1
Uniformly Valid Multiple Spatial-Temporal Scale Modeling for Wave Propagation in Heterogeneous Media
Jacob Fish and Wen ChenDepartment of Civil Engineering and Scientific Computation Research Center,
Rensselaer Polytechnic Institute, Troy, NY 12180, USA
Abstract: A novel dispersive model for wave propagation in heterogeneous media is developed. Themethod is based on a higher-order mathematical homogenization theory with multiple spatial and temporalscales. By this approach a fast spatial scale and a series of slow temporal scales are introduced to account forrapid spatial fluctuations of material properties as well as for the long-term behavior of the homogenizedsolution. The problem of secularity arising from the classical multiple spatial scale homogenization theoryfor wave propagation problems is resolved, giving rise to uniformly valid dispersive model. The proposeddispersive model is solved analytically and its solution has be found to be in good agreement with thenumerical solution of the source problem in a heterogeneous medium.
1. Introduction When a wavelength of a traveling signal in a heterogeneous medium is comparable tothe characteristic length of the microstructure, successive reflection and refraction of thewaves between the interfaces of the material lead to significant dispersion effects (see forexample [1][2][3]). The interest on the subject matter stems from the fact that the phenom-enon of dispersion cannot be captured by the classical homogenization theory.
The use of multiple-scale expansions as a systematic tool of homogenization for prob-lems other than elastodynamics can be traced to Sanchez-Palencia [4], Benssousan, Lionsand Papanicoulau [5], and Bakhvalov and Panasenko [6]. The role of higher order terms inthe asymptotic expansion has been investigated in statics by Gambin and Kroner [7], andBoutin [8]. In elastodynamics, Boutin and Auriault [9] demonstrated that the terms of ahigher order successively introduce effects of polarization, dispersion and attenuation.
For wave propagation in heterogeneous media, a single-frequency time dependence istypically assumed [10]. Notable exceptions are the recent articles of Fish and Chen[11][12], which investigated the initial-boundary value problem with rapidly varying coef-ficients. In [11] it has been shown that while higher-order multiple scale expansion inspace is capable of capturing the dispersion effect when the temporal scale of observationis small, it introduces secular terms which grow unbounded with time. In [12], a slow tem-poral scale was introduced to eliminated the secular terms up to the second order and tocapture the long-term behavior of the homogenized solution.
In an attempt to develop a uniformly valid dispersive model up to an arbitrary order, weextend the theory developed in [12] to fast spatial and a series of slow temporal scales.The fast spatial scale is designated to account for rapid spatial fluctuations of materialproperties and a series of slow temporal scales are aimed at capturing the long-term evolu-tion of the homogenization solutions. This results in a dispersive uniformly valid model,solution of which is obtained analytically and subsequently validated against the numeri-cal solution of the source problem in a heterogeneous medium.
2
2. Problem Statement
We consider wave propagation normal to layers of an array of elastic bilaminates in a
periodic arrangement with as a characteristic length as shown in Figure 1. The govern-ing elastodynamics equation of is given by
(1)
with appropriate boundary and initial conditions
, (2)
where denotes the displacement field; and the mass density and
elastic modulus, respectively; and denote the differentiation with respect to x
and time respectively; and in (1) denotes a rapid spatial variation of materialproperties.
The goal is to establish an effective homogeneous model in which local fluctuations intro-duced by material heterogeneity do not appear explicitly and the response of a heteroge-neous medium can be approximated by the response of the effective homogeneousmedium. This is facilitated by the method of multiple scale asymptotic expansion in spaceand time.
3. Asymptotic Analysis with Multiple Spatial and Temporal Scales
Under the premise that the macro domain is much larger than the unit cell
domain , i.e. , it is convenient to introduce a microscopicspatial length variable y [9][16] such that
(3)
where , and c are the macroscopic wavelength, the circular frequency, the wavenumber and the phase velocity of the macroscopic wave, respectively. In addition to thisfast spatial variable, we introduce multiple time scales
Ω
ρ x ε⁄( )u tt; E x ε⁄( )u x; x;
– 0=
u x 0,( ) f x( )= u t; x 0,( ) q x( )=
u x t,( ) ρ x ε⁄( ) E x ε⁄( )( ) x; ( ) t;
0 ε 1«<
Figure 1: A bilaminate with a periodic microstructure
L λ 2π( )⁄=
Ω Ω L⁄ ωΩ( ) c⁄ kΩ 1«= =
y x ε⁄=
λ ω k,
3
, (4)
where is the usual time coordinate and , k > 0 are various slow time scales. Since the
response quantities u and depend on x, , , , and , a multiple-scale
asymptotic expansion is employed to approximate the displacement and stress fields
,
(5)
Homogenization process consists of inserting the asymptotic expansions (5) into thegoverning equation (1), identifying the terms with the equal power of , and then solvingthe resulting problems.
Following the aforementioned procedure and expressing the spatial and temporal deriv-atives in terms of the fast and slow space-time coordinates
(6)
(7)
we obtain a series of equations in ascending power of starting with .
