non-local dispersive model for wave propagation in heterogeneous media: one-dimensional case

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2002; 54:331–346 (DOI: 10.1002/nme.423)

Non-local dispersive model for wave propagationin heterogeneous media: one-dimensional case

Jacob Fish∗;†, Wen Chen and Gakuji Nagai

Departments of Civil; Mechanical and Aerospace Engineering; Rensselaer Polytechnic Institute;Troy; NY 12180; U.S.A.

SUMMARY

Non-local dispersive model for wave propagation in heterogeneous media is derived from the higher-order mathematical homogenization theory with multiple spatial and temporal scales. In addition to theusual space–time co-ordinates, a fast spatial scale and a slow temporal scale are introduced to accountfor rapid spatial �uctuations of material properties as well as to capture the long-term behaviour of thehomogenized solution. By combining various order homogenized equations of motion the slow timedependence is eliminated giving rise to the fourth-order di�erential equation, also known as a ‘bad’Boussinesq problem. Regularization procedures are then introduced to construct the so-called ‘good’Boussinesq problem, where the need for C1 continuity is eliminated. Numerical examples are presentedto validate the present formulation. Copyright ? 2002 John Wiley & Sons, Ltd.

KEY WORDS: non-local; gradient; homogenization; multiple scales; dispersive; wave propagation

1. INTRODUCTION

The use of multiple-scale expansions as a systematic tool of averaging can be traced toSanchez-Palencia [1], Benssousan et al. [2], and Bakhvalov and Panasenko [3]. The role ofhigher-order terms in the asymptotic expansion has been investigated in statics by Gambinand Kroner [4], and Boutin [5]. In elastodynamics, Boutin and Auriault [6] demonstratedthe terms of a higher-order successively introduced e�ects of polarization, dispersion andattenuation. In References [6; 7], a single-frequency time dependence was assumed prior tothe homogenization process.A general setting for the initial-boundary value problem in heterogeneous media has been

presented by the �rst two authors in References [8; 9], where it was demonstrated that while

∗Correspondence to: Jacob Fish, Department of Civil Engineering and Scienti�c Computation Research Center,Rensselaer Polytechnic Institute, 110 8th Street, NY 12180-3590, U.S.A.

† E-mail: �[email protected]

Contract=grant sponsor: Sandia National Laboratories; contract=grant number: DE-AL04-94AL8500Contract=grant sponsor: O�ce of Naval Research; contract=grant number: N00014-97-1-0687Contract=grant sponsor: Japan Society for the Promotion of Science; contract=grant number: Heisei 11-nendo 06542

Received 10 February 2001Copyright ? 2002 John Wiley & Sons, Ltd. Revised 30 July 2001

332 J. FISH, W. CHEN AND G. NAGAI

the high-order homogenization is capable of capturing dispersion e�ects for a relatively shortobservation time window, it introduces secular terms which grow unbounded with time. Theproblem of secularity has been successfully resolved with the introduction of slow temporalscales to capture the long-term behaviour of the homogenized solution [8; 9].The primary objective of the present paper is to develop a non-local approach independent

of the slow time scale considered in References [8; 9]. We show that by adding various orderhomogenized equations of motion resulting from the higher mathematical homogenizationin space and time [8; 9], the slow time scale dependence can be eliminated. The resultingmacroscopic equation consists of a sequence of higher-order spatial derivatives similar to thoseobtained from the expansion of non-local �elds. Due to this resemblance we will subsequentlyrefer to the proposed model as a non-local model. If the terms containing spatial derivativesof �fth order and higher are neglected then the resulting model falls into the category ofgradient methods [10]. For the gradient version of the model the phase velocity is imaginaryin the case of higher frequency excitations giving rise to the so-called ‘bad’ Boussinesqequation (cf. References [11; 12]). The problem of imaginary phase velocities can be alleviatedby approximating the fourth-order spatial derivative with a mixed spatial-temporal derivativeresulting in what is often known as the ‘good’ Boussinesq equation. It contains only thesecond-order spatial-temporal derivatives thus eliminating the need for C1 continuous elements.The weak formulation and the �nite element approximation leads to the semi-discrete equationsof motion, which are then integrated by using standard time integration schemes.The outline of the paper is as follows. Problem description and high-order homogenization

with multiple space–time scales are reviewed in Sections 2 and 3, respectively. Non-localmodels are derived in Section 4. Finite element procedures for solving the non-local equationsof motion are formulated in Section 5. Section 6 gives numerical examples.

