Interface response theory of composite elastic media

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Interface response theory of composite elastic media

L. Dobrzynski, J. Mendialdua, A. Rodriguez, S. Bolibo, M. More

To cite this version:

Interface

_response

_theory

of

_composite

elastic media

L.

_Dobrzynski,

J. Mendialdua

_(*),

A.

_Rodriguez

_(*),

S. Bolibo and M. More

Equipe

Internationale de

_Dynamique

des Interfaces, Laboratoire de

_Dynamique

des Cristaux

Moléculaires,

U.R.A. No. 801 au CNRS, U.F.R. de

_Physique,

Université de Lille I, F-59655 Villeneuve

_d’Ascq

_Cedex, France

(Reçu

le 2

_février

_1989,

_accepté

sous

forme

définitive

le 21 mars

1989)

Résumé. 2014 Une théorie

générale

permettant d’étudier

_n’importe

_quel

matériau

_élastique

et

composite

est

_proposée.

Son

_application

aux

_composites

lamellaires est ensuite

_{développée.}

Ces résultats

_généraux

sont illustrés par des

exemples

de nouveaux modes de vibration localisés dans

des couches fluide et solide

comprises

entre deux autres solides semi-infinis. Abstract. 2014 A

general theory

for the

_study

of any

composite

elastic media is

_proposed.

It is then

applied

to

_{layered composites.}

These

general

results are illustrated

_{by examples}

of new localized

vibrations within fluid and solid slabs sandwiched between two other semi-infinite solids. Classification

Physics

Abstracts

68.25 - 68.65 - 68.15 - 68.48 - 64.45

1. Introduction.

Following

the

_pioneering

works

_[1]

of I. M. Lifshitz for

_phonons

and J. Friedel for electrons

on the

_study

of a

_point

defect in a

_solid,

the surface of a solid was studied as a

_planar

defect in an infinite solid

_[2].

Interfaces between two different media

_[3]

and then

_{superlattices}

_[4]

started then to be considered in a similar manner. However this

_approach

to interface

problems requires

one to know first the

_properties

of the free surface

_pieces

of matter out of which the

_composite

material is built. It was shown

_recently

[5]

that the

study

of any

composite

material can be undertaken

directly

from the bulk

_properties

of each of its

pieces

together

with the

_boundary

conditions at the _{interfaces. This interface response}

theory

[5]

deals in a unified manner with

_problems

formulated within discrete and continuous spaces. It

was

_applied

to

_phonons,

_{magnons, and electrons in the}

_{tight binding}

_[6]

_{approximation}

as well as to

_{electromagnetism}

and free electrons

_[7].

The

_present

_{paper shows how this}

_theory

_[5]

enables any

composite

elastic media to be

studied,

and

_gives

new results for localized elastic vibrations within a fluid and a solid slab sandwiched between two other semi-infinite solids.

The next _{section of this paper}

_gives

the

_{general equations enabling}

_any

_composite

elastic material to be studied. Section 3 illustrates this

_{general theory by}

an

_application

to

_layered

composite

materials.

_Finally,

we

_{give specific}

new results for localized elastic vibrations within

a fluid and a solid slab sandwiched between two other semi-infinite solids.

2. Général

_theory

for any elastic

composite

material.

2.1 BULK AND FREE SURFACE EQUATIONS. - The

equation

of motion for the

_{displacements}

ua, a =

1, 2,

3 of _a

point

of an infinite

_homogeneous

3-dimensional elastic material is

where p is the mass

_density,

the vibrational

_frequency,

_{T al3}

the stress tensor

C af3 J.L JI

are the elastic constants and

_{’Tl J.L JI}

the deformation tensor

Then the bulk

_equation

of motion

_(2.1)

can be rewritten as

Suppose

now that the elastic matter is limited

_by

a free surface whose

_position

in the infinite 3-dimensional space is

given by

Then the elastic constants are

where the Heavside

step

function is such that

The use in

_{equation (2.6)}

of 0

[X3 - f (xl, X2)]

means that the elastic medium

occupies

the

space x3 >

[(X1,X2).

Define with the

_help

of

_equations

_(2.4)

and

_(2.6)

the bulk

_operator

such that

Note that after differentiation of the

_right

hand side of

_equation

_(2.8b),

one has

Define then a surface response

_operator

A

such that its elements are

where the bulk response function

G

is defined

_by

~ ~

Knowing

the bulk response function G and the surface response

operator A,

one can obtain

the response function

gs

of the

_homogeneous

elastic material bounded

_by

the free surface defined

_{by equation}

_(2.5)

from the

_{general equation}

_[5]

where

Ï

is the

_identity

matrix.

