Brett A. Bednarcyk

Glenn Research Center, Cleveland, Ohio

Jacob Aboudi

Tel-Aviv University, Ramat-Aviv, Israel

Steven M. Arnold and James A. Pennline 

Glenn Research Center, Cleveland, Ohio

Integrated Structural/Acoustic 

Modeling of Heterogeneous Panels

NASA/TM—2012-217728

October 2012
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Brett A. Bednarcyk

Glenn Research Center, Cleveland, Ohio

Jacob Aboudi

Tel-Aviv University, Ramat-Aviv, Israel

Steven M. Arnold and James A. Pennline 

Glenn Research Center, Cleveland, Ohio

Integrated Structural/Acoustic 

Modeling of Heterogeneous Panels

NASA/TM—2012-217728

October 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

Available electronically at http://www.sti.nasa.gov

This work was sponsored by the Fundamental Aeronautics Program 

at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2012-217728 1 

Integrated Structural/Acoustic Modeling of Heterogeneous Panels

Brett A. Bednarcyk

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135 

Jacob Aboudi 

Tel-Aviv University

Ramat-Aviv 69978, Israel

Steven M. Arnold and James A. Pennline

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135 

Introduction

A continuum model is developed for the analysis and prediction of the dynamic response of 

multiphase panels that are subjected to impulsive loading. The multiphase medium consists of any 

number of different elastic anisotropic constituents including the case of voids. The impulsive loading is 

applied on the external boundaries of panel in the form of time-dependent tractions and/or displacements.

Acoustic loading can thus be modeled as a time-dependent pressure wave. The derived continuum theory 

results in a system of second-order equations in time. This system represents the governing elastodynamic 

equations for the discretized regions of the multiphase medium, the interfacial continuity conditions 

between these regions, and the externally applied boundary conditions. This system is solved by an 

explicit step-by-step procedure in time, and the resulting time-dependent displacement and stress response 

can be recorded at any point within the multiphase medium. In addition, parts of the discretized geometry 

may contain fluids. Thus, the air surrounding the multiphase medium can be simulated, with the acoustic 

loading applied to the air rather than directly on the multiphase panel. This is useful as it more 

realistically simulated the impact of a pressure wave on the surface of the panel, as wave reflection can be 

captured. Various cases of porous absorbers can be modeled by employing this theory. Examples include 

limp absorbers, where the solid ligaments are so thin and compliant that they can be assumed to have 

near-zero stiffness; rigid absorbers, where the ligaments are so stiff that they can be assumed to be 

motionless (with very high stiffness); and poroelastic absorbers, where both the ligaments and the air 

within the considered domain participate in the wave propagation. In the present investigation, the latter 

case is mainly considered. 

Theoretical model development is first presented. Then, results are given for the structural/acoustic 

performance of several foam core sandwich panels. Finally, heterogeneous panels, consisting of a stiff 

phase and a very compliant phase, are examined to determine the effect of the arrangement of the phases 

on the panel acoustic behavior. 

Full Three-Dimensional Theory

In the present section a three-dimensional continuum theory is developed for the analysis of wave 

propagation in multiphase materials. To this end, consider a three-dimensional domain defined by 0 ≤ x1 ≤

D, 0 ≤ x2 ≤ H, and 0 ≤ x3 ≤ L. This region is divided into Nα, Nβ and Nγ subcells in the x1, x2 and x3

directions, respectively. Each subcell (αβγ), whose dimensions are dα, hβ and lγ (α = 1,..., Nα, β = 1,...,

Nβ, γ = 1,... Nγ), can be filled by a distinct anisotropic elastic material (solid or fluid) or is left empty thus 

forming a pore, see Figure 1. 
NASA/TM—2012-217728 2 

Figure 1.—Model geometry.

The derivation of this theory starts from the elastodynamic equations of motion 

3,2,1,,)(

2

2

)()(

, ρ=σ = αβγ αβγ αβγ jiu dt

d

ji j i

(1)

where αβγ )( σ ji , αβγ )(

i u , ρ(αβγ)

 and t denote the stresses, displacements, mass density of the material in subcell 

(αβγ) and time. The stresses αβγ )( σij are related to the strains αβγ )( εij by the anisotropic Hooke's law 

3,2,1,,,,)()()( ε=σ = αβγ αβγαβγ C lkji ij ijkl kl (2)

where αβγ )( Cijkl are the components of the 4th-order stiffness tensor of the material in the subcell.

The displacement field in the subcell (αβγ) is approximated by a second-order expansion in the local 

coordinates )(

1

α x , )(

2

β x and )(

3

γ x , the origins of which are located at the center of the subcells, as follows: 

)(

)002(1

2

2)(

3

)(

)020(1

2

2)(

2

)(

)200(1

2

2)(

1

)(

)001(1

)(

3

)(

)010(1

)(

2

)(

)100(1

)(

1

)(

)000(1

)(

1

4

3

2

1

4

3

2

1

4

3

2

1 α α αβγ β β αβγ γ γ αβγ

αβγ αβγ α αβγ β αβγ γ αβγ

















+ − 















+ − 















+ −

+= + +

W

l

W x

h

W x

d

x

WxWxWxWu

(3)
NASA/TM—2012-217728 3 

)(

)002(2

2

2)(

3

)(

)020(2

2

2)(

2

)(

)200(2

2

2)(

1

)(

)001(2

)(

3

)(

)010(2

)(

2

)(

)100(2

)(

1

)(

)000(2

)(

2

4

3

2

1

4

3

2

1

4

3

2

1 α α αβγ β β αβγ γ γ αβγ

αβγ αβγ α αβγ β αβγ γ αβγ

















+ − 















+ − 















+ −

+= + +

W

l

W x

h

W x

d

x

WxWxWxWu

(4)

)(

)002(3

2

2)(

3

)(

)020(3

2

2)(

2

)(

)200(3

2

2)(

1

)(

)001(3

)(

3

)(

)010(3

)(

2

)(

)100(3

)(

1

)(

)000(3

)(

3

4

3

2

1

4

3

2

1

4

3

2

1 α α αβγ β β αβγ γ γ αβγ

αβγ αβγ α αβγ β αβγ γ αβγ

















+ − 















+ − 















+ −

+= + +

W

l

W x

h

W x

d

x

WxWxWxWu

(5)

where the time-dependent )(

)000(

αβγ Wi , (i = 1,2,3), are the volume-average displacements in the subcell which 

together with the higher-order time-dependent terms )(

)(

αβγ Wi lmn ; (l,m,n) = 1,2; must be determined. Thus, for 

example, )(

)020(

αβγ Wi denotes the coefficient in the expansion of αβγ )( ui the quadratic term associated with the 

2 x -direction. The resulting strain components in subcell (αβγ) are given by 

( )

( )

( ))(

)200(2

)(

1

)(

)100(2

)(

)020(1

)(

2

)(

)010(1

)(

12

)(

)200(3

)(

1

)(

)100(3

)(

)002(1

)(

3

)(

)001(1

)(

13

)(

)020(3

)(

2

)(

)010(3

)(

)002(2

)(

3

)(

)001(2

)(

23

)(

)002(3

)(

3

)(

)001(3

)(

33

)(

)020(2

)(

2

)(

)010(2

)(

22

)(

)200(1

)(

1

)(

)100(1

)(

11

3 3

2

1

3 3

2

1

3 3

2

1

3

3

3

αβγ αβγ β αβγ αβγ α αβγ

αβγ αβγ γ αβγ αβγ α αβγ

αβγ αβγ γ αβγ αβγ β αβγ

αβγ αβγ γ αβγ

αβγ αβγ β αβγ

αβγ αβγ α αβγ

=ε + ++

=ε + ++

=ε + ++

+=ε

+=ε

+=ε

WxWWxW

WxWWxW

WxWWxW

WxW

WxW

WxW

(6)

By averaging Equation (1) over the volume of subcell (αβγ), the following three relations are 

obtained 

3,2,1,)(

)000(

)()(

)000(3

)(

)000(2

)(

)000(1 ρ=++ = αβγ αβγ αβγ αβγ αβγ

i i KJI i i iW (7)

where )(

)000(1

αβγ

i I , )()000(2

αβγ

i J and )()000(3

αβγ K i which were originally introduced by Aboudi (1986, 1987) are given 

in terms of the surface-average of the tractions evaluated at the surfaces 2 )(

1 α

α ±= dy , 2 )(

2 β

β ±= hy and 

2 )(

3 γ

γ ±= ly , respectively. Thus 















 = −

+ αβγ − αβγ

α

αβγ )( )( )1()1(

)(

)000(1

1

i i itt

d

I (8)















 = −

+ αβγ − αβγ

β

αβγ )( )( )2()2(

)(

)000(2

1

i i itt

h

J (9)
NASA/TM—2012-217728 4 















 = −

+ αβγ − αβγ

γ

αβγ )( )( )3()3(

)(

)000(3

1

i i itt

l

K (10)

where the surface-average of the tractions are given by 

)(

3

)(

2

)(

1

)(

1

2

2

2

2

)1(

2

1 )(

αβγ α α γβ

−− γβ

±





 



 = σ ±= ∫∫

γ

γ

β

β

αβγ

xdxd dx

lh

t i

l

l

h

h i (11)

)(

3

)(

1

)(

2

)(

2

2

2

2

2

)2(

2

1 )(

αβγ β β γα

− − γα

±















 = σ ±= ∫∫

γ

γ

α

α

αβγ

xdxd

h

x

ld

t i

l

l

d

d i

(12)