2. PROBLEM DESCRIPTION

We consider wave propagation normal to the layers of an array of periodic elastic bilaminateswith l as a characteristic length (see Figure 1).

Figure 1. A bilaminate with periodic microstructure.

Copyright ? 2002 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 54:331–346

NON-LOCAL DISPERSIVE MODEL FOR WAVE PROPAGATION 333

The governing equation for the elastodynamics problem is given by

�(x=�)u; tt − {E(x=�)u; x}; x=0 (1)

with appropriate boundary conditions on the domain boundary @L

u(0; t)=0; u; x(L; t)=q(t)E(L)A

(2)

and initial conditions

u(x; 0)=f(x); u; t(x; 0)= g(x) (3)

where u(x; t) represents the displacement �eld; �(x=�), E(x=�), A and q(t) the mass density,elastic modulus, cross-sectional area and external load, respectively; ( ); x and ( ); t denotetotal derivatives with respect to space and time, respectively; 0¡��1 in (1) denotes therapid spatial variation of material properties. The goal is to establish an e�ective homogeneousmodel in which local �uctuations introduced by material heterogeneity do not appear explicitlyand the response of a heterogeneous medium can be approximated by the response of thee�ective homogeneous medium. This is facilitated by the method of multiple scale asymptoticexpansion in space and time.

3. ASYMPTOTIC ANALYSIS WITH MULTIPLE SPACE–TIME SCALES

Under the premise that the macroscopic domain L= �=(2�) is much larger than the unit celldomain l, i.e. l=L=(!l)=c=kl�1, it is convenient to introduce a microscopic spatial lengthvariable y [6; 13] such that

y= x=� (4)

where �; !; k and c are the macroscopic wavelength, the circular frequency, the wave numberand the phase velocity of the macroscopic wave, respectively. In addition to the fast spatialvariable, we introduce multiple time scales [8; 9]

�= t; �= �2t (5)

where � denotes the usual time scale and � is a slow time scale. Since the response quantitiesu and � depend on x, y, � and �, a multiple-scale asymptotic expansion is employed toapproximate the displacement and stress �elds

u(x; y; �; �)=n∑i=0�iui(x; y; �; �); �(x; y; �; �)=

n∑i=−1

�i�i(x; y; �; �) (6)

Asymptotic analysis consists of inserting the asymptotic expansions (6) into the governingequation (1), identifying the terms with equal power of �, and �nally, solving the resultingequations corresponding to each power of �.Following the aforementioned procedure and expressing the spatial and temporal derivatives

by the chain rule

u; x= u; x + �−1u;y; u; t = u;� + �2u;� (7)



we obtain a series of equations in ascending power of � starting at �−2. We successively equatethe factors of each of these powers to zero; u; x and u;y denote partial usual and fast spatialderivatives, respectively; u;� and u;� correspond to partial usual and slow temporal derivatives,respectively.

3.1. O(1) homogenization

At O(�−2), we have

{E(y)u0;y};y=0 (8)

The general solution to the above equation is

u0 = a1(x; �; �)∫ y0+y

y0

1E(�)

d�+ a2(x; �; �) (9)

where a1(x; �; �) and a2(x; �; �) are functions of macro co-ordinates. To ensure periodicityof u0 over the unit cell domain, l= l=� ((ˆ) denotes quantities in the stretched co-ordinatesystem y), a1 must vanish, implying that the leading-order displacement depends only on themacroscale

u0 = u0(x; �; �) (10)

At O(�−1), the perturbation equation is

{E(y) (u0; x + u1;y)};y=0 (11)

Due to linearity of the above equation, the general solution to u1 is decomposed as

u1(x; y; �; �)=U1(x; �; �) +H (y)u0; x (12)

Substituting (12) into (11) yields

{E(y) (1 +H;y)};y=0 (13)

For a l-periodic function G(x; y; �; �), we de�ne an averaging operator

〈G〉= 1

|l|

∫lG(x; y; �; �) dy (14)

