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One-dimensional linear elastic waves at moving property interface
Wave Motion 51 (2014) 1179–1192
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One-dimensional linear elastic waves at moving property interface Lang-Quan Shui ∗ , Zhu-Feng Yue, Yong-Shou Liu, Qing-Chang Liu, Jiao-Jiao Guo Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, PR China
highlights • • • • •
Proposed a moving property interface model. Proposed a novel method to study the waves at moving property interface. Property interface motion and wave velocity can influence the wave propagation. There may exist shock waves at moving property interface. Property interface motion has an impact on the wave frequency and energy.
article
info
Article history: Received 23 April 2014 Received in revised form 5 July 2014 Accepted 10 July 2014 Available online 21 July 2014 Keywords: Time-varying Moving property interface Elastic wave Shock
abstract Smart materials exhibit time-varying properties while time-varying external field is applied. To investigate the one-dimensional (1-D) homogeneous time-varying properties, a moving property interface (MPI) model is proposed, and the propagation of linear elastic waves at 1-D MPI is studied in this paper. Based on the idea of weak solutions and an infinity approximation, a novel method to deal with the difficulties in using the continuities to study the waves at MPI is also proposed. Some interesting phenomena are revealed: (i) besides wave impedance, the property interface motion and wave velocity are also very important factors that influence the wave propagation; (ii) at MPI, there may exist shock waves; (iii) the property interface motion has a significant impact on the wave frequency and energy. This research provides a theoretical viewpoint in the study of smart materials with a time-dependent mechanical properties at different loading conditions. © 2014 Elsevier B.V. All rights reserved.
1. Introduction For their smart features, such as sensing, feedback, information recognition and accumulation, self-recovery, selfadjusting and so on, smart materials are widely used in the multi-sensor, precision driving, vibration isolation/excitation and many other fields [1,2]. Smart materials exhibit time-varying properties while time-varying external field is applied, such as the shear modulus of the piezoelectric material [3] and electro rheological (ER) materials [4] experiencing timevarying electric field, material damping of the magnetostrictive material [5] and magneto rheological fluid [6] experiencing time-varying magnetic field, and modulus of the shape-memory material (SMA) [7] experiencing time-varying temperature field. The time-varying properties of smart materials can be realized through precise control of the external fields. Using the time-varying properties, the smart composites and structures gained a lot of new applications. Ruzzene, Baz et al. [8–13] applied the SMA and the piezoelectric material to the design of period structures considering the timevarying modulus. Such structures demonstrate excellent active vibration performances. Wu [14] and Yang [15] respectively
∗
Corresponding author. Tel.: +86 15902902159. E-mail address: [email protected] (L.-Q. Shui).
http://dx.doi.org/10.1016/j.wavemoti.2014.07.005 0165-2125/© 2014 Elsevier B.V. All rights reserved.
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Fig. 1. The moving interface in smart materials: (a) cell division of 1-D structure; (b) distribution of the property at time t0 ; (c) distribution of the property at time t1 ; (d) distribution of the property at time t2 .
used the dielectric elastomer with time-varying properties to design the 1-D and 2-D phononic crystals. These phononic crystals have an adjustable band-gap. Wright and Cobbold [16,17] studied the band-gap characteristics of the phononic crystals with time-varying scatterer, and extended the multiple scattering theory to time-varying structures. Luo and Wang [18,19] designed left-handed metamaterials with actively modulated left-handed transmission peak and electrically tunable negative permeability metamaterials based on ER fluids considering time-varying properties. Most of these researches are focused on applications of a specific smart material with time-varying properties. Few of them involve general theoretical study. Generally, there are two methods, which are wave propagation method and frequency domain analysis method, to research the mesoscopic physical mechanism and the transient response of smart structures with time-varying properties. The variance of the wave frequency in materials with time-varying properties makes the frequency domain analysis method unavailable. Thus the wave propagation method is a better choice to study the time-varying materials. A series of work has been reported on waves in time-varying materials [20–23]. This work on wave propagation in materials with time-varying properties will reveal interesting phenomenon different from the waves in static or moving media. This paper focuses on the propagation of linear elastic waves at 1-D moving property interface (MPI). MPI is a model that can sufficiently reflect the characteristics of a material with 1-D time-varying properties. Usually, the continuities are powerful and sufficient in gaining the wave propagation law at property interface. However, there are obstacles in using the continuities to study the waves at MPI. Hence we proposed a novel method, which is based on the idea of weak solutions and infinity approximation, to deal with the difficulties, and the propagation law of linear elastic waves at 1-D MPI is achieved. Specifically, the MPI model will be described based on the property variability of smart materials in Section 2. The movement mechanism of the property will be revealed in this section. In Section 3, according to the propagation characteristics of the wave when the property interface moving with different velocities, the wave propagation at MPI is classified. A complete discussion of the propagation law of the waves at MPI will be given in Sections 4 and 5. In Section 6, simulations will be conducted to verify the results obtained in previous sections. Section 7 is the summary of the whole work in this paper. In addition, the wavelength, wave frequency and energy characteristics of the waves on the MPI will be analyzed in Section 7. In Section 8, i.e. the last section, some conclusions will be made. 2. Moving property interface (MPI) We will find out a representative model which can sufficiently reflect the characteristics of materials with time-varying properties. This needs to take an in-deep understanding of the materials with time-varying properties, first. As well known, the application of the smart material is mainly based on the property controllability. The properties variance of smart materials is due to the time-varying external field. On the other hand, the edge effect, device-specific or human-induced inhomogeneity of external field, and some other factors, may lead to the spatial inhomogeneity of the property of smart materials. As a result, variable properties of smart materials are functions of time and space. A general variable property should be inhomogeneous in time and space. However, a homogeneous variable property is a basic model which can be used to describe an inhomogeneous one by integration. In the following, we put forward a 1-D wave model to describe the 1-D homogeneous variable property which named MPI. As shown in Fig. 1(a), the space in the 1-D structure is divided into a series of small cells: e1 , e2 , . . . , ei , . . . , en , and li−1,i represents the interface between the adjacent cells ei−1 , ei . Assume that the property of the material is P1 when the external field A acts on the material, and the property is P2 when field B is applied. P1 and P2 in Fig. 1(b)–(d) are expressed by pink and yellow respectively. At the initial time t0 (Fig. 1(b)), field A acts on cells e1 , . . . , ei−1 and field B on ei , . . . , en . After a time interval, at time t1 (Fig. 1(c)), the field on cell ei switches from B to A. Hence the property of cell ei switches from P2 to P1 , which means that the property interface moves rightward by a cell length and reaches li,i+1 . So repeatedly the MPI can be formed. Considering infinitesimal cells, the movement of the interface will be continuous.
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Fig. 2. Characteristic lines of waves at MPI (|V | ̸= c ): (a) |V | > cmax (b) |V | < cmin (c) c2 < −V < c1 (d) c1 < V < c2 , c2 < V < c1 and c1 < −V < c2 .
Let V represent the move velocity. When V = 0, the MPI degenerates into the well-known static property interface. When V = ∞, the MPI degenerates into a property transient. It should be noticed that the property moves without the transfer of matter. It should be emphasized that the property movement, which is induced by homogeneous spatio-temporal distribution of the controllable property, should be the most representative characteristics of practical materials with timevarying properties. 3. The classification of the waves at MPI A series of conventions are listed below:
√
√
E ρ . The wave velocity is c = E /ρ . Let cmax = max {c1 , c2 } , cmin = min {c1 , c2 } . V is the moving velocity of the property interface; • by symmetry, it can be assumed that the incident wave is coming from the material with property P1 . The incident wave propagates along the positive direction of the x axis; ˆ − ωt . The signed wavenumber, signed wave length, signed • a wave is expressed as W . And the displacement of W is u kx
• the wave impedance is z =
wave velocity, and signed wave impedance are defined as
ˆk = 2π /λ (W travels along the positive direction of x axis) −2π /λ (W travels along the negative direction of x axis),
λˆ =
2π kˆ
cˆ = c
λˆ λ
zˆ = z
λˆ λ
in which λ is the wavelength. The wave frequency ω = cˆ kˆ ; • the symbols that directly relate to the material properties (such as the wave velocity c, the material density ρ ) have subscripts that range from 1 to 2. The same subscript refers to the same meaning. This means that c1 is the wave velocity of the material with property P1 . And the material density, modulus and wave impedance corresponding to c1 are ρ1 , E1 and√ z1 . The subscripts are omitted if it will not be confused. I.e. c1 c = sin c represents c12 = sin c1 and c1 c2 = sin c2 ; c = E /ρ represents √ √ c1 = E1 /ρ1 and c2 = E2 /ρ2 . The free linear elastic wave equation is
σ ′ − p˙ = 0 σ = Eu′ , p = ρ u˙
(1)
˙ = ∂ (·) /∂ t. E and ρ are material modulus and density, respectively. u is the where the operators (·)′ = ∂ (·) /∂ x, (·) displacement of the particle. x and t represent space and time, respectively. According to the theory of characteristics of second-order 1-D wave equation, the vibration at a point can be determined by the disturbances coming along two characteristic lines, then the vibration at this point will spread out along another two characteristics lines. The two waves that spread out are unknown. For a posed continuous wave solution, u in Eq. (1) has two continuous conditions, which can be used to solve the unknown waves. Therefore, only if the number of characteristic lines leaving a point is two, the vibration near this point can be uniquely determined. When the property interface moving speed is not equal to the wave velocity, all the possible relationships of the characteristic lines and MPI are listed in Fig. 2. The red lines in Fig. 2 represent the MPI in spatio-temporal coordinates Oxt. And the black lines are characteristic lines. P1 , P2 are the properties of the materials on two sides of the MPI. W1 is incident wave. W2 and W3 represent the emitted waves with the different and same directions comparing with the incident wave respectively. It should be noted that the incident situation of W1a , W1b , W1c correspond to the emitted situation of W2a , W2b , W3c , successively, in Fig. 2(d). The cases in Fig. 2(a), (b) are the ones with the conditions |V | > cmax and |V | < cmin respectively. In these two cases there are two characteristic lines leaving the MPI, which indicate that the wave solution is well-posed. These cases are named as Usual Cases. The moving velocity of the property interface in Fig. 2(c) satisfies c2 < −V < c1 . In this case the number of the characteristic lines leaving the interface is three, which is larger than the number of the continuous conditions. In this situation, the solution of Eq. (1) may not be unique. There are three rest cases, i.e. c2 < V < c1 , c1 < V < c2 and c1 < −V < c2 . Each case has only one emitted wave. They can be simultaneously shown in Fig. 2(d). The three cases in
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V
Figure
Usual case 1 Usual case 2 Unusual case 1 Unusual case 2 Unusual case 3 Unusual case 4 Critical case
|V | > cmax |V | < cmin c2 < −V < c1 c2 < V < c1 c1 < V < c2 c1 < −V < c2 |V | = c1 , c2
Fig. 2(a) Fig. 2(b) Fig. 2(c) Fig. 2(d)-W1a Fig. 2(d)-W1b Fig. 2(d)-W1c Fig. 4
Fig. 2(d) each has only one characteristic line leaving the interface. Thus the solution of Eq. (1) may not exist considering the continuous conditions. The cases in Fig. 2(c) and (d), which cannot be solved directly by the two continuous conditions, are named as Unusual Cases. When the property interface moving speed is equal to the wave velocity, we name the case as Critical Case. According to above discussions, all cases in Fig. 2 and the Critical Case are listed in Table 1. It is obviously that the Unusual Cases will disappear when c1 = c2 . 4. Wave propagation in Usual cases and Critical cases 4.1. Continuous conditions of the waves According to the analysis in Section 3, the Usual Cases can be solved by continuous conditions. Therefore, the key to solve these cases is to obtain the continuous conditions. Usually the linear elastic waves have two continuous scalar. They are represented as Cu and C∂ u . Cu is the displacement and C∂ u is a variable associated with the first-order differential of the displacement. Let the basic assumption of continuum mechanics, C u = u.
