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Inversion for the complex elastic modulus of material from spherical wave propagation data in free field
Journal of Sound and Vibration 459 (2019) 114851
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Inversion for the complex elastic modulus of material from spherical wave propagation data in free field Qiang Lu*, Zhan-jiang Wang**, Yang Ding Northwest Institute of Nuclear Technology, Xi'an, 710024, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 September 2018 Received in revised form 29 June 2019 Accepted 9 July 2019 Available online 14 July 2019 Handling Editor: P. Tiso
A spherical wave experiment is conducted for polymethyl methacrylate (PMMA), with a spherical explosive charge of 0.125 g TNT as the source. The aim is to investigate the dynamic mechanical behavior of a viscoelastic material from the multiple measured radial particle velocities during the spherical wave propagation in a free field. Without using conventional mathematical models, a brief and effective method is suggested to determine the complex elastic modulus of the material through the combination of the viscoelastic spherical stress wave theory and the measured radial particle velocities at different radii in a free field. The complex elastic modulus, determined from the spherical wave data under different stress states, is compared with those evaluated through the conventional mathematical models or experimental results selected from the literature. The results show that PMMA behaves viscoelastically under spherical explosion loading. With increasing the propagation distance of spherical wave, the radial stress decreases from 682 to 40 MPa, and the radial strain decreases from 59.5 103 to 4.2 103. Meanwhile, the average storage modulus of PMMA, at high frequencies after approximately 300 kHz, decreases from 7.14 to 5.97 GPa. This indicates that the complex elastic modulus of PMMA is dependent on the stressestrain state, and the published parameters of viscoelastic materials, determined by different experimental techniques under different stressestrain states, should be used with caution. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Viscoelastic Spherical stress wave Propagation coefficient Complex elastic modulus
1. Introduction It is well known that the propagating character of stress waves is strongly dependent on the dynamic constitutive behavior of materials. The strain-rate effects become important for investigating the propagation of stress wave under explosion or impact loading. Thus, the viscosity of the material should be taken into account. In a split Hopkinson pressure bar (SHPB) apparatus, viscoelastic bars were recommended as the incident and transmitted bars for investigating the dynamic mechanical behavior of the low impedance material. This technique can obtain sufficiently high transmitted signals and increase the signal-to-noise ratio [1]. Owing to the attenuation and dispersion effects of a viscoelastic material, the measured strain by the strain gage at the middle of the incident or transmitted bar is not the same as that of the interfaces of the specimen and the bars. Thus, the viscoelastic behavior of the bars should initially be investigated
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (Q. Lu), [email protected] (Z.-j. Wang), [email protected] (Y. Ding). https://doi.org/10.1016/j.jsv.2019.114851 0022-460X/© 2019 Elsevier Ltd. All rights reserved.
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to determine the contact forces and displacements on the interfaces of the specimen and bars. When considering the geometric effects, Butt et al. [2,3], Jiang et al. [4], Fan et al. [5], Ahonsi et al. [6], Bussac et al. [7], and Bacon [8] recommended that the wave propagation coefficient can be used to determine the viscoelastic models for the bars. Generally, stress waves in viscoelastic bars (e.g., nylon or PMMA) for SHPB tests are only of the order of tens of MPa. Thus, viscoelastic models were identified from these tests under lower stress states. To determine the viscoelastic parameters of a material from spherical wave data, without using conventional mathematical models, Lee [9] presented the solution of a viscoelastic spherical wave equation using the more generally defined physical constants (such as the loss angle and frequency). Further, the “resonance” and “off-resonance” methods were proposed for determining the dilatational constants through the measurable vibration data of two points inside the medium. By this method, it is possible to obtain the relationship between the loss angle and frequency using a series of experiments by adjusting the input frequency of the spherical source. However, Lee only discussed the theoretical results, and there was no experimental validation. Wang et al. [10] conducted a spherical wave experiment in PMMA with a spherical charge of 0.125 g TNT as the source, and obtained the measured radial particle velocities within the radius of 5e30 mm. Based on the ZWT nonlinear viscoelastic constitutive model and the measured radial particle velocities at different radii, they used the characteristics method to investigate the propagation of a nonlinear viscoelastic spherical stress wave in PMMA, and proposed a new method for determining the nonlinear viscoelastic constitutive parameters of PMMA from the spherical wave data in a free field. According to their work, the viscoelastic model was determined from this spherical wave test under higher stress states of hundreds of MPa. In the present work, a spherical wave experiment for PMMA with a spherical explosive charge of 0.125gTNT as spherical source is carried out to investigate the dynamic mechanical behavior of a viscoelastic material. The measured radial particle velocities are within the radius of 10e80 mm, where the stress states are from tens of MPa to hundreds of MPa. A brief and effective method, through combining the viscoelastic spherical stress wave theory and the measured radial particle velocities at different radii in a free field, is proposed to determine the complex elastic modulus of material. Meanwhile, the complex elastic modulus, determined from spherical wave data under different stress states, is compared with those evaluated through the conventional mathematical models or experimental results selected from the literature.
2. Theory 2.1. Frequency equations of viscoelastic spherical stress wave The reduced displacement potential jðr; tÞ is a solution of an elastic or viscoelastic spherical stress wave propagation in an infinite medium. The relationship between the particle velocity vr ðr; tÞ and the reduced displacement potential jðr; tÞ in the Laplace domain can be expressed as [11].
evr ðr; sÞ ¼
1 1 e bðsÞ þ 2 sj ðr; sÞ r r
(1)
where r is the radius of Lagrange coordinates, s is the Laplace variable, bðsÞ is the propagation coefficient in the Laplace domain, and the superscript tilde denotes the Laplace transform of each function. According to Eq. (1), the particle velocity e vr ðr; sÞ at r1 and r2 can be written as follows:
evr ðr1 ; sÞ ¼
! 1 1 e ðr1 ; sÞ bðsÞ þ 2 sj r1 r1
(2)
evr ðr2 ; sÞ ¼
! 1 1 e bðsÞ þ 2 sj ðr2 ; sÞ r2 r2
(3)
where assuming that r2 > r1 . The transfer function between e vr ðr1 ; sÞ and e vr ðr2 ; sÞ is defined by
! evr ðr2 ; sÞ ¼ Hvr ðr2 ; r1 ; sÞ ¼ evr ðr1 ; sÞ
1 r2
bðsÞ þ r12 2
1 r1
bðsÞ þ r12 1
e ðr ; sÞ j 2 !
(4) e ðr ; sÞ j 1
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e ðr ; sÞ and j e ðr ; sÞ is expressed as According to Ref. [11], the relationship between j 2 1
e ðr ; sÞebðsÞðr2 r1 Þ e ðr ; sÞ ¼ j j 2 1
(5)
Thus, Eq. (4) can be formulated by
! Hvr ðr2 ; r1 ; sÞ ¼
evr ðr2 ; sÞ ¼ evr ðr1 ; sÞ
1 r2 bðsÞ
þ r12
1 r1 bðsÞ
þ r12 1
2
!ebðsÞðr2 r1 Þ
(6)
Let s ¼ ui; then, the frequency response function of the particle velocity is obtained by
! Hvr ðr2 ; r1 ; uiÞ ¼
evr ðr2 ; uiÞ ¼ evr ðr1 ; uiÞ
1 1 r2 bðuiÞ þ r 22
!ebðuiÞðr2 r1 Þ
(7)
1 1 r1 bðuiÞ þ r 21
where u is the angular frequency related to the frequency f by u ¼ 2pf , and i ¼ The propagation coefficient bðuiÞ in the frequency domain is written as [8].
bðuiÞ ¼ aðuÞ þ kðuÞi
pffiffiffiffiffiffiffi 1 is the imaginary unit.
