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

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2002; 54:347–363 (DOI: 10.1002/nme.424)

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

Jacob Fish∗;†, Wen Chen and Gakuji Nagai

Department of Civil Engineering and Scienti�c Computation Research Center;Rensselaer Polytechnic Institute; Troy; NY 12180; U.S.A.

SUMMARY

Three non-dispersive models in multi-dimensions have been developed. The �rst model consists of aleading-order homogenized equation of motion subjected to the secularity constraints imposing uniformvalidity of asymptotic expansions. The second, non-local model, contains a fourth-order spatial derivativeand thus requires C1 continuous �nite element formulation. The third model, which is limited to theconstant mass density and a macroscopically orthotropic heterogeneous medium, requires C 0 continuityonly and its �nite element formulation is almost identical to the classical local approach with theexception of the mass matrix. The modi�ed mass matrix consists of the classical mass matrix (lumpedor consistent) perturbed with a sti�ness matrix whose constitutive matrix depends on the unit cellsolution. Numerical results are presented to validate the present formulations. Copyright ? 2002 JohnWiley & Sons, Ltd.

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

1. INTRODUCTION

The primary objective of this manuscript is to extend the one-dimensional non-local modeldeveloped in Reference [1] to multi-dimensions. We start by developing a mathematicalhomogenization theory up to the second order with multiple spatial and temporal scales. Avariant of the dispersive model consisting of the leading-order homogenized equations of mo-tion subjected to the secularity conditions imposing uniformly valid asymptotic expansion isformulated �rst. A non-local dispersive model is developed by adding together three sets ofhomogenized equations of motion. The resulting equation is independent of slow time scales,but contains fourth-order spatial derivative and thus requires C1 continuous �nite element

∗Correspondence to: Jacob Fish, Department of Civil Engineering and Scienti�c Computation Research Center,Rensselaer Polytechnic Institute, 110 8th Street, Troy, 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

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

formulation. For the case of constant mass density and macroscopically orthotropic hetero-geneous medium the fourth-order spatial derivative term can be approximated by a mixedsecond-order derivative in space and time. The coe�cients of the mixed derivative term canbe constructed from the solution of the unit cell boundary value problem. Finite element for-mulation of this model is almost identical to the classical zero-order homogenization theorywith an exception of the mass matrix. The modi�ed mass matrix consists of the classical massmatrix (lumped or consistent) perturbed by a sti�ness matrix term whose constitutive matrixcoe�cients depend on the unit cell solution.

2. NON-LOCAL MODEL

We consider waves propagating in elastic heterogeneous solid with a periodic microstructure.The problem of elastodynamics on the scale of material heterogeneity can be stated as follows:

� �ui − �ij; j=0; �ij =Cijklekl; eij = 12(ui; j + uj; i) on �

BCs: ui= gi on �u; �ijnj= hi on ��

ICs: ui(x; t=0)=fi(x); u̇i(x; t=0)= gi(x) on �

(1)

where � denotes the macroscopic domain of interest with boundary �; �u and �� are bound-ary portions where displacements gi and tractions hi are prescribed, respectively, such that�u ∩��= ∅ and �=�u ∪��; ni denotes the normal vector on �; ui is the displacement vec-tor, eij the small strain tensor, �ij the stress tensor, Cijkl the elasticity tensor and � the massdensity. The elasticity tensor and the mass density are locally periodic. We assume that micro-constituents possess homogeneous properties. The superposed dot denotes di�erentiation withrespect to time, such that u̇i; �ui are velocity and acceleration vectors, respectively. The commafollowed by a subscript variable denotes the partial derivative. Summation convention overrepeated subscripts is adopted, except for subscripts x and y (Bold face letters denote eithervector or tensor quantities).