(2)
The continuous scalar associated with its first-order differential can be obtained according to the momentum theorem. ˜ and V is the moving The property variability as shown in Fig. 1 is considered. Assume that the area of the interface is A, velocity of the property interface. During interval dt, the length within the property transform is V dt. The momentum before
˜ dt u˙ 1 . The amount of the momentum change is ˜ dt u˙ 2 . And the one after the transform is ρ1 AV the transform is ρ2 AV ˜ 2 u′2 − AE ˜ dt u˙ 1 − ρ2 AV ˜ dt u˙ 2 . Meanwhile the force on the material is AE ˜ 1 u′1 . According to the momentum theorem, ρ1 AV it can be obtained that
˜ 2 u′2 − AE ˜ 1 u′1 dt . ˜ dt u˙ 2 = AE ˜ dt u˙ 1 − ρ AV ρ1 AV
(3)
2
˜ on both hands of Eq. (3), it can be simplified as Divided by Adt E1 u′1 + ρ1 V u˙ 1 = E2 u′2 + ρ2 V u˙ 2 .
(4)
Let C = Eu + ρ V u˙ , Eq. (4) can be rewritten as ′
C1 = C2 .
(5)
It is obvious that C∂ u = C = Eu′ + ρ V u˙ .
(6)
The following processing is to simplify Eq. (6). A traveling wave in 1-D homogeneous medium is considered. The displacement of the wave is u x − cˆ t . During the time interval dt, the particles disturbed by the wave have a spatial length of cˆ dt. And the mass is ρ Acˆ dt . The particles get
˜ ′ , it can be obtained that the momentum of ρ A˜ cˆ dt u˙ . Considering that the particles are acted by the force −AEu ˜ ′ dt . ρ A˜ cˆ dt u˙ = −AEu
(7)
˜ on both hands of Eq. (7), it can be simplified as Divided by Adt u˙ = −ˆc u′ .
(8)
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Fig. 3. Geometric relationships of wavelengths: λ1 /λ2 = AC /DF = (V − c1 )/(V − c2 ).
Substitute Eq. (8) into Eq. (6), the continuous scalar can be written as
C∂ u = −z V − cˆ ku´
(9)
ˆ − ωt . where we used a symbol u´ = ∂ u/∂ kx The characteristic lines of Eq. (1) along one direction are clusters of parallel lines. The wavelength can be defined as the spatial distance of any two characteristic lines with phase difference in one period. The particles on the MPI disturbed by the incident waves produce new waves. The new waves may distribute on any sides of the MPI according to the analysis in Section 3. One case, in which the emitted wave is not on the same side of the incident wave, is shown in Fig. 3. The angle between the MPI and t-axis is ̸ BDC = arctan (−V ). And for the incident wave and emitted wave, the characteristic lines and the t-axis have angles of ̸ BDA = arctan c1 and ̸ ECF = arctan c2 . The geometric relationship indicates that the wavelength ratio of the incident wave and emitted wave is λ1 /λ2 = AC /DF = (V − c1 ) / (V − c2 ). It is not difficult to verify that the waves on any sides of the MPI satisfy the relation below
λˆ ∝ V − cˆ .
(10)
Substitute Eq. (10) into Eq. (9), and ignore the constant coefficient, the simplified continuous scalar associated with the first-order differential of the displacement can be written as C∂ u = zˆ u´ .
(11)
In sum, Eqs. (2) and (11) are the two components of the continuous scalars for the continuous elastic waves on the MPI. 4.2. Usual case 1 As shown in Fig. 2(a), the incident wave W1 and emitted wave W2 , W3 are on either side of the MPI respectively. Thus, based on Eqs. (2) and (11), the continuities can be expressed as
u1 kˆ 1 x − ω1 t = u2 kˆ 2 x − ω2 t + u3 kˆ 3 x − ω3 t zˆ1 u´ 1 kˆ 1 x − ω1 t = zˆ2 u´ 2 kˆ 2 x − ω2 t + zˆ3 u´ 3 kˆ 3 x − ω3 t .
(12)
Integrating the second equation in Eq. (12) along the MPI, it can be simplified as z1 kˆ 1 V − cˆ1
u1 kˆ 1 V − cˆ1 t = −
z2 kˆ 2 V − cˆ2
u2 kˆ 2 V − cˆ2 t +
z2 kˆ 3 V − cˆ3
u3 kˆ 3 V − cˆ3 t .
(13)
Let the displacement near the MPI be Ui (i = 1, 2, 3), Eq. (12) can be expressed as
U1 = U2 + U3 z1 U1 = −z2 U2 + z2 U3 .
(14)
By solving Eq. (14), the displacement transmission coefficients of the forward transmitted wave W3 and backward transmitted wave W2 can be obtained as
U3 z2 + z1 = F = U1
2z2
U2 z2 − z1 B = = . U1
2z2
(15)
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Fig. 4. Characteristic lines of the critical case: (a) V = c2 < c1 (b) V = c2 > c1 (c) −V = c2 > c1 (d) −V = c1 < c2 (e) −V = c2 < c1 (f) −V = c1 > c2 . Table 2 The displacement propagation coefficients of Critical case. Case
V
Case basis
U2 /U1
U3 /U1
a b c d
V = c2 < c1 V = c2 > c1 −V = c2 > c1 −V = c1 < c2
|V | < cmin |V | > cmax |V | > cmax |V | < cmin
R B
∆=T ∆=F
∆=B ∆=R
F T
4.3. Usual case 2 In this case the reflected wave and incident wave are on the same side, and the transmitted wave is on the other side. Thus the continuities become
u1 kˆ 1 x − ω1 t + u2 kˆ 2 x − ω2 t = u3 kˆ 3 x − ω3 t zˆ1 u´ 1 kˆ 1 x − ω1 t + zˆ2 u´ 2 kˆ 2 x − ω2 t = zˆ3 u´ 3 kˆ 3 x − ω3 t
(16)
which can be simplified similarly to Eq. (12) as
U1 + U2 = U3 z1 U1 − z1 U2 = z2 U3 .
(17)
The displacement reflection coefficient R and transmission coefficient T can be obtained as
z1 − z2 U2 = R = U1 z1 + z2 U3 2z1 T = = . U1 z1 + z2
(18)
4.4. Critical case The Critical case can be considered as a transition of the Usual cases and Unusual cases. It is obvious that the incident ˆ → 0 as wave may not touch the MPI when V = c1 . Thus this case is neglected. Considering Eq. (10), it can be seen that λ V → cˆ , i.e. the characteristic lines in front of the MPI would be compressed into the MPI. Thus the wave may exhibit strong discontinuity on the MPI, i.e. the MPI receives a shock traveling with itself. Specifically, there are 6 cases contained in the critical case (Fig. 4). In Fig. 4, the red lines are the MPI, the black lines are characteristic lines. Generally, considering the limits of Usual case as V → ±c, the propagation law of these cases can be obtained. It is obvious that the propagation law in Usual case has nothing to do with V . Thus the Critical case has the same propagation coefficients with the Usual case. When the wavelength becomes zero, corresponding propagation coefficients would be the shock discontinuous coefficient ∆. The details are listed in Table 2. The cases in Fig. 4(e) and (f) are the same as the Usual case. No shock exists in these two cases. Thus they do not appear in Table 2. The symbols in Table 2 can be found in Eqs. (15) and (18). It should be noted that the discussions in Section 4.4 are the process of taking a limit of the Usual cases as V → ±c. The results are also ruled by the assumption of linear elasticity. More detailed analysis in Section 7.3 indicate that the discussion about the Critical case is meaningless because of corresponding infinite energy which do not exist in reality. However in a theoretical sense the discussion is still useful for the following Unusual cases. 5. Wave propagation in Unusual cases In these cases, the number of the unknown waves is not equal to that of the continuous conditions. Thus the waves cannot be determined directly by the continuity. To get over the difficulty, some improvements need to be sought. The effective method is to transform these cases into the Usual cases or Critical cases.
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Fig. 5. Interface approximation: (a) an approximation of the moving interface with step lines; (b) repeated interaction of characteristic lines and tiny interfaces.