The boundary conditions for the unit cell problem described by (13) are

(a) Periodicity

u1(y=0)= u1(y= l); �0(y=0)=�0(y= l)

(b) Continuity

[u1(y= l)]=0; [�0(y= l)]=0

(c) Normalization

〈u1(x; y; �; �)〉=U1(x; �; �)⇒ 〈H (y)〉=0 (15)



where 0661 is the volume fraction of the unit cell; [ ] is the jump operator and

�i=E(y)(ui; x + ui+1;y); i=0; 1; : : : ; n (16)

Equation (13) together with the boundary conditions (15) de�nes the unit cell boundaryvalue problem from which H (y) can be uniquely determined:

H1(y)=(1− )(E2 − E1)(1− )E1 + E2

{y − l

2

}; H2(y)=

(E1 − E2)(1− )E1 + E2

{y − (1 + )l

2

}(17)

At O(1), the perturbation equation is

�(y)u0; �� − {E(y)(u0; x + u1;y)}; x − {E(y)(u1; x + u2;y)};y=0 (18)

Applying the averaging operator de�ned in (14) to the above equation and accounting forthe periodicity of �1, yields the macroscopic equation of motion at O(1):

�0u0; �� − E0u0; xx=0 (19)

where

�0 = 〈�〉= �1 + (1− )�2; E0 = 〈E(y)(1 +H;y)〉= E1E2(1− )E1 + E2 (20)

3.2. O(�) homogenization

u2 is determined from the O(1) perturbation equation (18). Substituting (12) and (19) into(18) gives

{E(y)(u2;y +U1; x +Hu0; xx)};y= {((y)− 1)E0}u0; xx (21)

where

(y)=�(y)=�0 (22)

Linearity suggests that u2 may be sought in the form

u2(x; y; �; �)=U2(x; �; �) +H (y)U1; x + P(y)u0; xx (23)

Substituting the above expression into (21) yields

{E(y)(H + P;y)};y=((y)− 1)E0 (24)

The boundary conditions for the above equation are: periodicity and continuity of u2 and �1as well as the normalization condition 〈P(y)〉=0. P(y) can be uniquely determined by solvingthe unit cell boundary value problem de�ned by (24). For details we refer to Reference [8].Once P(y) is found, we can calculate

〈�H 〉=0; 〈E(H + P;y)〉=0 (25)

Next we consider the O(�) equilibrium equation

�(y)u1; �� − {E(y)(u1; x + u2;y)}; x − {E(y)(u2; x + u3;y)};y=0 (26)



Applying the averaging operator to (26) and exploiting the periodicity of �2, yields themacroscopic equation of motion at O(�).

�0U1; �� − E0U1; xx=0 (27)

3.3. O(�2) homogenization

u3 is determined from O(�) perturbation equation (26). Substituting (12), (23) into (26) andmaking use of the macroscopic equations of motion (19) and (27), we have

{E(y)(u3;y +U2; x +HU1; xx + Pu0; xxx)};y= {(y)E0H − E(y)(H + P;y)}u0; xxx + {((y)− 1)E0}U1; xx (28)

Owing to linearity of (28), the general solution to u3 is as follows:

u3(x; y; �; �)=U3(x; �; �) +H (y)U2; x + P(y)U1; xx +Q(y)u0; xxx (29)

Substituting (29) into (28) gives

{E(y)(P +Q;y)};y= (y)E0H − E(y)(H + P;y) (30)

The above equation, together with periodicity and continuity of u3 and �2 as well as thenormalization condition 〈Q(y)〉=0, fully determine Q(y).Consider the equilibrium equation of O(�2):

�(y)(u2; �� + 2u0; ��)− {E(y)(u2; x + u3;y)}; x − {E(y)(u3; x + u4;y)};y=0 (31)

Applying the averaging operator to (31) and taking into account periodicity of �3 yields

〈�u2; ��〉+ 2�0u0; �� − {〈E(y)(u2; x + u3;y)〉}; x=0 (32)

Substituting (23) and (29) into (32) yields

�0U2; �� − E0U2; xx= {〈E(y)(P +Q;y)〉 − 〈�P〉E0=�0}u0; xxxx − 2�0u0; �� (33)

Inserting the expressions for P(y) and Q(y) into (33) and averaging over the unit celldomain yields the macroscopic equation of motion of second order:

�0U2; �� − E0U2; xx= 1�2Edu0; xxxx − 2�0u0; �� (34)

where

Ed =[(1− )]2(E1�1 − E2�2)2E0l212�20[(1− )E1 + E2]2

(35)

where Ed characterizes the e�ect of the microstructure on the macroscopic behaviour. It isproportional to the square of the dimension of the unit cell l. Note that for the homogeneous



material, =0 or 1, and in the case where the impedances of two materials are identicalr= z2=z1 = 1 (z=

√E�), Ed vanishes.

