubbers, soft plastics, and foams where bar distortion effects and dampening is significantly higher than in elastic pressure bar systems. It remains a popular method for predicting strain signals along the bars, correcting for dispersion effects and estimating the average (engineering) dynamic response of soft materials under impact forces, in the absence of advanced measuring techniques such as digital image correlation (DIC), for measuring specimen true strain, and piezoelectric force transducers, for measuring interface forces.


References

[1] H. Kolsky, “An Investigation of Mechanical Properties of Materials at Very High Strain Rates of Loading,” Proc. Phys. Soc. Lon., vol. 62, no. B, pp. 676–700, 1949.

[2] V. P. Group, “Strain Gages and Instruments Tech Note TN-507-1: Errors Due to Wheatstone Bridge Nonlinearity,” 19 November 2010. [Online]. Available: http://www.vishaypg.com/docs/11057/tn5071.pdf. [Accessed December 2017].

[3] M. Bolduc and R. Arsenault, “Improving accuracy in SHPB,” Defence R&D Canada – Valcartier; TR 2005-380, 2008.

[4] M. Kaiser, “Advancements in the Split Hopkinson Bar Test,” Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1998.

[5] V. P. Group, “Errors Due to Transverse Sensitivity in Strain Gages,” 28 June 2011. [Online]. Available: http://www.vishaypg.com/docs/11059/tn509tn5.pdf. [Accessed 25 April 2018].

[6] P. Follansbee and C. Frantz, “Wave propagation in the split Hopkinson pressure bar,” J. Eng. Mat. Tech, vol. 105, pp. 61–66, 1983.

[7] H. Zhao, G. Gary and J. Klepaczko, “On the use of a viscoelastic split hopkinson pressure bar,” International Journal of Impact Engineering, vol. 19, no. 4, pp. 319–330, 1997.

[8] A. Tyas and A. J. Watson, “An investigation of frequency domain dispersion correction of pressure bar signals,” International Journal of Impact Engineering, vol. 25, pp. 87–101, 2001.

[9] L. Pochhammer, “Über die fortpflanzungsgeschwindigkeiten kleiner schwingungen in einem unbegrenzten isotropen kreiszylinder (On the propagation velocities of small vibrations in an infinite isotropic circular cylinder),” Zeitschrift für Reine und Angewandte Mathematik (Journal for Pure and Appllied Mathematics), vol. 81, pp. 324–336, 1876. (in German).

[10] C. Chree, “The equation of an isotropic elastic solid in polar and cylindrical coordinates, their solution and applicaitons,” Transactions of the Cambridge Philosophical Society, vol. 14, pp. 250–369, 1889.

[11] A. E. H. Love, “A Treatise on the Mathematical Theory of Elasticity Vol. 1, Cambridge: C. J. Clay, M.A. and Sons, at The Univeristy Press, 1934.

[12] D. Bancroft, “The velocity of longitudinal waves in cylindrical bars,” Physical Review, vol. 59, pp. 588–593, 1941.

[13] J. Gong, L. Malvern and D. Jenkins, “Dispersion investigation in the split Hopkinson pressure bar,” J. Eng. Mater. Technol, vol. 112, no. 3, pp. 309–314, 1990.

[14] D. Gorham, “A numerical method for the correction of dispersion in pressure bar signals,” J. Phys. E. Sci.Instrum, vol. 16, no. 6, pp. 477–479, 1983.


[15] D. Gorham and P. F. J. Pope, “An improved method for compressive stress-strain measurements at very high strain-rates,” Proc. Roy. Soc. London A, vol. 438, no. 1902, pp. 153–170, 1992.

[16] Z. Li and J. Lambros, “Determination of the dynamic response of brittle composites by the use of the split Hopkinson pressure bar,” Composites Science and Technology, vol. 59, pp. 1097–1107, 1999.

[17] R. M. Davies, “A critical study of the Hopkinson pressure bar,” Philosopical Transactions of the Royal Society London A, vol. 240, pp. 375–457, 1948.

[18] L. Manes, L. Peroni, M. Scapin and M. Giglio, “Analysis of strain rate behavior of an Al 6061 T6 alloy,” Procedia Engineering, vol. 10, pp. 3477–3482, 2011.

[19] D. Robertson and J. Dowling, “Design and response of Butterowrth and critically damped digital filters,” Journal of Electromyography and Kinesiology, vol. 13, pp. 569–573, 2003.

[20] R. Benck, G. Filbey Jr. and E. Murray Jr., “Quasi-static Compression Stress-Strain Curves – IV, 2024-T3510 and 6061-T6 Aluminum Alloys,” USA Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland, 1976.