5.1. Unusual case 1 As shown in Fig. 5(a), the MPI has been resolved as polylines. The MPI would be the limits of the polylines as the scale of the polyline approaches infinitesimal. The segments in polylines always can be chosen to satisfy Usual cases or Critical cases. Thus the Unusual cases can be transformed into the Usual cases or Critical cases. An incident wave that reflects or transmits at a segment may reach another segment, and the reflection and transmission are repeated (Fig. 5(b)). Such repeated interaction may lead to tedious calculation to analyze displacement of the emitted wave. A good resolution method can greatly simplify the calculation. In dealing with the Unusual case 1, for simplicity, we use a resolution method as shown in Fig. 6, in which the green polylines are resolved interface. In such a resolution method, a characteristic line interacts with the polylines less than two times. In Fig. 6, a unit of the polylines is ABC , where AB has a slope of − arctan−1 c2 , BC is parallel to the x axis. It is obvious that wave propagation at AB and BC satisfies the case (e) of Critical cases (Fig. 4(e)) and Usual case 1 (Fig. 2(a)), respectively. In addition, Ws represents the transmitted wave with opposite direction of W1 in Fig. 6. According to the resolution method in Fig. 6, three steps are needed to obtain the propagation law. First, the wave source should be divided into several parts. In Fig. 6, they are EF , FM and MN. Waves from the same part should obey the same propagation law. It is not hard to get the wave propagation law for every part of incident wave W1 . But it is difficult to determine the relationship between the propagate law for a certain part and for the whole source EN. The second step focuses on this problem. We will find the limits of the law for every part as the scale of the polyline approaches infinitesimal. The last, the final results can be obtained based on the results in the first two steps. 5.1.1. The first step: the division of the wave source In Fig. 6, we use different colors to distinguish the different parts of wave source. The new color appearing in the emitted wave indicates that the waves come from different parts mixed, and the components of the mixture wave can be found according to the color mixture in Fig. 6. According to the interaction characteristics of the characteristics lines and the unit polyline, the wave source EN can be divided into three parts as MN , FM , EF (Fig. 6). Waves come from the same part have the same propagation path. Let the ratio of the length of MN , FM , EF to the length of EN be l1 , l2 , l3 , successively. According to the geometric relationships, the ratios can be obtained as
MN c1 + c2 c1 − |V | = l1 = | V | + c1 c1 − c2 EN FM c1 + c2 |V | − c2 l2 = = |V | + c1 c1 − c2 EN | EF V | − c2 l3 = = . | V | + c1 EN
(19)
5.1.2. The second step: the limits of the displacements in subdomains It is a key problem to obtain the limits of the displacements in subdomains as the scale of the polyline approaches infinitesimal. Let the length of EN approaches dx. Thus EN can be regarded as a ‘‘measurable point’’. And MN , FM , EF are three measurable subdomains of the ‘‘point’’. It is acceptable considering a weak solution that the function value in a subdomain is determined by its ratio of the measure of the subdomain to the measure of the ‘‘point’’. We may assume that Eq. (1) has unique solution. If two possible solutions f (x) , g (x) satisfy f (x) ϕ (x) dx = g (x) ϕ (x) dx, where ϕ (x) is a smooth test function in compactly supported subset of real domain, then f (x) and g (x) can be regarded as the same solution. Considering a function in the form
ξ (x) = lim l→0
1 x ∈ ∪+∞ −∞ (il, (i + θ ) l]
0 x ∈ ∪+∞ −∞ ((i + θ) l, (i + 1) l]
(i is an integer, 0 < θ < 1, l > 0)
(20)
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Fig. 6. Characteristics lines of Unusual case 1: c2 < −V < c1 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
it is easy to verify that if f (x) = u (x) ξ (x), then g (x) = θ u (x). Based on these ideals, the displacements in the subdomains can be directly set as uMN = l1 U1 uFM = l2 U1 uEF = l3 U1
(21)
where U1 is the displacement on EN. 5.1.3. The third step: calculation of the displacement propagation coefficients It is easy to obtain the propagation coefficients on AB and BC . According to Eqs. (15) and (18), we have
z1 − z2 RAB = z1 + z2 2z1 TAB = z1 + z2 z2 + z1 FBC =
(22)
2z2
(23)
BBC = z2 − z1 . 2z2
The ratio of the displacements of W2 , WS , W3 to the displacement of W1 are denoted by R˜ , S˜ , T˜ , respectively. In Fig. 6, the district of W2 is cyan. The wave W2 comes from MN and is formed by the reflection on AB. Based on Eqs. (19), (21) and (22), the reflection coefficient can be obtained as R˜ =
U2
=
U1
uMN RAB U1
=
z1 − z2 c1 + c2 c1 − |V | z1 + z2 c1 − c2 c1 + |V |
.
(24)
The district of WS is red. Considering the color mixture, red is a mix of yellow and magenta, which correspond to the waves from FM and EF . The waves from FM experience the reflection on BD and transmission on BC . Consequently, the transmitted wave should have displacement of uFM (−RAB ) FBC . The waves from EF transmits BC directly and should cause the displacement of uEF BBC . Based on Eqs. (19) and (21)–(23), the transmission coefficient can be obtained as S˜ =
U3
=
U1
1 U1
[uFM (−RAB ) FBC + uEF BBC ] =
z2 − z1 z2
|V | − c2 . c1 − c2 |V | + c1 c1
(25)
The district of W3 is yellow and magenta. Yellow and magenta district represents the waves come from FN and reflect on AB. Thus its displacement is (uMN + uFM ) TAB . The red district is similar to the situation of WS . And the corresponding displacement should be uFM (−RAB ) BBC + uEF FBC . According to Eqs. (19) and (21)–(23), the transmission coefficient can be expressed as T˜ =
=
US U1
=
1 U1
[(uMN + uFM ) TAB + uFM (−RAB ) BBC + uEF FBC ]
z2 + z1 |V | − c2 2z2
|V | + c1
2z1 c1 + c2 (z1 − z2 )2 c1 + c2 + 1 + . 2 z1 + z2 |V | + c1 (z1 + z2 ) c1 − c2
(26)
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It is easy to verify that Eqs. (24)–(26) satisfy
˜ ˜ ˜ 1+ R = S+ T z1 1 − R˜ = z2 −S˜ + T˜
(27)
which also can be obtained based on Eqs. (2) and (11). Eq. (27) in fact is the continuous conditions of the wave. It can be claimed that the wave is continuous in Unusual case 1. 5.2. Unusual cases 2–4 Unusual case 1 will become Unusual cases 2–4 by time reversal. For example, after a time reversal, W2 in Fig. 6 becomes W1a , and W1 becomes the reflected wave W2a , as described in Fig. 2(d). The displacement propagation coefficients of Unusual cases 2–4 are denoted by R˜ , S˜ , T˜ respectively. We omit the analysis that is similar to Unusual case 1 and directly give the coefficients as R˜ = S˜ =
T˜ =
U2a U1a U2b U1b U3c U1c
=
z1 − z2 c1 + c2 c1 − |V |
(28)
z1 + z2 c1 − c2 c1 + |V |
|V | − c1 z2 c2 − c1 |V | + c2 2 z2 + z1 |V | − c1 c1 + c2 z1 − z2 2z1 c1 + c2 = +1 + . 2z2 |V | + c2 c2 − c1 z1 + z2 z2 + z1 |V | + c2
=
z2 − z1
c2
(29)
(30)
Summing up the above, for all the Unusual cases, the displacement propagation coefficients can be expressed as
R˜ = S˜ = T˜ =
z1 − z2 c1 + c2 cmax − |V | z1 + z2 |c1 − c2 | cmax + |V | z2 − z1 cmax |V | − cmin
|c1 − c2 | |V | + cmax z2 2 z2 + z1 |V | − cmin c1 + c2 z1 − z2 2z2
|V | + cmax
|c1 − c2 |
z1 + z2
(31)
+1 +
2z1
c1 + c2
z2 + z1 |V | + cmax
.
It should be noted that only if −c1 ≤ V ≤ −c2 , Eq. (31) satisfies continuous conditions (Eq. (27)). With only one emitted wave, the discontinuity of displacement on the MPI is obvious. The discontinuity would lead to a shock company with the MPI. We point out that the shock we observed is significantly different from the traditional one. A traditional shock is always in company with supersonic flow or nonlinear steepening. An acoustic shock usually has higher velocity than the sound velocity. However, the shock we observed appears in a static media. Such shock has the same velocity as the property moving velocity. The shock discontinuous value is determined by the wave velocities and wave impedance on both sides of the MPI. It can be claimed that our discovery would give a new mechanism of shock waves. 6. Numerical simulation In order to verify the results of above discussions, some cases are simulated by using COMSOL Multiphysics [24] software. All the numerical simulations are based on Eq. (1). The modulus and density are in the form P1 (x ≤ x0 + Vt ) P2 (x > x0 + Vt )
P =
(32)
where P = (E , ρ) , x0 is the initial position of the MPI. By the built-in Heaviside function in COMSOL, Eq. (32) can be rewritten as P = P1 H (x0 + Vt − x) + P2 H (x − x0 − Vt ), where H (x) is Heaviside function. According to the discussion in Section 5, the results of Unusual cases contain the results of Usual cases or Critical cases. Thus it is only needed to verify the Unusual cases. In the simulation, we use a 1-D model that contains two property parts and a MPI as shown in Fig. 7. The property in Part 1 and Part 2 are P1 and P2 respectively. The values of the properties are listed in Table 3. In addition, the initial MPI position, interface moving velocity and the items to be verified are also listed in Table 3. The incident wave is a pulse in the form u1 = 0.002xe0.5−2x (m) . 2
(33)
The amplitude of the pulse is defined as the half of the displacement peak–peak. And the wavelength is defined as the distance between the peaks. Consequently, the amplitude and wavelength are 0.001 m and 1 m respectively.
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Fig. 7. The schematic diagram of the model.