2.1. Asymptotic analysis with multiple spatial-temporal scales

As usual in homogenization methods we assume the characteristic size of the macroscopicproblem L to be much larger than the dimension of the heterogeneities l, i.e. �= l=L�1.The existence of two distinct scales introduces two spatial variables x and y with y=x=�. Inaddition to two spatial scales and the usual time scale denoted as t0 = t, we introduce twoslow time scales

t1 = �t; t2 = �2t (2)

to capture the long-term behaviour of the homogenized solution and to resolve the problem ofsecularity [2; 3]. Using the chain rule, the spatial and temporal derivatives can be expressed as

( ); i=( ); xi + �−1( ); yi ; ˙( )= ( ); t0 + �( ); t1 + �

2( ); t2 (3)

The strain and stress tensors then take the following form:

eij(u)= exij(u) + �−1eyij(u); �ij =Cijkl[exkl(u) + �−1eykl(u)] (4)

Copyright ? 2002 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 54:347–363

NON-LOCAL DISPERSIVE MODEL FOR WAVE PROPAGATION 349

where ex and ey are symmetric gradients with respect to the variables x and y, respectively:

exij(u)= u(i; xj) =12 (ui; xj + uj; xi); eyij(u)= u(i; yj) =

12 (ui; yj + uj;yi) (5)

The equation of motion becomes

�(@@t0+ �

@@t1+ �2

@@t2

)(@ui@t0

+ �@ui@t1

+ �2@ui@t2

)− [�ij; xj + �−1�ij; yj ]= 0 (6)

Since the displacement �eld ui depends on x; y; t0; t1 and t2, a multiple-scale asymptoticexpansion is employed to approximate the solution

ui(x; y; t)= u0i (x; y; t0; t1; t2) + �u1i (x; y; t0; t1; t2) + �

2u2i (x; y; t0; t1; t2) + · · · (7)

Substituting the second equation in (4) into (6) yields

�[ui; t0t0 + 2�ui; t0t1 + �2(2ui; t0t2 + ui; t1t1) + 2�

3ui; t1t2 + �4ui; t2t2 ]

= �−2L−2(ui) + �−1L−1(ui) + L0(ui) (8)

where

L−2(ui) = [Cijkleykl(u)]; yj

L−1(ui) = [Cijkleykl(u)]; xj + [Cijklexkl(u)]; yj

L0(ui) = [Cijklexkl(u)]; xj (9)

The stress expansion can be obtained by substituting (7) into (4)

�ij = �−1�−1ij + �0ij + ��1ij + �

2�2ij + · · · (10)

where

�−1ij =Cijkleykl(u0); �sij =Cijkl[exkl(us) + eykl(us+1)]; s=0; 1; 2; : : : (11)

Inserting the asymptotic expansion (7) into (8) yields the following equations of motionfor various orders:

O(�−2) : L−2(u0i )=0 (12)

O(�−1) : L−2(u1i ) + L−1(u0i )=0 (13)

O(1) : �u0i; t0t0 =L−2(u2i ) + L

−1(u1i ) + L0(u0i ) (14)

O(�) : �(u1i; t0t0 + 2u0i; t0t1)=L

−2(u3i ) + L−1(u2i ) + L

0(u1i ) (15)

O(�2) : �(u2i; t0t0 + 2u1i; t0t1 + 2u

0i; t0t2 + u

0i; t1t1)=L

−2(u4i ) + L−1(u3i ) + L

0(u2i ) (16)



where based on (9) and (11), the following holds:

L−2(u0i )=�−1ij; yj ; L−2(u1i ) + L

−1(u0i )=�0ij; yj + �

−1ij; xj

L−2(us+2i ) + L−1(us+1i ) + L0(usi )=�s+1ij; yj + �

sij; xj ; s=0; 1; 2

(17)

2.2. Resolution of problems at di�erent orders

Consider the O(�−2) equilibrium equation (12) �rst. Permultiplying it by u0i , integrating overthe unit cell domain Y and subsequently integrating by parts yields

∫@Yu0i �

−1ij nj ds−

∫Yu0(i; yj)Cijklu

0(k; yl) dY =0 (18)

The boundary integral term in (18) vanishes due to Y -periodicity on the unit cell boundary@Y . Furthermore, since Cijkl is a positive de�nite fourth-order tensor, we have

u0(i; yj) = 0 ⇒ u0i = u0i (x; t0; t1; t2) and �−1ij =0 (19)

We proceed to the O(�−1) equilibrium Equation (13). From (11), (17) and (19) follows:

�0ij; yj + �−1ij; xj =[Cijkl(eykl(u

1) + exkl(u0))]; yj =0 (20)

As a consequence of linearity, the general solution to u1 takes the following form:

uli (x; y; t0; t1; t2)=U1i (x; t0; t1; t2) +Hikl(y)exkl(u

0) (21)

where Hikl is a Y -periodic third-rank tensor, which is symmetric with respect to indices kand l. Substituting (21) into (20) yields