3.1 Homogenization
At , we have
(8)
The general solution of (8) is
(9)
where and are functions of macro coordinates and
multiple temporal scales. To ensure periodicity of over the unit cell domain
in the stretched coordinate system y, must vanish, implying that the leading-order dis-
placement depends only on the macroscale
(10)
tk ε2kt= k 0 1 2 … m, , , ,=( )
t0 tkσ y x ε⁄= t0 t1 … tm
u x y t0 t1 … tm, , , , ,( ) εiui x y t0 t1 … tm, , , , ,( )
i 0=
nn
∑=
σ x y t0 t1 … tm, , , , ,( ) εiσi x y t0 t1 … tm, , , , ,( )i 1–=
nn
∑=
ε
u x; u x, ε 1–u y,+=
u t; u t0, ε2u t1, ε4
u t2, … ε2mu tm,+ + + +=
ε ε 2–
O 1( )
O ε 2–( )
E y( )u0 y, y,
0=
u0 a1 x t0 t1 … tm, , , ,( ) 1E y( )----------- y a2 x t0 t1 … tm, , , ,( )+d
y0
y0 y+
∫=
a1 x t0 t1 … tm, , , ,( ) a2 x t0 t1 … tm, , , ,( )
u0 Ωˆ
Ω ε⁄=
a1
u0 u0 x t0 t1 … tm, , , ,( )=
4
At order , the perturbation equation is
(11)
Due to linearity of the above equation, the general solution of is
(12)
Substituting (12) into (11) yields
(13)
For a -periodic function , we define an averaging operator
(14)
The boundary conditions for the unit cell problem described by (13) are
(a) Periodicity: ,
(b) Continuity: ,
(c) Normalization: (15)
where is the volume fraction of the unit cell; is the jump operator; and
, (16)
Equation (13) together with the boundary conditions (15) define the unit cell boundaryvalue problem from which can be uniquely determined as
, (17)
At , the perturbation equation is
(18)
Applying the averaging operator defined in (14) to the above equation and taking intoaccount periodicity of , we get the macroscopic equation of motion at :
O ε 1–( )
E y( ) u0 x, u1 y,+( ) y, 0=
u1
u1 x y t0 t1 … tm, , , , ,( ) U1 x t0 t1 … tm, , , ,( ) L y( )u0 x,+=
E y( ) 1 L y,+( ) y, 0=
Ωˆ
g x y t0 t1 … tm, , , , ,( )
g⟨ ⟩ 1
Ωˆ------- g x y t0 t1 … tm, , , , ,( ) yd
Ω
∫=
u1 y 0=( ) u1 y Ωˆ
=( )= σ0 y 0=( ) σ0 y Ωˆ
=( )=
u1 y αΩˆ
=( ) 0= σ0 y αΩˆ
=( ) 0=
u1 x y t0 t1 … tm, , , , ,( )⟨ ⟩ U1 x t0 t1 … tm, , , ,( )= ⇒ L y( )⟨ ⟩ 0=
0 α 1≤ ≤
σi E y( ) ui x, ui 1 y,++( )= i 0 1 … nn, , ,=
L y( )
L1 y( )1 α–( ) E2 E1–( )1 α–( )E1 αE2+
----------------------------------------- yαΩ
ˆ
2--------–= L2 y( )
α E1 E2–( )1 α–( )E1 αE2+
---------------------------------------- y1 α+( )Ω
ˆ
2-----------------------–=
O 1( )
ρ y( )u0 t0t0, E y( ) u0 x, u1 y,+( ) x,– E y( ) u1 x, u2 y,+( ) y,– 0=
σ1 O 1( )
5
(19)
where
, (20)
The above macroscopic equation of motion is non-dispersive. In order to capture disper-sion effects, we next consider higher-order equilibrium equations.
3.2 Homogenization
is determined from perturbation equation (18). Substituting (12) and (19) into
(18), yields
(21)
where
(22)
We seek for the solution of in the form of
(23)
Substituting (23) into (21) yields
(24)
The boundary conditions for the above equation are: periodicity and continuity of
and as well as the normalization condition . Once the solution of is
obtained it can be easily verified that satisfies
, (25)
Consider the equilibrium equation:
(26)
Applying the averaging operator to the above equation, and exploiting (25) together withperiodicity of yields
(27)
3.3 Homogenization
ρ0u0 t0t0, E0u0 xx,– 0=
ρ0 ρ⟨ ⟩ αρ1 1 α–( )ρ2+= = E0 E y( ) 1 L y,+( )⟨ ⟩E1E2
1 α–( )E1 αE2+----------------------------------------= =
O ε( )
u2 O 1( )
E y( )u2 y, y, E0 θ y( ) 1–( ) E y( )L( ) y,– u0 xx, E y( )U1 x,
y,–=
θ y( ) ρ y( ) ρ0⁄=
u2
u2 x y t0 t1 … tm, , , , ,( ) U2 x t0 t1 … tm, , , ,( ) L y( )U1 x, M y( )u0 xx,+ +=
E y( ) L M y,+( ) y, E0 θ y( ) 1–( )=
u2
σ1 M y( )⟨ ⟩ 0= M y( )
M y( )
ρL⟨ ⟩ 0= E L M y,+( )⟨ ⟩ 0=
O ε( )
ρ y( )u1 t0t0, E y( ) u1 x, u2 y,+( ) x,
– E y( ) u2 x, u3 y,+( ) y,