)(

2

)(

1

)(

3

)(

3

2

2

2

2

)3(

2

1 )(

αβγ γ γ βα

−− βα

±















 = σ ±= ∫∫

β

β

α

α

αβγ

xdxd

l

x

hd

t i

h

h

d

d i

(13)

Next, by consecutively multiplying Equation (1) by )(

1

α x , )(

2

β x and )(

3

γ x and averaging the results over the 

volume of the subcell (αβγ) and integrating by parts, the following three set of three relations follow 

3,2,1, 12

)(

)100(

2

)()(

)000(1

)(

100(1 ) ρ=− = αβγ αβγ αβγ α αβγ iW d SI i i i

 (14)

3,2,1, 12

)(

)010(

2

)()(

)000(2

)(

)010(2 ρ=− = αβγ αβγ αβγ β αβγ iW

h

SJ i i i

 (15)

3,2,1, 12

)(

)001(

2

)()(

)000(3

)(

)001(3 ρ=− = αβγ αβγ αβγ γ αβγ iW

l

i SK i i

 (16)

In these equations, the quantities )(

)100(1

αβγ

i I , )(

)010(2

αβγ

i J and )()001(3

αβγ K i can be expressed in terms of the surfaceaverage of tractions as follows 















 = +

+ αβγ − αβγ

αβγ )( )( )1()1(

)(

)100(1 2

1

i i i I tt (17)















 = +

+ αβγ − αβγ

αβγ )( )( )2()2(

)(

)010(2 2

1

i i i J tt (18)















 = +

+ αβγ − αβγ

αβγ )( )( )3()3(

)(

)001(3 2

1

i i i K tt (19)

In addition, )(

)000(

αβγ

ij S denote the volume-average of the stress αβγ )( σij over the subcell namely, 
NASA/TM—2012-217728 5 

)(

3

)(

2

)(

1

)( 2

2

2

2

2

2

)(

)000(

1 αβγ γβα

−−− γβα

αβγ = σ ∫∫∫

γ

γ

β

β

α

α

xdxdxd

lhd

S ij

l

l

h

h

d

d ij (20)

Let the vector )(

)000(

αβγ S be defined by the following six components of )(

)000(

αβγ

ij S : 

[ ] )(

)000(12)000(13)000(23)000(33)000(22)000(11

)(

)000( ,,,,, αβγ αβγ S = SSSSSS (21)

In terms of the reduced matrix representation Cpq; p,q = 1,...,6; of the stiffness tensor Cijkl, it can be easily 

verified by employing the strain components expressions in Equation (6) that its components are given by 

( )

( ) ( ))(

)100(2

)(

)010(1

)(

6

)(

)100(3

)(

)001(1

)(

5

)(

)010(3

)(

)001(2

)(

4

)(

)001(3

)(

3

)(

)010(2

)(

2

)(

)100(1

)(

1

)(

)000(

αβγ αβγ αβγ αβγ αβγ αβγ

αβγ αβγ αβγ αβγ αβγ αβγ αβγ αβγ αβγ αβγ

+ ++ +

= + + + +

WWCWWC

WWCWCWCWCS

p p

p p p p p (22)

Next, by consecutively multiplying Equation (1) by )(2

1

α x , )(2

2

β x and )(2

3

γ x and averaging the result 

over the volume of the subcell (αβγ) and integrating by parts, the following three set of three relations 

follow 

[ ] 3,2,1, 12 10

2

4 12

)(

)200(

2

)(

)000(

2

()( )

)100(1

)(

)000(3

)(

)000(2

2

)(

)000(1

2

 =



 





+ ρ=−+ + α αβγ α αβγ αβγ αβγ αβγ α αβγ α αβγ W i d W d KJ S d I d

i i i i i i   (23)

[ ] 3,2,1, 12 10

2

4 12

)(

)020(

2

)(

)000(

2

)()(

)010(2

)(

)000(3

)(

)000(1

2

)(

)000(2

2

=

















+ ρ=−+ + β αβγ β αβγ αβγ αβγ αβγ β αβγ β αβγ W i

h

W

h

KI S

h

J

h

i i i i i i

 (24)

[ ] 3,2,1, 12 10

2

4 12

)(

)002(

2

)(

)000(

2

)()(

)001(3

)(

)000(2

)(

)000(1

2

)(

)000(3

2

=

















+ ρ=−+ + γ αβγ γ αβγ αβγ αβγ αβγ γ αβγ γ αβγ W i

l

W

l

SJI

l

K

l

i i i i i i

 (25)

In these equations, )(

)(

αβγ

ij lmn S are the volume-average first moments of the stresses which are given by 

( ) ( ) ( ) )(

3

)(

2

)(

1

)()(

3

)(

2

)(

1

2

2

2

2

2

2

)(

)(

1 γβα αβγ γβα

−−− ββα

αβγ = σ ∫∫∫

γ

γ

γ

β

α

α

xxx xdxdxd

hhd

S ij

l nml

l

l

h

d

d ij lmn (26)

It can be easily shown that the relevant components of the vector )(

)(

αβγ

lmn S , l + m + n ≠ 0, that is defined by 

[ ] )(

)(12)(13)(23)(33)(22)(11

)(

)( ,,,,, αβγ αβγ S lmn = lmn lmn lmn lmn lmn SSSSSS lmn

(27)

are given by 

[ ])(

)200(3

)(

5

)(

)200(2

)(

6

)(

)200(1

)(

1

2

)(

)100( 4

αβγ α αβγ αβγ αβγ αβγ αβγ αβγ = + + WCWCWC d Sp p p p

(28)
NASA/TM—2012-217728 6 

[ ])(

)020(3

)(

4

)(

)020(2

)(

2

)(

)020(1

)(

6

2

)(

)010( 4

αβγ β αβγ αβγ αβγ αβγ αβγ αβγ = + + WCWCWC

h

S p p p p

(29)

[ ])(

)002(3

)(

3

)(

)002(2

)(

4

)(

)002(1

)(

5

2

)(

)001( 4

αβγ γ αβγ αβγ αβγ αβγ αβγ αβγ = + + WCWCWC

l

S p p p p

(30)

where p = 1,...,6.

By substituting Equations (7) in (23), (24) and (25) one obtains, respectively, that 

)(

)200(

2

)()(

2 )100(1

)(

)000(1 20

12 αβγ αβγ α αβγ

α

αβγ = ρ+ i i Wi

d S

d

I  (31)

)(

)020(

2

)()(

2 )010(2

)(

)000(2 20

12 αβγ αβγ β αβγ

β

αβγ

i = i ρ+ Wi

h

S

h

J  (32)

)(

)002(

2

)()(

2 )001(3

)(

)000(3 20

12 αβγ αβγ γ αβγ

γ

αβγ

i = i ρ+ Wi

l

S

l

K  (33)

Hence substitution of Equations (31) to (33) in (7) yields 

















+ + ρ= − − − αβγ αβγ αβγ α αβγ β αβγ γ αβγ

γ

αβγ

β

αβγ

α

)(

)002(

2

)(

)020(

2

)(

)200(

2

)(

)000(

)()(

2 )001(3

)(

2 )010(2

)(

2 )100(1 20 20 20

12 12 12

i i i i i i Wi

l

W

h

W d S W

l

S

h

S

d

  (34)

Just like the surface-average of tractions: 

)( )1( ± αβγ

it , )( )2( ± αβγ

it and 

)( )3( ± αβγ

it which were defined by Equations 

(11) to (13), let us define the surface-average of displacements: 

)( )1( ± αβγ

ui , )( )2( ± αβγui and 

)( )3( ± αβγ

ui as follows 

)(

3

)(

2

)(

1

)( 2

2

2

2

)1(

2

1 )(

αβγ α α γβ

−− γβ

±





 



 = ±= ∫∫

γ

γ

β

β

αβγ

xdxd dxu

lh

u i

l

l

h

h i

(35)

)(

3

)(

1

)(

2

)( 2

2

2

2

)2(

2

1 )(

αβγ β β γα

−− γα

±















 = ±= ∫∫

γ

γ

α

α

αβγ

xdxd

h

xu

ld

u i

l

l

d

d i (36)

)(

2

)(

1

)(

3

)( 2

2

2

2

)3(

2

1 )(

αβγ γ γ βα

−− βα

±















 = ±= ∫∫

β

β

α

α

αβγ

xdxd

l

xu

hd

u i

h

h

d

d i

(37)

In the following, these surface-average displacements will be related to the microvariables )(

)(

αβγ Wi lmn ; 

(lmn) = 0,1,2; in the displacements expansions. To this end, by substituting the displacements expansion 

(3) to (5) in (35) to (37), the following relations are obtained 
NASA/TM—2012-217728 7 

)(

)200(

2

)(

)100(

)(

)000(

)1(

2 4

)(

αβγ α αβγ α αβγ

±

±= +

αβγ

i i i Wi

d W d Wu (38)

)(

)020(

2

)(

)010(

)(

)000(

)2(

2 4

)(

αβγ β αβγ β αβγ

±

±= +

αβγ

i i i Wi

h

W

h

Wu (39)

)(

)002(

2

)(

)001(

)(

)000(

)3(

2 4

)(

αβγ γ αβγ γ αβγ

±

±= +

αβγ

i i i Wi

l

W

l

Wu (40)

Subtraction of the pair of two equations in (38), (39) and (40) yields, respectively, that 