[Cijkl(Gklmn + �mk�nl)]; yj exmn(u0)=0 (22)

where

Gklmn(y)= 12(Hkmn; yl +Hlmn; yk )=H(k; yl)mn (23)

�km is the Kronecker delta. Since (22) is valid for an arbitrary combination of macroscopicstrain �eld exmn(u0), we get the governing equation over the unit cell domain

[Cijkl(Gklmn + �km�ln)]; yj =0 (24)

The solution Hkmn is sought in the space W de�ned by

W = {w|w Y -periodic; 〈w〉=0} (25)

where

〈·〉= |Y |−1∫Y· dY (26)



is the averaging operator. Thus, the unit cell boundary value problem can be stated as

C 0ijmn; yj = 0; C 0ijmn(y)=Cijkl(Gklmn + �km�ln); 〈Hkmn(y)〉=0 (27)

For complex microstructures the �nite element method is employed for discretization ofHikl(y), which yields a set of linear algebraic equations with six right-hand sides in 3D [4].

2.2.1. O(1) Homogenization. Based on (14) and (17), the O(1)perturbation equation canbe written as

�u0i; t0t0 =�1ij; yj + �

0ij; xj (28)

Applying the averaging operator de�ned in (26) to (28) and taking into account Y -periodicityof �1ij gives

〈�〉u0i; t0t0 − 〈�0ij; xj〉=0 (29)

From (11) and (21), we have

�0ij =C0ijmn(y)exmn(u

0) (30)

Inserting (30) into (29) yields the O(1) macroscopic equation of motion:

�0u0i; t0t0 −D0ijmn(exmn(u0)); xj =0 (31)

where

�0 = 〈�〉; D0ijmn= 〈C 0ijmn(y)〉 (32)

The O(1) macroscopic equation of motion is non-dispersive. In order to capture dispersione�ects, higher-order terms will be considered in the subsequent sections.

2.2.2. O(�) Homogenization. Combining (11), (21) and (31), yields

{Cijkl[eykl(u2) + exkl(U1) +Hkmn(exmn(u0)); xl]}; yj=[(�(y)D0ijmn − C 0ijmn(y))exmn(u0)]; xj (33)

where

�(y)=�(y)=�0 (34)

Linearity suggests that u2i may be sought in the form

u2i (x; y; t0; t1; t2)=U2i (x; t0; t1; t2) +Hikl(y)exkl(U

1) + Pijmn(y)(exmn(u0)); xj (35)

where Pijmn(y) is a Y -periodic fourth-rank tensor. Substituting (35) into (33) yields

[Cijkl(Bklpmn +Hkmn�lp)]; yj(exmn(u0)); xp =[�(y)D

0ipmn − C 0ipmn(y)](exmn(u0)); xp (36)



where

Bklpmn(y)= 12(Pkpmn; yl + Plpmn; yk )=P(k; yl)pmn (37)

Equation (36) is valid for any combination of macroscopic strain gradients (exmn(u0)); xp .This yields the following unit cell boundary value problem:

C1ijpmn; yj = �(y)D0ipmn − C 0ipmn(y); C1ijpmn(y)=Cijkl(Bklpmn +Hkmn�lp); 〈Pijmn(y)〉=0 (38)

from which Pijmn(y) can be determined. The third equation in (38) is the normalization con-dition similar to (27) for Hkmn. Based on (17) the O(�) perturbation equation (15) can bewritten as

�(u1i; t0t0 + 2u0i; t0t1)=�

2ij; yj + �

1ij; xj (39)

Substituting (21) into (39) and applying the averaging operator as well as taking intoaccount Y -periodicity of �2ij , yields

�0U 1i; t0t0 + 〈�(y)Hikl(y)〉(exkl(u0)); t0t0 + 2�0u0i; t0t1 = 〈�1ij; xj〉 (40)

From (11), (21) and (35), we can derive

�1ij =C0ijmn(y)exmn(U

1) + C1ijpmn(y)(exmn(u0)); xp (41)

Inserting (41) into (40) yields the macroscopic equations of motion at O(�):

�0U 1i; t0t0 −D0ijmn(exmn(U1)); xj =D1ijkmn(exmn(u0)); xk xj −〈�(y)Hikl(y)〉(exkl(u0)); t0t0 − 2�0u0i; t0t1