Fig. 8. Wave propagation in Unusual case 1: there are three emitted waves. The propagation coefficients are listed in Table 4. Table 3 Parameters of the simulations. No. of Unusual case
E1 (GPa)
E2 (GPa)
ρ1 (kg/m) ρ2 kg/m3 V (m/s)
1 2 3 4
200 200 54 54
54 54 200 200
8000 8000 6000 6000
6000 6000 8000 8000
x0 (m)
The items to be verified
4000 4000 −4000
5 2 −2 3
Eqs. (10), (24)–(26) Eqs. (10) and (31) Eqs. (10) and (29) Eqs. (10) and (30)
−4000
Table 4 Wave peaks and wavelengths under an Unusual case 1. Data types
R˜
λˆ 2 /λˆ 1
S˜
λˆ S /λˆ 1
T˜
λˆ 3 /λˆ 1
Theoretical values Simulation values Errors
0.1686 0.1859 10.26%
−0.1111 −0.1199
−0.3395 −0.3309 −2.533%
0.1111 0.1221 10.80%
1.508 1.555 3.117%
0.7778 0.7919 1.813%
7.923%
6.1. Simulation of Unusual case 1 Using the parameters in Table 3, Unusual case 1 is simulated. The propagation details are shown in Fig. 8. It can be seen that one incident wave leads to three emitted waves, which meets the situation in Figs. 2(c) and 6. The waveform is unchanged as displayed in Fig. 8. Thus the wave amplitude and wavelength can be calculated as done for the incident wave. Based on the simulation data, we listed the amplitudes and wavelengths in Table 4. The theoretical values in Table 4 are calculated by Eqs. (10) and (31). As listed in Table 4, the theory values are verified with errors less than 11%. R˜ and S˜ are much smaller than T˜ , which can be seen in Fig. 8. So the same error may exhibit less influence on T˜ . Thus it can be seen that the reflected wave and backward transmitted wave have larger errors as shown in Table 4. The theoretical values in Table 4 satisfy the continuity. Meanwhile the continuity and smoothness of the waveform are displayed in Fig. 8. Hence the continuity results in Eq. (27) are verified. 6.2. Simulation of Unusual cases 2–4 Shock may exist in Unusual cases 2–4 according to the analysis in Section 5.2. The simulation becomes a little difficult because of the discontinuity on the MPI. In these simulations, we attenuated the high frequency components of the waves
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Fig. 9. Wave propagation in Unusual case 2: there is only one emitted wave. The propagation coefficients are listed in the 2, 3 column in Table 5.
Fig. 10. Wave propagation in Unusual case 3: there is only one emitted wave. The propagation coefficients are listed in the 4, 5 column in Table 5.
Fig. 11. Wave propagation in Unusual case 4: there is only one emitted wave. The propagation coefficients are listed in the 6, 7 column in Table 5.
Table 5 Wave peaks and wavelengths under an Unusual cases 2–4. Data types
R˜
λˆ 2 /λˆ 1
S˜
λˆ S /λˆ 1
T˜
λˆ 3 /λˆ 1
Theoretical values Simulation values Errors
0.1686 0.1801 6.821%
−9.000 −9.102
0.1528 0.1648 7.853%
9.000 9.640 7.111%
0.6786 0.6805 0.2800%
1.286 1.290 0.3110%
1.133%
to reduce the data fluctuation near the MPI. The propagation scenarios based on the simulations are shown in Figs. 9–11. And the values of amplitudes and wavelength are listed in Table 5. The results indicate that in Unusual cases 2–4 there exist unique strong discontinuous waves on the MPI. The locally fluctuating of displacement is due to the numerical errors near the MPI. As listed in Table 5, the errors between the theory values and simulated values are less than 8%.
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L.-Q. Shui et al. / Wave Motion 51 (2014) 1179–1192 Table 6 The displacement propagation coefficients of waves on MPI. Case
V
U2 /U1
U3 /U1
U˜ /U1
U∆ /U1
Usual case 1 Usual case 2 Unusual case 1 Unusual case 2 Unusual case 3 Unusual case 4
|V | > cmax |V | ≤ cmin −c1 ≤ V ≤ −c2 c2 ≤ V < c1 c1 < V ≤ c2 −c2 ≤ V ≤ −c1
B R R˜ R˜ / /
F T S˜ / S˜ /
/ / T˜ / / T˜
/ / / 1 + R˜ 1 − S˜ 1 − T˜
Fig. 12. Curves of Tr −u − V : (a) E1 = 54 GPa, E2 = 200 GPa, ρ1 = 6000 kg/m3 , ρ2 = 8000 kg/m3 (b) E1 = 200 GPa, E2 = 54 GPa, ρ1 = 8000 kg/m3 , ρ2 = 6000 kg/m3 .
7. Discussion 7.1. The displacement propagation coefficients analysis The displacement propagation coefficients for all cases are listed in Table 6, in which the Critical cases are merged into Usual and Unusual cases. In Usual cases, the displacement propagation coefficients have nothing to do with the interface moving velocity, and they are only determined by the wave impedance. However, the displacement propagation coefficients of Unusual cases are determined not only by the wave impedance but also by the wave velocities and the interface moving velocity. Based on Table 6, the curves of displacement propagation coefficients (Tr −u ) versus the interface moving velocity are shown in Fig. 12, in which the green and black solid lines represent Unusual cases and Usual cases respectively. The dashed lines correspond to the Critical cases. It can be seen and verified that four couples of coefficients are continuously ˜ and S˜ − B. monotonous about the interface moving velocity. They are F − T˜ − T , R˜ − R − R˜ , B − S, 7.2. The wave frequency analysis Eq. (10) gives the wavelength that applies to all cases. So the wave frequency can be written as
ω = cˆ kˆ ∝
cˆ V − cˆ
.
(34)
Based on Eq. (34), the curves of the frequency versus the interface moving velocity are displayed in Fig. 13, where four independent smooth curves are shown. The green and black part in the curves correspond to Unusual cases and Usual cases, respectively. It can be seen that the frequencies depend on the interface moving velocity continuously. The frequency holds invariance only when V = 0, i.e. in the static case, which is consistent with the existing cognitive. 7.3. The mechanical energy analysis Suppose that the displacement of the incident wave is u1 ( x ) =
f (x) ̸≡ 0 0
(|x| ≤ 1) (|x| > 1) .
(35)
Thus the emitted waves are in the form u (x) =
Tr −u f x/λˆ 0
|x| ≤ λˆ |x| > λˆ .
(36)
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Fig. 13. Curves of ω − V : (a) E1 = 54 GPa, E2 = 200 GPa, ρ1 = 6000 kg/m3 , ρ2 = 8000 kg/m3 (b) E1 = 200 GPa, E2 = 54 GPa, ρ1 = 8000 kg/m3 , ρ2 = 6000 kg/m3 .
Fig. 14. Curves of ϑ − V : (a) E1 = 54 GPa, E2 = 200 GPa, ρ1 = 6000 kg/m3 , ρ2 = 8000 kg/m3 (b) E1 = 200 GPa, E2 = 54 GPa, ρ1 = 8000 kg/m3 , ρ2 = 6000 kg/m3 .
According to Eqs. (8) and (35), the mechanical energy of incident wave can be expressed as
E1 =
1
1
E1 u′12 + ρ1 u˙ 21 dx = E1
2 −1
1
f ′2 (x) dx.
(37)
−1
Similarly, based on Eqs. (8), (10), (36) and (37), the mechanical energy of emitted waves is ˆ λ
2 Tr −u ˆ
E =E
−λ
df
2
ˆ x/λ
dx
ˆ dx = E V − c1 Tr2−u E1 . E V − cˆ
(38)
1
The larger the emitted wave energy is, the more energy the waves gain from the MPI, and it is more difficult to change the material properties near the MPI. Thus we define the ratio of emitted wave energy to incident wave energy as the resistance of the property interface motion, which is in the form
ϑ=
E
E1
V − cˆ1 E 2 = V − cˆ Tr −u E1
(39)
where the symbol Σ represents the sum of all emitted waves. A larger ϑ means greater resistance. ϑ = 1 indicates the energy conservation. It should be noted that Eq. (39) indicates that our theory may fail near the Critical cases, because ϑ is singular near the Critical cases. Such phenomena break the assumption of linear elasticity. Thus our theory only can be applied to the cases appropriately away from the Critical cases. A curve of ϑ − V is given in Fig. 14. It can be seen that Unusual cases 2 and 3 have very low emitted energy, which indicate that the emitted wave may disappear if the dissipation is considered in reality. 8. Conclusion The physical model of 1-D MPI, which should be the most representative characteristics of practical materials with 1-D time-varying properties, is proposed based on the property variability of smart materials. The linear elastic wave propagation law at 1-D MPI is obtained based on a novel method beyond the continuities. The law holds only if the interface moving velocity appropriately deviates from the wave velocities. It is revealed that the property interface motion has significant
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influence on wave frequency and energy. The wave propagation coefficients are determined not only by the wave impedance but also by the property interface motion and wave velocity. Additionally, we discovered shock waves at the MPI. All in all, this research provides a theoretical viewpoint in the study of smart materials with a time-dependent mechanical properties at different loading conditions. Acknowledgment This work is supported by the NPU Foundation for Fundamental Research (Grant No. JC20110255), to whom the authors express their deep gratitude. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
E.B. Murphy, F. Wudl, The world of smart healable materials, Prog. Polym. Sci. 35 (2010) 223–251. D.G. Anderson, J.A. Burdick, R. Langer, Materials science — smart biomaterials, Science 305 (2004) 1923–1924. E.F. Crawley, J. Deluis, Use of piezoelectric actuators as elements of intelligent structures, AIAA J. 25 (1987) 1373–1385. T.C. Halsey, Electrorheological fluids, Science 258 (1992) 761–766. E.W. Lee, Magnetostriction and magnetomechanical effects, Rep. Progr. Phys. 18 (1995) 184–229. M.R. Jolly, J.W. Bender, J.D. carlson, Properties and applications of commercial magnetorheological fluids, J. Intell. Mater. Syst. Struct. 10 (1999) 5–13. C. Liu, H. Qin, P.T. Mather, Review of progress in shape-memory polymers, J. Mater. Chem. 17 (2007) 1543–1558. M. Ruzzene, A. Baz, Attenuation and localization of wave propagation in periodic rods using shape memory inserts, Smart Mater. Struct. 9 (2000) 805–816. M. Ruzzene, A. Baz, Control of wave propagation in periodic composite rods using shape memory inserts, J. Vib. Acoust. 122 (2000) 151–159. T. Chen, M. Ruzzene, A. Baz, Control of wave propagation in composite rods using shape memory inserts, J. Vib. Control 6 (2000) 1065–1081. O. Thorp, M. Ruzzene, A. Baz, Attenuation and localization of wave propagation in rods with periodic shunted piezoelectric patches, Smart Mater. Struct. 10 (2001) 979–989. A. Singh, D.J. Pines, A. Baz, Active/passive reduction of vibration of periodic one-dimensional structures using piezoelectric actuators, Smart Mater. Struct. 13 (2004) 698–711. Y. Kim, A. Baz, Active control of a two-dimensional periodic structure, Proc. SPIE 5386 (2004) 329–339. L.-Y. Wu, M.-L. Wu, L.-W. Chen, The narrow pass band filter of tunable 1D phononic crystals with a dielectric elastomer layer, Smart Mater. Struct. 18 (2009) 015011. W.-P. Yang, L.-W. Chen, The tunable acoustic band gaps of two-dimensional phononic crystals with a dielectric elastomer cylindrical actuator, Smart Mater. Struct. 17 (2008) 015011. D.W. Wright, R.S.C. Cobbold, Acoustic wave transmission in time-varying phononic crystals, Smart Mater. Struct. 18 (2009) 015008. D.W. Wright, R.S.C. Cobbold, Two-dimensional phononic crystals with time-varying properties: a multiple scattering analysis, Smart Mater. Struct. 19 (2010) 045006. C.R. Luo, L.S. Wang, J.Q. Guo, Y. Huang, X.P. Zhao, Tunable effects of connective dendritic left-handed metamaterials based on electrorheological fluids, Acta Phys. Sin. 58 (2009) 3214–3219. L.S. Wang, C.R. Luo, Y. Huang, X.P. Zhao, Electrically tunable negative permeability metamaterials based on electrorheological fluids, Acta Phys. Sin. 57 (2008) 3571–3577. M. Rousseau, G.A. Maugin, M. Berezovski, Elements of study on dynamic materials, Arch. Appl. Mech. 81 (2011) 925–942. J.S. Jensen, Space–time topology optimization for one-dimensional wave propagation, Comput. Methods Appl. Math. 198 (2009) 705–715. K.A. Lurie, Introduction to the Mathematical Theory of Dynamic Materials, Springer, New York, 2007. G.A. Maugin, Sixty years of configurational mechanics (1950–2010), Mech. Res. Commun. 50 (2013) 39–49. Equation-based modeling, COMSOL Multiphysics, COMSOL Inc. Version 4.2.1.110 (2008).