 = −

+ αβγ − αβγ

α

αβγ )( )( )1()1(

)(

)100(

1

i i uu i d

W (41)















 = −

+ αβγ − αβγ

β

αβγ )( )( )2()2(

)(

)010(

1

i i uu i h

W (42)















 = −

+ αβγ − αβγ

γ

αβγ )( )( )3()3(

)(

)001(

1

i i uu i l

W (43)

Similarly, adding the pair of two equations in (38), (39) and (40) yields, respectively, that 

)(

2 )000(

)1()1(

2

)(

)200(

2 4 )( )(

αβγ

α

+ −

α

αβγ −















 = +

αβγ αβγ

i i i Wi d

uu

d

W (44)

)(

2 )000(

)2()2(

2

)(

)020(

2 4 )( )(

αβγ

β

+ −

β

αβγ −















 = +

αβγ αβγ

i i i Wi h

uu

h

W (45)

)(

2 )000(

)3()3(

2

)(

)002(

2 4 )( )(

αβγ

γ

+ −

γ

αβγ −















 = +

αβγ αβγ

i i i Wi l

uu

l

W (46)

Substitution of Equations (44) to (46) in (34) provides the following expression for the second time 

derivative )(

)000(

αβγ Wi

 in terms of the stress moments and the second time derivatives of the surface-average 

displacements: 
NASA/TM—2012-217728 8 

















+ +

ρ

+















 = +++++

αβγ

γ

αβγ

β

αβγ

α

αβγ

+ − + − + −

αβγ αβγ αβγ αβγ αβγ αβγ αβγ

)(

2 )001(3

)(

2 )010(2

)(

2)( )100(1

)3()3()2()2()1()1(

2

2

)(

)000(

1 1 1

2

15

16

1 )( )( )( )( )( )(

i i i

i i i i i i i

S

l

S

h

S

d

uuuuuu

dt

d W

(47)

Similarly, the expressions for )(

)200(

αβγ Wi

 , )(

)020(

αβγ Wi

 and )(

)002(

αβγ Wi

 can be obtained by substituting Equation (47) 

in (44) to (46) yielding respectively 

















+ +

ρ −















 −−−−













 = +

αβγ

γ

αβγ

β

αβγ

αα

αβγ

+ − + − + −

α

αβγ αβγ αβγ αβγ αβγ αβγ αβγ

)(

2 )001(3

)(

2 )010(2

)(

22)( )100(1

)1()1( )3()3()2()2(

2

2

2

)(

)200(

130 1 1

7

4

1 )( )( )( )( )( )(

i i i

i i i i i i i

S

l

S

h

S

dd

uuuuuu

dt

d

d

W

(48)

















+ +

ρ −















 −−













 = +−− +

αβγ

γ

αβγ

β

αβγ

αβ

αβγ

+ − + − + −

β

αβγ αβγ αβγ αβγ αβγ αβγ αβγ

)(

2 )001(3

)(

2 )010(2

)(

22)( )100(1

)1()1( )2()2( )3()3(

2

2

2

)(

)020(

130 1 1

7

4

1 )( )( )( )( )( )(

i i i

i i i i i i i

S

l

S

h

S

dh

uuuuuu

dt

d

h

W

(49)

















+ +

ρ −































 = +−−−− +

αβγ

γ

αβγ

β

αβγ

αγ

αβγ

+ − + − + −

γ

αβγ αβγ αβγ αβγ αβγ αβγ αβγ

)(

2 )001(3

)(

2 )010(2

)(

22)( )100(1

)2()2()1()1( )3()3(

2

2

2

)(

)002(

130 1 1

7

4

1 )( )( )( )( )( )(

i i i

i i i i i i i

S

l

S

h

S

dl

uuuu uu

dt

d

l

W

(50)

With the established values of )(

)(

αβγ Wi lmn  given above in terms of the second-order time derivatives of the 

surface-average displacements and volume-average stress, moments one can proceed and express the 

values of )(

)(1

αβγ

i lmn I , )()(2

αβγ

i lmn J and )()(3

αβγ K i lmn in these terms as follows. Equations (31) and (14), (32) and (15) 

and (33) and (16) provide, respectively, that 

)(

2 )001(3

)(

2 )010(2

)(

2 )100(1

)1()1( )3()3()2()2(

2

2)(

)(

)000(1

2

3

2

3

2

21

7

80

)( )( )( )( )( )(

αβγ

γ

αβγ

β

αβγ

α

αβγ + − + − + −

αβγ

+ − −















 −−−−















+ ρ = αβγ αβγ αβγ αβγ αβγ αβγ

i i i

i i i i i i i

S

l

S

h

S

d

uuuuuu

dt

d I

(51)

)(

)000(1

)1()1(

2

2)(

)(

)100(1

)( )(

12

αβγ

+ −

α

αβγ αβγ +















 − ρ = αβγ αβγ

i i i i Suu

dt

dd I (52)
NASA/TM—2012-217728 9 

)(

2 )001(3

)(

2 )010(2

)(

2 )100(1

)1()1( )2()2( )3()3(

2

2)(

)(

)000(2

2

3

2

21

2

3

7

80

)( )( )( )( )( )(

αβγ

γ

αβγ

β

αβγ

α

αβγ + − + − + −

αβγ

− + −















 −−















+−− + ρ = αβγ αβγ αβγ αβγ αβγ αβγ

i i i

i i i i i i i

S

l

S

h

S

d

uuuuuu

dt

d J

(53)

)(

)000(2

)2()2(

2

)( 2

( )

)010(2

)( )(

12

αβγ

+ − β

αβγ

αβγ +















 − ρ = αβγ αβγ

i i i i Suu

dt

h d J (54)

and 

)(

2 )001(3

)(

2 )010(2

)(

2 )100(1

)2()2()1()1( )3()3(

2

2)(

)(

)000(3

2

21

2

3

2

3

7

80

)( )( )( )( )( )(

αβγ

γ

αβγ

β

αβγ

α

αβγ + − + − + −

αβγ

− − +

































+−−−− + ρ = αβγ αβγ αβγ αβγ αβγ αβγ

i i i

i i i i i i i

S

l

S

h

S

d

uuuu uu

dt

d K

(55)

)(

)000(3

)3()3(

2

)( 2

)(

)001(3

)( )(

12

αβγ

+ − γ

αβγ

αβγ +















 − ρ = αβγ αβγ

i i i i Suu

dt

l d K (56)

From Equations (8) to (10) and (17) to (19), the surface-average tractions can be obtained as 

)(

)100(1

)(

)000(1

)1(

2

)(

α αβγ αβγ

±

±= +

αβγ

i i i II d

t (57)

)(

)010(2

)(

)000(2

)2(

2

)( β αβγ αβγ

±

±= +

αβγ

i i i JJ

h

t (58)

)(

)001(3

)(

)000(3

)3(

2

)(

γ αβγ αβγ

±

±= +

αβγ

i i KK i

l

t (59)

Hence, it is possible to express these surface-average tractions in terms of the second-order time 

derivatives of the surface-average displacements and volume-average stresses and stress moments by 

substituting in Equations (57) to (59) the expression of )(

)(1

αβγ

i lmn I , )()(2

αβγ

i lmn J and )()(3

αβγ K i lmn which are given by 

Equations (51) to (56) yielding 

)(

(1 )000

)(

2 )001(3

)(

2 )010(2

)(

)100(1

)1()1(

2

2)(

)1()1( )2( ()3()2( )3

2

2)(

)1(

4

3

4

3

4

21

12

7

160

)( )(

)( )( )( ( ) )( )(

)(

αβγ αβγ

γ

αβγ α

β

αβγ α

α

+ −

α

αβγ

+ − + − + −

α

αβγ

±

+

















± − − 













 − ρ +

















−−−+















+ ρ ±

=

αβγ αβγ

αβγ αβγ αβγ αβγ αβγ αβγ

αβγ

i i i i i i

i i i i i i

i

SS

l

d S

h

d S

d

uu

dt

dd

uuu uuu

dt

dd

t

(60)
NASA/TM—2012-217728 10

)(

)000(2

)(

2 )001(3

)(

)010(2

)(

2 )100(1

)2()2(

2

)( 2

)1()1( )2()2( )3()3(

2

)( 2

)2(

4

3

4

21

4

3

12

7

160

)( )(

)( )( )( )( )( )(

)(

αβγ αβγ

γ

αβγ β

β

αβγ

α

β + − β

αβγ

+ − + − + − β

αβγ

±

+

















−± + − 













 − ρ

+















 −−















+−− +

ρ ±

=

αβγ αβγ

αβγ αβγ αβγ αβγ αβγ αβγ

αβγ

i i i i i i

i i i i i i

i

SS

l

h

S

h

S

d

h

uu

dt

h d

uuuuuu

dt

h d

t

(61)

)(

)000(3

)(

)001(3

)(

2 )010(2

)(

2 )100(1

)3()3(

2

)( 2

)2()2()1()1( )3()3(

2

)( 2

)3(

4

21

4

3

4

3

12

7

160

)( )(

)( )( )( )( )( )(

)(

αβγ αβγ

γ

αβγ

β

αβγ γ

α

γ + − γ

αβγ

+ − + − + − γ

αβγ

±

+

















−± − +















 − ρ

+

































+−−−− +

ρ ±

=

αβγ αβγ

αβγ αβγ αβγ αβγ αβγ αβγ

αβγ

i i i i i i

i i i i i i

i

SS

l

S

h

l

S

d

l

uu

dt

l d

uuuu uu

dt

l d

t

(62)