(42)

where

D1ijkmn= 〈C1ijkmn(y)〉 (43)

2.2.3. O(�2) Homogenization. u3i can be determined from O(�) perturbation equation (39).Combining (11), (21), (35), (31) and (42), yields

{Cijkl[eykl(u3) + exkl(U2) +Hkmn(exmn(U1)); xl + Pkrmn(exmn(u0)); xr xl]}; yj

=[�(y)D0ijmn − C 0ijmn(y)](exmn(U1)); xj + {[�(y)D1ijlmn − C1ijlmn(y)]

+ �(y)[Hikl − �−10 〈�Hikl〉]D0kjmn}(exmn(u0)); xj xl (44)

Due to the linearity of (44) u3i can be sought in the form

u3i (x; y; t0; t1; t2) =U3i (x; t0; t1; t2) +Hikl(y)exkl(U

2) + Pijkl(y)(exkl(U1)); xj

+Qijkmn(y)(exmn(u0)); xk xj (45)



Substituting (45) into (44) yields

[Cijkl(Aklprmn + Pkrmn�lp)]; yj(exmn(u0)); xr xp = {[�(y)D1irpmn − C1irpmn(y)]

+ �(y)[Hikp − �−10 〈�Hikp〉]D0krmn}(exmn(u0)); xr xp(46)

where

Aklprmn(y)= 12(Qkprmn; yl +Qlprmn; yk )=Q(k; yl)prmn (47)

Equation (46) is valid for any combination of macroscopic strain gradients (exmn(u0)); xr xp .Thus Qijkmn(y) can be determined from the solution of the boundary value problem on theunit cell domain

C2ijprmn; yj = [�(y)D1irpmn − C1irpmn(y)] + �(y)[Hikp − �−10 〈�Hikp〉]D0krmn

C2ijprmn(y) =Cijkl(Aklprmn + Pkrmn�lp); 〈Qijkmn(y)〉=0 (48)

Where the last equation in (48) is the normalization condition. The O(�2) perturbation equa-tion (16) can be rewritten as

�(u2i; t0t0 + 2u1i; t0t1 + 2u

0i; t0t2 + u

0i; t1t1)=�

3ij; yj + �

2ij; xj (49)

Substituting (21) and (35) into (49) and applying the averaging operator yields

�0U 2i; t0t0 + 〈�Hikl〉(exkl(U1)); t0t0 + 〈�Pijmn〉(exmn(u0)); xj t0t0 + 2�0U 1

i; t0t1

+ 2〈�Hikl〉(exkl(u0)); t0t1 + 2�0u0i; t0t2 + �0u0i; t1t1 = 〈�2ij; xj〉 (50)

From (11), (35) and (45), we have

�2ij =C0ijmn(y)exmn(U

2) + C1ijrmn(y)(exmn(U1)); xr + C

2ijprmn(y)(exmn(u

0)); xr xp (51)

Substituting (51) into (50) gives the O(�2) macroscopic equations of motion

�0U 2i; t0t0 −D0ijmn(exmn(U2)); xj =D

2ijprmn(exmn(u

0)); xr xp xj +D1ijrmn(exmn(U

1)); xr xj

−〈�Hikl〉(exkl(U1)); t0t0 − 〈�Pijmn〉(exmn(u0)); xj t0t0 − 2�0U 1i; t0t1

− 2〈�Hikl〉(exkl(u0)); t0t1 − 2�0u0i; t0t2 − �0u0i; t1t1 (52)

where

D2ijprmn= 〈C2ijprmn(y)〉 (53)



Remark 1. The expressions (38), (43), (48), (53) and (35) for D1; D2 and u2i show that

D1 =O(Cl̂); D2 =O(Cl̂2); 〈�H〉=O(�l̂); 〈�P〉=O(�l̂2) (54)

where l̂= l=� is the unit cell size in the stretched co-ordinate system y. This implies that�D1 =O(Cl); �2D2 =O(Cl2); �〈�H〉=O(�l) and �2〈�P〉=O(�l2) can be directly calculatedfrom known geometric and material properties of micro-constituents, independent of the valueof �. Furthermore, when the mass density is constant within the unit cell, the tensors 〈�H〉and 〈�P〉 vanish and the unit cell boundary value problems can be greatly simpli�ed.