(8)
where aðuÞ is the attenuation coefficient, and kðuÞ is the wave number. aðuÞ and kðuÞ are frequency-dependent continuous function. aðuÞ is a positive even function, and kðuÞ is an odd function. aðuÞ and kðuÞ can describe how a wave changes its shape as it travels in a viscoelastic medium. Eq. (7) can be formulated by the modulus and argument of Hvr ðr2 ; r1 ; uiÞ as
Hvr ðr2 ; r1 ; uiÞ ¼ jHvr ðr2 ; r1 ; uiÞjei4ðr2 ;r1 ;uÞ
(9)
where jHvr ðr2 ; r1 ; uiÞj and 4ðr2 ; r1 ; uÞ are the modulus and argument of, respectively. From Eqs. (7)e(9), the modulus and argument of Hvr ðr2 ; r1 ; uiÞ are expressed in terms of aðuÞ and kðuÞ as follows
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u aðuÞ þ r1 þ k2 ðuÞ aðuÞðr2 r1 Þ r1 u 2 e jHvr ðr2 ; r1 ; uiÞj ¼ t 2 r2 aðuÞ þ r1 þ k2 ðuÞ 1 4ðr2 ; r1 ; uÞ ¼ kðuÞðr2 r1 Þ þ tan1
kðuÞ aðuÞ þ r1 2
! tan1
(10) kðuÞ aðuÞ þ r1 1
! (11)
It can be seen that the modulus jHvr ðr2 ; r1 ; uiÞj and the argument 4ðr2 ; r1 ; uÞ are affected by the geometric effects and the viscosity of the material. If the modulus jHvr ðr2 ; r1 ; uiÞj and the argument 4ðr2 ; r1 ; uÞ are known, aðuÞ and kðuÞ can be obtained by solving simultaneously Eqs. (10) and (11). The propagation coefficient bðuiÞ for a viscoelastic spherical stress wave is defined in terms of the complex bulk modulus e uÞ by Ref. [11]. e uÞ and the complex shear modulus Gð Kð
b2 ðuiÞ ¼
r0 u2
e uÞ e uÞ þ 4 Gð Kð 3
(12)
where r0 is the density of the material, which is assumed to be constant. When considering the propagation of a viscoelastic spherical stress wave in a low loss material, such as PMMA, the frequency dependence of the Poisson's ratio m can be neglected from a practical point of view. Thus, the Poisson's ratio m is assumed to be constant, following the work of Pritz [12]. e uÞ are defined in terms of the complex elastic modulus e e uÞ and Gð Kð EðuÞ and the Poisson's ratio m as follows:
e uÞ ¼ Kð
e EðuÞ 3ð1 2mÞ
(13)
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e uÞ ¼ Gð
e EðuÞ 2ð1 þ mÞ
(14)
From Eqs. (12)e(14), we have
e EðuÞ ¼
ð1 mÞr0 u2 ð1 2mÞð1 þ mÞb2 ðuiÞ
¼ E0 ðuÞ þ E00 ðuÞi
(15)
where E0 ðuÞ is storage modulus and E00 ðuÞ is loss modulus. Eq. (15) shows that if the density r0, the Poisson's ratio m, and the propagation coefficient bðuiÞ are known, then we can obtain the frequency-dependent complex elastic modulus e EðuÞ. 2.2. Method for calculating the complex elastic modulus using the measured radial particle velocities at different radii Suppose that vr at r1 and r2 are measured in a free field; then, the experimental frequency response function H mpv vr ðr2 ; r1 ; uiÞ can be calculated by
Z
∞
0 H mpv vr ðr2 ; r1 ; uiÞ ¼ Z ∞ 0
vr ðr2 ; tÞeuit dt vr ðr1 ; tÞeuit dt
M P
¼
k¼1 N P
vr ðr2 ; tk Þeuitk Dt (16) vr ðr1 ; tk Þeuitk Dt
k¼1
where Dt is the sample interval, M and N are the sampling numbers, the superscript “mpv” of H mpv vr ðr2 ; r1 ; uiÞ denotes that the function is calculated by the measured particle velocity vr at r1 and r2 . The experimental frequency response function H mpv vr ðr2 ; r1 ; uiÞ is equal to the theoretical frequency response function Hvr ðr2 ; r1 ; uiÞ, thus, a function gðr2 ; r1 ; bðr2 ; r1 ; uiÞÞ is defined as
gðr2 ; r1 ; bðuiÞÞ ¼ Hvr ðr2 ; r1 ; uiÞ H mpv vr ðr2 ; r1 ; uiÞ ! 1 1 bðr2 ; r1 ; uiÞ þ 2 r2 r 2 bðr2 ;r1 ;uiÞðr2 r1 Þ !e H mpv ¼ vr ðr2 ; r1 ; uiÞ ¼ 0 1 1 bðr2 ; r1 ; uiÞ þ 2 r1 r1
(17)
In the present case, the complex derivative of g 0 ðr2 ; r1 ; bðuiÞÞ can be calculated analytically; therefore, an iterative formula of Newton type was chosen to solve the nonlinear Eq. (17) by
bnþ1 ðr2 ; r1 ; uiÞ ¼ bn ðr2 ; r1 ; uiÞ
gðr2 ; r1 ; bn ðr2 ; r1 ; uiÞÞ g 0 ðr2 ; r1 ; bn ðr2 ; r1 ; uiÞÞ
(18)
The convergence of this numerical iteration method depends on the initial value b0 ðr2 ; r1 ; uiÞ. According to Eq. (12), it seems that we can take the initial estimate b0 ðr2 ; r1 ; uiÞ by
b20 ðr2 ; r1 ; uiÞ ¼
r0 u2
e uÞ e uÞ þ 4 Gð Kð 3
(19)
e uÞ. In view of this, we used Eq. e uÞ and Gð following the work of Zhao [13]. Unfortunately, in our case, we know nothing about Kð (11) to estimate the wave number k0 ðr2 ; r1 ; uÞ of the propagation coefficient b0 ðr2 ; r1 ; uiÞ by
k0 ðr2 ; r1 ; uÞ z
4mpv ðr2 ; r1 ; uÞ ðr2 r1 Þ
(20)
where 4mpv ðr2 ; r1 ; uÞ is the argument of H mpv vr ðr2 ; r1 ; uiÞ, and the last two terms on the left of Eq. (11) are neglected. The substitution of k0 ðr2 ; r1 ; uÞ into Eq. (10) and the attenuation coefficient a0 ðr2 ; r1 ; uÞ can be obtained by solving the following equation as
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vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u mpv r1 u a0 ðr2 ; r1 ; uÞ þ r1 þ k20 ðr2 ; r1 ; uÞ a0 ðr2 ;r1 ;uÞðr2 r1 Þ 2 H ðr2 ; r1 ; uiÞ ¼ t e vr 2 r2 a ðr ; r ; uÞ þ r1 þ k2 ðr ; r ; uÞ 0
2
1
0
1
2
(21)
1
The initial estimate b0 ðr2 ; r1 ; uiÞ is determined by
b0 ðr2 ; r1 ; uiÞ ¼ a0 ðr2 ; r1 ; uÞ
4mpv ðr2 ; r1 ; uÞ i ðr2 r1 Þ
(22)
When the convergence conditions of Eq. (18) are met, we can obtain the propagation coefficient bðr2 ; r1 ; uiÞ. Thus, the complex elastic modulus can be defined according to Eq. (15) by
e Eðr2 ; r1 ; uÞ ¼
ð1 mÞr0 u2 ð1 2mÞð1 þ mÞb2 ðr2 ; r1 ; uiÞ
¼ E0 ðr2 ; r1 ; uÞ þ E00 ðr2 ; r1 ; uÞi
(23)
The calculation process of the complex elastic modulus e Eðr2 ; r1 ; uÞ is summarized in Fig. 1. It should be noted that the frequency equations of a viscoelastic spherical stress wave are based on the linear viscoelastic model without using conventional mathematical formulas. Thus, the theory presented in this paper does not take into account the nonlinear effects of spherical wave propagation. Unfortunately, the nonlinear effects of spherical wave propagation in the experimental case always exist, especially under high stressestrain state. For the ideal linear viscoelastic material, there is no nonlinearity in the measured waves, and the complex elastic modulus e Eðr ; r ; uÞ of the material can be obtained by Eq. (23) using the two measured particle velocities at any radii. When 2
1
considering the nonlinear effects of spherical wave propagation, the complex elastic modulus e Eðr2 ;r1 ; uÞ, which is calculated by Eq. (23) using the measured particle velocities at r1 and r2 , represents the local equivalent complex elastic modulus in the range of r1 to r2 . 2.3. Numerical examples and discussions To illustrate the method for calculating the complex elastic modulus from spherical wave data in a free field, the particular case of a viscoelastic spherical wave propagating in PMMA is discussed. The generalized Maxwell model (GMM) parameters published in the literature for PMMA are provided in Table 1. The GMM model, shown in Fig. 2, is composed of one elastic spring acting in parallel with two Maxwell elements. The complex bulk modulus and the complex shear modulus for PMMA in the frequency domain are expressed as [11].
e uÞ ¼ Kð
1 E q ui E q ui E0 þ 1 1 þ 2 2 3ð1 2mÞ q1 ui þ 1 q2 ui þ 1
(24)
e uÞ ¼ Gð
1 E q ui E q ui E0 þ 1 1 þ 2 2 2ð1 þ mÞ q1 ui þ 1 q2 ui þ 1
(25)
where E0 denotes the initial elastic modulus, qj ¼ hj =Ej (j ¼ 1,2) is the relaxation time of the jth Maxwell element, hj (j ¼ 1,2) is the viscosity coefficient of the jth Maxwell element, and Ej (j ¼ 1,2)is the elastic modulus of the jth Maxwell element. As shown in Ref. [11], the radial particle velocities at r ¼ 25, 50, 75, and 100 mm for the viscoelastic spherical stress wave in PMMA were numerically calculated by using the GMM parameters of Butt's model І presented in Table 1. Because Butt's model І is the linear viscoelastic model, there is no nonlinearity in the waves shown in Ref. [11], and the inversion of the complex modulus can be calculated by using the particle velocities at any two locations of r ¼ 25, 50, 75, and 100 mm. For example, following the first step shown in Fig. 1, the frequency response function H mpv vr ðr2 ; r1 ; uiÞ is calculated by Eq. (16) using the particle velocities at r1 ¼ 25 mm and r2 ¼ 50 mm shown in Ref. [11], and the real and imaginary part of H mpv vr ðr2 ; r1 ; uiÞ are shown in Fig. 3a and b, respectively. The modulus H mpv ðr2 ; r1 ; uiÞ and the argument 4mpv ðr2 ; r1 ; uÞ of H mpv ðr2 ; r1 ;
uiÞ are shown in Fig. 3c and d, respectively.
vr
vr
Following the second and third step shown in Fig. 1, the inversion results of the attenuation coefficient aðuÞ and the wave number kðuÞ, which are the real and imaginary part of the propagation coefficient bðr2 ; r1 ; uiÞ, are shown in Fig. 3e and f, respectively. Meanwhile, the theoretical results of the attenuation coefficient aðuÞ and wave number kðuÞ for Butt's model І can be obtained by Eq. (12), and are also shown in Fig. 3e and f.
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Fig. 1. The inversion method of the complex elastic modulus of materials through the combination of the viscoelastic spherical stress wave theory and the measured radial particle velocities at different radii in a free field.
Following the fourth step shown in Fig. 1, the inversion results of the storage modulus E0 ðuÞ and loss modulus E00 ðuÞ, which are the real and imaginary part of the complex modulus e Eðr2 ; r1 ; uÞ, are shown in Fig. 3g and h, respectively. Meanwhile, the
theoretical results of the storage modulus E0 ðuÞ and loss modulus E00 ðuÞ for Butt's model І can be obtained by Eq. (23), and are also shown in Fig. 3g and h. It can be seen from Fig. 3(eeh) that the inversion and theoretical results are in good agreements. This means that the inversion method for the complex elastic modulus presented in the preceding paragraphs is valid.
Table 1 The GMM parameters for PMMA. Reference Butt's model І [2] Butt's model ІІ [2] Wang's model [10]
r0 (kg/m3) 3
1.20 10 1.20 103 1.19 103
E0 (GPa)
E1 (GPa)
E2 (GPa)
q1 ¼
5.15 5.14 2.04
0.41 0.41 0.90
0.55 0.45 3.87
95.3 95.1 ∞
h1 E1
(ms)
q2 ¼ 1.88 2.34 2.05
h2 E2
(ms)
m 0.35 0.35 0.35
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Fig. 2. Rheological model for generalized Maxwell visco-elastic constitutive equation, consisting of a linear spring, a Maxwell element I and a Maxwell element II in parallel.
3. Experimental setup To investigate the dynamic mechanical behavior of a viscoelastic material, a spherical wave experiment in PMMA will be introduced. The experimental test setup and the PMMA sample are shown in Fig. 4a and b, respectively. A spherical pentaerythrite tetranitrate (PETN) explosive charge, with a diameter of 5 mm, density of 1.5 g/cm3, and equivalent to 0.125 g TNT, is placed at the center of the PMMA sample. The PMMA sample, with a diameter of 250 mm, height of 250 mm and density of 1180 kg/m3, is placed at the center of a solenoid. The wire loops with diameters of 0.22 mm are placed within the sample and are concentric with the spherical explosive charge at the mid-plane. The vertically oriented magnetic field generated by the solenoid surrounding the sample is applied at the mid-plane. When the divergent spherical stress wave generated by the spherical explosive charge travels in PMMA, the wire loops travel with the particle motion caused by the wave motion. The motion of the wire loops through the magnetic field flux lines generates a measurable electromagnetic induction voltage e in volts. Therefore, the particle velocity vr ðr; tÞ can be given by Faraday's law
vr ðr; tÞ ¼
e Bl
(26)
where B is the magnetic induction intensity, and l is the length of the particle velocity gage related to the radius r by l ¼ 2pr. Owing to the border effect of the sample, the waves traveling in the directions of increasing and decreasing r would be superposed at each wire loop. In Section 2, the method for determining the complex elastic modulus is given by the theory of viscoelastic spherical stress wave in an infinite medium; thus, to obtain the spherical stress wave data in a free field, the reflecting wave from the border of the sample is eliminated by using data processing technique referred to Bacon's work [8].