Equations (60) to (62) can be written in the compact form 

[ ] [ ]

)(

)001(33

)001(32

)001(31

)010(23

)010(22

)010(21

)100(13

)100(12

)100(11

)000(33

)000(32

)000(31

)000(23

)000(22

)000(21

)000(13

)000(12

)000(11

)(

)(

)3(

3

)3(

3

)3(

2

)3(

2

)3(

1

)3(

1

)2(

3

)2(

3

)2(

2

)2(

2

)2(

1

)2(

1

)1(

3

)1(

3

)1(

2

)1(

2

)1(

1

)1(

1

2

2

)(

)(

)3(

3

)3(

3

)3(

2

)3(

2

)3(

1

)3(

1

)2(

3

)2(

3

)2(

2

)2(

2

)2(

1

)2(

1

)1(

3

)1(

3

)1(

2

)1(

2

)1(

1

)1(

1

αβγ

αβγ

αβγ

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

αβγ

αβγ

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+





















































































































+





















































































































































=





















































































































































S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

L

u

u

u

u

u

u

u

u

u

u

u

u

u

u

u

u

u

u

dt

d K

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

(63)
NASA/TM—2012-217728 11

where α = 1,..., Nα; β = 1,..., Nβ; γ = 1,... Nγ and K(αβγ)is a 18×18 matrix whose elements depend on the 

dimensions of the subcell (αβγ) and the properties of the material filling this subcell, whereas L(αβγ)

is a 

matrix of the same dimensions whose elements depend on the dimensions of the subcell.

The continuity of the interfacial tractions between neighboring subcells implies that 

α β γ

+ −

= = =β−=α =γ

αβγ γβ+α

i i itt N N ,...,1,,...1,1,...,1,3,2,1, N

)( ),,1( )1( )1( (64)

1,...,1,,...,1,,...,1,3,2,1,

)( )1,,( )3( )3(

= = =α =β −=γ α β γ

+ αβγ − +γβα

i i itt N N N (65)

1,...,1,,...,1,,...,1,3,2,1,

)( )1,,( )3( )3(

= = =α =β −=γ α β γ

+ αβγ − +γβα

i i itt N N N (66)

In addition, the continuity of the displacements at the interfaces of the subcells yields 

α β γ

+ −

= = =β−=α =γ

αβγ γβ+α

i i iuu N N ,...,1,,...1,1,...,1,3,2,1, N

)( ),,1( )1( )1( (67)

α β γ

+ −

= = =α =γ−=β

αβγ γ+βα

i i iuu N N ,...,1,1,...,1,,...,1,3,2,1, N

)( ),1,( )2( )2( (68)

1,...,1,,...,1,,...,1,3,2,1,

)( )1,,( )3( )3(

= = =α =β −=γ α β γ

+ αβγ − +γβα

i i iuu N N N (69)

For imperfect bonding between the interfaces, Equations (67) to (69) take the form 

α β γ

+

αβγ

+ −

= =β−=α =γ

−− = αβγ γβ+α αβγ

i N N N

ttRuui i i i

,...,1,,...1,1,...,1,3,2,1

,0)(

)( ),,1( )( )1(

)(

1

)1( )1(

(70)

α β γ

+

αβγ

+ −

= =α =γ−=β

− − = αβγ γ+βα αβγ

i N N N

ttRuui i i i

,...,1,1,...,1,,...,1,3,2,1

,0)(

)( ),1,( )( )2(

)(

2

)2( )2(

(71)

1,...,1,,...1,,...,1,3,2,1

,0)(

)( )1,,( )( )3(

)(

3

)3( )3(

= =α =β −=γ

− − =

α β γ

+

αβγ

+ αβγ − +γβα αβγ

i N N N

ttRuui i i i (72)

where )()(

1i tR αβγ , )()(2i tR αβγ and )()(3i tR αβγ denote the time depending debonding functions that describe 

the behavior of the interfacial debonding of the subcell in the three directions. Wave propagation with 

imperfect bonding between the phases has been previously investigated by Aboudi (1988a). Not that, in 

the presence of perfect bonding, these functions are identically zero. 

Next, the following time-dependent boundary conditions must be imposed depending on whether the 

tractions or displacements are prescribed at the surfaces of the subcells at x1 = 0, x1= D; x2 = 0, x2 = H and 

x3 = 0, x3 = L. 
NASA/TM—2012-217728 12

( ) ( )

β γ

γβ

−

γβ

−

=β= =γ

= = βγ βγ

i N N

txxft txxfu ii ii

,...,1,,...1,3,2,1

or,,, ,,, )(

3

)(

2

)1(

)1(

)(

3

)(

2

)1(

)1( )1( )1(

(73)

and 

( ) ( )

β γ

γβ

+

γβ

+

=β= =γ

= = αβγ αβγ

i N N

ii txxgt ii txxgu N N

,...,1,,...1,3,2,1

or,,, ,,, )(

3

)(

2

)1(

)1(

)(

3

)(

2

)1(

)1( )( )(

(74)

( ) ( )

α γ

γα

−

γα

−

= =α =γ

= = γα γα

i N N

txxft txxfu ii ii

,...,1,,...1,3,2,1

or,,, ,,, )(

3

)(

1

)2(

)2(

)(

3

)(

1

)2(

)2( )1( )1(

(75)

and 

( ) ( )

α γ

γα

+

γα

+

= =α =γ

= =

α βγ α βγ

i N N

i i txxgt ii txxgu

N N

,...,1,,...1,3,2,1

or,,, ,,, )(

3

)(

1

)2(

)2(

)(

3

)(

1

)2(

)2( )( )(

(76)

Finally, 

( ) ( )

α β

βα

−

βα

−

= =α =β

= = αβ αβ

i N N

txxft txxfu ii ii

,...,1,,...1,3,2,1

or,,, ,,, )(

2

)(

1

)3(

)3(

)(

2

)(

1

)3(

)3( )1( )1(

(77)

and 

( ) ( ) α β

βα

+

βα

+

= =α =β

= =

αβ γ αβ γ

i N N

i i txxgt ii txxgu

N N

,...,1,,...1,3,2,1

or,,, ,,, )(

2

)(

1

)3(

)3(

)(

2

)(

1

)3(

)3( )( )(

(78)

In these equations, k )(

i f and k )(i g , i,k = 1,2,3, are time-dependent functions that describe the temporal 

form of the various applied loadings at the specific locations of boundaries of the region.

There are 18NαNβNγ unknowns 

)(

2

2 )1( ± αβγ

i dt

d u , )(

2

2 )2( ± αβγ

i dt

d u , )(

2

2 )3( ± αβγ

i dt

d u , i = 1,2,3. On the other hand, the 

interfacial traction and displacement provide 6(NαNβNγ – NβNγ), 6(NαNβNγ – NαNγ) and 6(NαNβNγ – NαNβ) 

which are respectively given by Equations (64) and (67) (or (70)), (65) and (68) (or (71)), and (66) and 

(69) (or (72)). In addition, the boundary conditions (73) and (74), (75) and (76) and (77) and (78) form 

another 6NβNγ, 6NαNγ and 6NαNβ relations, respectively. Thus the total number of equations is 18NαNβNγ. 

The resulting system of the 18NαNβNγ equations can be formally represented by 

= BXA tt )()( (79)

where A is a matrix of dimension 18NαNβNγ ×18NαNβNγ whose elements are the material parameters and 

geometric dimensions, and B(t) consists of time-dependent elements that involve the stresses and the 

imposed boundary conditions at time t. This equation represents the entire multiphase medium that result 
NASA/TM—2012-217728 13

from filling the subcells with various types of elastic materials (including pores and fluids). It also 

represents the elastodynamic equations that govern the motion of the material within the subcells, the 

various interfacial conditions between the subcells, and the applied time-dependent boundary conditions. 

A central finite difference of a second order accuracy in the time increment ∆t reduces this ordinary 

differential equation to the following explicit form 

)()()()(2)( 12 X XX BA ttttttt − ∆+∆−−=∆+ (80)

from which the variables can be computed at time t + ∆t from their known values at times t and t – ∆t. 

This procedure is continued to the next time increment. Stability of this finite difference procedure is 

ensured by a proper choice of the value of the time increment ∆t. It should be noted that in the analysis of 

porous materials, the tractions at the walls of the empty subcell are equal to zero, so that the number of 

equations can be reduced by excluding this type of subcell from the system of equations. 

In the special case of multi-layered composites, the accuracy of the model was verified by 

comparison with the ray theory prediction (Aboudi, 1988b). Furthermore, in this latter special situation of 

multi-layered composites, extensive applications and verifications of the accuracy of the model were 

recently presented by Clements et al. (1996, 1997a, 1997b, 1998) including comparisons with measured 

data. In addition, these authors introduced more developments and made more refinements to the model. 

Also, the analysis of harmonic wave propagation in laminated composites was presented by Aboudi and 

Hevroni (1991). 

Specialization to Two-Dimensional Theory for Thermoelastic Panels

Consider the case of two-dimensional wave propagation such that there is no dependence of any field 

variable on the direction x1. This situation corresponds to the case in which the direction of the 

propagating waves is perpendicular to the x1-direction. For example, wave propagation in fiber reinforced 

materials in which the waves propagate in the perpendicular direction to the continuous fibers can be 

analyzed by present two-dimensional theory. Wave propagation in a panel in the through-thickness 

direction can be analyzed by the present two-dimensional theory for wave traveling in the x2-direction. 