3. DISPERSIVE MODELS

The macroscopic equations of motion are given in (31), (42) and (52). The initial and bound-ary conditions for the above equations are given as

ICs: u0i (x; 0; 0; 0)=fi(x); u̇0i (x; 0; 0; 0)= gi(x)

Usi (x; 0; 0; 0)=0; U̇ s

i (x; 0; 0; 0)=0 s=1; 2(55)

BCs: u0i = gi on �u; [D0ijklexkl(u0)]nj= hi on ��

U si =0 on �u; [D0ijklexkl(U

s)]nj=0 on �� s=1; 2(56)

It has been shown in References [2; 3] that the right-hand-side terms in (42) and (52) giverise to secular asymptotic expansions, i.e. higher-order terms grow unbounded in time. In orderto resolve the problem of secularity, we set the right-hand-side terms to zero. The secularityfree solution can be obtained by solving the leading-order macroscopic equations (31)

�0u0i; t0t0 −D0ijmn(exmn(u0)); xj =0 (57)

subjected to the secularity constraints:

D1ijkmn(exmn(u0)); xk xj − 〈�(y)Hikl(y)〉(exkl(u0)); t0t0 − 2�0u0i; t0t1 = 0 (58)

D2ijprmn(exmn(u0)); xr xp xj − 〈�Pijmn〉(exmn(u0)); xj t0t0

−2〈�Hikl〉(exkl(u0)); t0t1 − 2�0u0i; t0t2 − �0u0i; t1t1 = 0 (59)

Analytical solution of (57)–(59) for one-dimensional problems has been given in Refer-ences [2; 3].

3.1. Non-local equations of motion

In this section we develop an alternative approach by which the three sets of macroscopicequations are combined into a single equation and the dependence on slow time is eliminated.



We start by de�ning the mean displacement as

Ui(x; t)= 〈ui(x; y; t)〉= u0i + �U 1i + �

2U 2i + · · · (60)

Multiplying (42) and (52) by � and �2, respectively, and then adding the resulting equationsto the leading-order macroscopic equations (31) gives

�0(u0i + �U1i + �

2U 2i ); t0t0 −D0ijmn(exmn(u0 + �U1 + �2U2)); xj

= �2D2ijprmn(exmn(u0)); xr xp xj + �D

1ijkmn(exmn(u

0 + �U1)); xk xj

− �〈�Hikl〉(exkl(u0 + �U1)); t0t0 − 2�2〈�Hikl〉(exkl(u0)); t0t1− �2〈�Pijmn〉(exmn(u0)); xj t0t0 − 2��0(u0i + �U 1

i ); t0t1 − �2�0(u0i; t1t1 + 2u0i; t0t2) (61)

Exploiting the relations

u0i + �U1i + �

2U 2i =Ui +O(�

3); �2u0i = �2Ui +O(�3); �(u0i + �U

1i )= �Ui +O(�

3)

Ui; t0t0 + 2�Ui; t0t1 + �2(Ui; t1t1 + 2Ui; t0t2)= �Ui +O(�3)

�(Ui; t0t0 + 2�Ui; t0t1)= � �Ui +O(�3); �2Ui; t0t0 = �

2 �Ui +O(�3)

(62)

and neglecting terms of order O(�3) and higher in (61), we get one set of macroscopicequations of motion with respect to the mean displacement

�0 �Ui −D0ijmn(exmn(U)); xj − �D1ijkmn(exmn(U)); xk xj − �2D2ijprmn(exmn(U)); xr xp xj+ �〈�Hikl〉exkl( �U) + �2〈�Pijmn〉(exmn( �U)); xj =0 (63)

For the case of constant mass density within the unit cell, 〈�Hikl〉= 〈�Pijmn〉=0 and conse-quently Equation (63) can be simpli�ed as

� �Ui −D0ijmn(exmn(U)); xj − �D1ijkmn(exmn(U)); xk xj − �2D2ijprmn(exmn(U)); xr xp xj =0 (64)

Moreover, for the case of macroscopically orthotropic materials D1ijkmn=0 and Equation (64)can be further simpli�ed as

� �Ui −D0ijmn(exmn(U)); xj − �2D2ijprmn(exmn(U)); xr xp xj =0 (65)

In the remainder of this paper, attention is restricted to the approximation and numericalimplementation of (65). This equation contains the sixth-rank tensor, which can be evaluatedby solving the unit cell boundary value problems up to the second order. The highest spa-tial derivatives appearing in (65) is fourth order and therefore C1 continuity is required forthe �nite element implementation. Moreover, the two sets of physically meaningful bound-ary conditions are insu�cient to de�ne a well-posed initial-boundary value problem. Theone-dimensional counterpart of (65) is known as a ‘bad’ Boussinesq equation, which yieldsmeaningless solution for the case of oscillatory loading [1]. To resolve these di�culties we



will attempt to approximate the fourth-order spatial derivative in terms of the mixed second-order spatial–temporal derivative. The resulting approximation will be termed as a ‘Good’Boussinesq problem.