[21] G. Johnson and W. Cook, “A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures,” in Proceedings of the Seventh International Symposium on Ballistics, The Hague, The Netherlands, 1983.

[22] B. Corbett, “Numerical simulations of target hole diameters for hypervelocity impacts into elevated and room temperature bumpers,” International Journal of Impact Engineering, vol. 33, pp. 431–440, 2006.

[23] A. Elsen, M. Ludwig, R. Schaefer and P. Groche, "Fundametnals of EMPT-Welding, “in Proceedings of 4th International Conference on High Speed Forming, Columbus, OH, 2010.

[24] S. Rigby and A. Barr, “A review of Pochhammer-Chree dispersion in the Hopkinson bar,” Engineering and Computational Mechanics, pp. 1–16, 2017.


List of symbols/abbreviations/acronyms/initialisms

DND Department of National Defence

DRDC Defence Research and Development Canada

DTFT Discrete-Time Fourier Transform

IDTFT Inverse Discrete-Time Fourier Transform

SHPB Split Hopkinson Pressure Bar

ε Strain

σ Stress

Vexc Bridge excitation voltage

Ag Amplifier gain

GF Gage factor

C0 Fundamental speed of sound in a material

Cω Phase velocity

E Young’s modulus

ν Poisson’s ratio

λ Lame’s first parameter

ρ Density

ω Angular frequency

γ Wave number

Λ Wavelength

Lg Gage length

Jn Bessel function of the first kind, order n

a Bar radius

φ Phase shift

DOCUMENT CONTROL DATA *Security markings for the title, authors, abstract and keywords must be entered when the document is sensitive

1. ORIGINATOR (Name and address of the organization preparing the document. A DRDC Centre sponsoring a contractor's report, or tasking agency, is entered in Section 8.)

DRDC – Valcartier Research CentreDefence Research and Development Canada2459 route de la BravoureQuébec (Québec) G3J 1X5Canada

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3. TITLE (The document title and sub-title as indicated on the title page.)

On spectral wave analysis with distortion corrections applied to elastic Hopkinson bar systems

4. AUTHORS (Last name, followed by initials – ranks, titles, etc., not to be used)

Polyzois, I.

5. DATE OF PUBLICATION (Month and year of publication of document.)

August 2019

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45

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24

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Scientific Report

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DRDC – Valcartier Research CentreDefence Research and Development Canada2459 route de la BravoureQuébec (Québec) G3J 1X5Canada

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DRDC-RDDC-2019-R147

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12. KEYWORDS, DESCRIPTORS or IDENTIFIERS (Use semi-colon as a delimiter.)

Spectral wave analysis; Hopkinson pressure bar; Impact Dynamics; Material Characterisation;aluminum 6061-T6

13. ABSTRACT (When available in the document, the French version of the abstract must be included here.)

The dynamic mechanical properties of materials used in defense applications, under high strain rate and shock loading, are necessary in designing and optimizing their capabilities under such conditions. The split Hopkinson pressure bar (SHPB) technique has been a popular test method for characterizing the properties of materials under dynamic loading at moderate to high strain rates by predicting the propagation of pressure waves in long slender bars. This paper outlines the procedure for predicting the propagation of pressure waves in an elastic split Hopkinson bar system and correcting for dispersion and radial inertia effects. The procedure is then used to characterize the high strain rate properties of aluminum alloy 6061-T6 under impact and validated against the literature.

Pour concevoir des matériaux employés aux fins d’applications de défense impliquant des vitesses de déformation élevées et de puissants chocs de chargement, ainsi que pour optimiser les capacités de ces matériaux dans telles conditions, il est important de tenir compte de leurs propriétés mécaniques dynamiques. La technique de la barre de pression Hopkinson (Split Hopkinson Pressure Bar – SHPB) constitue une méthode répandue de caractérisation des propriétés de matériaux soumis à des charges dynamiques impliquant des vitesses de déformation moyennes à élevées, grâce à la prévision de la propagation d’ondes de pression dans de longues barres effilées. Le présent article donne un aperçu d’une procédure suivie pour prévoir la propagation d’ondes de pression dans un système élastique de barre Hopkinson, de même que pour corriger des effets de dispersion et d’inertie radiale. La procédure sert ensuite à caractériser les propriétés de vitesse de déformation élevée de l’alliage d’aluminium 6061-T6 lorsqu’il subit des impacts, ainsi qu’à valider ces propriétés en fonction de ce qui figure dans d’autres articles.

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l wave sis with distortion ms · DRDC-RDDC-2019-R147 i Abstract The dynamic mechanical properties of materials used in defense applications, under high strain rate and shock loading,