Remark 1It can be easily shown that in the case of equal mass density, �1 =�2, the following identityholds:

Ed = 〈E(y)(P +Q;y)〉= 〈H (y)H (y)〉E0 (36)

where Ed

Ed = �2Ed (37)

The signi�cance of (36) is that Ed can be evaluated by averaging over the leading-orderunit cell solution without consideration of the higher-order boundary value problem in theunit cell. This feature will greatly simplify numerical computations in the multi-dimensionalcase [14].

4. NON-LOCAL MODELS

The macroscopic equations of motion are stated in (19), (27) and (34). The initial andboundary conditions for the above equations of motion are prescribed as

ICs: u0(x; 0; 0) = f(x); u0;t(x; 0; 0) = g(x)

Ui(x; 0; 0) = 0; Ui ;t(x; 0; 0) = 0 (i=1; 2)(38)

BCs: u0(0; �; �) = 0; u0; x(L; �; �) =q(t)E0(A)

Ui(0; �; �) = 0; Ui; x(L; �; �) = 0 (i=1; 2)

(39)

The above initial-boundary value problem is well-posed. From the equation of motion(27) and the initial and boundary conditions (38) and (39), we can readily deduce thatU1(x; �; �)≡ 0.In Reference [9] we have shown that the dispersive solution can be obtained by solving

the leading-order macroscopic equation of motion (19) subjected to the secularity constraint

Ed�2u0; xxxx − 2�0u0; ��=0 (40)

The above formulation necessitates simultaneous solution of two sets of partial di�eren-tial equations with two temporal co-ordinates. In this section, we will derive an alternativeformulation in an attempt to eliminate the dependence on the slow time co-ordinate.We start by de�ning the mean displacement U (x; t) as

U (x; t)= 〈u(x; y; �; �)〉= u0(x; �; �) + �U1(x; �; �) + �2U2(x; �; �) + · · · (41)



Adding (19), (27) and (34), and neglecting terms of order O(�3) and higher, we obtain anequation of motion for the mean displacement

�0 �U − E0U;xx − EdU;xxxx=0 (42)

where �U =U;tt denotes the second total time derivative of the mean displacement. Equa-tion (42) is fourth order in space. It necessitates four boundary conditions to de�ne a well-posed boundary value problem. However, for the problem under consideration, there are onlytwo physically meaningful boundary conditions for the mean displacement. Mathematically,similar equation to (42) arises in �uid dynamics of shallow water theory and crystal–latticetheory, and is often known as a ‘bad’ Boussinesq equation (cf. References [11; 12]).To study the characteristics of the so-called ‘bad’ Boussinesq equation (42), we consider

a model problem with an initial disturbance in the displacement �eld. One end of the rod is�xed and the other is load free. We add two arti�cial boundary conditions

U;xx(0; t)=0; U; xxx(L; t)=0 (43)

The solution to the above initial-boundary value problem can be obtained in closed form

U (x; t)=∞∑n=1Bn sin

pnxLcos

{[√1− Ed

E0L2(pn)2

]pnct=L

}(44)

where

c=√E0=�0; pn=(2n− 1)�2 (45)

Bn=2L

∫ L

0f(x) sin

pnxLdx (n=1; 2; 3; : : :) (46)

Solution (44) is coined as ‘bad’ to re�ect the fact that the phase velocity is imaginary forhigher frequency excitations.In comparison, the exact solution of macroscopic equation of motion (34) subjected to

secularity constraint (40) is given as [9]

ud(x; t)=∞∑n=1Bn sin

pnxLcos{(1− Ed

2E0L2(pn)2

)pnct=L

}(47)

The Taylor’s series expansion of square root term in (44) is given as√1− Ed

E0L2(pn)2 =1 +

(− Ed2E0L2

(pn)2)− 18

(EdE0L2

(pn)2)2

− · · · (48)

Note that Ed=E0L2 =O(l2=L2) and thus if pn=O{(L=l)�} with �¡1; then the higher-orderterms in (48) can be neglected and the gradient approximation (42) coincides with the exactsolution of (34) and (40). Moreover, if the excitation function is of low frequency only,for which Ed=E0L2(pn)2 is of order one, then the two formulations will also provide similarresults.