– 0=
σ2
ρ0U1 t0t0, E0U1 xx,– 0=
O ε2( )
6
is determined from the perturbation equation (26). Inserting (12) and (23) into
(26) and making use of the macroscopic equations of motion (19) and (27), gives
(28)
Due to linearity of (28) the general solution of is as follows:
(29)
Substituting (29) into (28) gives
(30)
The above equation, together with the periodicity and continuity of and as well as
the normalization condition , fully determine . After is solved for,we can calculate
(31)
(32)
Consider the equilibrium equation of :
(33)
Applying the averaging operator to the above equation, and exploiting periodicity of
and making use of (31) and (32) leads to
(34)
where
(35)
u3 O ε( )
E y( )u3 y, y, E0θ y( )L E y( ) L M y,+( )– E y( )M( ) y,– u0 xxx, +=
E0 θ y( ) 1–( ) E y( )L( ) y,– U1 xx, E y( )U2 x, y,–
u3
u3 x y t0 t1 … tm, , , , ,( ) U3 x t0 t1 … tm, , , ,( ) L y( )U2 x,+ +=
M y( )U1 xx, N y( )u0 xxx,+
E y( ) M N y,+( ) y, E0Lθ y( ) E y( ) L M y,+( )–=
u3 σ2
N y( )⟨ ⟩ 0= N y( ) N y( )
ρM⟨ ⟩α 1 α–( )[ ]2 ρ2 ρ1–( ) E1ρ1 E2ρ2–( )E0Ω
ˆ 2
12ρ0E1E2-----------------------------------------------------------------------------------------------------=
E M Ny,+( )⟨ ⟩α 1 α–( )E0Ω
ˆ 2
12ρ0-----------------------------------
E2 E1–( ) α2ρ1 1 α–( )2ρ2–[ ] E0ρ0+
1 α–( )E1 αE2+------------------------------------------------------------------------------------------- ρ0–
–=
O ε2( )
ρ y( ) u2 t0t0, 2u0 t0t1,+( ) E y( ) u2 x, u3 y,+( ) x,– E y( ) u3 x, u4 y,+( ) y,– 0=
σ3
ρ0U2 t0t0, E0U2 xx,–1
ε2-----Edu0 xxxx, 2ρ0u0 t0t1,–=
Ed
α 1 α–( )[ ]2E1ρ1 E2ρ2–( )2
E0Ω2
12ρ02 1 α–( )E1 αE2+[ ]2
---------------------------------------------------------------------------------=
7
characterizes the effect of the microstructure on the macroscopic behavior. It is pro-
portional to the square of the dimension of the unit cell . Note that for homogeneous
materials (i.e., or ) and in the case of impedance ratio
( ) equal to unity, vanishes.
3.4 Homogenization
is determined from perturbation equation (33). Substituting (12), (23) and
(29) into (33) and making use of (19), (27) and (34) yields
(36)
Due to linearity, the general solution of can be sought in the form
(37)
Substituting (37) into (36) yields
(38)
The above equation, together with the periodicity and continuity of and as well as
the normalization condition , uniquely determines . The solution of
satisfies
, (39)
The equilibrium equation at is:
(40)
Applying the averaging operator to (40), exploiting periodicity of and making use of
(39) yields
(41)
Ed
Ωα 0= α 1= r z2 z1⁄=
z Eρ= Ed
O ε3( )
u4 O ε2( )
E y( )u4 y, y, θ y( ) E0M Ed ε2⁄+( ) E y( ) M N y,+( )– E y( )N( ) y,– u0 xxxx, +=
E0θ y( )L E y( ) L M y,+( )– E y( )M( ) y,– U1 xxx, +
E0 θ y( ) 1–( ) E y( )L( ) y,– U2 xx, E y( )U3 x, y,
–
u4
u4 x y t0 t1 … tm, , , , ,( ) U4 x t0 t1 … tm, , , ,( ) L y( )U3 x,+ +=
M y( )U2 xx, N y( )U1 xxx, P y( )u0 xxxx,+ +
E y( ) N Py,+( ) y, θ y( ) E0M Ed ε2⁄+( ) E y( ) M N y,+( )–=
u4 σ3
P y( )⟨ ⟩ 0= P y( )P y( )
ρN⟨ ⟩ 0= E N Py,+( )⟨ ⟩ 0=
O ε3( )
ρ y( ) u3 t0t0, 2u1 t0t1,+( ) E y( ) u3 x, u4 y,+( ) x,– E y( ) u4 x, u5 y,+( ) y,– 0=
σ4
ρ0U3 t0t0, E0U3 xx,–1
ε2-----EdU1 xxxx, 2ρ0U1 t0t1,–=
8
3.5 Homogenization
is determined from perturbation equation (40). Substituting (12), (23), (29)
and (37) into (40) and making use of (19), (27), (34) and (41) yields
(42)
Due to linearity, the general solution of can be sought in the form
(43)
Substituting (43) into (42) gives
(44)
The above equation, together with the periodicity and continuity of and as well as
the normalization condition , uniquely determines . After is
solved for, expressions for and can be derived.