4. Results 4.1. Attenuation of peak velocity of spherical stress wave in PMMA Fig. 5 shows the measured particle velocity in a free field from the spherical wave experiment for PMMA. It can be seen that, owing to the reflecting wave from the border of the sample, the farther the gaging point is from the center of the explosion, the shorter the effective duration of the particle velocity. Fig. 6 shows that the peak particle velocity decreases with increasing wave propagation distance. Generally, the peak particle velocityedistance law can be expressed as [14].
vr
max
¼ Arh
(27)
where A and h are the fitted values. The power exponent h is the attenuation index, and it represents how fast the peak particle velocity changes with the wave propagation distance. Through the method of least squares, the experimental results are fitted by Eq. (27). The fitted curve is also shown in Fig. 6. Comparing the experimental results and the fitted curve, we can find out that the fitted curve, with fitted attenuation index h ¼ 1.33, is in good agreement with the experimental results at r ¼ 10e80 mm. The attenuation index h > 1 means that the attenuation of spherical wave propagation in PMMA includes not only the geometric effects but also the viscous effects as explained in Ref. [10].
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Fig. 3. The calculation process of the complex modulus Eðr2 ; r1 ; uÞ for Butt's model I. (a). the real part of Hvr ðr2 ;r1 ; uiÞ. (b). The imaginary part of Hvr ðr2 ;r1 ; uiÞ. (c). the modulus of Hvr ðr2 ;r1 ; uiÞ. (d). The argument of Hvr ðr2 ;r1 ; uiÞ. (e) comparison between the inversion and theoretical results of aðuÞ. (f). Comparison between the inversion and theoretical results of kðuÞ. (g). Comparison between the inversion and theoretical results of E0 ðuÞ. (h). Comparison between the inversion and theoretical results of E00 ðuÞ.
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Fig. 4. The spherical wave experiment. (a). Scheme of the spherical wave experiment, where the particle velocity waves are measured by magneto-electric velocimeter. (b). The PMMA sample.
Fig. 5. Experimental measured spherical waves vr ðr; tÞ in a free field for PMMA. (a) r ¼ 10, 20, 30, and 40 mm (b). r ¼ 50, 60, 70, and 80 mm.
4.2. The upper and lower limiting frequency of the response function of particle velocity The arguments 4mpv ðr2 ; r1 ; uÞ of H mpv vr ðr2 ;r1 ; uiÞ, determined by the measured particle velocities in the case of r1 ¼ 10 mm and r2 ¼ 20, 40, 60, and 80 mm, are shown in Fig. 7a. It can be seen that the argument 4mpv ðr2 ; r1 ; uÞ with different r1 and r2 becomes ambiguous at higher frequencies. We take the frequency at the beginning of the unreasonable change in the argument 4mpv ðr2 ; r1 ; uÞ as the upper limiting frequency referred as fmax ðr2 ;r1 Þ. As shown in Fig. 7a, when r1 ¼ 10 mm and r2 ¼ 20, 40, 60, and 80 mm, fmax ðr2 ; r1 Þ is accurate up to about 1900, 1750, 1600, and 1300 kHz, respectively. We can find that when r1 remains unchanged, fmax ðr2 ; r1 Þ decreases as r2 increases. It means that the determined upper limiting frequency fmax ðr2 ; r1 Þ of H mpv vr ðr2 ; r1 ; uiÞ is decided by the measured particle velocity away from the center of the explosion. The main reasons for argument 4mpv ðr2 ; r1 ; uÞ to become ambiguous after fmax ðr2 ; r1 Þ are that the frequency components of the particle velocity after fmax ðr2 ; r1 Þ become very weak owing to the fast decrease in the high frequency components caused by the viscous effects and they are inundated in the background noise. When considering the lower limiting frequency fmin ðr2 ; r1 Þ of the transfer function H mpv vr ðr2 ; r1 ; uiÞ shown in Fig. 7b, the border effect of the sample must be considered. Owing to the border effect of the sample, the wavelength of vr ðr; tÞ at the gage point is equal or lesser than the distance from the gage point to the border of the sample. Thus, the maximum wavelength lmax of vr ðr; tÞ can be estimated by
lmax ðrÞ ¼ Rb r where Rb is the radius of the sample. The corresponding lower limiting frequency fmin of vr ðr; tÞ can be estimated by
(28)
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Fig. 6. Comparison of the measured peak particle velocities and the fitted curve.
Fig. 7. The arguments 4mpv ðr2 ; r1 ; uÞ of H mpv vr ðr2 ; r1 ; uiÞ determined by the measured particle velocities in the case of r ¼ 10 mm and r ¼ 20, 40, 60, and 80 mm. (a) The upper limiting frequency of 4mpv ðr2 ; r1 ; uÞ is approximately accurate up to fmax ðr2 ; r1 Þ (b) the lower limiting frequency of 4mpv ðr2 ; r1 ; uÞ is approximately accurate up to fmin ðr2 ; r1 Þ.
fmin ðrÞ ¼
C
lmax ðrÞ
¼
C Rb r
(29)
where C is the sound speed of PMMA. Eq. (29) means that the spectrum components of vr ðr; tÞ below approximately fmin ðrÞ are unbelievable. As for the transfer function H mpv vr ðr2 ;r1 ; uiÞ, the lower limiting frequency fmin ðr2 ; r1 Þ is decided by fmin ðr1 Þ and fmin ðr2 Þ. It can be seen from Eq. (29) that fmin ðr2 Þ > fmin ðr1 Þ if r2 > r1 . Thus, fmin ðr2 ; r1 Þ can be estimated by
fmin ðr2 ; r1 Þ ¼ fmin ðr2 Þ ¼
C Rb r2
(30)
The sound speed C of PMMA is 2882.8 m/s and the radius Rb of the sample is 125 mm. When r2 ¼ 20, 40, 60, and 80 mm, fmin ðr2 ; r1 Þ is accurate up to about 28, 34, 44, and 64 kHz, respectively. It can be observed from Fig. 7 that fmax ðr2 ; r1 Þ decreases as r2 increases, and fmin ðr2 ; r1 Þ increases as r2 increases. This means that the upper and lower limiting frequency of each H mpv vr ðr2 ; r1 ; uiÞ are all decided by the measured particle velocity away from the center of the explosion, and the effective frequency band Df ¼ fmax ðr2 ; r1 Þ fmin ðr2 ; r1 Þ decreases as r2 increases.
Q. Lu et al. / Journal of Sound and Vibration 459 (2019) 114851
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Fig. 8. The calculated storage modulus E0 ðr2 ; r1 ; uÞ (a). r1 ¼ 10 mm, r2 ¼ 20 mm. (b). r1 ¼ 20 mm, r2 ¼ 30 mm. (c) r1 ¼ 30 mm, r2 ¼ 40 mm. (d) r1 ¼ 40 mm, r2 ¼ 50 mm. (e) r1 ¼ 50 mm, r2 ¼ 60 mm. (f) r1 ¼ 60 mm, r2 ¼ 70 mm, and r1 ¼ 70 mm, r2 ¼ 80 mm.
4.3. Complex elastic modulus determined by the measured particle velocity in free field The complex elastic modulus e Eðr2 ; r1 ; uÞ with different r1 and r2 are calculated by the measured particle velocities at different radii according to the calculation process shown in Fig. 1. Figs. 8 and 9 show the storage modulus E0 ðr ; r ; uÞ and the loss modulus E00 ðr ; r ; uÞ of the complex elastic modulus e Eðr ; 2
1
2
1
2
r1 ; uÞ, which are calculated by the particle velocities at two adjacent gage points where r1 and r2 satisfy the relation by r2 r1 ¼ 10 mm, respectively. It can be seen from Figs. 8 and 9 that the storage modulus E0 ðr2 ; r1 ; uÞ (approximately above 300 kHz) tends to constant value, and the loss modulus E00 ðr2 ; r1 ; uÞ decreases to zero with the increase in frequency. Meanwhile, there
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Q. Lu et al. / Journal of Sound and Vibration 459 (2019) 114851
Fig. 9. The calculated loss modulus E00 ðr2 ; r2 ; uÞ (a). r1 ¼ 10 mm, r2 ¼ 20 mm. (b). r1 ¼ 20 mm, r2 ¼ 30 mm. (c) r1 ¼ 30 mm, r2 ¼ 40 mm. (d) r1 ¼ 40 mm, r2 ¼ 50 mm. (e) r1 ¼ 50 mm, r2 ¼ 60 mm. (f) r1 ¼ 60 mm, r2 ¼ 70 mm, and r1 ¼ 70 mm, r2 ¼ 80 mm.
is nonlinearity in the measured spherical waves because the complex elastic modulus e Eðr2 ; r1 ; uÞ in the region near the explosion center is not the same as that in the region away from the explosion center. For example, E0 ðr2 ; r1 Þf > 300kHz , the average storage modulus above 300 kHz of each curve shown in Fig. 8 is defined by f ¼¼u=2 Pp¼fmax 0
E ðr2 ; r1 Þf > 300kHz ¼
f ¼u=2p¼300kHz
E0 ðr2 ; r1 ; uÞ
L
where L is the number of E0 ðr2 ; r1 ; uÞ when f > 300 kHz.
(31)
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13
Fig. 10. The average storage modulus E0 ðr2 ; r1 Þf > 300kHz at higher frequencies after 300 kHz.