Elimination of high frequency excitation modes can also be accomplished by using a su�-ciently coarse spatial discretization (or coarse �nite element mesh) for numerical approxima-tion of the non-local or gradient initial-boundary value problem.

Remark 2It is appropriate to note that the gradient model (42) contains the information on the charac-teristic length. This can be seen by comparing the classical non-local continuum theory (seefor example Reference [15]) with the current model. Assuming a constant weight function,the spatial derivative of the non-local strain, �, [15] is given by

ddx��=1�

∫ �=2

−�=2u; xx(x + s) ds= u; xx +

�2

24u; xxxx + · · · (49)

where � is a characteristic length. Matching the coe�cients of the fourth-order term in (49)and (42) yields Ed = �2E0=24. For a volume fraction equal to 0.5 the relation between thecharacteristic size and the unit cell dimension is

�l=

√2|(�2E2=�1E1)− 1|

(1 + (�2=�1))(1 + (E2=E1))(50)

An upper bound of this ratio (obtained by maximizing the ratio with respect to �2=�1and E2=E1) is equal to

√2. It can easily be shown that when the weight function is

(�=2�) cos(s�=�), the upper bound of the ratio �=l is approximately 30 per cent higher. Forexample, in concrete it has been observed [16] that the characteristic length is approximatelythree aggregates. Assuming that a typical volume fraction is approximately 0.5, the charac-teristic size approximately contains 1.5 unit cells. This agrees well with our model whichpredicts the characteristic length to be in the range of 1.4–1.8 unit cells depending on thechoice of weight functions.To this end we propose an alternative non-local approach by which the second-order spa-

tial derivative (42) is approximated by a second-order temporal derivative by exploiting thefollowing relation (for details see Reference [14]):

U;xx=�0E0

�U +O(�3) (51)

which yields

�0 �U − E0U;xx − �0Ek �U;xx=0 (52)

where

Ek =EdE0=[(1− )]2(E1�1 − E2�2)2l212�20[(1− )E1 + E2]2

(53)

Note that Equation (52) is second order in space, so two boundary conditions are su�cient.Since the highest spatial derivative appearing in this equation is second order, the need forC1 continuity is eliminated.



Equation (52) can be solved analytically for the aforementioned model problem by separa-tion of variables, which yields

U (x; t)=∞∑n=1Bn sin

pnxLcos

(pnct=L√

1 + (Ed=E0L2)(pn)2

)(54)

It can be readily observed that solution (54) is well-behaved and convergent with increaseof terms in the Fourier series. Comparison to the source problem in a heterogeneous mediumwill be conducted in Section 6.

5. NUMERICAL PROCEDURES

In this section, we focus on the �nite element implementation of the ‘good’ Boussinesqequation (52). Based on the usual weighted-residual approach, the weak statement of theproblem is as follows: For t ∈ (0; T0], �nd U (x; t)∈H 1

0 (0; L), such that∫ L

0�0Aw �U dx +

∫ L

0E0Aw;xU;x dx +

∫ L

0�0EkAw;x �U;x dx=E0AU;xw(L) + �0EkA �U;xw(L)

(55)

and

U (x; 0)=f(x); U (x; 0)= g(x) (56)

for all admissible test functions w(x)∈H 10 (0; L), where

H 10 (0; L) = {w(x)∈H 1(0; L)|w(0)=0} (57)

H 1(0; L) = {q= q(x); x∈ (0; L)|q; q; x ∈L2(0; L)} (58)

with L2(0; L) denoting the set of square-integrable functions over (0; L).Introducing the �nite element approximation into the above weak form leads to the semi-

discrete equations of motion

M �d+Kd=F (59)

where d(t) is the vector of nodal values of displacements and F is the load vector; M andK are the mass and sti�ness matrices, respectively:

M=Ne∑e=1me; K=

Ne∑e=1ke (60)

me =∫Le�0ANTN dx +

∫Le�0EkANT; xN; x dx; ke=

∫LeE0ANT; xN; x dx (61)

F={q(t) +

�0EkE0

�q(t)}x=L

(62)



where me, ke and N are element mass matrix, sti�ness matrix and shape function, respectively.Equation (59) can be integrated in the time domain by using standard time integration schemes.