Contents lists available at ScienceDirect
Wave Motion journal homepage: www.elsevier.com/locate/wavemoti
One-dimensional linear elastic waves at moving property interface Lang-Quan Shui ∗ , Zhu-Feng Yue, Yong-Shou Liu, Qing-Chang Liu, Jiao-Jiao Guo Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, PR China
highlights • • • • •
Proposed a moving property interface model. Proposed a novel method to study the waves at moving property interface. Property interface motion and wave velocity can influence the wave propagation. There may exist shock waves at moving property interface. Property interface motion has an impact on the wave frequency and energy.
article
info
Article history: Received 23 April 2014 Received in revised form 5 July 2014 Accepted 10 July 2014 Available online 21 July 2014 Keywords: Time-varying Moving property interface Elastic wave Shock
abstract Smart materials exhibit time-varying properties while time-varying external field is applied. To investigate the one-dimensional (1-D) homogeneous time-varying properties, a moving property interface (MPI) model is proposed, and the propagation of linear elastic waves at 1-D MPI is studied in this paper. Based on the idea of weak solutions and an infinity approximation, a novel method to deal with the difficulties in using the continuities to study the waves at MPI is also proposed. Some interesting phenomena are revealed: (i) besides wave impedance, the property interface motion and wave velocity are also very important factors that influence the wave propagation; (ii) at MPI, there may exist shock waves; (iii) the property interface motion has a significant impact on the wave frequency and energy. This research provides a theoretical viewpoint in the study of smart materials with a time-dependent mechanical properties at different loading conditions. © 2014 Elsevier B.V. All rights reserved.
1. Introduction For their smart features, such as sensing, feedback, information recognition and accumulation, self-recovery, selfadjusting and so on, smart materials are widely used in the multi-sensor, precision driving, vibration isolation/excitation and many other fields [1,2]. Smart materials exhibit time-varying properties while time-varying external field is applied, such as the shear modulus of the piezoelectric material [3] and electro rheological (ER) materials [4] experiencing timevarying electric field, material damping of the magnetostrictive material [5] and magneto rheological fluid [6] experiencing time-varying magnetic field, and modulus of the shape-memory material (SMA) [7] experiencing time-varying temperature field. The time-varying properties of smart materials can be realized through precise control of the external fields. Using the time-varying properties, the smart composites and structures gained a lot of new applications. Ruzzene, Baz et al. [8–13] applied the SMA and the piezoelectric material to the design of period structures considering the timevarying modulus. Such structures demonstrate excellent active vibration performances. Wu [14] and Yang [15] respectively
∗
Corresponding author. Tel.: +86 15902902159. E-mail address: [email protected] (L.-Q. Shui).
http://dx.doi.org/10.1016/j.wavemoti.2014.07.005 0165-2125/© 2014 Elsevier B.V. All rights reserved.
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Fig. 1. The moving interface in smart materials: (a) cell division of 1-D structure; (b) distribution of the property at time t0 ; (c) distribution of the property at time t1 ; (d) distribution of the property at time t2 .
used the dielectric elastomer with time-varying properties to design the 1-D and 2-D phononic crystals. These phononic crystals have an adjustable band-gap. Wright and Cobbold [16,17] studied the band-gap characteristics of the phononic crystals with time-varying scatterer, and extended the multiple scattering theory to time-varying structures. Luo and Wang [18,19] designed left-handed metamaterials with actively modulated left-handed transmission peak and electrically tunable negative permeability metamaterials based on ER fluids considering time-varying properties. Most of these researches are focused on applications of a specific smart material with time-varying properties. Few of them involve general theoretical study. Generally, there are two methods, which are wave propagation method and frequency domain analysis method, to research the mesoscopic physical mechanism and the transient response of smart structures with time-varying properties. The variance of the wave frequency in materials with time-varying properties makes the frequency domain analysis method unavailable. Thus the wave propagation method is a better choice to study the time-varying materials. A series of work has been reported on waves in time-varying materials [20–23]. This work on wave propagation in materials with time-varying properties will reveal interesting phenomenon different from the waves in static or moving media. This paper focuses on the propagation of linear elastic waves at 1-D moving property interface (MPI). MPI is a model that can sufficiently reflect the characteristics of a material with 1-D time-varying properties. Usually, the continuities are powerful and sufficient in gaining the wave propagation law at property interface. However, there are obstacles in using the continuities to study the waves at MPI. Hence we proposed a novel method, which is based on the idea of weak solutions and infinity approximation, to deal with the difficulties, and the propagation law of linear elastic waves at 1-D MPI is achieved. Specifically, the MPI model will be described based on the property variability of smart materials in Section 2. The movement mechanism of the property will be revealed in this section. In Section 3, according to the propagation characteristics of the wave when the property interface moving with different velocities, the wave propagation at MPI is classified. A complete discussion of the propagation law of the waves at MPI will be given in Sections 4 and 5. In Section 6, simulations will be conducted to verify the results obtained in previous sections. Section 7 is the summary of the whole work in this paper. In addition, the wavelength, wave frequency and energy characteristics of the waves on the MPI will be analyzed in Section 7. In Section 8, i.e. the last section, some conclusions will be made. 2. Moving property interface (MPI) We will find out a representative model which can sufficiently reflect the characteristics of materials with time-varying properties. This needs to take an in-deep understanding of the materials with time-varying properties, first. As well known, the application of the smart material is mainly based on the property controllability. The properties variance of smart materials is due to the time-varying external field. On the other hand, the edge effect, device-specific or human-induced inhomogeneity of external field, and some other factors, may lead to the spatial inhomogeneity of the property of smart materials. As a result, variable properties of smart materials are functions of time and space. A general variable property should be inhomogeneous in time and space. However, a homogeneous variable property is a basic model which can be used to describe an inhomogeneous one by integration. In the following, we put forward a 1-D wave model to describe the 1-D homogeneous variable property which named MPI. As shown in Fig. 1(a), the space in the 1-D structure is divided into a series of small cells: e1 , e2 , . . . , ei , . . . , en , and li−1,i represents the interface between the adjacent cells ei−1 , ei . Assume that the property of the material is P1 when the external field A acts on the material, and the property is P2 when field B is applied. P1 and P2 in Fig. 1(b)–(d) are expressed by pink and yellow respectively. At the initial time t0 (Fig. 1(b)), field A acts on cells e1 , . . . , ei−1 and field B on ei , . . . , en . After a time interval, at time t1 (Fig. 1(c)), the field on cell ei switches from B to A. Hence the property of cell ei switches from P2 to P1 , which means that the property interface moves rightward by a cell length and reaches li,i+1 . So repeatedly the MPI can be formed. Considering infinitesimal cells, the movement of the interface will be continuous.
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Fig. 2. Characteristic lines of waves at MPI (|V | ̸= c ): (a) |V | > cmax (b) |V | < cmin (c) c2 < −V < c1 (d) c1 < V < c2 , c2 < V < c1 and c1 < −V < c2 .