Fig. 10 shows the average storage modulus E0 ðr2 ; r1 Þf > 300kHz . It can be seen that E0 ðr2 ; r1 Þf > 300kHz decreases from 7.14 to 5.97 GPa with increasing distance from the explosion center. e ; r ; uÞ at lower frequencies, we can observe in Figs. 8 and 9 that the When considering the complex elastic modulus Eðr 2 1 scattering of the complex elastic modulus increases because the particle velocity spectrum components at lower frequencies are not adequately developed owing to the interference that is caused by the wave reflecting from the border of the sample. Eq. (30) can provide the estimation for the lower limiting frequency of e Eðr ; r ; uÞ. For example, when r ¼ 70 mm and r 2
1
1
2
¼ 80 mm, the lower limiting frequency of e Eðr2 ; r1 ; uÞ is approximately 64 kHz. It means that a e Eðr2 ; r1 ; uÞ below 64 kHz is unbelievable. 5. Discussions 5.1. Stressestrain state dependence of the complex elastic modulus of PMMA Fig. 5 shows that the rising edges of the measured particle velocities at r ¼ 10e80 mm are only of the order of 0.6e1.0 ms. Suppose that the propagation of the peak particle velocity is consistent with the theory of strong discontinuous viscoelastic spherical waves, following the work of Wang [10]. Taking compression as positive, we have
½vr ¼ D½εr
(32)
½sr ¼ r0 D½vr
(33)
where D is the shock speed calculated by using the time of arrival in Fig. 5, sr is the radial stress, εr is the radial strain, and [] denotes the discontinuous jump across the wave front. According to Eqs. (32) and (33), the radial stress sr and the radial strain εr at different radii are shown in Fig. 11. It can be seen that the radial stress sr decreases from 682 to 40 MPa, and the radial strain εr decreases from 59.5 103 to 4.2 103. Meanwhile, Fig. 11 compares the average storage modulus E0 ðr2 ; r1 Þf > 300kHz with the radial stress sr and the radial strain εr at different radii. It shows that, when the radial stress sr and radial strain εr are reduced with the increasing distance of the spherical wave, the average storage modulus E0 ðr2 ; r1 Þf > 300kHz is also reduced and tends to 5.97 GPa. On the basis of the results shown in Fig. 11, we can know that the average storage modulus E0 ðr2 ; r1 Þf > 300kHz is dependent on the stressestrain
state. Take the average storage modulus E0 ðr2 ; r1 Þf > 300kHz ¼ 5.97 GPa when r1 ¼ 70 mm and r2 ¼ 80 mm as the reference value,
and define the function c by
c¼
E0 ðr2 ; r1 Þf > 300kHz E0 ð80mm; 70mmÞf > 300kHz E0 ð80mm; 70mmÞf > 300kHz
where c represents the deviation of E0 ðr2 ; r1 Þf > 300kHz relative to E0 ð80mm; 70mmÞf > 300kHz ¼ 5.97 GPa.
(34)
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Fig. 11. The average storage modulus E0 ðr2 ; r1 Þf > 300kHz changes with the stressestrain state owing to the nonlinear effects of spherical wave propagation. The bar chart represents the average storage modulus E0 ðr2 ; r1 Þf > 300kHz between two adjacent gage points. The solid line with a symbol of a circle or square represents the radial stress sr or the radial strain εr at different radii, respectively.
Fig. 12. The storage modulus E0 ðuÞ, determined through the spherical stress wave propagation data under different stress-strain states, is compared to Butt's and Wang's models. The solid line represents the storage modulus E0 ðr2 ; r1 ; uÞ determined by the velocities at r1 ¼ 10 mm and r2 ¼ 20 mm or r1 ¼ 70 mm and r2 ¼ 80 mm, respectively. The solid line with a symbol of a circle, square or triangle represents Butt's model I, Butt's model II or Wang's model, respectively.
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15
Fig. 13. The loss modulus E00 ðuÞ, determined through the spherical stress wave propagation data under different stress-strain state, is compared to Butt's and Wang's models. The solid line represents the storage modulus E00 ðr1 ; r2 ; uÞ determined by the velocities at r1 ¼ 10 mm and r2 ¼ 20 mm or r1 ¼ 70 mm and r2 ¼ 80 mm, respectively. The solid line with a symbol of a circle, square or triangle represents Butt's model I, Butt's model II or Wang's model, respectively.
Fig. 14. The storage modulus E0 ðuÞ, determined through the spherical stress wave propagation data at r1 ¼ 70 mm and r2 ¼ 80 mm, is compared to Mousavi's results.
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Within the radius of 10e80 mm, c is reduced from 19.6% to 0% with the radial stress sr decreasing from 682 to 40 MPa, and radial strain εr decreasing from 59.5 103 to 4.2 103. When r 30 mm, the radial stress sr 176 MPa, radial strain εr 17.8 103, and relative deviation c 3.0%. From a practical point of view, the stressestrain state dependence of the complex elastic modulus can be neglected as the radial stress sr 176 MPa and radial strain εr 17.8 103.
5.2. Comparison of the complex elastic modulus of PMMA determined in different ways This section presents a comparison between the experimentally determined complex elastic modulus of PMMA from the spherical wave data in a free field and those evaluated through the GMM or experimental results selected from the literature. Butt et al. [2,3] used a PMMA bar with a diameter of 30 mm and length of 2 m to investigate the dynamic mechanical behavior of a viscoelastic material. In their work, the peak strain in the PMMA bar is approximately 6 103 or less. The corresponding peak stress in PMMA bar, estimated by s ¼ Eε where E is assumed to be 6 GPa, is approximately 36 MPa or less. Wang et al. [10] used the characteristics method to investigate the propagation of the spherical stress wave in the case of the tamped explosion with 0.125 g TNT in PMMA. The spherical PETN explosive charge used in their work is the same as that used in this study. Wang et al. analyzed the propagating characteristic of the measured particle velocity at r ¼ 5e30 mm, where the radial stress sr was approximately 176e1700 MPa, and the radial strain εr was approximately 17.8 103e180 103. The GMM parameters of PMMA have been determined from the stress wave data in a lower stressestrain state by Butt et al. [2], and that in a higher stressestrain state by Wang et al. [10]. These GMM parameters are presented in Table 1. e ; r ; uÞ determined through the spherical stress wave propaFigs. 12 and 13 compare the complex elastic modulus Eðr 2
1
gation data in a free field with that calculated by Butt's and Wang's models. It can be seen from Fig. 12 that Butt's models are closer to the storage modulus E0 ðr2 ; r1 ; uÞ calculated by two particle velocities at r1 ¼ 70 mm and r2 ¼ 80 mm, and Wang's model is closer to the storage modulus E0 ðr2 ; r1 ; uÞ calculated by two particle velocities at r1 ¼ 10 mm and r2 ¼ 20 mm. Fig. 13 shows that the loss modulus E00 ðr2 ; r1 ; uÞ from Wang's model is greater than that from Butt's models, and the loss modulus E00 ðr2 ; r1 ; uÞ, determined from the measured radial particle velocities in this work, is closer to Butt's models. This indicates that the published parameters of viscoelastic materials, determined from different experimental techniques under different stressestrain states, should be used with caution.
Fig. 15. The loss modulus E00 ðuÞ, determined through the spherical stress wave propagation data at r1 ¼ 70 mm and r2 ¼ 80 mm, is compared to Mousavi's results.
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17
Mousavi et al. [15] used PMMA bars with diameters of 10 mm and 20 mm to identify the complex elastic modulus. Their results, shown in Figs. 14 and 15, are very accurate up to approximately 40 kHz for a F10 mm PMMA bar, and 15 kHz for a F20mm PMMA bar. Meanwhile, Mousavi's results are also compared with the complex elastic modulus determined through the two particle velocities at r1 ¼ 70 mm and r2 ¼ 80 mm. For PMMA, it can be seen that the complex elastic modulus from the measured strains on an impacted bar is accurate up to approximately tens of kHz. However, the complex elastic modulus, determined through the spherical stress wave propagation data in a free field, is accurate approximately from tens of kHz to thousands of kHz in the present study. For one-dimensional stress wave propagation in a bar, according to Ref. [16], the upper limiting frequency where the inertia effects become dominant can be estimated by
fmax
1D
¼
1 10d
sffiffiffiffiffi Er
r
(35)
where d is the diameter of the bar, and Er can be considered as the storage modulus at high frequencies. For the case of the bar, to improve the accuracy of the complex elastic modulus from the measured strains on an impacted bar at high frequencies, we should use a thinner bar according to Eq. (35). In addition, for the case of the spherical wave experiment, we should use a larger sample, as shown in Fig. 4, to obtain the particle velocities in the region far away from the explosion center. This allows improving the accuracy of the complex elastic modulus determined through the spherical stress wave propagation data in a free field at low frequencies. 6. Conclusions In the present study, without using conventional mathematical models, a brief and effective method is proposed to determine the complex elastic modulus of a viscoelastic material. This is achieved by the combining the theory of viscoelastic spherical stress wave and the measured radial particle velocities at different radii in a free field. From the above results the following conclusions can be drawn: (1) Based on the theory of the viscoelastic spherical wave, the frequency response function of particle velocity between two points is given in terms of the propagation coefficient and the wave propagation distance. Meanwhile, the frequency response function can be obtained by the ratio of the particle velocity spectrums of two measurable points from a spherical wave experiment. By solving the frequency response equation shown in Eq. (17), the local equivalent complex elastic modulus can be obtained numerically. (2) The results show that the PMMA behaves viscoelastically under spherical explosion loading, and when the radial stress and radial strain are reduced with the increasing distance of the spherical wave, the average storage modulus of PMMA at high frequencies after approximately 300 KHz, is also reduced from 7.14 to 5.97 GPa. This indicates that the complex elastic modulus of PMMA is dependent on the stressestrain state, and there is nonlinearity in the measured spherical waves at higher stressestrain states. (3) Through the comparisons between the determined complex elastic modulus from spherical wave data and those evaluated through the GMM or experimental results selected from the literature, the published parameters of viscoelastic material, determined from different experimental techniques under different stress-strain states, should be used with caution. For PMMA, from a practical point of view, when the stress and strain are less than 176 MPa and 17.8 103, respectively, the stressestrain state dependence of the complex elastic modulus can be neglected.
Acknowledgments The authors acknowledge the financial support from the National Natural Science Foundation of China under Grant no.11172244. References [1] L.L. Wang, K. Labibes, Z. Azari, Generalization of split Hopkinson bar technique to use viscoelastic bars, Int. J. Impact Eng. 15 (1994) 669e686. [2] H.S.U. Butt, P. Xue, T.Z. Jiang, B. Wang, Parametric identification for material of viscoelastic SHPB from wave propagation data incorporating geometrical effects, Int. J. Mech. Sci. 91 (2015) 46e64. [3] H.S.U. Butt, P. Xue, Determination of the wave propagation coefficient of viscoelastic SHPB: significance for characterization of cellular materials, Int. J. Impact Eng. 74 (2014) 83e91. [4] T.Z. Jiang, P. Xue, H.S.U. Butt, Pulse shaper design for dynamic testing of viscoelastic materials using polymeric SHPB, Int. J. Impact Eng. 79 (2015) 45e52. [5] L.F. Fan, L.N.Y. Wong, G.W. Ma, Experimental investigation and modeling of viscoelastic behavior of concrete, Constr. Build. Mater. 48 (2013) 814e821. [6] B. Ahonsi, J.J. Harrigan, M. Aleyaasin, On the propagation coefficient of longitudinal stress waves in viscoelastic bars, Int. J. Impact Eng. 45 (2012) 39e51. [7] M.N. Bussac, P. Collet, G. Gary, B. Lundberg, S. Mousavi, Viscoelastic impact between a cylindrical striker and a long cylindrical bar, Int. J. Impact Eng. 35 (2008) 226e239.