Remark 3For (62) to be valid the load q(t) must be twice di�erentiable in time or approximated assuch. Based on (35) it can be easily shown that

�0EkE0

6�T 2

12(63)

where �T is the time required for the wave to travel through the unit cell. In other words,for the second term in (62) to be comparable in magnitude to the �rst term the load periodshould be � �T=

√12. In the present manuscript attention is restricted to load frequencies which

are signi�cantly larger than T and thus the second term in (62) is neglected.

6. NUMERICAL RESULTS AND DISCUSSION

To validate the proposed non-local model, we consider two problems with di�erent loadings.For each problem, a reference solution of the source problem for the corresponding hetero-geneous solid is constructed by utilizing a �ne mesh with element size su�ciently smallcompared to the size of the microconstituents.

Problem 1Macro-domain L=40m; unit cell l=0:2m composed of two material constituents with E1 =120GPa, E2 = 6GPa, �1 = 8000 kg=m3, �2 = 3000 kg=m3; volume fraction =0:5. At x=20man initial disturbance f(x) in displacement �eld f(x)=f0a0[x− (x0− )]2[x− (x0 + )]2{1−h[x − (x0 + )]} · {1 − h(x0 − − x)} is applied; a0 = 1= 4 and h(x) denote the Heavisidestep function; f0, x0 and are the magnitude, the location of the maximum value and thehalf-width of the initial pulse, respectively. The pulse is similar in shape to the Gaussiandistribution function.The calculated homogenized material properties are: E0 = 11:43GPa, �0 = 5500 kg=m3 and

Ed = 1:76× 107 N. In this case, E1=E2 = 20 and the ratio of impedance of the two materialconstituents is r=7:30. The magnitude of the initial pulse is f0 = 1:0m.We plot the time-varying displacement at x=30m in Figures 2 and 3, for the cases of

=1:4 and 0:6m, respectively. The corresponding ratios between the pulse width and the unitcell dimension 2 =l are 14 and 6, respectively. In each of the �gures, we show three timewindows and compare the reference solution of the heterogeneous solid, the �nite elementsolution predicted by the classical homogenization U0, and the analytical and the �nite elementsolutions of the ‘good’ Boussinesq problem.

Problem 2The size of the domain and that of the unit cell are L=1 m and l=0:02 m, respectively.The material properties of the two constituents are E1 = 200 GPa, E2 = 5 GPa, �1 =�2 =8000 kg=m3, and =0:5. The calculated homogenized material properties are: E0 = 9:76 GPa,�0 = 8000 kg=m

3 and Ed = 7:36× 104 N. In this case, E1=E2 = 40.



Figure 2. The response with an initial disturbance in displacement ( =1:4m).

The bar is subjected to an impact load q(t)= q0a0t4(t−T )4[1− h(t−T )] at l=1m, whereT is the duration of the impact pulse, q0 =−50 KN and a0 is scaled in such a way that06q(t)6q0. The time-varying displacements at x=0:5 m are plotted in Figures 4 and 5,which correspond to pulse duration T =62:83 and 15:71 �s, respectively. In each of the two�gures, there are three responses corresponding to the reference solution, the �nite elementsolution predicted by the classical homogenization U0 and the �nite element solution of the‘good’ Boussinesq equation.



Figure 3. The domain with an initial disturbance in displacement ( =0:6m).