In this two-dimensional theory all variables depend only on the x2- and x3-directions, in addition to 

the time, t. The plane x2–x3 is divided this time into Nβ × Nγ subcells with β = 1,..., Nβ, γ = 1,... Nγ. In 

addition, )(

2

β x and )(

3

β x are local coordinates whose origin is located at the center of the subcell (βγ), see

Figure 2. Hence the displacement expansions, Equations (3) to (5), reduce in this special case to 

)(

)02(1

2

2)(

3

)(

)20(1

2

2)(

2

)(

)01(1

)(

3

)(

)10(1

)(

2

)(

)00(1

)(

1

4

3

2

1

4

3

2

1 β β βγ γ γ βγ

βγ βγ β βγ γ βγ

















+ − 















+ −

+= +

W

l

W x h x

WxWxWu

(1)

)(

)02(2

2

2)(

3

)(

)20(2

2

2)(

2

)(

)01(2

)(

3

)(

)10(2

)(

2

)(

)00(2

)(

2

4

3

2

1

4

3

2

1 β β βγ γ γ βγ

βγ βγ β βγ γ βγ

















+ − 















+ −

+= +

W

l

W x h x

WxWxWu

(2)

)(

)02(3

2

2)(

3

)(

)20(3

2

2)(

2

)(

)01(3

)(

3

)(

)10(3

)(

2

)(

)00(3

)(

3

4

3

2

1

4

3

2

1 β β βγ γ γ βγ

βγ βγ β βγ γ βγ

















+ − 















+ −

+= +

W

l

W x

h

x

WxWxWu

(3)
NASA/TM—2012-217728 14

The resulting strain components in subcell (βγ) are given by 

( )

( )

( ))(

)20(1

)(

2

)(

)10(1

)(

12

)(

)02(1

)(

3

)(

)01(1

)(

13

)(

)20(3

)(

2

)(

)10(3

)(

)02(2

)(

3

)(

)01(2

)(

23

)(

)02(3

)(

3

)(

)01(3

)(

33

)(

)20(2

)(

2

)(

)10(2

)(

22

)(

11

3

2

1

3

2

1

3 3

2

1

3

3

0

βγ βγ β βγ

βγ βγ γ βγ

βγ βγ γ βγ βγ β βγ

βγ βγ γ βγ

βγ βγ β βγ

βγ

+=ε

+=ε

+=ε ++

+=ε

+=ε

=ε

WxW

WxW

WxWWxW

WxW

WxW

(4)

The governing equations that correspond to Equations (7), (15) and (16) are given by 

3,2,1,)(

)00(

)()(

)00(3

)(

)00(2 ρ=+ = βγ βγ βγβγ

i KJ i i iW (5)

3,2,1, 12

)(

)10(

2

)()(

)00(2

)(

)10(2 ρ=− = βγ βγ βγ β βγ iW

h

SJ i i i

 (6)

3,2,1, 12

)(

)01(

2

)()(

)00(3

)(

)01(3 ρ=− = βγ βγ βγ γ βγ iW

l

i SK i i

 (7)

Figure 2.—Geometry of the model specialized to two dimensions.
NASA/TM—2012-217728 15

Here )(

)(2

βγ

i mn J and )(

)(3

βγ K i mn are related to the surface-average tractions in the following manner 















 = −

+ βγ − βγ

β

βγ )( )( )2()2(

)(

)00(2

1

i i itt

h

J (8)















 = −

+ βγ − βγ

γ

βγ )( )( )3()3(

)(

)00(3

1

i ii tt

l

K (9)

and 















 = +

+ βγ − βγ

βγ )( )( )2()2()(

)10(2 2

1

i i i J tt (10)















 = +

+ βγ − βγ

βγ )( )( )3()3()(

)01(3 2

1

i ii K tt (11)

where the surface-average tractions are given by 

)(

3

)(

2

)(

2

2

2

)2(

2

1 )(

βγ β β γ

− γ

±















 ±=σ= ∫

γ

γ

βγ

xd

h

x

l

t i

l

l i (12)

)(

2

)(

3

)(

3

2

2

)3(

2

1 )(

βγ γ γ β

− β

±















 ±=σ= ∫

β

β

βγ

xd

l

x

h

t i

h

h i

(13)

Furthermore, the volume-average stresses and stress moments are given by 

( ) ( ) 3,2,1,, 1 )(

3

)(

2

)()(

3

)(

2

2

2

2

2

)(

)( = σ = γβ βγ γβ

−− γβ

βγ ∫∫

γ

γ

β

β

jixdxdxx lhS ij

l nm

l

h

h ij mn (14)

Let 

[ ] )(

)(12)(13)(23)(33)(22)(11

)(

)( ,,,,, βγ βγ S mn = mn mn mn mn mn SSSSSS mn

(15)

In conjunction with strain components Equations (4), (14) yields 

= + + ( )++ + θΓ− βγ βγβγ βγβγ βγβγ βγ βγβγ βγβγ βγ )()(

)10(1

)(

6

)(

)01(1

)(

5

)(

)10(3

)(

)01(2

)(

4

)(

)01(3

)(

3

)(

)10(2

)(

2

)(

p )00( p p p p p WCWCWWCWCWCS p

(16)

[ ] )(

)20(3

)(

4

)(

)20(2

)(

2

)(

)20(1

)(

6

2

)(

)10( 4

βγ β βγβγ βγβγ βγβγ = + + WCWCWC

h

Sp p p p

(17)
NASA/TM—2012-217728 16

[ ])(

)02(3

)(

3

)(

)02(2

)(

4

)(

)02(1

)(

5

2

)(

)01( 4

βγ γ βγβγ βγβγ βγβγ = + + WCWCWC

l

Sp p p p

(18)

where p = 1,...,6 and βγ )( Γp are the components of the thermal stresses and θ is the temperature deviation 

from a reference temperature, TR. 

The two equations that correspond to Equations (24) and (25) are presently given by 

[ ] 3,2,1, 12 10

2

4 12

)(

)20(

2

)(

)00(

2

)()(

)10(2

)(

)00(3

2

)(

)00(2

2

=

















+ ρ=− + β βγ β βγ βγ βγ β βγ β βγ iW

h

W

h

SK

h

J

h

i i i i i

 (19)

[ ] 3,2,1, 12 10

2

4 12

)(

)02(

2

)(

)00(

2

)()(

)01(3

)(

)00(2

2

)(

)00(3

2

=

















+ ρ=− + γ βγ γ βγ βγ βγ γ βγ γ βγ iW

l

W

l

SJ

l

K

l

i i i i i

 (20)

Hence by substituting Equation (5) in these two equations, the following relations (that correspond to 

Equations (32) and (33)) can be presently established 

)(

)20(

2

)()(

2 )10(2

)(

)00(2 20

12 βγ βγ β βγ

β

βγ

i = i ρ+ Wi

h

S

h

J  (21)

)(

)02(

2

)()(

2 )01(3

)(

)00(3 20

12 βγ βγ γ βγ

γ

βγ

i = i ρ+ Wi

l

S

l

K  (22)

Hence, substituting Equations (21) and (22) in (5) yields 

















+ ρ= −− βγ βγβγ β βγ γ βγ

γ

βγ

β

)(

)02(

2

)(

)20(

2

)(

)00(

)()(

2 )01(3

)(

2 )10(2 20 20

12 12

i i i i Wi

l

W

h

S W

l

S

h

 (23)

The surface-average displacements 

)( )2( ± βγ

i u and 

)( )3( ± βγ

i u are defined by 

)(

3

)(

2

)( 2

2

)2(

2

1 )(

βγ β β γ

− γ

±















 = ±= ∫

γ

γ

βγ

xd

h

xu

l

u i

l

l i

(24)

)(

2

)(

3

)( 2

2

)3(

2

1 )(

βγ γ γ β

− β

±















 = ±= ∫

β

β

βγ

xd

l

xu

h

u i

h

h i

(25)

Substitution of the displacement expansions (1) to (3) in Equations (24) and (25) reveals that these 

surface-average displacements are related to the microvariables )(

)(

βγ Wi mn as follows 
NASA/TM—2012-217728 17

)(

)20(

2

)(

)10(

)(

)00(

)2(

42

)(

βγ β βγ β βγ

±

+±=

βγ

i i i Wi

h

W

h

Wu (26)

)(

)02(

2

)(

)01(

)(

)00(

)3(

42

)(

βγ γ βγ γ βγ

±

+±=

βγ

i i i Wi

l

W

l

Wu (27)

Manipulation of every pair in these equations results in the following 















 = −

+ βγ − βγ

β

βγ )( )( )2()2(

)(

)10(

1

i i uu i h

W (28)















 = −

+ βγ − βγ

γ

βγ )( )( )3()3(

)(

)01(

1

i i uu i l

W (29)

)(

2 )00(

)2()2(

2

)(

)20(

2 4 )( )(

βγ

β

+ −

β

βγ −















 = +

βγ βγ

i i i Wi h

uu

h

W (30)

)(

2 )00(

3( )3()

2

)(

)02(

2 4 )( )(

βγ

γ

+ −

γ

βγ −















 = +

βγ βγ

i ii Wi l

uu

l

W (31)

Substitution of Equations (30) and (31) in (23) yields 

















+

ρ

+















 = +++ βγ

γ

βγ

β

βγ

+ − + −

βγ βγ βγ βγ βγ

)(

2 )01(3

)(

2)( )10(2

)3()3()2()2(

2

2

)(

)00(

11

7

60

14

1 )( )( )( )(

i i iii i i S

l

S

h

uuuu

dt

d W (32)