The equilibrium equation is:
(45)
Applying the averaging operator to the above equation and taking into account periodic-ity of , gives
(46)
where
O ε4( )
u5 O ε3( )
E y( )u5 y, y, θ y( ) E0N LEd ε2⁄+( ) E y( ) N Py,+( )– E y( )P( ) y,– u0 xxxxx, +=
θ y( ) E0M Ed ε2⁄+( ) E y( ) M N y,+( )– E y( )N( ) y,– U1 xxxx, +
E0θ y( )L E y( ) L M y,+( )– E y( )M( ) y,– U2 xxx, +
E0 θ y( ) 1–( ) E y( )L( ) y,– U3 xx, E y( )U4 x, y,–
u5
u5 x y t0 t1 … tm, , , , ,( ) U5 x t0 t1 … tm, , , ,( ) L y( )U4 x,+ +=
M y( )U3 xx, N y( )U2 xxx, P y( )U1 xxxx, Q y( )u0 xxxxx,+ + +
E y( ) P Qy,+( ) y, θ y( ) E0N LEd ε2⁄+( ) E y( ) N Py,+( )–=
u5 σ4
Q y( )⟨ ⟩ 0= Q y( ) Q y( )ρP⟨ ⟩ E P Qy,+( )⟨ ⟩
O ε4( )
ρ y( ) u4 t0t0, 2u2 t0t1, 2u0 t0t2, u0 t1t1,+ + +( ) E y( ) u4 x, u5 y,+( ) x,– –
E y( ) u5 x, u6 y,+( ) y, 0=
σ5
ρ0t02
2
∂
∂ U4 E0x
2
2
∂
∂ U4–Ed
ε2------
x4
4
∂
∂ U2 Eg
ε4------
x6
6
∂
∂ u0 2ρ0 t0 t∂ 1
2
∂∂ U2 2ρ0 t0 t2∂
2
∂∂ u0– ρ0
t12
2
∂
∂ u0–––=
9
(47)
3.6 Homogenization
Next we determine the value of from perturbation equation (45). Substituting
(12), (23), (29), (37) and (43) into (45) and making use of (19), (27), (34), (41) and (46),yields
(48)
Due to linearity of (48) the general solution of can be sought in the form
(49)
Substituting the above expression into (48) yields
(50)
The above equation, together with the periodicity and continuity of and as well as
the normalization condition , uniquely determines . After issolved for, it can be easily shown that
, (51)
The equilibrium equation at is:
Eg
α 1 α–( )[ ]2E1ρ1 E2ρ2–( )2
E0Ω4
360ρ04 1 α–( )E1 αE2+[ ]4
--------------------------------------------------------------------------------- α2E2
2 2α2ρ12 1 α–( )2ρ2
2– +[=
6α 1 α–( )ρ1ρ2 ] 2+ α 1 α–( )E1E2 3α2ρ12 3 1 α–( )2ρ2
2+ +[
11α 1 α–( )ρ1ρ2] 1 α–( )2E1
2 α2ρ12
2 1 α–( )2ρ22
6α 1 α–( )ρ1ρ2 ] ––[–
O ε5( )
u6 O ε4( )
E y( )u6 y, y, θ y( ) E0P MEd ε2Eg ε4⁄–⁄+( ) E y( ) P Qy,+( )– –=
E y( )Q( ) y, u0 xxxxxx, θ y( ) E0N LEd ε2⁄+( ) E y( ) N Py,+( )– –+
E y( )P( ) y, U1 xxxxx, θ y( ) E0M Ed ε2⁄+( ) E y( ) M N y,+( ) ––+
E y( )N( ) y, U2 xxxx, E0θ y( )L E y( ) L M y,+( )– E y( )M( ) y,– U3 xxx,+ +
E0 θ y( ) 1–( ) E y( )L( ) y,– U4 xx, E y( )U5 x, y,–
u6
u6 x y t0 t1 … tm, , , , ,( ) U6 x t0 t1 … tm, , , ,( ) L y( )U5 x,+ +=
M y( )U4 xx, N y( )U3 xxx, P y( )U2 xxxx, Q y( )U1 xxxxx, R y( )u0 xxxxxx,+ + + +
E y( ) Q Ry,+( ) y, θ y( ) E0P MEd ε2⁄ Eg ε4⁄–+( ) E y( ) P Qy,+( )–=
u6 σ5
R y( )⟨ ⟩ 0= R y( ) R y( )
ρQ⟨ ⟩ 0= E Q Ry,+( )⟨ ⟩ 0=
O ε5( )
10
(52)
Applying the averaging operator to the above equation, exploiting periodicity of and
making use of (51) yields
(53)
3.7 Higher Order Homogenization and Summary of Macroscopic Equations
The homogenization process described in the previous section can be systematicallygeneralized to an arbitrary order. In this section we summarize various order macroscopicequations of motion and state the initial and boundary conditions.