It is useful to use the

_following

notation : D for the space in which the elastic matter is defined and M _{for the space of its free surface. Then}

_{equation (2.13)}

_provides

_[5]

where 3-

_{(MM )}

is the inverse matrix of

A useful

_particular

value of

_equation

_(2.14)

is

where

_{Us 1 (MM)}

and

_(MM)

are the inverse matrices of

gs (MM )

and

G (MM ),

respectively.

Let us stress that in order to calculate the elements of

_gs

from

_equations

_(2.14)

and

_(2.16),

one has in

_general

to _{discretize the surface space}

_M,

in order to calculate the inverse matrices defined above.

2.2 THE GENERAL EQUATIONS FOR A COMPOSITE MATERIAL. - Let

us now consider any

composite

material contained in its space of definition D and formed out of N different

homogeneous pieces

situated in their spaces

Di.

Each

_piece

is bounded

_by

an interface

Ml, adjacent

in

_general

_{to j (1, j , J)}

other

_pieces

_through

subinterface domains

Mij.

The ensemble of all these interface spaces

Mi

will be called the interface space M of the

Considering

first each

_piece

with its free

surface,

one can calculate from

_equation

(2.16)

its

Then the

_{corresponding}

_g-1

_{(MM )}

for the

_composite

material is obtained

_[5]

in the

_following

manner

All the

_boundary

conditions at the interfaces are satisfied

_through

these

_{equations (2.18).}

By

taking

a discrete number of

points

in the total interface space, one calculates then the inverse

_{g (MM )}

of

_g-1

_{1 (MM).}

Let us define here

G (DD )

as a block

_diagonal

reference

response function constructed out ot the elements of the _{bulk response} functions

G(D,D,).

All the elements of the response

function g

of the

_composite

material can be obtained

_[5]

from

The new interface states can be calculated from

and the total variation of the

_density

of states

dN (lù 2)

between the

_composite

material and the reference one was shown

_[5]

to be

where the

_phase

_{shift ’TJ ( úJ 2)}

is

_{given by}

For

_perturbation

_defects,

the

_{equation (2.21)}

reduces to that first

_proposed

in 1952

_by

Lifshitz for

_phonons

around

_point

defects and

_by

Friedel

[1]

for free electrons around

impurities

in metals and used

_by

him to derive the well known Friedel sum rule. The results

(2.18-2.22)

in

_slightly

different forms were obtained

also,

within the Surface Green’s Function

Matching Theory

[8]

for materials with one, two and two

periodic

interfaces. The

generalization

to _any

_composite

material was achieved afterwards within the Interface

Response Theory

[5].

Let us stress

_finally

that

_if

_U(D))

represents

an

eigenvector

of the reference

system,

equation (2.19)

enables one to calculate the

_eigenvectors

_{1 u (D )}

of the

_composite

material

In

_{particular if}

_U(D))

_represents

a bulk wave launched in one

_homogeneous

_piece

of the

composite

material,

equation

(2.23)

enables to calculate all the waves reflected and

transmitted

_by

the interfaces.

Rather than

_expanding

more on these

_general

_results,

let us illustrate them

_by

a few

applications

to

_{layered composite}

elastic materials.

3.

Layered composite

materials.

Any

layered composite

material can be built out of semi-infinite and slab

_pieces.

We will therefore first show here how to obtain from

_equation

_(2.17)

the

_{corresponding}

_{g;- 1 (Mi}

_{Mi )}

for a semi-infinite solid and then for a slab. Then we will indicate how these results can be also used for viscous and non viscous fluids.

_Finally

we will describe how to use these results for any

layered composite.

In all what follows we assume the interfaces

_{perpendicular}

to _{the X3 axis. Then the function}

f (xl, X2)

vanishes

_(see

_Eq.

_(2.5)).

_{Taking advantage}

of the infinitesimal translational invariance of this

_{layered composite}

in directions

_parallel

to the

interfaces,

one can Fourier

analyze

all

_operators,

and in

_particular

_{the response functions}

_g

and

_G,

_according

to

where

and

il

and

_{i2 being}

unit vectors in the 1- and

2-directions,

respectively. Using

equations (2.8a)

and

(2.12),

one finds that the Fourier coefficient

_Gal3

(lq 1 X3

_{x3 )}

of the bulk response function

G

is the solution of the

_following

_system

of

_ordinary

differential

_equations

Let us now choose an

_isotropic

elastic medium for which the bulk response function

G

can be calculated in closed form.