3.2. The ‘Good’ Boussinesq problem

We start by establishing the relation between D2 and D0. From the leading-order unit cellboundary value problem (27), we have the following integral equation over the unit celldomain: ∫

YQiprklC 0

ijmn; yj dY =∫@YQiprklC 0

ijmnnj ds−∫YQiprkl ; yjC

0ijmn dY =0 (66)

The boundary integral vanishes due to periodicity, and Equation (66) becomes∫YQiprkl ; yjCijst(Gstmn + �ms�nt) dY =0 (67)

from which we have ∫YAijprklCijmn dY =−

∫YAijprklCijstGstmn dY (68)

where A is the symmetric gradient of Q. From (48) and (68) we get∫YC2ijprkl dY =−

∫YAmnprklCmnstGstij dY +

∫YCijpqPqrkl dY (69)

Similarly, from the second-order unit cell boundary value problem (48), we have thefollowing integral equation:

∫YHistC2ijprmn; yj dY =

∫YHist{[�(y)D1irpmn − C1irpmn(y)]

+ �(y)[Hikp − �−10 〈�Hikp〉]D0krmn} dY (70)

Integrating the left-hand-side of (70) by parts with consideration of periodicity and insertingthe expression for C2 gives

−∫YAmnprklCmnstGstij dY =

∫Y[HsijC2stprkl ; yt +GstijCstpqPqrkl] dY (71)

Substituting (71) into (69) and (70), we have

∫YC2ijprkl dY =

∫YPqrklC 0

pqij dY +∫YHsij{[�(y)D1srpkl − C1srpkl(y)]

+ �(y)[Hspq − �−10 〈�Hspq〉]D0qrkl} dY (72)



From the �rst-order unit cell boundary value problem (38), we have the following integralequation:

∫YPirklC1ijpmn; yj dY =

∫YPirkl[�(y)D0ipmn − C 0

ipmn(y)] dY (73)

Similarly∫YPirklC 0

ipmn(y) dY =∫Y�(y)PirklD0ipmn dY +

∫YBijrklC1ijpmn dY (74)

Substituting (74) into (72) and using the notation for the averaging operator 〈〉 gives

D2ijprkl = �−10 〈�Pqrkl〉D0pqij + �−10 〈�Hsij〉D1srpkl + 〈BmnrklC1mnpij〉

− 〈HsijC1srpkl〉+ �−10 〈�HsijHspq〉D0qrkl − �−20 〈�Hsij〉〈�Hspq〉D0qrkl (75)

In the case of constant mass density, Equation (75) reduces to

D2ijprkl = 〈BmnrklC1mnpij −HsijC1srpkl〉+ 〈HsijHspq〉D0qrkl (76)

Using least square approximation of the �rst term in (76), 〈BmnrklC1mnpij − HsijC1srpkl〉 can beapproximated by VijpqD0qrkl and thus the relation between D

2 and D0 can be expressed as

D2ijprkl =[Vijpq + 〈HsijHspq〉]D0qrkl (77)

Utilizing approximation (77) yields

�2D2ijprmn(exmn(U)); xr xp xj = �2[Vijpq + 〈HsijHspq〉]D0qrmn(exmn(U)); xr xp xj

= �2[Vijpq + 〈HsijHspq〉](D0qrmnexmn(u0)); xr xp xj +O(�3) (78)

From the leading-order macroscopic equations of motion (31) and the relations in (62), wehave

�2D0qrmn(exmn(u0)); xr = �

2�0u0q; t0t0 = �2�0Uq; t0t0 +O(�

3)= �2�0 �Uq +O(�3) (79)