Let V represent the move velocity. When V = 0, the MPI degenerates into the well-known static property interface. When V = ∞, the MPI degenerates into a property transient. It should be noticed that the property moves without the transfer of matter. It should be emphasized that the property movement, which is induced by homogeneous spatio-temporal distribution of the controllable property, should be the most representative characteristics of practical materials with timevarying properties. 3. The classification of the waves at MPI A series of conventions are listed below:
√
√
E ρ . The wave velocity is c = E /ρ . Let cmax = max {c1 , c2 } , cmin = min {c1 , c2 } . V is the moving velocity of the property interface; • by symmetry, it can be assumed that the incident wave is coming from the material with property P1 . The incident wave propagates along the positive direction of the x axis; ˆ − ωt . The signed wavenumber, signed wave length, signed • a wave is expressed as W . And the displacement of W is u kx
• the wave impedance is z =
wave velocity, and signed wave impedance are defined as
ˆk = 2π /λ (W travels along the positive direction of x axis) −2π /λ (W travels along the negative direction of x axis),
λˆ =
2π kˆ
cˆ = c
λˆ λ
zˆ = z
λˆ λ
in which λ is the wavelength. The wave frequency ω = cˆ kˆ ; • the symbols that directly relate to the material properties (such as the wave velocity c, the material density ρ ) have subscripts that range from 1 to 2. The same subscript refers to the same meaning. This means that c1 is the wave velocity of the material with property P1 . And the material density, modulus and wave impedance corresponding to c1 are ρ1 , E1 and√ z1 . The subscripts are omitted if it will not be confused. I.e. c1 c = sin c represents c12 = sin c1 and c1 c2 = sin c2 ; c = E /ρ represents √ √ c1 = E1 /ρ1 and c2 = E2 /ρ2 . The free linear elastic wave equation is
σ ′ − p˙ = 0 σ = Eu′ , p = ρ u˙
(1)
˙ = ∂ (·) /∂ t. E and ρ are material modulus and density, respectively. u is the where the operators (·)′ = ∂ (·) /∂ x, (·) displacement of the particle. x and t represent space and time, respectively. According to the theory of characteristics of second-order 1-D wave equation, the vibration at a point can be determined by the disturbances coming along two characteristic lines, then the vibration at this point will spread out along another two characteristics lines. The two waves that spread out are unknown. For a posed continuous wave solution, u in Eq. (1) has two continuous conditions, which can be used to solve the unknown waves. Therefore, only if the number of characteristic lines leaving a point is two, the vibration near this point can be uniquely determined. When the property interface moving speed is not equal to the wave velocity, all the possible relationships of the characteristic lines and MPI are listed in Fig. 2. The red lines in Fig. 2 represent the MPI in spatio-temporal coordinates Oxt. And the black lines are characteristic lines. P1 , P2 are the properties of the materials on two sides of the MPI. W1 is incident wave. W2 and W3 represent the emitted waves with the different and same directions comparing with the incident wave respectively. It should be noted that the incident situation of W1a , W1b , W1c correspond to the emitted situation of W2a , W2b , W3c , successively, in Fig. 2(d). The cases in Fig. 2(a), (b) are the ones with the conditions |V | > cmax and |V | < cmin respectively. In these two cases there are two characteristic lines leaving the MPI, which indicate that the wave solution is well-posed. These cases are named as Usual Cases. The moving velocity of the property interface in Fig. 2(c) satisfies c2 < −V < c1 . In this case the number of the characteristic lines leaving the interface is three, which is larger than the number of the continuous conditions. In this situation, the solution of Eq. (1) may not be unique. There are three rest cases, i.e. c2 < V < c1 , c1 < V < c2 and c1 < −V < c2 . Each case has only one emitted wave. They can be simultaneously shown in Fig. 2(d). The three cases in
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V
Figure
Usual case 1 Usual case 2 Unusual case 1 Unusual case 2 Unusual case 3 Unusual case 4 Critical case
|V | > cmax |V | < cmin c2 < −V < c1 c2 < V < c1 c1 < V < c2 c1 < −V < c2 |V | = c1 , c2
Fig. 2(a) Fig. 2(b) Fig. 2(c) Fig. 2(d)-W1a Fig. 2(d)-W1b Fig. 2(d)-W1c Fig. 4
Fig. 2(d) each has only one characteristic line leaving the interface. Thus the solution of Eq. (1) may not exist considering the continuous conditions. The cases in Fig. 2(c) and (d), which cannot be solved directly by the two continuous conditions, are named as Unusual Cases. When the property interface moving speed is equal to the wave velocity, we name the case as Critical Case. According to above discussions, all cases in Fig. 2 and the Critical Case are listed in Table 1. It is obviously that the Unusual Cases will disappear when c1 = c2 . 4. Wave propagation in Usual cases and Critical cases 4.1. Continuous conditions of the waves According to the analysis in Section 3, the Usual Cases can be solved by continuous conditions. Therefore, the key to solve these cases is to obtain the continuous conditions. Usually the linear elastic waves have two continuous scalar. They are represented as Cu and C∂ u . Cu is the displacement and C∂ u is a variable associated with the first-order differential of the displacement. Let the basic assumption of continuum mechanics, C u = u.
(2)
The continuous scalar associated with its first-order differential can be obtained according to the momentum theorem. ˜ and V is the moving The property variability as shown in Fig. 1 is considered. Assume that the area of the interface is A, velocity of the property interface. During interval dt, the length within the property transform is V dt. The momentum before
˜ dt u˙ 1 . The amount of the momentum change is ˜ dt u˙ 2 . And the one after the transform is ρ1 AV the transform is ρ2 AV ˜ 2 u′2 − AE ˜ dt u˙ 1 − ρ2 AV ˜ dt u˙ 2 . Meanwhile the force on the material is AE ˜ 1 u′1 . According to the momentum theorem, ρ1 AV it can be obtained that
˜ 2 u′2 − AE ˜ 1 u′1 dt . ˜ dt u˙ 2 = AE ˜ dt u˙ 1 − ρ AV ρ1 AV
(3)
2
˜ on both hands of Eq. (3), it can be simplified as Divided by Adt E1 u′1 + ρ1 V u˙ 1 = E2 u′2 + ρ2 V u˙ 2 .
(4)
Let C = Eu + ρ V u˙ , Eq. (4) can be rewritten as ′
C1 = C2 .
(5)
It is obvious that C∂ u = C = Eu′ + ρ V u˙ .
(6)
The following processing is to simplify Eq. (6). A traveling wave in 1-D homogeneous medium is considered. The displacement of the wave is u x − cˆ t . During the time interval dt, the particles disturbed by the wave have a spatial length of cˆ dt. And the mass is ρ Acˆ dt . The particles get
˜ ′ , it can be obtained that the momentum of ρ A˜ cˆ dt u˙ . Considering that the particles are acted by the force −AEu ˜ ′ dt . ρ A˜ cˆ dt u˙ = −AEu
(7)
˜ on both hands of Eq. (7), it can be simplified as Divided by Adt u˙ = −ˆc u′ .
(8)
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Fig. 3. Geometric relationships of wavelengths: λ1 /λ2 = AC /DF = (V − c1 )/(V − c2 ).
Substitute Eq. (8) into Eq. (6), the continuous scalar can be written as
C∂ u = −z V − cˆ ku´
(9)
ˆ − ωt . where we used a symbol u´ = ∂ u/∂ kx The characteristic lines of Eq. (1) along one direction are clusters of parallel lines. The wavelength can be defined as the spatial distance of any two characteristic lines with phase difference in one period. The particles on the MPI disturbed by the incident waves produce new waves. The new waves may distribute on any sides of the MPI according to the analysis in Section 3. One case, in which the emitted wave is not on the same side of the incident wave, is shown in Fig. 3. The angle between the MPI and t-axis is ̸ BDC = arctan (−V ). And for the incident wave and emitted wave, the characteristic lines and the t-axis have angles of ̸ BDA = arctan c1 and ̸ ECF = arctan c2 . The geometric relationship indicates that the wavelength ratio of the incident wave and emitted wave is λ1 /λ2 = AC /DF = (V − c1 ) / (V − c2 ). It is not difficult to verify that the waves on any sides of the MPI satisfy the relation below
λˆ ∝ V − cˆ .
(10)
Substitute Eq. (10) into Eq. (9), and ignore the constant coefficient, the simplified continuous scalar associated with the first-order differential of the displacement can be written as C∂ u = zˆ u´ .
(11)
In sum, Eqs. (2) and (11) are the two components of the continuous scalars for the continuous elastic waves on the MPI. 4.2. Usual case 1 As shown in Fig. 2(a), the incident wave W1 and emitted wave W2 , W3 are on either side of the MPI respectively. Thus, based on Eqs. (2) and (11), the continuities can be expressed as
u1 kˆ 1 x − ω1 t = u2 kˆ 2 x − ω2 t + u3 kˆ 3 x − ω3 t zˆ1 u´ 1 kˆ 1 x − ω1 t = zˆ2 u´ 2 kˆ 2 x − ω2 t + zˆ3 u´ 3 kˆ 3 x − ω3 t .
(12)
Integrating the second equation in Eq. (12) along the MPI, it can be simplified as z1 kˆ 1 V − cˆ1
u1 kˆ 1 V − cˆ1 t = −
z2 kˆ 2 V − cˆ2
u2 kˆ 2 V − cˆ2 t +
z2 kˆ 3 V − cˆ3
u3 kˆ 3 V − cˆ3 t .
(13)
Let the displacement near the MPI be Ui (i = 1, 2, 3), Eq. (12) can be expressed as
U1 = U2 + U3 z1 U1 = −z2 U2 + z2 U3 .
(14)
By solving Eq. (14), the displacement transmission coefficients of the forward transmitted wave W3 and backward transmitted wave W2 can be obtained as
U3 z2 + z1 = F = U1
2z2
U2 z2 − z1 B = = . U1
2z2
(15)
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Fig. 4. Characteristic lines of the critical case: (a) V = c2 < c1 (b) V = c2 > c1 (c) −V = c2 > c1 (d) −V = c1 < c2 (e) −V = c2 < c1 (f) −V = c1 > c2 . Table 2 The displacement propagation coefficients of Critical case. Case
V
Case basis
U2 /U1
U3 /U1
a b c d
V = c2 < c1 V = c2 > c1 −V = c2 > c1 −V = c1 < c2
|V | < cmin |V | > cmax |V | > cmax |V | < cmin
R B
∆=T ∆=F
∆=B ∆=R
F T
4.3. Usual case 2 In this case the reflected wave and incident wave are on the same side, and the transmitted wave is on the other side. Thus the continuities become
u1 kˆ 1 x − ω1 t + u2 kˆ 2 x − ω2 t = u3 kˆ 3 x − ω3 t zˆ1 u´ 1 kˆ 1 x − ω1 t + zˆ2 u´ 2 kˆ 2 x − ω2 t = zˆ3 u´ 3 kˆ 3 x − ω3 t
(16)
which can be simplified similarly to Eq. (12) as
U1 + U2 = U3 z1 U1 − z1 U2 = z2 U3 .