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[8] C. Bacon, An experimental method for considering dispersion and attenuation in a viscoelastic Hopkinson bar, Exp. Mech. 4 (1998) 242e249. [9] T.M. Lee, Coordination of in situ vibration tests and spherical viscoelastic waves, IEEE Trans. Son. Ultrason. SU 13 (1966) 130e135. [10] L.L. Wang, H.W. Lai, Z.J. Wang, L.M. Yang, Studies on nonlinear visco-elastic spherical waves by characteristics analyses and its application, Int. J. Impact Eng. 55 (2013) 1e10. [11] Q. Lu, Z.J. Wang, Studies of the propagation of viscoelastic spherical divergent stress waves based on the generalized Maxwell model, J. Sound Vib. 371 (2016) 183e195. [12] T. Pritz, The Poisson's loss factor of solid viscoelastic materials, J. Sound Vib. 306 (2007) 790e802. [13] H. Zhao, G. Gary, A three dimensional analytical solution of the longitudinal wave propagation in an infinite linear viscoelastic cylindrical bar. application to experimental techniques, J. Mech. Phys. Solids 43 (1995) 1335e1348. [14] L. Sambuelli, Theoretical derivation of a peak particle velocityedistance law for the prediction of vibrations from blasting, Rock Mech. Rock Eng. 42 (2009) 547e556. [15] S. Mousavi, D.F. Nicolas, B. Lundberg, Identification of complex moduli and Poisson's ratio from measured strains on an impacted bar, J. Sound Vib. 277 (2004) 971e986. € m, L. Abrahamsson, Parametric identification of viscoelastic materials from time and frequency domain data, Inverse Probl. [16] M. Mossberg, L. Hillstro Eng. 9 (2001) 645e670.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Inversion for the complex elastic modulus of material from spherical wave propagation data in free field Qiang Lu*, Zhan-jiang Wang**, Yang Ding Northwest Institute of Nuclear Technology, Xi'an, 710024, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 September 2018 Received in revised form 29 June 2019 Accepted 9 July 2019 Available online 14 July 2019 Handling Editor: P. Tiso
A spherical wave experiment is conducted for polymethyl methacrylate (PMMA), with a spherical explosive charge of 0.125 g TNT as the source. The aim is to investigate the dynamic mechanical behavior of a viscoelastic material from the multiple measured radial particle velocities during the spherical wave propagation in a free field. Without using conventional mathematical models, a brief and effective method is suggested to determine the complex elastic modulus of the material through the combination of the viscoelastic spherical stress wave theory and the measured radial particle velocities at different radii in a free field. The complex elastic modulus, determined from the spherical wave data under different stress states, is compared with those evaluated through the conventional mathematical models or experimental results selected from the literature. The results show that PMMA behaves viscoelastically under spherical explosion loading. With increasing the propagation distance of spherical wave, the radial stress decreases from 682 to 40 MPa, and the radial strain decreases from 59.5 103 to 4.2 103. Meanwhile, the average storage modulus of PMMA, at high frequencies after approximately 300 kHz, decreases from 7.14 to 5.97 GPa. This indicates that the complex elastic modulus of PMMA is dependent on the stressestrain state, and the published parameters of viscoelastic materials, determined by different experimental techniques under different stressestrain states, should be used with caution. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Viscoelastic Spherical stress wave Propagation coefficient Complex elastic modulus
1. Introduction It is well known that the propagating character of stress waves is strongly dependent on the dynamic constitutive behavior of materials. The strain-rate effects become important for investigating the propagation of stress wave under explosion or impact loading. Thus, the viscosity of the material should be taken into account. In a split Hopkinson pressure bar (SHPB) apparatus, viscoelastic bars were recommended as the incident and transmitted bars for investigating the dynamic mechanical behavior of the low impedance material. This technique can obtain sufficiently high transmitted signals and increase the signal-to-noise ratio [1]. Owing to the attenuation and dispersion effects of a viscoelastic material, the measured strain by the strain gage at the middle of the incident or transmitted bar is not the same as that of the interfaces of the specimen and the bars. Thus, the viscoelastic behavior of the bars should initially be investigated
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (Q. Lu), [email protected] (Z.-j. Wang), [email protected] (Y. Ding). https://doi.org/10.1016/j.jsv.2019.114851 0022-460X/© 2019 Elsevier Ltd. All rights reserved.
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to determine the contact forces and displacements on the interfaces of the specimen and bars. When considering the geometric effects, Butt et al. [2,3], Jiang et al. [4], Fan et al. [5], Ahonsi et al. [6], Bussac et al. [7], and Bacon [8] recommended that the wave propagation coefficient can be used to determine the viscoelastic models for the bars. Generally, stress waves in viscoelastic bars (e.g., nylon or PMMA) for SHPB tests are only of the order of tens of MPa. Thus, viscoelastic models were identified from these tests under lower stress states. To determine the viscoelastic parameters of a material from spherical wave data, without using conventional mathematical models, Lee [9] presented the solution of a viscoelastic spherical wave equation using the more generally defined physical constants (such as the loss angle and frequency). Further, the “resonance” and “off-resonance” methods were proposed for determining the dilatational constants through the measurable vibration data of two points inside the medium. By this method, it is possible to obtain the relationship between the loss angle and frequency using a series of experiments by adjusting the input frequency of the spherical source. However, Lee only discussed the theoretical results, and there was no experimental validation. Wang et al. [10] conducted a spherical wave experiment in PMMA with a spherical charge of 0.125 g TNT as the source, and obtained the measured radial particle velocities within the radius of 5e30 mm. Based on the ZWT nonlinear viscoelastic constitutive model and the measured radial particle velocities at different radii, they used the characteristics method to investigate the propagation of a nonlinear viscoelastic spherical stress wave in PMMA, and proposed a new method for determining the nonlinear viscoelastic constitutive parameters of PMMA from the spherical wave data in a free field. According to their work, the viscoelastic model was determined from this spherical wave test under higher stress states of hundreds of MPa. In the present work, a spherical wave experiment for PMMA with a spherical explosive charge of 0.125gTNT as spherical source is carried out to investigate the dynamic mechanical behavior of a viscoelastic material. The measured radial particle velocities are within the radius of 10e80 mm, where the stress states are from tens of MPa to hundreds of MPa. A brief and effective method, through combining the viscoelastic spherical stress wave theory and the measured radial particle velocities at different radii in a free field, is proposed to determine the complex elastic modulus of material. Meanwhile, the complex elastic modulus, determined from spherical wave data under different stress states, is compared with those evaluated through the conventional mathematical models or experimental results selected from the literature.
2. Theory 2.1. Frequency equations of viscoelastic spherical stress wave The reduced displacement potential jðr; tÞ is a solution of an elastic or viscoelastic spherical stress wave propagation in an infinite medium. The relationship between the particle velocity vr ðr; tÞ and the reduced displacement potential jðr; tÞ in the Laplace domain can be expressed as [11].
evr ðr; sÞ ¼
1 1 e bðsÞ þ 2 sj ðr; sÞ r r
(1)
where r is the radius of Lagrange coordinates, s is the Laplace variable, bðsÞ is the propagation coefficient in the Laplace domain, and the superscript tilde denotes the Laplace transform of each function. According to Eq. (1), the particle velocity e vr ðr; sÞ at r1 and r2 can be written as follows:
evr ðr1 ; sÞ ¼
! 1 1 e ðr1 ; sÞ bðsÞ þ 2 sj r1 r1
(2)
evr ðr2 ; sÞ ¼
! 1 1 e bðsÞ þ 2 sj ðr2 ; sÞ r2 r2
(3)
where assuming that r2 > r1 . The transfer function between e vr ðr1 ; sÞ and e vr ðr2 ; sÞ is defined by
! evr ðr2 ; sÞ ¼ Hvr ðr2 ; r1 ; sÞ ¼ evr ðr1 ; sÞ
1 r2
bðsÞ þ r12 2
1 r1
bðsÞ þ r12 1
e ðr ; sÞ j 2 !
(4) e ðr ; sÞ j 1
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3
e ðr ; sÞ and j e ðr ; sÞ is expressed as According to Ref. [11], the relationship between j 2 1
e ðr ; sÞebðsÞðr2 r1 Þ e ðr ; sÞ ¼ j j 2 1
(5)
Thus, Eq. (4) can be formulated by
! Hvr ðr2 ; r1 ; sÞ ¼
evr ðr2 ; sÞ ¼ evr ðr1 ; sÞ
1 r2 bðsÞ
þ r12
1 r1 bðsÞ
þ r12 1
2
!ebðsÞðr2 r1 Þ
(6)
Let s ¼ ui; then, the frequency response function of the particle velocity is obtained by
! Hvr ðr2 ; r1 ; uiÞ ¼
evr ðr2 ; uiÞ ¼ evr ðr1 ; uiÞ
1 1 r2 bðuiÞ þ r 22
!ebðuiÞðr2 r1 Þ
(7)
1 1 r1 bðuiÞ þ r 21
where u is the angular frequency related to the frequency f by u ¼ 2pf , and i ¼ The propagation coefficient bðuiÞ in the frequency domain is written as [8].
bðuiÞ ¼ aðuÞ þ kðuÞi
pffiffiffiffiffiffiffi 1 is the imaginary unit.