The phenomenon of dispersion can be clearly observed in Figures 2–5. When the widthof the initial disturbance in displacement is much larger than the unit cell size or the impactpulse duration is comparatively long, which corresponds to the cases in Figures 2 and 4; thepulse almost maintains its initial shape except for some small wiggles at the wavefront. Inthis case, the classical (leading-order) homogenization provides a crude approximation to theresponse of the heterogeneous solid. However, when the pulse width of the initial disturbanceis comparable to the dimension of the unit cell or the impact pulse duration is very short,



Figure 4. The response with an impact load (pulse duration T =62:83 �s).

which are the cases in Figures 3 and 5, the wave becomes strongly dispersive and the classicalhomogenization errs badly. On the other hand, the non-local dispersive model proposed inthis paper provides a good approximation to the response of heterogeneous solid.



Figure 5. The domain with an impact load (pulse duration T =15:71 �s).

ACKNOWLEDGEMENTS

This work was supported by the Sandia National Laboratories under Contract DE-AL04-94AL8500,the O�ce of Naval Research through grant number N00014-97-1-0687, and the Japan Society for thePromotion of Science under contract number Heisei 11-nendo 06542.



REFERENCES

1. Sanchez-Palencia E. Non-homogeneous Media and Vibration Theory. Springer: Berlin, 1980.2. Benssousan A, Lions JL, Papanicoulau G. Asymptotic Analysis for Periodic Structures. North-Holland:Amsterdam, 1978.

3. Bakhvalov NS, Panasenko GP. Homogenization: Averaging Processes in Periodic Media. Kluwer: Dordrecht,1989.

4. Gambin B, Kroner E. High order terms in the homogenized stress–strain relation of periodic elastic media.Physica Status Solidi 1989; 51:513–519.

5. Boutin C. Microstructural e�ects in elastic composites. International Journal of Solids and Structures 1996;33(7):1023–1051.

6. Boutin C, Auriault JL. Rayleigh scattering in elastic composite materials. International Journal of EngineeringScience 1993; 31(12):1669–1689.

7. Kevorkian J, Bosley DL. Multiple-scale homogenization for weakly nonlinear conservation laws with rapidspatial �uctuations. Studies in Applied Mathematics 1998; 101:127–183.

8. Fish J, Chen W. Higher-order homogenization of initial=boundary-value problem. ASCE Journal of EngineeringMechanics 2001; 127:1223–1230.

9. Chen W, Fish J. A dispersive model for wave propagation in periodic heterogeneous media based onhomogenization with multiple spatial and temporal scales. ASME Journal of Applied Mechanics 2001; 68:153–161.

10. Lasry D, Belytschko T. Localization limiters in transient problems. International Journal of Solids andStructures 1988; 24(6):581–597.

11. Maugin GA. Physical and mathematical models of nonlinear waves in solids. In Nonlinear Waves in Solids,Je�rey A, Engelbrecht J (eds). Springer: Wien-New York, 1994.

12. Whitham GB. Linear and Nonlinear Waves. Wiley: New York, 1974.13. Boutin C, Auriault JL. Dynamic behavior of porous media saturated by a viscoelastic �uid. Application to

bituminous concretes. International Journal of Engineering Science 1990; 28(11):1157–1181.14. Fish J, Chen W, Nagai G. Non-local dispersive model for wave propagation in heterogeneous media: multi-

dimensional case. International Journal for Numerical Methods in Engineering 2002; 54:347–363.15. Bazant Z, Belytschko TB, Chang TP. Continuum theory for strain-softening. Journal of Engineering Mechanics

1984; 110(12):1666–1692.16. Pijaudier-Cabot G, Bazant ZP. Nonlocal damage theory. Journal of Engineering Mechanics 1987; 113(10):

1512–1533.17. Santosa F, Symes WW. A dispersive e�ective medium for wave propagation in periodic composites. SIAM

Journal of Applied Mathematics 1991; 51(4):984–1005.18. Francfort GA. Homogenization and linear thermoelasticity. SIAM Journal on Mathematical Analysis 1983;

14(4):696–708.19. Mei CC, Auriault JL, Ng CO. Some applications of the homogenization theory. In Advances in Applied

Mechanics, Hutchinson JW, Wu TY (eds), vol. 32. Academic Press: Boston, 1996; 277–348.20. Carey GF, Oden JT. Finite Elements; A Second Course, vol. II. Prentice-Hall: NJ, 1983.21. Carey GF, Oden JT. Finite Elements; Mathematical Aspects, vol. IV. Prentice-Hall: NJ, 1983.


non-local dispersive model for wave propagation in heterogeneous media: one-dimensional case

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