The macroscopic equations of motion are:
: (19)
: (27)
: (34)
: (41)
: (46)
: (53)
and higher:
ρ y( ) u5 t0t0, 2u3 t0t1, 2u1 t0t2, u1 t1t1,+ + +( ) E y( ) u5 x, u6 y,+( ) x,
– –
E y( ) u6 x, u7 y,+( ) y,
0=
σ6
ρ0t02
2
∂
∂ U5 E0x
2
2
∂
∂ U5–Ed
ε2------
x4
4
∂
∂ U3 Eg
ε4------
x6
6
∂
∂ U1 2ρ0 t0 t1∂
2
∂∂ U3 2ρ0 t0 t2∂
2
∂∂ U1– ρ0
t12
2
∂
∂ U1–––=
O 1( ) ρ0u0 t0t0, E0u0 xx,– 0=
O ε( ) ρ0U1 t0t0, E0U1 xx,– 0=
O ε2( ) ρ0U2 t0t0, E0U2 xx,–1
ε2-----Edu0 xxxx, 2ρ0u0 t0t1,–=
O ε3( ) ρ0U3 t0t0, E0U3 xx,–1
ε2-----EdU1 xxxx, 2ρ0U1 t0t1,–=
O ε4( ) ρ0t02
2
∂
∂ U4 E0x
2
2
∂
∂ U4–Ed
ε2------
x4
4
∂
∂ U2 Eg
ε4------
x6
6
∂
∂ u0 2ρ0 t0 t1∂
2
∂∂ U2 2ρ0 t0 t∂ 2
2
∂∂ u0– ρ0
t12
2
∂
∂ u0–––=
O ε5( ) ρ0t02
2
∂
∂ U5 E0x
2
2
∂
∂ U5–Ed
ε2------
x4
4
∂
∂ U3 Eg
ε4------
x6
6
∂
∂ U1 2ρ0 t0 t1∂
2
∂∂ U3 2ρ0 t0 t∂ 2
2
∂∂ U1– ρ0
t12
2
∂
∂ U1–––=
O ε6( )
ρ0t02
2
∂
∂ U2m E0x
2
2
∂
∂ U2m–Ed
ε2------
x4
4
∂
∂ U2 m 1–( ) Eg
ε4------
x6
6
∂
∂ U2 m 2–( )– +=
11
(54)
(55)
where ; ; , , , can be evaluated
using higher-order homogenization process.
Subsequently, we consider the following model problem: a domain composed of anarray of bilaminates with fixed boundary at and free boundary at subjected
Es1
ε6--------
x8
8
∂
∂ U2 m 3–( ) Es2
ε8--------
x10
10
∂
∂ U2 m 4–( )– … 1–( )m 1+ Es m 2–( )
ε2m-------------------
x2 m 1+( )
2 m 1+( )
∂
∂ u0 –+ +
2ρ0 t0 tm∂
2
∂∂ u0
t1 tm 1–∂
2
∂∂ u0 … 1
2---
tm 2⁄2
2
∂
∂ u0+ + +
–
2ρ0 t0 tm 1–∂
2
∂∂ U2
t1 tm 2–∂
2
∂∂ U2 … 1
2---
t m 1–( ) 2⁄2
2
∂
∂ U2+ + +
–
2ρ0 t0 tm 2–∂
2
∂∂ U4
t1 tm 3–∂
2
∂∂ U4 … 1
2---
t m 2–( ) 2⁄2
2
∂
∂ U4+ + +
… 2ρ0 t0 t1∂
2
∂∂ U2 m 1–( )––
ρ0t02
2
∂
∂ U2m 1+ E0x
2
2
∂
∂ U2m 1+–Ed
ε2------
x4
4
∂
∂ U2m 1– Eg
ε4------
x6
6
∂
∂ U2m 3–– +=
Es1
ε6--------
x8
8
∂
∂ U2m 5– Es2
ε8--------
x10
10
∂
∂ U2m 7–– … 1–( )m 1+ Es m 2–( )
ε2m-------------------
x2 m 1+( )
2 m 1+( )
∂
∂ U1 –+ +
2ρ0 t0 tm∂
2
∂∂ U1
t1 tm 1–∂
2
∂∂ U1 … 1
2---
tm 2⁄2
2
∂
∂ U1+ + +
–
2ρ0 t0 tm 1–∂
2
∂∂ U3
t1 tm 2–∂
2
∂∂ U3 … 1
2---
t m 1–( ) 2⁄2
2
∂
∂ U3+ + +
–
2ρ0 t0 tm 2–∂
2
∂∂ U5
t1 tm 3–∂
2
∂∂ U5 … 1
2---
t m 2–( ) 2⁄2
2
∂
∂ U5+ + +
… 2ρ0 t0 t1∂
2
∂∂ U2m 1–––
m 3 4 5 …, , ,= tk k 0 1 2 … m, , , ,=( ) Es1 Es2 … Es m 2–( )
x 0= x l=
12
to an initial disturbance in the displacement field. At , the displacement field isdetermined by the equation of motion (19) and the following initial and boundary condi-tions
ICs: , (56)
BCs: , (57)
The calculation of is obtained by solving the equation of motion
(27). The initial and boundary conditions applied to must be such that the global field
meets macroscopic initial conditions and conditions imposed on the boundary:
Taking into account (56) and (57), the initial and boundary conditions for become
ICs: ,
BCs: ,
Similarly, the macroscopic field is determined from the equation
of motion (34), with the initial and boundary conditions for such that the global
field should satisfy the macroscopic initial and boundary conditions.