Consequently, Equations (28) to (31) establish the following expressions 

















+

ρ −















 −−













 = + βγ

γ

βγ

ββ

βγ

+ − + −

β

βγ βγ βγ βγ βγ

)(

2 )01(3

)(

22)( )10(2

)2()2( )3()3(

2

2

2

)(

)20(

11

7

240 6

7

2 )( )( )( )(

i i i ii i i S

l

S

hh

uuuu

dt

d

h

W (33)

















+

ρ −































 = ++−− βγ

γ

βγ

βγ

βγ

+ − + −

γ

βγ βγ βγ βγ βγ

)(

2 )01(3

)(

22)( )10(2

)2()2( )3()3(

2

2

2

)(

)02(

11

7

240 6

7

2 )( )( )( )(

i i i ii i i S

l

S

hl

uuuu

dt

d

l

W (34)

With the established values of )(

)(

βγ Wi mn  , one obtains from Equations (21) and (6) that 

)(

2 )01(3

)(

2 )10(2

)2()2( )3()3(

2

2)(

)(

)00(2 7

12

7

72 6

70

)( )( )( )(

βγ

γ

βγ

β

βγ + − + −

βγ + − 













 −−















+ ρ = βγ βγ βγ βγ

i i i ii i i S

l

S

h

uuuu

dt

d J (35)

and 
NASA/TM—2012-217728 18

)(

)00(2

)2()2(

2

)( 2

)(

)10(2

)( )(

12

βγ

+ − β

βγ

βγ +















 − ρ = βγ βγ

i i i i Suu

dt

h d J (36)

Similarly, Equations (22) and (7) result in 

)(

2 )01(3

)(

2 )10(2

)2()2( )3()3(

2

2)(

)(

)00(3 7

72

7

12 6

70

)( )( )( )(

βγ

γ

βγ

β

βγ + − + −

βγ − +

































++−− ρ = βγ βγ βγ βγ

i i i ii i i S

l

S

h

uuuu

dt

d K (37)

and 

)(

)00(3

)3()3(

2

)( 2

)(

)01(3

)( )(

12

βγ

+ − γ

βγ

βγ +















 − ρ = βγ βγ

i ii i Suu

dt

l d K (38)

From Equations (8) to (11), the surface-average tractions can be obtained as 

)(

)10(2

)(

)00(2

)2(

2

)( β βγ βγ

±

±= +

βγ

i i i JJ

h

t (39)

)(

)01(3

)(

)00(3

)3(

2

)(

γ βγ βγ

±

+±=

βγ

i i KK i

l

t (40)

Consequently, Equations (35) to (38) yield that 

)(

)00(2

)(

2 )01(3

)(

)10(2

)2()2(

2

)( 2

)2()2( )3()3(

2

)2( )( 2

7

6

7

36

12

6

140

)( )(

)( )( )( )( )(

βγ βγ

γ

βγ β

β

+ − β

βγ

+ − + − β± βγ

+

















± − 













 − ρ

+















 −−















+

ρ ±=

βγ βγ

βγ βγ βγ βγ βγ

i i i i i

i i i ii

SS

l

h

S

h

uu

dt

h d

uuuu

dt

h d

t

(41)

and 

)(

)00(3

)(

)01(3

)(

2 )10(2

)3()3(

2

)( 2

)2()2( )3()3(

2

)3( )( 2

7

36

7

6

12

6

140

)( )(

)( )( )( )( )(

βγ βγ

γ

βγ

β

γ + − γ

βγ

+ − + − γ± βγ

+

















−± +















 − ρ

+

































++−−

ρ ±=

βγ βγ

βγ βγ βγ βγ βγ

ii i i i

i i i ii

SS

l

S

h

l

uu

dt

l d

uuuu

dt

l d

t

(42)

These two sets of 12 equations provide the desired relations between the surface-average tractions and the 

second-order time derivatives of the surface-average displacements and the volume-average stresses and 

stress moments.

These equations can summarized in the compact form 
NASA/TM—2012-217728 19

[ ] [ ]

)(

)01(33

)01(32

)01(31

)10(23

)10(22

)10(21

)00(33

)00(32

)00(31

)00(23

)00(22

)00(21

)(

1

)(

)3(

3

)3(

3

)3(

2

)3(

2

)3(

1

)3(

1

)2(

3

)2(

3

)2(

2

)2(

2

)2(

1

)2(

1

2

2

)(

1

)(

)3(

3

)3(

3

)3(

2

)3(

2

)3(

1

)3(

1

)2(

3

)2(

3

)2(

2

)2(

2

)2(

1

)2(

1

βγ

βγ

βγ

−

+

−

+

−

+

−

+

−

+

−

+

βγ

βγ

−

+

−

+

−

+

−

+

−

+

−

+













































































+





































































































=





































































































S

S

S

S

S

S

S

S

S

S

S

S

L

u

u

u

u

u

u

u

u

u

u

u

u

dt

d K

t

t

t

t

t

t

t

t

t

t

t

t

(43)

where β = 1,..., Nβ, γ = 1,... Nγ and [K1]

(βγ)

is a 12×12 matrix whose elements depend on the dimensions 

of the subcell (βγ) and the properties of the material filling this subcell, whereas [L1]

(βγ)

is a matrix of the 

same dimensions whose elements depend on the dimensions of the subcell. In the present twodimensional case there are 12NβNγ unknown surface-average displacements 

)( )2( ± βγ

ui and 

)( )3( ± βγ

i u . 

The interfacial traction and displacements (assuming perfect bonding) between the subcells are 

β γ

+ −

= = =γ−=β

βγ γ+β

i i itt N ,...,1,1,...,1,3,2,1, N

)( ),1( )2( )2( (44)

1,...,1,,...,1,3,2,1,

)( )1,( )3( )3(

= =β= −=γ β γ

+ βγ − +γβ

i i itt N N (45)

and 

β γ

+ −

= =γ−=β=

βγ γ+β

i i iuu N ,...,1,1,...,1,3,2,1, N

)( ),1( )2( )2( (46)

1,...,1,,...,1,3,2,1,

)( )1,( )3( )3(

= =β= −=γ β γ

+ βγ − +γβ

i i iuu N N (47)

Finally, the boundary conditions at x2 = 0 and x2 = H are 

( ) ( )

γ

γ

−

γ

−

=γ=

= = γ γ

i N

txft txfu ii ii

,...,1,3,2,1

or,, ,,)(

3

)2(

)2(

)(

3

)2(

)2( )1( )1(

(48)
NASA/TM—2012-217728 20

and 

( ) ( )

γ

γ

+

γ

+

=γ=

= = βγ βγ

i N

ii txgt ii txgu

N N

,...,1,3,2,1

or,, ,,)(

3

)2(

)2(

)(

3

)2(

)2( )( )(

(49)

At x3 = 0 and x3 = L the boundary conditions are 

( ) ( )

β

β

−

β

−

=β=

= = β β

i N

ii txft ii txfu

,...,1,3,2,1

or,, ,,)(

2

)3(

)3(

)(

2

)3(

)3( )1( )1(

(50)

and 

( ) ( )

β

β

+

β

+

=β=

= =

β γ β γ

i N

ii txgt ii txgu

N N

,...,1,3,2,1

or,, ,,)(

2

)3(

)3(

)(

2

)3(

)3( )( )(

(51)

In summary, just like the three-dimensional continuum theory, the interfacial tractions, Equations (44) 

and (45), and displacements, Equations (46) and (47), continuity conditions as well as the boundary 

conditions, Equations (48) to (51), provide the requested system of equations, given by Equation (79), of 

12NβNγ equations to be solved at any time yielding the surface-average displacements at that time.

Extension to Inelastic Constituent Materials

It is possible to analyze and model wave propagation in inelastic composites. Here it is assumed that 

the applied impact is sufficiently strong enough to cause the inelastic material to exhibit inelastic flow. To 

this end, the constitutive equation of the material filling subcell (αβγ) is given instead of Equation (2) by 

[ ] 3,2,1,,,, )()()()( ε−ε=σ = αβγ αβγ αβγ αβγ C lkji I

ij ijkl kl kl

(1)

where αβγ )( εI

ij are the inelastic strain components. It is assumed that the inelasticity is governed by an 

isotropic flow rule. Thus for plasticity, the evolution equation of αβγ )( εI

ij is given by 

αβγ αβγαβγ )()()( σΛ=ε ij

I

ij   (2)

where αβγ )( Λ is the proportionality function. For viscoplastic constituents, the flow rule is given by 

αβγ αβγαβγ )()()( σΛ=ε ij

I

ij  (3)

Since the inelastic flow rule of the inelastic phase is isotropic, it must be that the stiffness tensor αβγ )( Cijkl of 

the material is isotropic as well, namely 

( ) Cijkl ij kl δδ+δδμ+δδλ= jkiljlik

αβγ αβγ )()( αβγ )( (4)

where λ(αβγ) and μ(αβγ)are the Lame' constants of the material.