Inserting (79) into (78) gives

�2D2ijprmn(exmn(U)); xr xp xj = �2�0[Vijpq + 〈HsijHspq〉] �Uq; xp xj +O(�3) (80)

Substituting (80) into (65) and neglecting terms of O(�3) and higher yields the so-called‘Good’ Boussinesq problem in multi-dimensions

�0 �Ui −D0ijkl(exkl(U)); xj − �0Ekijkl(exkl( �U)); xj =0 (81)



where

Ekijpq= �2[Vijpq + 〈HsijHspq〉] (82)

Remark 2. For macroscopically isotropic materials the �rst term in (82) is much smallerthan the second and thus Ek can be approximated in terms of the leading-order unit cell bound-ary value solution and thus eliminating the need for solving higher-order unit cell boundaryvalue problems. It can be seen that Ek introduces the length-scale into the macroscopic equa-tions of motion.

4. FINITE ELEMENT FORMULATION

In this section, we focus on the �nite element semi-discretization of Equation (81). Sincethe highest spatial derivatives appearing in (81) is second order, the usual C0 �nite elementapproximation is su�cient. The weak statement of the problem is formulated as follows. Foreach t ∈ (0; T0], �nd Ui(x; t)∈H 1(�), such that Ui(x; t)= gi on �u and∫

��0wi �Ui d�−

∫�wiD0ijkl(exkl(U)); xj d�−

∫��0wiEkijkl(exkl( �U)); xj d�=0 (83)

Ui(x; 0)=fi(x); U̇i(x; 0)= gi(x) (84)

for all admissible test functions wi(x)∈H 10 (�), where H

1(�) is the Sobolev space de�ned as

H 1(�)= {v(x)∈L2(�); v; xi ∈L2(�)} (85)

with L2(�) denoting the set of square-integrable functions over �, and

H 10 (�)= {w(x)∈H 1(�)|w(x)=0 on �u} (86)

Integrating (83) by parts and accounting for major symmetry of D0 and Ek , we have theweak form

∫��0wi �Ui d� +

∫�exij(w)D0ijklexkl(U) d� +

∫��0exij(w)Ekijklexkl( �U) d�

=∫��wihi ds+

∫��0winjEkijklexkl( �U) ds (87)

Following Remark 3 in Reference [1] the second boundary term in (87) can be neglectedprovided that the wavelengths are signi�cantly larger than the unit cell size. Otherwise itcontributes a non-symmetric term to the mass matrix.Finite element approximation of the above weak form leads to the semi-discrete equations

of motion

M �d+Kd=F (88)



where d(t) is the vector of nodal displacements; M; K and F are the system mass and sti�nessmatrices as well as the load vector, respectively:

M=Ne∑e=1me; K=

Ne∑e=1ke; F=

Ne∑e=1f e (89)

me =∫�e�0NTN d� +

∫�e�0BTEkB d�; ke =

∫�eBTD0B d�; f e =

∫��eNTh ds (90)

where N and B are the shape function and the symmetric gradient of N; me;ke; f e the elementmass matrix, sti�ness matrix, and force vector, respectively; D0 the homogenized elasticitymatrix and Ek the matrix constructed from the elements of Ekijkl .Equation (88) is integrated in the time domain using standard time integration schemes.

5. NUMERICAL RESULTS

To validate the proposed non-local model, two-dimensional and three-dimensional problemsare considered. Numerical examples are restricted to macroscopically isotropic medium withconstant mass density. Problems involving randomly or periodically distributed particles, suchas concrete, dense polycrystals and short �bre composites, fall into this category.

Problem 1To compare the solution of the non-local model to the solutions of the classical homogeniza-tion model and the source heterogeneous problem, a two-dimensional plane strain problem asshown in Figure 1 is considered. The left edge is �xed while the right edge is subjected toimpact load q(t)= a0(t−�=2)t4(t−T )4[1− h(t−T )] · a0 is scaled so that −16q(t)61; T isthe duration of the impact pulse, and h(t) denotes the Heaviside step function. The functionq(t) generates a Gaussian-like shape pulse.The microstructure consists of hexagonally arranged circular �bres embedded in matrix

material. This con�guration is macroscopically isotropic. The volume fraction ratio of �bresis 0.60. The Young’s modulus of �bres and matrix are Ea = 50, and Em =1, respectively; thePoisson’s ratios are �a = �m =0:2, and mass densities are �a =�m =1. Homogenized properties

Figure 1. Two-dimensional plane strain multi-scale problem.