With this in mind, we obtain the initial and boundary conditions for different order equa-tions of motion:
ICs: ,
, (58)
BCs: , (59)
From the above equations of motion and the initial-boundary conditions, we can readilydeduce that
f x( ) O 1( )
u0 x 0 0 … 0, , , ,( ) f x( )= u0 t; x 0 0 … 0, , , ,( ) q x( ) 0= =
u0 0 t0 t1 … tm, , , ,( ) 0= u0 x, l t0 t1 … tm, , , ,( ) 0=
εU1 x t0 t1 … tm, , , ,( )
εU1
u0 εU1+
u0 x 0 0 … 0, , , ,( ) εU1 x 0 0 … 0, , , ,( )+ f x( )=
u0 t; x 0 0 … 0, , , ,( ) εU1 t; x 0 0 … 0, , , ,( )+ 0=
u0 0 t0 t1 … tm, , , ,( ) εU1 0 t0 t1 … tm, , , ,( )+ 0=
u0 x, l t0 t1 … tm, , , ,( ) εU1 x, l t0 t1 … tm, , , ,( )+ 0=
εU1
εU1 x 0 0 … 0, , , ,( ) 0= εU1 t; x 0 0 … 0, , , ,( ) 0=
εU1 0 t0 t1 … tm, , , ,( ) 0= εU1 x, l t0 t1 … tm, , , ,( ) 0=
ε2U2 x t0 t1 … tm, , , ,( )
ε2U2
u0 εU1 ε2U2+ +
u0 x 0 0 … 0, , , ,( ) f x( )= u0 t; x 0 0 … 0, , , ,( ) q x( ) 0= =
Ui x 0 0 … 0, , , ,( ) 0= Ui t; x 0 0 … 0, , , ,( ) 0= i( 1 2 3 …, , ),=
Ui 0 t0 t1 … tm, , , ,( ) 0= Ui x, l t0 t1 … tm, , , ,( ) 0= i( 0 1 2 …, ), ,=
13
, (60)
4. Solution of Macroscopic Equations We start with the zero-order equation of motion (19), the solution of which can be soughtby means of separation of variables in the form
(61)
Substituting the above equation into (19) and dividing by the product yields
(62)
where is the separation constant and
(63)
The resulting differential equations and corresponding solutions are
, (64)
(65)
(66)
where and are constants of integration; and are
undetermined functions.
Substituting (61), (65) and (66) into (59) gives
, (67)
The second equation in (67) yields
, (68)
Due to linearity of the differential equation, the total solution consists of the sum of indi-vidual solutions. Hence, we may write
U2m 1+ x t0 t1 … tm, , , ,( ) 0≡ m( 0 1 2 …, ), ,=
u0 x t0 t1 … tm, , , ,( ) X x( )T t0 t1 … tm, , ,( )=
X T⋅
1T---
t02
2
∂∂ T
c2X′′
X------- λ2–= =
λ
c E0 ρ0⁄=
X′′ λ2
c2
-----X+ 0=t02
2
∂∂ T λ2
T+ 0=
X x( ) A1λxc
------ A2λxc
------cos+sin=
T t0 t1 … tm, , ,( ) D t1 t2 … tm, , ,( ) λt0( ) F t1 t2 … tm, , ,( ) λt0( )cos+sin=
A1 A2 D t1 t2 … tm, , ,( ) F t1 t2 … tm, , ,( )
A2 0= A1λlc-----cos 0=
λn 2n 1–( )πc2l------= n( 1 2 3 … ), , ,=
14
(69)
which can be shown to satisfy the boundary conditions. Inserting (69) into the secondorder macroscopic equation of motion (34) yields
(70)
The right-hand-side in (70) is also a solution to the corresponding homogeneous equa-tion, and therefore, it generates secular terms. In order to eliminate secular terms and toavoid unbounded resonance of , the source term must vanish, i.e.
, (71)
Let
(72)
Then (71) can be written as
, (73)
Differentiating the first equation in (73) and inserting the second equation into the result-ing equation leads to
(74)
Likewise, differentiating the second equation in (73) and inserting the first equation intothe resulting equation yields
u0 x t0 t1 … tm, , , ,( )λnx
c-------- Dn t1 t2 … tm, , ,( ) λnt0( ) +sin[sin
n 1=
∞
∑=
Fn t1 t2 … tm, , ,( ) λnt0( ) ]cos
t02
2
∂
∂ U2 c2
x2
2
∂
∂ U2–λnx
c--------
Ed
ε2ρ0
-----------λn
c-----
4
Dn 2λn t1∂∂Fn+ λnt0( )sin +
sin
n 1=
∞
∑=
Ed
ε2ρ0
-----------λn
c-----
4
Fn 2λn t1∂∂Dn– λnt0( ) cos
U2 x t0 t1 … tm, , , ,( )
Ed
ε2ρ0
-----------λn
c-----
4
Dn 2λn t1∂∂Fn+ 0=
Ed
ε2ρ0
-----------λn
c-----
4
Fn 2λn t1∂∂Dn– 0=
ωn
Ed
2cρ0------------
λn
c-----
3
=
ε2
t1∂∂Fn ωnDn+ 0= ε2
t1∂∂Dn ωnFn– 0=
ε4
t12
2
∂
∂ Fn ωn2Fn+ 0=
15
(75)
The general solutions to (74) and (75) are
(76)
(77)
where , , and are
undetermined functions.