With the isotropic assumption of the inelastic flow rule, Equation (1) implies that 
NASA/TM—2012-217728 21

)()()()()( 2 αβγ αβγαβγ αβγ αβγ εμ−ε=σ I

ij Cijkl kl ij

(5)

In the perfectly elastic case, the quadratic displacement expansions, Equations (3) to (5), produce 

linear variation in strains and stresses at each point within a given subcell. In the presence of inelastic 

effects, however, a linear strain field generated by these expansions does not imply the linearity of the 

stress field due to the path-dependent deformation. Thus the displacement field microvariables must 

depend implicitly on the inelastic strain distributions, giving rise to a higher-order stress field than the 

linear strain field generated from the assumed displacement field representation. In the presence of 

inelastic effects, this higher order stress field is represented by a higher order Legendre polynomial 

expansion in the local coordinates. Therefore, the strain field generated from the assumed displacement 

field, and the resulting mechanical stress field, must be expressed in terms of Legendre polynomials as 

( ) ( ) ( ))(

3

)(

2

)(

1

)(

)(

000

)( αβγ α γβ ∞

=

∞

=

∞

=

αβγ =ε ∑∑∑ Γ ζζζ lmn ij lmn l m n

nml

ij PPPe (6)

( ) ( ) ( ))(

3

)(

2

)(

1

)(

)(

000

)( αβγ α γβ ∞

=

∞

=

∞

=

αβγ =σ ∑∑∑ lmn ij lmn l m n ζζζτΓ

nml

ij PPP (7)

where lmn 12( )( 12 )( nml +++=Γ )12 , and the nondimensionalized variables ζi’s defined in the interval 

–1 ≤ ζi ≤ 1, are given in terms of the local subcell coordinates as ( ) 2 )()(

1 α

αα =ζ dx , ( ) 2 )()(

2 β

ββ =ζ hx , 

( ) 2 )()(

3 γ

γγ =ζ lx . 

For the given displacement field representation the upper limits on the summation in the strains 

expansion (6) becomes 1. The upper limits on the summations in the stresses expansion (7) on the other 

hand are chosen so that an accurate representation of the stress fields is obtained within each subcell, 

which depends on the amount of inelastic flow. The coefficients )(

)(

αβγ

ij lmn e and )(

)(

αβγ τij lmn in the above 

expansions are determined as follows.

The strain coefficients )(

)(

αβγ

ij lmn e are explicitly determined in terms of the displacement field 

microvariables using orthogonal properties of Legendre polynomials. For example )(

)100(1

)(

)000(

αβγ αβγ = Weij and 

3 2 )(

)200(1

)(

)100(

αβγ

α

αβγ e = Wd ij . 

The stress coefficients )(

)(

αβγ τij lmn are expressed in terms of strain coefficients and the unknown inelastic 

strain distributions by first substituting the Legendre polynomial representations for αβγ )( εij and αβγ)( σij into 

the constitutive equations and then utilizing the orthogonality of Legendre polynomials as 

)(

)(

)(

)(

)()(

)(

αβγ αβγαβγ αβγ

ij lmn =τ ijko ko lmn − ReC ij lmn (8)

The )(

)(

αβγ Rij lmn terms represent inelastic stress distributions and are calculated in the following manner: 

( ) ( ) ( ) )(

3

)(

2

)(

1

)(

3

)(

2

)(

1

)( 1

1

1

1

1

1 )((

)(

)( 4

αβγ α γβαγβ

−−− αβγ

αβγ ζζζζζζε Γ

μ= ∫∫∫ R l m n dddPPP I

ij

lmn

ij lmn

(9)
NASA/TM—2012-217728 22

With the above expression, the volume-average stresses and stress moments )(

)(

αβγ

ij lmn S in the present case of 

inelastic constituents can be readily evaluated by employing Equation (26) in conjunction with (7) to (9). 

This yields the following expressions which respectively replace Equations (22) and (28) to (30): 

( )

( ) ( ) )(

)000(

)(

)100(2

)(

)010(1

)(

6

)(

)100(3

)(

)001(1

)(

5

)(

)010(3

)(

)001(2

)(

4

)(

)001(3

)(

3

)(

)010(2

)(

2

)(

)100(1

)(

1

)(

)000(

αβγ αβγ αβγ αβγ αβγ αβγ αβγ

αβγ αβγ αβγ αβγ αβγ αβγ αβγ αβγ αβγ αβγ

+ ++ −+

= + + + +

p p p

p p p p p

RWWCWWC

WWCWCWCWCS (10)

[ ] )(

)100(

)(

)200(3

)(

5

)(

)200(2

)(

6

)(

)200(1

)(

1

2

)(

)100( 4 32

αβγ α αβγ αβγ αβγ αβγ αβγ αβγ α αβγ

p = p + p + p − Rp

d WCWCWC d S (11)

[ ] )(

)010(

)(

)020(3

)(

4

)(

)020(2

)(

2

)(

)020(1

)(

6

2

)(

)010( 4 32

αβγ β αβγ αβγ αβγ αβγ αβγ αβγ β αβγ

p = p + p + p − Rp

h

WCWCWC

h

S (12)

[ ] )(

)001(

)(

)002(3

)(

3

)(

)002(2

)(

4

)(

)002(1

)(

5

2

)(

)001( 4 32

αβγ γ αβγ αβγ αβγ αβγ αβγ αβγ γ αβγ

p = p + p + p − Rp

l

WCWCWC

l

S (13)

where p = 1,...,6. The components )(

)(

αβγ Rp lmn of the vector )(

)(

αβγ R lmn are given by 

[ ] )(

)(33)(22)(11 )(12)(13)(23

)(

)( ,,,,, αβγ αβγ R lmn = lmn lmn lmn lmn lmn RRRRRR lmn

(14)

These inelastic stress contributions are obtained by integrating the relevant flow rule to yield the inelastic 

strain at time t. 

In conclusion, the integration of the inelastic flow rule at the current time t provides the inelastic 

strain I αβγ )( ε from which the stress terms )(

)(

αβγ R lmn can be determined according to Equation (9). Hence, 

one can compute the volume-average stress and stress moments )(

)(

αβγ

lmn S from Equations (10) to (13). The 

latter can be used in Equation (63) to obtain the various surface tractions 

)( )1( ± αβγ

t , )( )2( ± αβγ

t and 

)( )3( ± αβγ

t . 

These can be employed in conjunction with the interfacial and boundary conditions to construct the 

current system of ordinary differential Equation (79) from which X at time t + ∆t can be obtained. 

Structural/Acoustic Evaluation of Foam Sandwich Panels

The above two-dimensional wave propagation theory has been applied to model the acoustic 

attenuation properties of structural panels of the type used in rotary aircraft cabins. These simulations 

were performed using a “virtual acoustic test chamber”, which defines the geometry in which the acoustic 

properties of the panels are evaluated. The main issue with a simulated test chamber is eliminating the 

effects of spurious reflections on the “measured” signal. For instance, in Figure 3, the applied cyclic 

loading, which simulates an acoustic signal, is applied on the lower boundary of the analysis space. The 

wave travels through the air and then impacts the bottom of the panel. The wave travels through the panel 

and into the air, where the wave is recorded at a “virtual sensor” location 0.5 cm above the panel surface.

The wave then continues through the air to the boundary of the analysis space and is reflected back, 

eventually making its way back to the top of the panel. Since we are primarily interested in the effects of 

the panel on the propagating waves without reflected waves from the rear surfaces, we wish to exclude 

the effects of these latter waves. In order to avoid recording pressures that include the effects of these 
NASA/TM—2012-217728 23

reflected waves, the dimension of the virtual test chamber behind the panel must be sufficiently long. It 

was determined that a length of 21 cm was sufficiently long for an applied wave frequency of 10,000 Hz. 

Note that, as the frequency decreases, this length will increase as a longer time is needed for several 

waves to pass though the panel to be recorded, but this longer time requirement can allow reflected waves 

enough time to interfere. It should be noted that noise issues in many rotorcraft cabins tend to occur at 

frequencies in the range of 1000 to 3000 Hz. Examining these frequencies would require a longer virtual 

test chamber. Herein, air was assumed to be a fluid possessing no ability to transmit shear stresses whose 

bulk modulus is given by, 

2

p ρ= cK (15)

where cp is the velocity of sound in the fluid. The properties employed for air are, ρ = 0.00129 g/cm3 and 

cp = 346 m/s. 

Figure 4 is a plot versus time of an applied 10,000 Hz signal (i.e., pressure wave) and the response at 

the location of the virtual pressure sensor (0.5 cm above the panel top surface location in the middle of the 

chamber) in the case of an empty chamber (filled only with air) and in the case of a typical panel. It takes 

approximately 250 μs for the wave to reach the virtual sensor location. Clearly, a large decrease in the 

wave amplitude is caused by the presence of the panel.

Figure 3.—Virtual acoustic test chamber used to simulate the acoustic behavior 

of structural panels using the wave propagation theory. (a) Dimensions.

(b) Representative discretization and boundary conditions.
NASA/TM—2012-217728 24

Figure 4.—Example pressure waves recorded at the virtual 

sensor location 0.5 cm above the panel top surface.

A facesheet thickness study was performed on four different sandwich panels, with Al and IM7/8552 

facesheets and two Rohacell foam core densities, using the virtual test chamber (Figure 3). As shown in 

Figure 5, the total thickness of the panel was kept constant at 2 cm, while the facesheet (FS) thickness 

was varied between 0.8 and 1.6 mm. The relevant properties of the panel constituent materials are given 

in Table 1. Note that effective (homogenized) properties are used for the composite facesheet and foam 

materials, as opposed to modeling individual plies, fiber/matrix constituents, or pores within the foam.