(8)
where aðuÞ is the attenuation coefficient, and kðuÞ is the wave number. aðuÞ and kðuÞ are frequency-dependent continuous function. aðuÞ is a positive even function, and kðuÞ is an odd function. aðuÞ and kðuÞ can describe how a wave changes its shape as it travels in a viscoelastic medium. Eq. (7) can be formulated by the modulus and argument of Hvr ðr2 ; r1 ; uiÞ as
Hvr ðr2 ; r1 ; uiÞ ¼ jHvr ðr2 ; r1 ; uiÞjei4ðr2 ;r1 ;uÞ
(9)
where jHvr ðr2 ; r1 ; uiÞj and 4ðr2 ; r1 ; uÞ are the modulus and argument of, respectively. From Eqs. (7)e(9), the modulus and argument of Hvr ðr2 ; r1 ; uiÞ are expressed in terms of aðuÞ and kðuÞ as follows
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u aðuÞ þ r1 þ k2 ðuÞ aðuÞðr2 r1 Þ r1 u 2 e jHvr ðr2 ; r1 ; uiÞj ¼ t 2 r2 aðuÞ þ r1 þ k2 ðuÞ 1 4ðr2 ; r1 ; uÞ ¼ kðuÞðr2 r1 Þ þ tan1
kðuÞ aðuÞ þ r1 2
! tan1
(10) kðuÞ aðuÞ þ r1 1
! (11)
It can be seen that the modulus jHvr ðr2 ; r1 ; uiÞj and the argument 4ðr2 ; r1 ; uÞ are affected by the geometric effects and the viscosity of the material. If the modulus jHvr ðr2 ; r1 ; uiÞj and the argument 4ðr2 ; r1 ; uÞ are known, aðuÞ and kðuÞ can be obtained by solving simultaneously Eqs. (10) and (11). The propagation coefficient bðuiÞ for a viscoelastic spherical stress wave is defined in terms of the complex bulk modulus e uÞ by Ref. [11]. e uÞ and the complex shear modulus Gð Kð
b2 ðuiÞ ¼
r0 u2
e uÞ e uÞ þ 4 Gð Kð 3
(12)
where r0 is the density of the material, which is assumed to be constant. When considering the propagation of a viscoelastic spherical stress wave in a low loss material, such as PMMA, the frequency dependence of the Poisson's ratio m can be neglected from a practical point of view. Thus, the Poisson's ratio m is assumed to be constant, following the work of Pritz [12]. e uÞ are defined in terms of the complex elastic modulus e e uÞ and Gð Kð EðuÞ and the Poisson's ratio m as follows:
e uÞ ¼ Kð
e EðuÞ 3ð1 2mÞ
(13)
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e uÞ ¼ Gð
e EðuÞ 2ð1 þ mÞ
(14)
From Eqs. (12)e(14), we have
e EðuÞ ¼
ð1 mÞr0 u2 ð1 2mÞð1 þ mÞb2 ðuiÞ
¼ E0 ðuÞ þ E00 ðuÞi
(15)
where E0 ðuÞ is storage modulus and E00 ðuÞ is loss modulus. Eq. (15) shows that if the density r0, the Poisson's ratio m, and the propagation coefficient bðuiÞ are known, then we can obtain the frequency-dependent complex elastic modulus e EðuÞ. 2.2. Method for calculating the complex elastic modulus using the measured radial particle velocities at different radii Suppose that vr at r1 and r2 are measured in a free field; then, the experimental frequency response function H mpv vr ðr2 ; r1 ; uiÞ can be calculated by
Z
∞
0 H mpv vr ðr2 ; r1 ; uiÞ ¼ Z ∞ 0
vr ðr2 ; tÞeuit dt vr ðr1 ; tÞeuit dt
M P
¼
k¼1 N P
vr ðr2 ; tk Þeuitk Dt (16) vr ðr1 ; tk Þeuitk Dt
k¼1
where Dt is the sample interval, M and N are the sampling numbers, the superscript “mpv” of H mpv vr ðr2 ; r1 ; uiÞ denotes that the function is calculated by the measured particle velocity vr at r1 and r2 . The experimental frequency response function H mpv vr ðr2 ; r1 ; uiÞ is equal to the theoretical frequency response function Hvr ðr2 ; r1 ; uiÞ, thus, a function gðr2 ; r1 ; bðr2 ; r1 ; uiÞÞ is defined as
gðr2 ; r1 ; bðuiÞÞ ¼ Hvr ðr2 ; r1 ; uiÞ H mpv vr ðr2 ; r1 ; uiÞ ! 1 1 bðr2 ; r1 ; uiÞ þ 2 r2 r 2 bðr2 ;r1 ;uiÞðr2 r1 Þ !e H mpv ¼ vr ðr2 ; r1 ; uiÞ ¼ 0 1 1 bðr2 ; r1 ; uiÞ þ 2 r1 r1
(17)
In the present case, the complex derivative of g 0 ðr2 ; r1 ; bðuiÞÞ can be calculated analytically; therefore, an iterative formula of Newton type was chosen to solve the nonlinear Eq. (17) by
bnþ1 ðr2 ; r1 ; uiÞ ¼ bn ðr2 ; r1 ; uiÞ
gðr2 ; r1 ; bn ðr2 ; r1 ; uiÞÞ g 0 ðr2 ; r1 ; bn ðr2 ; r1 ; uiÞÞ
(18)
The convergence of this numerical iteration method depends on the initial value b0 ðr2 ; r1 ; uiÞ. According to Eq. (12), it seems that we can take the initial estimate b0 ðr2 ; r1 ; uiÞ by
b20 ðr2 ; r1 ; uiÞ ¼
r0 u2
e uÞ e uÞ þ 4 Gð Kð 3
(19)
e uÞ. In view of this, we used Eq. e uÞ and Gð following the work of Zhao [13]. Unfortunately, in our case, we know nothing about Kð (11) to estimate the wave number k0 ðr2 ; r1 ; uÞ of the propagation coefficient b0 ðr2 ; r1 ; uiÞ by
k0 ðr2 ; r1 ; uÞ z
4mpv ðr2 ; r1 ; uÞ ðr2 r1 Þ
(20)
where 4mpv ðr2 ; r1 ; uÞ is the argument of H mpv vr ðr2 ; r1 ; uiÞ, and the last two terms on the left of Eq. (11) are neglected. The substitution of k0 ðr2 ; r1 ; uÞ into Eq. (10) and the attenuation coefficient a0 ðr2 ; r1 ; uÞ can be obtained by solving the following equation as
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vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u mpv r1 u a0 ðr2 ; r1 ; uÞ þ r1 þ k20 ðr2 ; r1 ; uÞ a0 ðr2 ;r1 ;uÞðr2 r1 Þ 2 H ðr2 ; r1 ; uiÞ ¼ t e vr 2 r2 a ðr ; r ; uÞ þ r1 þ k2 ðr ; r ; uÞ 0
2
1
0
1
2
(21)
1
The initial estimate b0 ðr2 ; r1 ; uiÞ is determined by
b0 ðr2 ; r1 ; uiÞ ¼ a0 ðr2 ; r1 ; uÞ
4mpv ðr2 ; r1 ; uÞ i ðr2 r1 Þ
(22)
When the convergence conditions of Eq. (18) are met, we can obtain the propagation coefficient bðr2 ; r1 ; uiÞ. Thus, the complex elastic modulus can be defined according to Eq. (15) by
e Eðr2 ; r1 ; uÞ ¼
ð1 mÞr0 u2 ð1 2mÞð1 þ mÞb2 ðr2 ; r1 ; uiÞ
¼ E0 ðr2 ; r1 ; uÞ þ E00 ðr2 ; r1 ; uÞi
(23)
The calculation process of the complex elastic modulus e Eðr2 ; r1 ; uÞ is summarized in Fig. 1. It should be noted that the frequency equations of a viscoelastic spherical stress wave are based on the linear viscoelastic model without using conventional mathematical formulas. Thus, the theory presented in this paper does not take into account the nonlinear effects of spherical wave propagation. Unfortunately, the nonlinear effects of spherical wave propagation in the experimental case always exist, especially under high stressestrain state. For the ideal linear viscoelastic material, there is no nonlinearity in the measured waves, and the complex elastic modulus e Eðr ; r ; uÞ of the material can be obtained by Eq. (23) using the two measured particle velocities at any radii. When 2
1
considering the nonlinear effects of spherical wave propagation, the complex elastic modulus e Eðr2 ;r1 ; uÞ, which is calculated by Eq. (23) using the measured particle velocities at r1 and r2 , represents the local equivalent complex elastic modulus in the range of r1 to r2 . 2.3. Numerical examples and discussions To illustrate the method for calculating the complex elastic modulus from spherical wave data in a free field, the particular case of a viscoelastic spherical wave propagating in PMMA is discussed. The generalized Maxwell model (GMM) parameters published in the literature for PMMA are provided in Table 1. The GMM model, shown in Fig. 2, is composed of one elastic spring acting in parallel with two Maxwell elements. The complex bulk modulus and the complex shear modulus for PMMA in the frequency domain are expressed as [11].
e uÞ ¼ Kð
1 E q ui E q ui E0 þ 1 1 þ 2 2 3ð1 2mÞ q1 ui þ 1 q2 ui þ 1
(24)
e uÞ ¼ Gð
1 E q ui E q ui E0 þ 1 1 þ 2 2 2ð1 þ mÞ q1 ui þ 1 q2 ui þ 1
(25)
where E0 denotes the initial elastic modulus, qj ¼ hj =Ej (j ¼ 1,2) is the relaxation time of the jth Maxwell element, hj (j ¼ 1,2) is the viscosity coefficient of the jth Maxwell element, and Ej (j ¼ 1,2)is the elastic modulus of the jth Maxwell element. As shown in Ref. [11], the radial particle velocities at r ¼ 25, 50, 75, and 100 mm for the viscoelastic spherical stress wave in PMMA were numerically calculated by using the GMM parameters of Butt's model І presented in Table 1. Because Butt's model І is the linear viscoelastic model, there is no nonlinearity in the waves shown in Ref. [11], and the inversion of the complex modulus can be calculated by using the particle velocities at any two locations of r ¼ 25, 50, 75, and 100 mm. For example, following the first step shown in Fig. 1, the frequency response function H mpv vr ðr2 ; r1 ; uiÞ is calculated by Eq. (16) using the particle velocities at r1 ¼ 25 mm and r2 ¼ 50 mm shown in Ref. [11], and the real and imaginary part of H mpv vr ðr2 ; r1 ; uiÞ are shown in Fig. 3a and b, respectively. The modulus H mpv ðr2 ; r1 ; uiÞ and the argument 4mpv ðr2 ; r1 ; uÞ of H mpv ðr2 ; r1 ;
uiÞ are shown in Fig. 3c and d, respectively.
vr
vr
Following the second and third step shown in Fig. 1, the inversion results of the attenuation coefficient aðuÞ and the wave number kðuÞ, which are the real and imaginary part of the propagation coefficient bðr2 ; r1 ; uiÞ, are shown in Fig. 3e and f, respectively. Meanwhile, the theoretical results of the attenuation coefficient aðuÞ and wave number kðuÞ for Butt's model І can be obtained by Eq. (12), and are also shown in Fig. 3e and f.
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Fig. 1. The inversion method of the complex elastic modulus of materials through the combination of the viscoelastic spherical stress wave theory and the measured radial particle velocities at different radii in a free field.
Following the fourth step shown in Fig. 1, the inversion results of the storage modulus E0 ðuÞ and loss modulus E00 ðuÞ, which are the real and imaginary part of the complex modulus e Eðr2 ; r1 ; uÞ, are shown in Fig. 3g and h, respectively. Meanwhile, the
theoretical results of the storage modulus E0 ðuÞ and loss modulus E00 ðuÞ for Butt's model І can be obtained by Eq. (23), and are also shown in Fig. 3g and h. It can be seen from Fig. 3(eeh) that the inversion and theoretical results are in good agreements. This means that the inversion method for the complex elastic modulus presented in the preceding paragraphs is valid.