Solutions (76) and (77) must satisfy (73). Inserting (76) and (77) into (73) gives
, (78)
Substituting (76), (77) and (78) into (69) yields
(79)
Since the source term of macroscopic equation of motion has been set to zero andconsidering the initial and boundary conditions described in Section 3.7, we deduce
(80)
Inserting (79) and (80) into the fourth order macroscopic equation of motion (46) yields
(81)
ε4
t12
2
∂
∂ Dn ωn2Dn+ 0=
Dn t1 t2 … tm, , ,( ) Gn t2 t3 … tm, , ,( )ωnt1
ε2-----------
Jn t2 t3 … tm, , ,( )ωnt1
ε2-----------
cos+sin=
Fn t1 t2 … tm, , ,( ) Kn t2 t3 … tm, , ,( )ωnt1
ε2-----------
Sn t2 t3 … tm, , ,( )ωnt1
ε2-----------
cos+sin=
Gn t2 t3 … tm, , ,( ) Jn t2 t3 … tm, , ,( ) Kn t2 t3 … tm, , ,( ) Sn t2 t3 … tm, , ,( )
Jn Kn–= Gn Sn=
u0 x t0 t1 … tm, , , ,( )λnx
c-------- Kn t2 t3 … tm, , ,( )
ωnt1
ε2----------- λnt0–
+sinsin
n 1=
∞
∑=
Sn t2 t3 … tm, , ,( )ωnt1
ε2----------- λnt0–
cos
O ε2( )
U2 x t0 t1 … tm, , , ,( ) 0=
t02
2
∂
∂ U4 c2
x2
2
∂
∂ U4–λnx
c-------- 1
ε4-----
Eg
ρ0------
λn
c-----
6
ωn2
+ Kn 2λn t2∂∂Sn–
ωnt1
ε2----------- λnt0–
sin +
sin
n 1=
∞
∑=
1
ε4-----
Eg
ρ0------
λn
c-----
6
ωn2+ Sn 2λn t2∂
∂Kn+ωnt1
ε2----------- λnt0–
cos
16
Again, the right-hand-side term in the above equation is the solution of the correspond-ing homogeneous equation and thus will generate secular terms. In order to eliminate sec-ularity arising from , we set
, (82)
Let
(83)
Equation (82) can be written as
, (84)
The general solutions to the above equations are
(85)
(86)
Substituting (85) and (86) into (79) gives
(87)
Since the source term of macroscopic equation of motion vanishes and consider-ing the initial and boundary conditions prescribed in Section 3.7, we conclude that
(88)
The above procedure can be systematically extended to higher order equations, whichyields
U4
1
ε4-----
Eg
ρ0------
λn
c-----
6
ωn2+ Kn 2λn t2∂
∂Sn– 0=1
ε4-----
Eg
ρ0------
λn
c-----
6
ωn2+ Sn 2λn t2∂
∂Kn+ 0=
γn1λn-----
Eg
2ρ0---------
λn
c-----
6 1
2---ωn
2+1
2cρ0------------ Eg
Ed2
4c2ρ0
--------------+ λn
c-----
5
= =
ε4
t2∂∂Sn γnKn– 0= ε4
t2∂∂Kn γnSn+ 0=
Kn t2 t3 … tm, , ,( ) Vn t3 t4 … tm, , ,( )γnt2
ε4---------
Wn t3 t4 … tm, , ,( )γnt2
ε4---------
cos+sin=
Sn t2 t3 … tm, , ,( ) Wn t3 t4 … tm, , ,( )γnt2
ε4---------
Vn t3 t4 … tm, , ,( )γnt2
ε4---------
cos–sin=
u0 x t0 t1 … tm, , , ,( )λnx
c-------- Wn t3 t4 … tm, , ,( )
γnt2
ε4---------
ωnt1
ε2-----------+ λnt0–
–sinsin
n 1=
∞
∑=
Vn t3 t4 … tm, , ,( )γnt2
ε4---------
ωnt1
ε2-----------+ λnt0–
cos
O ε4( )
U4 x t0 t1 … tm, , , ,( ) 0=
17
(89)
where and are constants of integration and
(90)
(91)
(92)
(93)
where and i is an integer.