(17)
The displacement reflection coefficient R and transmission coefficient T can be obtained as
z1 − z2 U2 = R = U1 z1 + z2 U3 2z1 T = = . U1 z1 + z2
(18)
4.4. Critical case The Critical case can be considered as a transition of the Usual cases and Unusual cases. It is obvious that the incident ˆ → 0 as wave may not touch the MPI when V = c1 . Thus this case is neglected. Considering Eq. (10), it can be seen that λ V → cˆ , i.e. the characteristic lines in front of the MPI would be compressed into the MPI. Thus the wave may exhibit strong discontinuity on the MPI, i.e. the MPI receives a shock traveling with itself. Specifically, there are 6 cases contained in the critical case (Fig. 4). In Fig. 4, the red lines are the MPI, the black lines are characteristic lines. Generally, considering the limits of Usual case as V → ±c, the propagation law of these cases can be obtained. It is obvious that the propagation law in Usual case has nothing to do with V . Thus the Critical case has the same propagation coefficients with the Usual case. When the wavelength becomes zero, corresponding propagation coefficients would be the shock discontinuous coefficient ∆. The details are listed in Table 2. The cases in Fig. 4(e) and (f) are the same as the Usual case. No shock exists in these two cases. Thus they do not appear in Table 2. The symbols in Table 2 can be found in Eqs. (15) and (18). It should be noted that the discussions in Section 4.4 are the process of taking a limit of the Usual cases as V → ±c. The results are also ruled by the assumption of linear elasticity. More detailed analysis in Section 7.3 indicate that the discussion about the Critical case is meaningless because of corresponding infinite energy which do not exist in reality. However in a theoretical sense the discussion is still useful for the following Unusual cases. 5. Wave propagation in Unusual cases In these cases, the number of the unknown waves is not equal to that of the continuous conditions. Thus the waves cannot be determined directly by the continuity. To get over the difficulty, some improvements need to be sought. The effective method is to transform these cases into the Usual cases or Critical cases.
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Fig. 5. Interface approximation: (a) an approximation of the moving interface with step lines; (b) repeated interaction of characteristic lines and tiny interfaces.
5.1. Unusual case 1 As shown in Fig. 5(a), the MPI has been resolved as polylines. The MPI would be the limits of the polylines as the scale of the polyline approaches infinitesimal. The segments in polylines always can be chosen to satisfy Usual cases or Critical cases. Thus the Unusual cases can be transformed into the Usual cases or Critical cases. An incident wave that reflects or transmits at a segment may reach another segment, and the reflection and transmission are repeated (Fig. 5(b)). Such repeated interaction may lead to tedious calculation to analyze displacement of the emitted wave. A good resolution method can greatly simplify the calculation. In dealing with the Unusual case 1, for simplicity, we use a resolution method as shown in Fig. 6, in which the green polylines are resolved interface. In such a resolution method, a characteristic line interacts with the polylines less than two times. In Fig. 6, a unit of the polylines is ABC , where AB has a slope of − arctan−1 c2 , BC is parallel to the x axis. It is obvious that wave propagation at AB and BC satisfies the case (e) of Critical cases (Fig. 4(e)) and Usual case 1 (Fig. 2(a)), respectively. In addition, Ws represents the transmitted wave with opposite direction of W1 in Fig. 6. According to the resolution method in Fig. 6, three steps are needed to obtain the propagation law. First, the wave source should be divided into several parts. In Fig. 6, they are EF , FM and MN. Waves from the same part should obey the same propagation law. It is not hard to get the wave propagation law for every part of incident wave W1 . But it is difficult to determine the relationship between the propagate law for a certain part and for the whole source EN. The second step focuses on this problem. We will find the limits of the law for every part as the scale of the polyline approaches infinitesimal. The last, the final results can be obtained based on the results in the first two steps. 5.1.1. The first step: the division of the wave source In Fig. 6, we use different colors to distinguish the different parts of wave source. The new color appearing in the emitted wave indicates that the waves come from different parts mixed, and the components of the mixture wave can be found according to the color mixture in Fig. 6. According to the interaction characteristics of the characteristics lines and the unit polyline, the wave source EN can be divided into three parts as MN , FM , EF (Fig. 6). Waves come from the same part have the same propagation path. Let the ratio of the length of MN , FM , EF to the length of EN be l1 , l2 , l3 , successively. According to the geometric relationships, the ratios can be obtained as
MN c1 + c2 c1 − |V | = l1 = | V | + c1 c1 − c2 EN FM c1 + c2 |V | − c2 l2 = = |V | + c1 c1 − c2 EN | EF V | − c2 l3 = = . | V | + c1 EN
(19)
5.1.2. The second step: the limits of the displacements in subdomains It is a key problem to obtain the limits of the displacements in subdomains as the scale of the polyline approaches infinitesimal. Let the length of EN approaches dx. Thus EN can be regarded as a ‘‘measurable point’’. And MN , FM , EF are three measurable subdomains of the ‘‘point’’. It is acceptable considering a weak solution that the function value in a subdomain is determined by its ratio of the measure of the subdomain to the measure of the ‘‘point’’. We may assume that Eq. (1) has unique solution. If two possible solutions f (x) , g (x) satisfy f (x) ϕ (x) dx = g (x) ϕ (x) dx, where ϕ (x) is a smooth test function in compactly supported subset of real domain, then f (x) and g (x) can be regarded as the same solution. Considering a function in the form
ξ (x) = lim l→0
1 x ∈ ∪+∞ −∞ (il, (i + θ ) l]
0 x ∈ ∪+∞ −∞ ((i + θ) l, (i + 1) l]
(i is an integer, 0 < θ < 1, l > 0)
(20)
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Fig. 6. Characteristics lines of Unusual case 1: c2 < −V < c1 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
it is easy to verify that if f (x) = u (x) ξ (x), then g (x) = θ u (x). Based on these ideals, the displacements in the subdomains can be directly set as uMN = l1 U1 uFM = l2 U1 uEF = l3 U1
(21)
where U1 is the displacement on EN. 5.1.3. The third step: calculation of the displacement propagation coefficients It is easy to obtain the propagation coefficients on AB and BC . According to Eqs. (15) and (18), we have
z1 − z2 RAB = z1 + z2 2z1 TAB = z1 + z2 z2 + z1 FBC =
(22)
2z2
(23)
BBC = z2 − z1 . 2z2
The ratio of the displacements of W2 , WS , W3 to the displacement of W1 are denoted by R˜ , S˜ , T˜ , respectively. In Fig. 6, the district of W2 is cyan. The wave W2 comes from MN and is formed by the reflection on AB. Based on Eqs. (19), (21) and (22), the reflection coefficient can be obtained as R˜ =
U2
=
U1
uMN RAB U1
=
z1 − z2 c1 + c2 c1 − |V | z1 + z2 c1 − c2 c1 + |V |
.
(24)
The district of WS is red. Considering the color mixture, red is a mix of yellow and magenta, which correspond to the waves from FM and EF . The waves from FM experience the reflection on BD and transmission on BC . Consequently, the transmitted wave should have displacement of uFM (−RAB ) FBC . The waves from EF transmits BC directly and should cause the displacement of uEF BBC . Based on Eqs. (19) and (21)–(23), the transmission coefficient can be obtained as S˜ =
U3
=
U1
1 U1
[uFM (−RAB ) FBC + uEF BBC ] =
z2 − z1 z2
|V | − c2 . c1 − c2 |V | + c1 c1
(25)
The district of W3 is yellow and magenta. Yellow and magenta district represents the waves come from FN and reflect on AB. Thus its displacement is (uMN + uFM ) TAB . The red district is similar to the situation of WS . And the corresponding displacement should be uFM (−RAB ) BBC + uEF FBC . According to Eqs. (19) and (21)–(23), the transmission coefficient can be expressed as T˜ =
=
US U1
=
1 U1
[(uMN + uFM ) TAB + uFM (−RAB ) BBC + uEF FBC ]
z2 + z1 |V | − c2 2z2
|V | + c1
2z1 c1 + c2 (z1 − z2 )2 c1 + c2 + 1 + . 2 z1 + z2 |V | + c1 (z1 + z2 ) c1 − c2
(26)
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It is easy to verify that Eqs. (24)–(26) satisfy
˜ ˜ ˜ 1+ R = S+ T z1 1 − R˜ = z2 −S˜ + T˜
(27)
which also can be obtained based on Eqs. (2) and (11). Eq. (27) in fact is the continuous conditions of the wave. It can be claimed that the wave is continuous in Unusual case 1. 5.2. Unusual cases 2–4 Unusual case 1 will become Unusual cases 2–4 by time reversal. For example, after a time reversal, W2 in Fig. 6 becomes W1a , and W1 becomes the reflected wave W2a , as described in Fig. 2(d). The displacement propagation coefficients of Unusual cases 2–4 are denoted by R˜ , S˜ , T˜ respectively. We omit the analysis that is similar to Unusual case 1 and directly give the coefficients as R˜ = S˜ =
T˜ =
U2a U1a U2b U1b U3c U1c
=
z1 − z2 c1 + c2 c1 − |V |
(28)
z1 + z2 c1 − c2 c1 + |V |
|V | − c1 z2 c2 − c1 |V | + c2 2 z2 + z1 |V | − c1 c1 + c2 z1 − z2 2z1 c1 + c2 = +1 + . 2z2 |V | + c2 c2 − c1 z1 + z2 z2 + z1 |V | + c2
=
z2 − z1
c2
(29)
(30)
Summing up the above, for all the Unusual cases, the displacement propagation coefficients can be expressed as
R˜ = S˜ = T˜ =
z1 − z2 c1 + c2 cmax − |V | z1 + z2 |c1 − c2 | cmax + |V | z2 − z1 cmax |V | − cmin
|c1 − c2 | |V | + cmax z2 2 z2 + z1 |V | − cmin c1 + c2 z1 − z2 2z2
|V | + cmax
|c1 − c2 |
z1 + z2
(31)
+1 +
2z1
c1 + c2
z2 + z1 |V | + cmax
.