Table 1 The GMM parameters for PMMA. Reference Butt's model І [2] Butt's model ІІ [2] Wang's model [10]
r0 (kg/m3) 3
1.20 10 1.20 103 1.19 103
E0 (GPa)
E1 (GPa)
E2 (GPa)
q1 ¼
5.15 5.14 2.04
0.41 0.41 0.90
0.55 0.45 3.87
95.3 95.1 ∞
h1 E1
(ms)
q2 ¼ 1.88 2.34 2.05
h2 E2
(ms)
m 0.35 0.35 0.35
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Fig. 2. Rheological model for generalized Maxwell visco-elastic constitutive equation, consisting of a linear spring, a Maxwell element I and a Maxwell element II in parallel.
3. Experimental setup To investigate the dynamic mechanical behavior of a viscoelastic material, a spherical wave experiment in PMMA will be introduced. The experimental test setup and the PMMA sample are shown in Fig. 4a and b, respectively. A spherical pentaerythrite tetranitrate (PETN) explosive charge, with a diameter of 5 mm, density of 1.5 g/cm3, and equivalent to 0.125 g TNT, is placed at the center of the PMMA sample. The PMMA sample, with a diameter of 250 mm, height of 250 mm and density of 1180 kg/m3, is placed at the center of a solenoid. The wire loops with diameters of 0.22 mm are placed within the sample and are concentric with the spherical explosive charge at the mid-plane. The vertically oriented magnetic field generated by the solenoid surrounding the sample is applied at the mid-plane. When the divergent spherical stress wave generated by the spherical explosive charge travels in PMMA, the wire loops travel with the particle motion caused by the wave motion. The motion of the wire loops through the magnetic field flux lines generates a measurable electromagnetic induction voltage e in volts. Therefore, the particle velocity vr ðr; tÞ can be given by Faraday's law
vr ðr; tÞ ¼
e Bl
(26)
where B is the magnetic induction intensity, and l is the length of the particle velocity gage related to the radius r by l ¼ 2pr. Owing to the border effect of the sample, the waves traveling in the directions of increasing and decreasing r would be superposed at each wire loop. In Section 2, the method for determining the complex elastic modulus is given by the theory of viscoelastic spherical stress wave in an infinite medium; thus, to obtain the spherical stress wave data in a free field, the reflecting wave from the border of the sample is eliminated by using data processing technique referred to Bacon's work [8].
4. Results 4.1. Attenuation of peak velocity of spherical stress wave in PMMA Fig. 5 shows the measured particle velocity in a free field from the spherical wave experiment for PMMA. It can be seen that, owing to the reflecting wave from the border of the sample, the farther the gaging point is from the center of the explosion, the shorter the effective duration of the particle velocity. Fig. 6 shows that the peak particle velocity decreases with increasing wave propagation distance. Generally, the peak particle velocityedistance law can be expressed as [14].
vr
max
¼ Arh
(27)
where A and h are the fitted values. The power exponent h is the attenuation index, and it represents how fast the peak particle velocity changes with the wave propagation distance. Through the method of least squares, the experimental results are fitted by Eq. (27). The fitted curve is also shown in Fig. 6. Comparing the experimental results and the fitted curve, we can find out that the fitted curve, with fitted attenuation index h ¼ 1.33, is in good agreement with the experimental results at r ¼ 10e80 mm. The attenuation index h > 1 means that the attenuation of spherical wave propagation in PMMA includes not only the geometric effects but also the viscous effects as explained in Ref. [10].
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Fig. 3. The calculation process of the complex modulus Eðr2 ; r1 ; uÞ for Butt's model I. (a). the real part of Hvr ðr2 ;r1 ; uiÞ. (b). The imaginary part of Hvr ðr2 ;r1 ; uiÞ. (c). the modulus of Hvr ðr2 ;r1 ; uiÞ. (d). The argument of Hvr ðr2 ;r1 ; uiÞ. (e) comparison between the inversion and theoretical results of aðuÞ. (f). Comparison between the inversion and theoretical results of kðuÞ. (g). Comparison between the inversion and theoretical results of E0 ðuÞ. (h). Comparison between the inversion and theoretical results of E00 ðuÞ.
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Fig. 4. The spherical wave experiment. (a). Scheme of the spherical wave experiment, where the particle velocity waves are measured by magneto-electric velocimeter. (b). The PMMA sample.
Fig. 5. Experimental measured spherical waves vr ðr; tÞ in a free field for PMMA. (a) r ¼ 10, 20, 30, and 40 mm (b). r ¼ 50, 60, 70, and 80 mm.
4.2. The upper and lower limiting frequency of the response function of particle velocity The arguments 4mpv ðr2 ; r1 ; uÞ of H mpv vr ðr2 ;r1 ; uiÞ, determined by the measured particle velocities in the case of r1 ¼ 10 mm and r2 ¼ 20, 40, 60, and 80 mm, are shown in Fig. 7a. It can be seen that the argument 4mpv ðr2 ; r1 ; uÞ with different r1 and r2 becomes ambiguous at higher frequencies. We take the frequency at the beginning of the unreasonable change in the argument 4mpv ðr2 ; r1 ; uÞ as the upper limiting frequency referred as fmax ðr2 ;r1 Þ. As shown in Fig. 7a, when r1 ¼ 10 mm and r2 ¼ 20, 40, 60, and 80 mm, fmax ðr2 ; r1 Þ is accurate up to about 1900, 1750, 1600, and 1300 kHz, respectively. We can find that when r1 remains unchanged, fmax ðr2 ; r1 Þ decreases as r2 increases. It means that the determined upper limiting frequency fmax ðr2 ; r1 Þ of H mpv vr ðr2 ; r1 ; uiÞ is decided by the measured particle velocity away from the center of the explosion. The main reasons for argument 4mpv ðr2 ; r1 ; uÞ to become ambiguous after fmax ðr2 ; r1 Þ are that the frequency components of the particle velocity after fmax ðr2 ; r1 Þ become very weak owing to the fast decrease in the high frequency components caused by the viscous effects and they are inundated in the background noise. When considering the lower limiting frequency fmin ðr2 ; r1 Þ of the transfer function H mpv vr ðr2 ; r1 ; uiÞ shown in Fig. 7b, the border effect of the sample must be considered. Owing to the border effect of the sample, the wavelength of vr ðr; tÞ at the gage point is equal or lesser than the distance from the gage point to the border of the sample. Thus, the maximum wavelength lmax of vr ðr; tÞ can be estimated by
lmax ðrÞ ¼ Rb r where Rb is the radius of the sample. The corresponding lower limiting frequency fmin of vr ðr; tÞ can be estimated by
(28)
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Fig. 6. Comparison of the measured peak particle velocities and the fitted curve.
Fig. 7. The arguments 4mpv ðr2 ; r1 ; uÞ of H mpv vr ðr2 ; r1 ; uiÞ determined by the measured particle velocities in the case of r ¼ 10 mm and r ¼ 20, 40, 60, and 80 mm. (a) The upper limiting frequency of 4mpv ðr2 ; r1 ; uÞ is approximately accurate up to fmax ðr2 ; r1 Þ (b) the lower limiting frequency of 4mpv ðr2 ; r1 ; uÞ is approximately accurate up to fmin ðr2 ; r1 Þ.
fmin ðrÞ ¼
C
lmax ðrÞ
¼
C Rb r
(29)
where C is the sound speed of PMMA. Eq. (29) means that the spectrum components of vr ðr; tÞ below approximately fmin ðrÞ are unbelievable. As for the transfer function H mpv vr ðr2 ;r1 ; uiÞ, the lower limiting frequency fmin ðr2 ; r1 Þ is decided by fmin ðr1 Þ and fmin ðr2 Þ. It can be seen from Eq. (29) that fmin ðr2 Þ > fmin ðr1 Þ if r2 > r1 . Thus, fmin ðr2 ; r1 Þ can be estimated by
fmin ðr2 ; r1 Þ ¼ fmin ðr2 Þ ¼
C Rb r2
(30)
The sound speed C of PMMA is 2882.8 m/s and the radius Rb of the sample is 125 mm. When r2 ¼ 20, 40, 60, and 80 mm, fmin ðr2 ; r1 Þ is accurate up to about 28, 34, 44, and 64 kHz, respectively. It can be observed from Fig. 7 that fmax ðr2 ; r1 Þ decreases as r2 increases, and fmin ðr2 ; r1 Þ increases as r2 increases. This means that the upper and lower limiting frequency of each H mpv vr ðr2 ; r1 ; uiÞ are all decided by the measured particle velocity away from the center of the explosion, and the effective frequency band Df ¼ fmax ðr2 ; r1 Þ fmin ðr2 ; r1 Þ decreases as r2 increases.
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Fig. 8. The calculated storage modulus E0 ðr2 ; r1 ; uÞ (a). r1 ¼ 10 mm, r2 ¼ 20 mm. (b). r1 ¼ 20 mm, r2 ¼ 30 mm. (c) r1 ¼ 30 mm, r2 ¼ 40 mm. (d) r1 ¼ 40 mm, r2 ¼ 50 mm. (e) r1 ¼ 50 mm, r2 ¼ 60 mm. (f) r1 ¼ 60 mm, r2 ¼ 70 mm, and r1 ¼ 70 mm, r2 ¼ 80 mm.
4.3. Complex elastic modulus determined by the measured particle velocity in free field The complex elastic modulus e Eðr2 ; r1 ; uÞ with different r1 and r2 are calculated by the measured particle velocities at different radii according to the calculation process shown in Fig. 1. Figs. 8 and 9 show the storage modulus E0 ðr ; r ; uÞ and the loss modulus E00 ðr ; r ; uÞ of the complex elastic modulus e Eðr ; 2
1
2
1
2
r1 ; uÞ, which are calculated by the particle velocities at two adjacent gage points where r1 and r2 satisfy the relation by r2 r1 ¼ 10 mm, respectively. It can be seen from Figs. 8 and 9 that the storage modulus E0 ðr2 ; r1 ; uÞ (approximately above 300 kHz) tends to constant value, and the loss modulus E00 ðr2 ; r1 ; uÞ decreases to zero with the increase in frequency. Meanwhile, there
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Fig. 9. The calculated loss modulus E00 ðr2 ; r2 ; uÞ (a). r1 ¼ 10 mm, r2 ¼ 20 mm. (b). r1 ¼ 20 mm, r2 ¼ 30 mm. (c) r1 ¼ 30 mm, r2 ¼ 40 mm. (d) r1 ¼ 40 mm, r2 ¼ 50 mm. (e) r1 ¼ 50 mm, r2 ¼ 60 mm. (f) r1 ¼ 60 mm, r2 ¼ 70 mm, and r1 ¼ 70 mm, r2 ¼ 80 mm.
is nonlinearity in the measured spherical waves because the complex elastic modulus e Eðr2 ; r1 ; uÞ in the region near the explosion center is not the same as that in the region away from the explosion center. For example, E0 ðr2 ; r1 Þf > 300kHz , the average storage modulus above 300 kHz of each curve shown in Fig. 8 is defined by f ¼¼u=2 Pp¼fmax 0
E ðr2 ; r1 Þf > 300kHz ¼
f ¼u=2p¼300kHz
E0 ðr2 ; r1 ; uÞ
L
where L is the number of E0 ðr2 ; r1 ; uÞ when f > 300 kHz.