Inserting , into (89) and using the initial conditions (58)
we can determine and as
, (94)
and thus a uniformly valid dispersive solution, denoted here as , is given as
u0 x t0 t1 … tm, , , ,( )λnx
c-------- An
β m 2–( )ntm
ε2m------------------------- …
β2nt4
ε8------------+ +
β1nt3
ε6------------
γnt2
ε4---------+ + +
sinsin
n 1=
∞
∑=
ωnt1
ε2----------- λnt0)– Bn
β m 2–( )ntm
ε2m------------------------- …
β2nt4
ε8------------
β1nt3
ε6------------
γnt2
ε4---------
ωnt1
ε2-----------+ λnt0 ) ]–+ + + +
cos+
An Bn
β1n1λn-----
Es1
2ρ0---------
λn
c-----
8
ωnγn+=
β2n1λn-----
Es2
2ρ0---------
λn
c-----
10
ωnβ1n12---γn
2+ +=
β3n1λn-----
Es3
2ρ0---------
λn
c-----
12
ωnβ2n γnβ1n+ +=
……
β m 2–( )n1λn-----
Es m 2–( )2ρ0
-------------------λn
c-----
2 m 1+( )
ωnβ m 3–( )n γnβ m 4–( )n+ + +=
β1nβ m 5–( )n β2nβ m 6–( )n … 12---β m 2⁄ 2–( )n
2 ]+ + +
βin i 1 2 … m 2–( ), , ,=( )
tk ε2kt= k 0 1 2 … m, , , ,=( )
An Bn
An 0= Bn2l--- f x( ) 2n 1–( )πx
2l--------------------------sin xd
0
l
∫=
ud
ud x t0 t1 … tm, , , ,( ) Bn
λnx
c-------- λnt0
ωnt1
ε2-----------
γnt2
ε4---------
β1nt3
ε6------------+ + +
–cossin
n 1=
∞
∑=
18
(95)
For function evaluation, we insert , , which yields
(96)
5. Numerical Results To assess the accuracy of the proposed formulation, we construct a reference solution byutilizing a very fine finite element mesh to discretize the problem domain. We consider thefollowing initial disturbance in displacements:
where and is the Heaviside step function; , and are the magni-
tude, the location of the maximum value and the half width of the initial pulse. Substitut-ing the initial disturbance into (94) and integrating analytically, we get
The material properties considered are: GPa, GPa, Kg/
m3, Kg/m3, and volume fraction . The dimension of the macro-
domain and that of the unit cell are set as m and m, respectively. The
homogenized material properties are calculated as GPa, Kg/m3
and N. In this case, and the ratio of the impedance of the
β2nt4
ε8------------ …+
β m 2–( )ntm
ε2m-------------------------
+
tk ε2kt= k 0 1 2 … m, , , ,=( )
ud x t,( ) Bn
λnx
c-------- λn ωn γn β1n β2n … β m 2–( )n ) ]t+ + + + +(–[ cossin
n 1=
∞
∑=
f x( ) f0a0 x x0 δ–( )–[ ]2x x0 δ+( )–[ ]2 1 H x x0 δ+( )–[ ]– ⋅=
1 H x0 δ– x–( )–[ ]
a0 1 δ4⁄= H x( ) f0 x0 δ
f x( )
Bn2l--- f0a0 x x0 δ–( )–[ ]2
x x0 δ+( )–[ ]2 2n 1–( )πx2l
--------------------------sin xd
x0 δ–
x0 δ+
∫=
256l2f0
δ4 2n 1–( )π[ ]5------------------------------------- 12l
2 2n 1–( )δπ( )2 ]2n 1–( )πx0
2l----------------------------- 2n 1–( )πδ
2l--------------------------- –sinsin–
=
6 2n 1–( )δπl2n 1–( )πx0
2l----------------------------- 2n 1–( )πδ
2l---------------------------
cossin
E1 120= E2 6= ρ1 8000=
ρ2 3000= α 0.5=
l 40= Ω 0.2=
E0 11.43= ρ0 5500=
Ed 1.76 107×= E1 E2⁄ 20=
19
two material constituents is . The initial pulse is centered at the midpoint of the
domain, i.e. m, with the magnitude m.
Figure 2: Displacements at for the initial half pulse width .
Evolution of displacements of the point ( m) is plotted in figures 2-4 for three
cases corresponding to m, m and m, respectively. In other
words, the ratios between the pulse width and the unit cell dimension are 8, 5 and3, respectively. Each of the figures 2-4 depicts four graphs corresponding to the finite ele-ment solution of the source problem, the analytical nondispersive solution , the
r 7.30=
x0 20= f0 1.0=
x 30m= δ 0.8m=
x 30=
δ 0.8= δ 0.5= δ 0.3=
2δ Ω⁄
u0 x t,( )
20
dispersive solution up to the second order and the dispersive solution up to the
fourth order.
Figure 3: Displacements at for the initial half pulse width
The phenomenon of dispersion can be clearly observed in Figures 2-4. Figure 2 corre-sponds to rather low frequency case, where the pulse almost maintains its initial shapeexcept for some minor wiggles at the wavefront. In this case, the leading-order homogeni-zation can give a reasonable approximation to the response of a heterogeneous medium.However, when the pulse width of the initial disturbance is comparable to the dimensionof the unit cell and the observation time is large, which are the cases shown in Figures 3and 4, the wave becomes strongly dispersive and the leading-order homogenization errsbadly. It can be readily observed that the dispersive solution provides a good
ud x t,( )
x 30m= δ 0.5m=
ud x t,( )
21
approximation to the response of the heterogeneous media even as the initial pulse widthis only 3 times of the unit cell dimension.
Figure 4: Displacements at for the initial half pulse width
6. Concluding Remarks Mathematical homogenization theory with multiple spatial and temporal scales havebeen investigated. This work is motivated by our recent studies [11][12] which suggestedthat in absence of multiple time scaling, higher order mathematical homogenizationmethod gives rise to secular terms which grow unbounded with time. In attempt todevelop a uniformly valid dispersive model up to an arbitrary order we extend the theorydeveloped in [12] to fast spatial scale and a series of slow temporal scales.
In our future work we will focus on the following two issues: (i) generalization to themultidimensional case, and (ii) a finite element implementation.
x 30m= δ 0.3m=
22
Acknowledgment This work was funded by Sandia National Laboratories. Sandia is a multi-programnational laboratory operated by the Sandia Corporation, a Lockheed Martin Company, forthe United States Department of Energy under Contract DE-AL04-94AL8500.
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