It should be noted that only if −c1 ≤ V ≤ −c2 , Eq. (31) satisfies continuous conditions (Eq. (27)). With only one emitted wave, the discontinuity of displacement on the MPI is obvious. The discontinuity would lead to a shock company with the MPI. We point out that the shock we observed is significantly different from the traditional one. A traditional shock is always in company with supersonic flow or nonlinear steepening. An acoustic shock usually has higher velocity than the sound velocity. However, the shock we observed appears in a static media. Such shock has the same velocity as the property moving velocity. The shock discontinuous value is determined by the wave velocities and wave impedance on both sides of the MPI. It can be claimed that our discovery would give a new mechanism of shock waves. 6. Numerical simulation In order to verify the results of above discussions, some cases are simulated by using COMSOL Multiphysics [24] software. All the numerical simulations are based on Eq. (1). The modulus and density are in the form P1 (x ≤ x0 + Vt ) P2 (x > x0 + Vt )
P =
(32)
where P = (E , ρ) , x0 is the initial position of the MPI. By the built-in Heaviside function in COMSOL, Eq. (32) can be rewritten as P = P1 H (x0 + Vt − x) + P2 H (x − x0 − Vt ), where H (x) is Heaviside function. According to the discussion in Section 5, the results of Unusual cases contain the results of Usual cases or Critical cases. Thus it is only needed to verify the Unusual cases. In the simulation, we use a 1-D model that contains two property parts and a MPI as shown in Fig. 7. The property in Part 1 and Part 2 are P1 and P2 respectively. The values of the properties are listed in Table 3. In addition, the initial MPI position, interface moving velocity and the items to be verified are also listed in Table 3. The incident wave is a pulse in the form u1 = 0.002xe0.5−2x (m) . 2
(33)
The amplitude of the pulse is defined as the half of the displacement peak–peak. And the wavelength is defined as the distance between the peaks. Consequently, the amplitude and wavelength are 0.001 m and 1 m respectively.
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Fig. 7. The schematic diagram of the model.
Fig. 8. Wave propagation in Unusual case 1: there are three emitted waves. The propagation coefficients are listed in Table 4. Table 3 Parameters of the simulations. No. of Unusual case
E1 (GPa)
E2 (GPa)
ρ1 (kg/m) ρ2 kg/m3 V (m/s)
1 2 3 4
200 200 54 54
54 54 200 200
8000 8000 6000 6000
6000 6000 8000 8000
x0 (m)
The items to be verified
4000 4000 −4000
5 2 −2 3
Eqs. (10), (24)–(26) Eqs. (10) and (31) Eqs. (10) and (29) Eqs. (10) and (30)
−4000
Table 4 Wave peaks and wavelengths under an Unusual case 1. Data types
R˜
λˆ 2 /λˆ 1
S˜
λˆ S /λˆ 1
T˜
λˆ 3 /λˆ 1
Theoretical values Simulation values Errors
0.1686 0.1859 10.26%
−0.1111 −0.1199
−0.3395 −0.3309 −2.533%
0.1111 0.1221 10.80%
1.508 1.555 3.117%
0.7778 0.7919 1.813%
7.923%
6.1. Simulation of Unusual case 1 Using the parameters in Table 3, Unusual case 1 is simulated. The propagation details are shown in Fig. 8. It can be seen that one incident wave leads to three emitted waves, which meets the situation in Figs. 2(c) and 6. The waveform is unchanged as displayed in Fig. 8. Thus the wave amplitude and wavelength can be calculated as done for the incident wave. Based on the simulation data, we listed the amplitudes and wavelengths in Table 4. The theoretical values in Table 4 are calculated by Eqs. (10) and (31). As listed in Table 4, the theory values are verified with errors less than 11%. R˜ and S˜ are much smaller than T˜ , which can be seen in Fig. 8. So the same error may exhibit less influence on T˜ . Thus it can be seen that the reflected wave and backward transmitted wave have larger errors as shown in Table 4. The theoretical values in Table 4 satisfy the continuity. Meanwhile the continuity and smoothness of the waveform are displayed in Fig. 8. Hence the continuity results in Eq. (27) are verified. 6.2. Simulation of Unusual cases 2–4 Shock may exist in Unusual cases 2–4 according to the analysis in Section 5.2. The simulation becomes a little difficult because of the discontinuity on the MPI. In these simulations, we attenuated the high frequency components of the waves
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Fig. 9. Wave propagation in Unusual case 2: there is only one emitted wave. The propagation coefficients are listed in the 2, 3 column in Table 5.
Fig. 10. Wave propagation in Unusual case 3: there is only one emitted wave. The propagation coefficients are listed in the 4, 5 column in Table 5.
Fig. 11. Wave propagation in Unusual case 4: there is only one emitted wave. The propagation coefficients are listed in the 6, 7 column in Table 5.
Table 5 Wave peaks and wavelengths under an Unusual cases 2–4. Data types
R˜
λˆ 2 /λˆ 1
S˜
λˆ S /λˆ 1
T˜
λˆ 3 /λˆ 1
Theoretical values Simulation values Errors
0.1686 0.1801 6.821%
−9.000 −9.102
0.1528 0.1648 7.853%
9.000 9.640 7.111%
0.6786 0.6805 0.2800%
1.286 1.290 0.3110%
1.133%
to reduce the data fluctuation near the MPI. The propagation scenarios based on the simulations are shown in Figs. 9–11. And the values of amplitudes and wavelength are listed in Table 5. The results indicate that in Unusual cases 2–4 there exist unique strong discontinuous waves on the MPI. The locally fluctuating of displacement is due to the numerical errors near the MPI. As listed in Table 5, the errors between the theory values and simulated values are less than 8%.
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V
U2 /U1
U3 /U1
U˜ /U1
U∆ /U1
Usual case 1 Usual case 2 Unusual case 1 Unusual case 2 Unusual case 3 Unusual case 4
|V | > cmax |V | ≤ cmin −c1 ≤ V ≤ −c2 c2 ≤ V < c1 c1 < V ≤ c2 −c2 ≤ V ≤ −c1
B R R˜ R˜ / /
F T S˜ / S˜ /
/ / T˜ / / T˜
/ / / 1 + R˜ 1 − S˜ 1 − T˜
Fig. 12. Curves of Tr −u − V : (a) E1 = 54 GPa, E2 = 200 GPa, ρ1 = 6000 kg/m3 , ρ2 = 8000 kg/m3 (b) E1 = 200 GPa, E2 = 54 GPa, ρ1 = 8000 kg/m3 , ρ2 = 6000 kg/m3 .
7. Discussion 7.1. The displacement propagation coefficients analysis The displacement propagation coefficients for all cases are listed in Table 6, in which the Critical cases are merged into Usual and Unusual cases. In Usual cases, the displacement propagation coefficients have nothing to do with the interface moving velocity, and they are only determined by the wave impedance. However, the displacement propagation coefficients of Unusual cases are determined not only by the wave impedance but also by the wave velocities and the interface moving velocity. Based on Table 6, the curves of displacement propagation coefficients (Tr −u ) versus the interface moving velocity are shown in Fig. 12, in which the green and black solid lines represent Unusual cases and Usual cases respectively. The dashed lines correspond to the Critical cases. It can be seen and verified that four couples of coefficients are continuously ˜ and S˜ − B. monotonous about the interface moving velocity. They are F − T˜ − T , R˜ − R − R˜ , B − S, 7.2. The wave frequency analysis Eq. (10) gives the wavelength that applies to all cases. So the wave frequency can be written as
ω = cˆ kˆ ∝
cˆ V − cˆ
.
(34)
Based on Eq. (34), the curves of the frequency versus the interface moving velocity are displayed in Fig. 13, where four independent smooth curves are shown. The green and black part in the curves correspond to Unusual cases and Usual cases, respectively. It can be seen that the frequencies depend on the interface moving velocity continuously. The frequency holds invariance only when V = 0, i.e. in the static case, which is consistent with the existing cognitive. 7.3. The mechanical energy analysis Suppose that the displacement of the incident wave is u1 ( x ) =
f (x) ̸≡ 0 0
(|x| ≤ 1) (|x| > 1) .
(35)
Thus the emitted waves are in the form u (x) =
Tr −u f x/λˆ 0
|x| ≤ λˆ |x| > λˆ .
(36)
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Fig. 13. Curves of ω − V : (a) E1 = 54 GPa, E2 = 200 GPa, ρ1 = 6000 kg/m3 , ρ2 = 8000 kg/m3 (b) E1 = 200 GPa, E2 = 54 GPa, ρ1 = 8000 kg/m3 , ρ2 = 6000 kg/m3 .
Fig. 14. Curves of ϑ − V : (a) E1 = 54 GPa, E2 = 200 GPa, ρ1 = 6000 kg/m3 , ρ2 = 8000 kg/m3 (b) E1 = 200 GPa, E2 = 54 GPa, ρ1 = 8000 kg/m3 , ρ2 = 6000 kg/m3 .
According to Eqs. (8) and (35), the mechanical energy of incident wave can be expressed as
E1 =
1
1
E1 u′12 + ρ1 u˙ 21 dx = E1
2 −1
1
f ′2 (x) dx.
(37)
−1
Similarly, based on Eqs. (8), (10), (36) and (37), the mechanical energy of emitted waves is ˆ λ
2 Tr −u ˆ
E =E
−λ
df
2
ˆ x/λ
dx
ˆ dx = E V − c1 Tr2−u E1 . E V − cˆ
(38)
1
The larger the emitted wave energy is, the more energy the waves gain from the MPI, and it is more difficult to change the material properties near the MPI. Thus we define the ratio of emitted wave energy to incident wave energy as the resistance of the property interface motion, which is in the form
ϑ=
E
E1
V − cˆ1 E 2 = V − cˆ Tr −u E1
(39)
where the symbol Σ represents the sum of all emitted waves. A larger ϑ means greater resistance. ϑ = 1 indicates the energy conservation. It should be noted that Eq. (39) indicates that our theory may fail near the Critical cases, because ϑ is singular near the Critical cases. Such phenomena break the assumption of linear elasticity. Thus our theory only can be applied to the cases appropriately away from the Critical cases. A curve of ϑ − V is given in Fig. 14. It can be seen that Unusual cases 2 and 3 have very low emitted energy, which indicate that the emitted wave may disappear if the dissipation is considered in reality. 8. Conclusion The physical model of 1-D MPI, which should be the most representative characteristics of practical materials with 1-D time-varying properties, is proposed based on the property variability of smart materials. The linear elastic wave propagation law at 1-D MPI is obtained based on a novel method beyond the continuities. The law holds only if the interface moving velocity appropriately deviates from the wave velocities. It is revealed that the property interface motion has significant
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influence on wave frequency and energy. The wave propagation coefficients are determined not only by the wave impedance but also by the property interface motion and wave velocity. Additionally, we discovered shock waves at the MPI. All in all, this research provides a theoretical viewpoint in the study of smart materials with a time-dependent mechanical properties at different loading conditions. Acknowledgment This work is supported by the NPU Foundation for Fundamental Research (Grant No. JC20110255), to whom the authors express their deep gratitude. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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