(31)
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Fig. 10. The average storage modulus E0 ðr2 ; r1 Þf > 300kHz at higher frequencies after 300 kHz.
Fig. 10 shows the average storage modulus E0 ðr2 ; r1 Þf > 300kHz . It can be seen that E0 ðr2 ; r1 Þf > 300kHz decreases from 7.14 to 5.97 GPa with increasing distance from the explosion center. e ; r ; uÞ at lower frequencies, we can observe in Figs. 8 and 9 that the When considering the complex elastic modulus Eðr 2 1 scattering of the complex elastic modulus increases because the particle velocity spectrum components at lower frequencies are not adequately developed owing to the interference that is caused by the wave reflecting from the border of the sample. Eq. (30) can provide the estimation for the lower limiting frequency of e Eðr ; r ; uÞ. For example, when r ¼ 70 mm and r 2
1
1
2
¼ 80 mm, the lower limiting frequency of e Eðr2 ; r1 ; uÞ is approximately 64 kHz. It means that a e Eðr2 ; r1 ; uÞ below 64 kHz is unbelievable. 5. Discussions 5.1. Stressestrain state dependence of the complex elastic modulus of PMMA Fig. 5 shows that the rising edges of the measured particle velocities at r ¼ 10e80 mm are only of the order of 0.6e1.0 ms. Suppose that the propagation of the peak particle velocity is consistent with the theory of strong discontinuous viscoelastic spherical waves, following the work of Wang [10]. Taking compression as positive, we have
½vr ¼ D½εr
(32)
½sr ¼ r0 D½vr
(33)
where D is the shock speed calculated by using the time of arrival in Fig. 5, sr is the radial stress, εr is the radial strain, and [] denotes the discontinuous jump across the wave front. According to Eqs. (32) and (33), the radial stress sr and the radial strain εr at different radii are shown in Fig. 11. It can be seen that the radial stress sr decreases from 682 to 40 MPa, and the radial strain εr decreases from 59.5 103 to 4.2 103. Meanwhile, Fig. 11 compares the average storage modulus E0 ðr2 ; r1 Þf > 300kHz with the radial stress sr and the radial strain εr at different radii. It shows that, when the radial stress sr and radial strain εr are reduced with the increasing distance of the spherical wave, the average storage modulus E0 ðr2 ; r1 Þf > 300kHz is also reduced and tends to 5.97 GPa. On the basis of the results shown in Fig. 11, we can know that the average storage modulus E0 ðr2 ; r1 Þf > 300kHz is dependent on the stressestrain
state. Take the average storage modulus E0 ðr2 ; r1 Þf > 300kHz ¼ 5.97 GPa when r1 ¼ 70 mm and r2 ¼ 80 mm as the reference value,
and define the function c by
c¼
E0 ðr2 ; r1 Þf > 300kHz E0 ð80mm; 70mmÞf > 300kHz E0 ð80mm; 70mmÞf > 300kHz
where c represents the deviation of E0 ðr2 ; r1 Þf > 300kHz relative to E0 ð80mm; 70mmÞf > 300kHz ¼ 5.97 GPa.
(34)
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Fig. 11. The average storage modulus E0 ðr2 ; r1 Þf > 300kHz changes with the stressestrain state owing to the nonlinear effects of spherical wave propagation. The bar chart represents the average storage modulus E0 ðr2 ; r1 Þf > 300kHz between two adjacent gage points. The solid line with a symbol of a circle or square represents the radial stress sr or the radial strain εr at different radii, respectively.
Fig. 12. The storage modulus E0 ðuÞ, determined through the spherical stress wave propagation data under different stress-strain states, is compared to Butt's and Wang's models. The solid line represents the storage modulus E0 ðr2 ; r1 ; uÞ determined by the velocities at r1 ¼ 10 mm and r2 ¼ 20 mm or r1 ¼ 70 mm and r2 ¼ 80 mm, respectively. The solid line with a symbol of a circle, square or triangle represents Butt's model I, Butt's model II or Wang's model, respectively.
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Fig. 13. The loss modulus E00 ðuÞ, determined through the spherical stress wave propagation data under different stress-strain state, is compared to Butt's and Wang's models. The solid line represents the storage modulus E00 ðr1 ; r2 ; uÞ determined by the velocities at r1 ¼ 10 mm and r2 ¼ 20 mm or r1 ¼ 70 mm and r2 ¼ 80 mm, respectively. The solid line with a symbol of a circle, square or triangle represents Butt's model I, Butt's model II or Wang's model, respectively.
Fig. 14. The storage modulus E0 ðuÞ, determined through the spherical stress wave propagation data at r1 ¼ 70 mm and r2 ¼ 80 mm, is compared to Mousavi's results.
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Within the radius of 10e80 mm, c is reduced from 19.6% to 0% with the radial stress sr decreasing from 682 to 40 MPa, and radial strain εr decreasing from 59.5 103 to 4.2 103. When r 30 mm, the radial stress sr 176 MPa, radial strain εr 17.8 103, and relative deviation c 3.0%. From a practical point of view, the stressestrain state dependence of the complex elastic modulus can be neglected as the radial stress sr 176 MPa and radial strain εr 17.8 103.
5.2. Comparison of the complex elastic modulus of PMMA determined in different ways This section presents a comparison between the experimentally determined complex elastic modulus of PMMA from the spherical wave data in a free field and those evaluated through the GMM or experimental results selected from the literature. Butt et al. [2,3] used a PMMA bar with a diameter of 30 mm and length of 2 m to investigate the dynamic mechanical behavior of a viscoelastic material. In their work, the peak strain in the PMMA bar is approximately 6 103 or less. The corresponding peak stress in PMMA bar, estimated by s ¼ Eε where E is assumed to be 6 GPa, is approximately 36 MPa or less. Wang et al. [10] used the characteristics method to investigate the propagation of the spherical stress wave in the case of the tamped explosion with 0.125 g TNT in PMMA. The spherical PETN explosive charge used in their work is the same as that used in this study. Wang et al. analyzed the propagating characteristic of the measured particle velocity at r ¼ 5e30 mm, where the radial stress sr was approximately 176e1700 MPa, and the radial strain εr was approximately 17.8 103e180 103. The GMM parameters of PMMA have been determined from the stress wave data in a lower stressestrain state by Butt et al. [2], and that in a higher stressestrain state by Wang et al. [10]. These GMM parameters are presented in Table 1. e ; r ; uÞ determined through the spherical stress wave propaFigs. 12 and 13 compare the complex elastic modulus Eðr 2
1
gation data in a free field with that calculated by Butt's and Wang's models. It can be seen from Fig. 12 that Butt's models are closer to the storage modulus E0 ðr2 ; r1 ; uÞ calculated by two particle velocities at r1 ¼ 70 mm and r2 ¼ 80 mm, and Wang's model is closer to the storage modulus E0 ðr2 ; r1 ; uÞ calculated by two particle velocities at r1 ¼ 10 mm and r2 ¼ 20 mm. Fig. 13 shows that the loss modulus E00 ðr2 ; r1 ; uÞ from Wang's model is greater than that from Butt's models, and the loss modulus E00 ðr2 ; r1 ; uÞ, determined from the measured radial particle velocities in this work, is closer to Butt's models. This indicates that the published parameters of viscoelastic materials, determined from different experimental techniques under different stressestrain states, should be used with caution.
Fig. 15. The loss modulus E00 ðuÞ, determined through the spherical stress wave propagation data at r1 ¼ 70 mm and r2 ¼ 80 mm, is compared to Mousavi's results.
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Mousavi et al. [15] used PMMA bars with diameters of 10 mm and 20 mm to identify the complex elastic modulus. Their results, shown in Figs. 14 and 15, are very accurate up to approximately 40 kHz for a F10 mm PMMA bar, and 15 kHz for a F20mm PMMA bar. Meanwhile, Mousavi's results are also compared with the complex elastic modulus determined through the two particle velocities at r1 ¼ 70 mm and r2 ¼ 80 mm. For PMMA, it can be seen that the complex elastic modulus from the measured strains on an impacted bar is accurate up to approximately tens of kHz. However, the complex elastic modulus, determined through the spherical stress wave propagation data in a free field, is accurate approximately from tens of kHz to thousands of kHz in the present study. For one-dimensional stress wave propagation in a bar, according to Ref. [16], the upper limiting frequency where the inertia effects become dominant can be estimated by
fmax
1D
¼
1 10d
sffiffiffiffiffi Er
r
(35)
where d is the diameter of the bar, and Er can be considered as the storage modulus at high frequencies. For the case of the bar, to improve the accuracy of the complex elastic modulus from the measured strains on an impacted bar at high frequencies, we should use a thinner bar according to Eq. (35). In addition, for the case of the spherical wave experiment, we should use a larger sample, as shown in Fig. 4, to obtain the particle velocities in the region far away from the explosion center. This allows improving the accuracy of the complex elastic modulus determined through the spherical stress wave propagation data in a free field at low frequencies. 6. Conclusions In the present study, without using conventional mathematical models, a brief and effective method is proposed to determine the complex elastic modulus of a viscoelastic material. This is achieved by the combining the theory of viscoelastic spherical stress wave and the measured radial particle velocities at different radii in a free field. From the above results the following conclusions can be drawn: (1) Based on the theory of the viscoelastic spherical wave, the frequency response function of particle velocity between two points is given in terms of the propagation coefficient and the wave propagation distance. Meanwhile, the frequency response function can be obtained by the ratio of the particle velocity spectrums of two measurable points from a spherical wave experiment. By solving the frequency response equation shown in Eq. (17), the local equivalent complex elastic modulus can be obtained numerically. (2) The results show that the PMMA behaves viscoelastically under spherical explosion loading, and when the radial stress and radial strain are reduced with the increasing distance of the spherical wave, the average storage modulus of PMMA at high frequencies after approximately 300 KHz, is also reduced from 7.14 to 5.97 GPa. This indicates that the complex elastic modulus of PMMA is dependent on the stressestrain state, and there is nonlinearity in the measured spherical waves at higher stressestrain states. (3) Through the comparisons between the determined complex elastic modulus from spherical wave data and those evaluated through the GMM or experimental results selected from the literature, the published parameters of viscoelastic material, determined from different experimental techniques under different stress-strain states, should be used with caution. For PMMA, from a practical point of view, when the stress and strain are less than 176 MPa and 17.8 103, respectively, the stressestrain state dependence of the complex elastic modulus can be neglected.
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