Figure 6 is a plot of sound pressure level (SPL) reduction caused by the presence of the panel in the 

virtual test chamber versus the panel areal mass for each of the four panel types with varying facesheet 

thickness. Note that, in this figure, the closer given panel design falls to the upper left, the better 

performance (as high SPL reduction with low mass is desirable).

TABLE 1.—MATERIAL PROPERTIES USED FOR 

PANEL ACOUSTIC ATTENUATION STUDIES

Property Aluminum Quasi-Isotropic 

IM7/8552

Rohacell 31A 

Foam

Rohacell 110A 

Foam

E, GPa 72.4 57 0.036 0.16

ν 0.33 0.32 0.38 0.38

ρ, g/cm3 2.8 1.578 0.032 0.11

The SPL is given by,















 =

ref

rms

p

p

10 SPL log20 (16)

where pref = 20 μPa is the standard reference pressure for airborne sound, and the root mean square 

pressure is given by, 
NASA/TM—2012-217728 25

∫ =

av t

av

dttp

t

p

0

2

rms ([ )] 1 (17)

p(t) is the pressure signal as a function of time, and tav is the duration of the pressure signal.

Figure 6 shows that the denser Rohacell 110 foam core and the aluminum facesheet provide the best 

overall SPL reduction, but the composite facesheet with the Rohacell 110 provides almost as much SPL 

reduction at a significantly lower mass. For the Rohacell 110 foam core, increasing the FS thickness does

not have much of effect on the SPL reduction. In contrast, the FS thickness has a significant impact on the 

SPL reduction for the Rohacell 31 core as the contribution of this lower density foam to the panel 

damping is much lower than in the case of the Rohacell 110 foam core panel. It is noteworthy that the 

composite facesheet (FS) curves appear as almost an extension of the aluminum FS curves as if the 

aluminum facesheets continued to become thinner.

Figure 5.—Foam core panels 

considered with three different 

facesheet thicknesses.

Figure 6.—Areal mass versus SPL reduction for the four panel types simulated.
NASA/TM—2012-217728 26

Figure 7.—(a) Panel bending/buckling performance index versus SPL reduction for the four panel types simulated. 

(b) Panel strength performance index versus SPL reduction for the four panel types simulated.

From Figure 6, it appears that the Rohacell 110 core with thin composite facesheets provides the best 

overall performance of the panels considered. However, this figure considers only the raw unit mass of 

the panel and does not account for the panel’s structural performance. If instead we plot the SPL 

reduction versus structural performance indices (Ashby, 2005), as shown in Figure 7(a) and (b), it is clear 

that the less dense Rohacell 31 core panel with composite facesheets is also quite competitive. The 

performance indices are associated with the structural efficiency of the panel per unit mass for panel 

bending and buckling, Figure 7(a), and panel in-plane strength, Figure 7(b). The plots provide a good 

measure of the trade-off between acoustic performance and structural performance of the panels for 

lightweight applications. This is particularly important for rotorcraft cabin panels, where interior noise is 

an important design consideration as is structural mass.

Acoustic Evaluation of Heterogeneous Panels 

Here we consider two panels consisting of 75 percent aluminum and 25 percent Rohacell 31A foam, 

as shown in Figure 8. The first includes two layers of Rohacell foam, while the second has discrete 

Rohacell 31A inclusions. These panels were simulated in the virtual test chamber shown in Figure 3 at a 

variety of frequencies. The resulting SPL reductions are plotted versus frequency in Figure 9. Also plotted 

is the “mass law” for transmission loss through a panel (Fahy, 1985), 

TL = log20 10 ( ) s fm − 47 (18) 

where ms is the panel areal mass in kg/m2

, f is the frequency in Hz, and TL is the transmission loss (i.e., 

SPL reduction) in dB. Both panels considered have an identical areal mass of 42.2 kg/m2

; all that is 

different is the arrangement of the foam and aluminum constituents. Equation (18) provides an 

engineering estimate of panel SPL reduction in frequencies governed by the mass law, where the material 

stiffness and microstructure do not play a roll. This is typically in the range of 100 to 1000 Hz, as shown 

in Figure 10. 

Figure 9 shows that the panel with discrete foam inclusions provides more SPL reduction than does 

the layered panel. This is expected as the discrete inclusions are more effective in converting waves from 

longitudinal to shear. Both panels considered provide SPL reductions that are lower than the mass law 

prediction (as expected, see Figure 10). However, at the lower frequencies considered, the predictions 
NASA/TM—2012-217728 27

associated with both panel types converge and begin to follow the mass law curve. This is because, at 

lower frequencies, the waves have longer wavelengths and thus interact with the microstructure to a lesser 

extent. It can also be shown that the results are independent of the stiffness of the panel at these lower 

frequencies, as predicted by the mass law. 

Figure 8.—Heterogeneous Rohacell 31A/aluminum panels considered in the 

acoustic study.

Figure 9.—Heterogeneous Rohacell 31A/aluminum panel SPL reductions at a 

variety of frequencies.
NASA/TM—2012-217728 28

Figure 10.—Typical frequency ranges for the acoustic attenuation properties of 

structural panels.

Conclusion

An integrated structural/acoustic model has been developed for the analysis of the dynamic response 

of multiphase panels that are subjected to impulsive loading, such as an acoustic pressure wave. Versions 

of the model applicable to three-dimensional bodies and two-dimensional bodies were presented. The 

geometry of the heterogeneous medium is discretized into an arbitrary number of subcells, each of which 

may contain a distinct solid material, fluid, or a void. By enforcing equilibrium and continuity conditions 

within and between these subcells, the dynamic response of the medium to time-dependent boundary 

conditions can be determined. An extension of the model to include inelastic material behavior was also 

discussed.

The model was applied to analyze the structural/acoustic response of foam core sandwich panels, 

which are candidates for application in the cabins of rotorcraft. This was accomplished using a virtual test 

chamber, in which the acoustic wave can be applied to the air surrounding the panel rather than directly to 

the panel surface. This captures the interaction of the wave with the panel surface rather than assuming a 

known boundary condition at the panel surface, and is thus more physically representative of the 

interaction of the structure with the acoustic environment. However, as in real acoustic testing, care must 

be taken to ensure that reflected waves do not pollute the results so that the true panel acoustic 

performance is obtained. Various panel designs were compared in terms of their acoustic and structural 

performance, and it was demonstrated that a trade-off exists between the desire to design panels with high 

transmission loss (better acoustic performance) and the desire to design panels with the lowest mass 

(better structural performance). 

It is clear that the presented model is capable of evaluating the impact of material and geometric 

design parameter changes on the structural/acoustic performance of lightweight panels for application in 

rotorcraft. It’s ease of use and efficiency for performing full dynamic simulations, like those presented 

herein, make it an attractive alternative to the finite element method for this purpose. Enhancements that 

would make the model more useful would be an automated ability to alter the applied acoustic wave 

frequency, while simultaneously adjusting the size of the virtual test chamber to ensure that reflected 

waves do not influencing the results. 
NASA/TM—2012-217728 29

References

Aboudi, J. (1986) Harmonic Waves in Composite Materials, Wave Motion, 8, 289-303. 

Aboudi, J. (1987) Transient Waves in Composite Materials, Wave Motion, 9, 141-156. 

Aboudi, J. (1988a) Wave Propagation in Damaged Composite Materials, Int. J. Solids and Structures. 24, 

117-138. 

Aboudi, J. (1988b) Nonlinear Wave Propagation in Laminated Composites, in: Wave Propagation in 

Structural Composites, ASME AMD-vol. 90, 133-140. 

Aboudi, J. and Hevroni, I. (1991) “Wave Propagation in Angle-Ply Laminates” J. Sound Vib., 150, 15-24. 

Ashby, M.F. (2005) Materials Selection in Mechanical Design, Elsevier Butterworth Heinemann, New 

York.

Clements, B. E., Johnson, J. N. and Hixon, R. S. (1996), Stress waves in composite materials. Phys. Rev. 

E, 54, 6876-6888. 

Clements, B. E., Johnson J. N. and Addessio, F. L. (1997a), Wave propagation in an epoxy-graphite 

laminate. J. Appl. Phys. 82, 4831-4837. 

Clements, B. E., Johnson, J. N., Addessio, F. L. and Hixon, R. S. (1997b), Impact produced stress waves 

in composites. J. de Physique IV 7 (Colloque C3), 767-772. 

Clements, B. E., Johnson, J. N. and Addessio, F. L. (1998), Wave propagation in laminates: a study of the 

nonhomogenized dynamic method of cells. Wave Motion 27, 193-209. 

Fahy, F. (1985) Sound and Structural Vibration Radiation, Transmission and Response, Academic Press, 

London. 
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Integrated Structural/Acoustic Modeling of Heterogeneous Panels 

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14. ABSTRACT

A model for the dynamic response of heterogeneous media is presented. A given medium is discretized into a number of subvolumes, each 

of which may contain an elastic anisotropic material, void, or fluid, and time-dependent boundary conditions are applied to simulate impact 

or incident pressure waves. The full time-dependent displacement and stress response throughout the medium is then determined via an 

explicit solution procedure. The model is applied to simulate the coupled structural/acoustic response of foam core sandwich panels as well 

as aluminum panels with foam inserts. Emphasis is placed on the acoustic absorption performance of the panels versus weight and the 

effects of the arrangement of the materials and incident wave frequency. 

15. SUBJECT TERMS

Wave propagation; Acoustic attenuation; Panels; Foams; Sandwich structures; Sound waves; Dynamic response; Elastodynamics; 

Transmission loss; Simulation microstructure; Micromechanics 

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