Figure 2. Comparison of responses at the centre point A.

are evaluated by solving the leading-order microscopic boundary value problem, which yields

D0 =

3:64 0:89 0:00

0:89 3:68 0:00

0:00 0:00 1:36

; Ek =

3:11 −0:10 0:00

−0:10 3:09 0:00

0:00 0:00 1:60

× 10−2

In this example, loading period duration is taken as T =7. The source heterogeneous prob-lem is discretized with 1560× 676 bilinear square-shape �nite elements. For the classicalhomogenization and non-local methods, the microstructure is discretized with 60× 104 bi-linear square-shape �nite elements, whereas the macro-problem is discretized with 104× 52bilinear square-shape �nite elements.Figure 2 shows the time-varying displacement u1 at point A (in Figure 1) corresponding

to the centre of macro-domain. Good agreement between the solution of the non-local modeland the reference solution of the source problem can be seen. On the other hand, the solutionof the classical homogenization model shows signi�cant deviation from the reference solution.



Figure 3. Three-dimensional multi-scale problem of concrete.

Figure 4. Comparison of responses at the centre point A (concrete sample).

Problem 2As an example of various three-dimensional problems, an isotropically damaged concretebeam of 450× 150× 150 mm3 shown in Figure 3 is considered. The left face is �xed andthe right face is subjected to the impact load q(t) perpendicular to it. A quarter region is



discretized with 90× 15× 15 trilinear cubic-shape �nite elements due to the symmetry. Theunit cell model is reconstructed from a three-dimensional digital imaging process [5]. Theunit cell consists of 75 mm-cubic region, which is three times larger than the maximum sizeof an aggregate. The unit cell is discretized with 1503 trilinear cubic-shape �nite elements.The volume fraction of aggregates is 0.49; The Young’s modulus of aggregates and damagedmortar are Ea = 55GPa; Edm =2:6GPa, respectively. The Poisson’s ratios �a = 0:15; �m =0:19;and the mass densities �a =�m =2:2× 10−6 kg=mm3. Load period duration time is T =100�s.Figure 4 plots the time-varying displacement u1 and the normal stress �1 as obtained withthe non-local and the classical homogenization model at the centre A in the Figure 3. Plate1 shows the maximum principal stresses in the unit cell at the center A (t=915 �s). Thestresses in the unit cell are approximated up to O(1). No comparison to the reference solutionhas been made as that would involve over 100 million degrees-of-freedom.

6. SUMMARY AND FUTURE RESEARCH DIRECTIONS

Three dispersive models for wave propagation in heterogeneous media have been developedfor one- and multi-dimensional problems. This work is motivated by our recent studies for one-dimensional problems [2; 6] which suggested that in absence of multiple time scaling, higher-order homogenization method gives rise to secular terms which grow unbounded with timeand the problem of secularity can be successively resolved with the introduction of slow timescales. The �rst model consists of a leading-order homogenized equation of motion subjectedto secularity constraints imposing uniform validity of asymptotic expansions. The second, non-local model, contains fourth-order spatial derivative and thus requires C1 continuous �niteelement formulation. The third model, which has been implemented for constant mass densityand macroscopically isotropic heterogeneous medium, requires C0 continuity only and its �niteelement formulation is almost identical to the classical local approach with the exception ofthe mass matrix. The modi�ed mass matrix consists of the classical mass matrix (lumped orconsistent) perturbed with a sti�ness matrix whose constitutive matrix depends on the unitcell solution.Several issues, however, have not been addressed:

1. Only the special case with constant mass density and macroscopically isotropicheterogeneous medium have been implemented and validated. For general macro-scopically anisotropic materials D1 = 0 and thus the issue of C1 continuity has to beresolved.

2. For macroscopically orthotropic materials higher-order unit cell problems have to besolved.

3. The present model requires implicit time integration. Lumping of the additional termin the mass matrix,

∫� �0B

TEkB d�, is identically zero. Therefore, various mass matrixsplitting procedures have to be investigated in the context of explicit methods.

4. For the �nite element formulation of the dispersive model involving secularity con-straints (57)–(59), time integration procedures with multiple time scales have to bedeveloped.

These issues will be investigated in our future work.


Plate 1. Comparison of maximum stresses in the unit cell at t=915 �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.

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

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