3.1 BULK RESPONSE FUNCTION FOR AN ISOTROPIC ELASTIC SOLID. - An

_isotropic

elastic medium is a medium whose

_properties

are

_isotropic

in all directions of space. For such a

medium

The _square of the

_longitudinal

and shear

_plane

wave velocities are

_respectively

Such a material is also

_isotropic

within the

(Xl, X2)

_plane.

It is

possible

to choose in

equations (3.2)

and

_{(3.4) k2}

= 0. The results obtained for

G(k11

x3 x3 )

can be used for any

other direction of

_k¡1,

after a rotation of the Xi and X2 axes such that

_k¡1

lies

along

xl. As in this

case

1 k¡, 1

=

kll

=

kl,

we will write them as function of

_kll

rather than

kl.

Let us define also

and

Another

_interesting

_property

due to the

_isotropy

in the

_{(xl, x2)}

_plane

is the

_decoupling

of the transverse vibrations

_{polarized along x2}

from the

_sagittal

vibrations

_polarized

in the

(x1, x3 )

plane.

This

_decoupling

will remain for

_{all layered composites}

and it is

_possible

to

study separately the X2 polarized

transverse vibrations and the

_sagittal

ones in such

composites.

The elements of

Let us remark that _{a t =}

(klr- úJ2/cf)1I2

is real for

úJ :ctkll

and that we choose

a

t = - i

( cv

2/ cf -

kir )112

for úJ :> Ct

kll.

The

negative sign

in this last result

corresponds

in the response functions to

_outgoing

waves at _x3 = _{± 00.} The _same consideration

applies

to

3.2 ONE SEMI-INFINITE SOLID. - Let

us consider now a semi-infinite solid such that x3 r 0. For the

isotropic

solid described above and for

kp -

*Ikll ,

the elements of the

cleavage

operator

defined

_{by equation}

_(2.10)

become

With the

_help

of

_equations

_(2.11),

_{(2.15), (2.16)}

and

_(3.12)

one obtains

Let us note that the

_{gs 1 (00 )}

for the same _{elastic medium but such that X3 -- 0 has the} same

expression

(3.14)

but with a

_change

of

_sign

in the

_off-diagonal

elements

_coupling

the xi and x3

polarisations.

Note also that

from

which,

with the

_help

of

_equation

_(2.20)

one obtains the well known

_Rayleigh

wave

dispersion

relation.

With the

_help

of

_equation

_(2.14),

one also recovers

[10]

the response function

Us

of the semi-infinite solid.

3.3 A FREE SURFACE SOLID SLAB. - Let

us now consider an

_isotropic

elastic slab such that

The interface space M is now formed out of these two _{surfaces X3} = _± a. The

_cleavage

operator

has contributions

_proportional

to a

_(X3

+

_{a )}

and to 6

_{(x3 - a ) ;}

the coefficients of the first one are

_equal

to those

_{given by equation}

_(3.13),

the

_signs

of the coefficients of the second contribution are

_{just changed.}

As

_above,

one can calculate the

corresponding

Us

(MM).

The calculation of the

_sagittal

contribution to

_gs

1 (MM)

can be obtained more

_easily,

with the

_help

of the reflection

_symmetry

_through

the middle of the slab. This

_decouples

the

corresponding

4 x 4 matrix in two

_{(2 x 2)}

matrices:

_{g-- ’(MM)}

for the

_symmetrical

coordinates

and

_gsÂs

for the

_{antisymmetrical}

coordinates

respectively.

Their

_expressions

are

where

and

where

The above

_expressions

are

_particularly

useful for

composites keeping

this reflexion

symmetry

through

the middle of a central slab. In

general,

however,

we will need the

(4 x 4 )

expression

of the slab

_{g s 1 (MM )}

for the normal

_{coordinates {I- a,}

_{Xl)’ 1-}

_{a, X3),}

where

3.4 THE SPECIAL CASE OF FLUIDS. -

It was shown before

[11]

that the motion of a fluid

governed by

the linearized Navier-Stokes

_equation

can be studied with the

_help

of the

equations

that describe the motion of a solid

_providing

that

and

where co is the

longitudinal speed

of sound in the

fluid,

po its

density,

and g

and

1£ ’ are

the coefficients of shear and dilatation

_viscosity.

A non-viscous fluid can also be studied from the above

_{equations, by taking}

the limit

ct - 0 in them.

3.5 ANY LAYERED ELASTIC COMPOSITE. - We have

now all the

_ingredients

necessary for the

study

of the elastic

_properties

_{of any}

_{layered composite}

with fluid and solid

_parts.

Once the

s ’(Mi

Mi )

for each different

_part

is

determined,

then the

_{g- 1 (MM)}

of the

_composite

is obtained

_by

linear

_{superposition}

of these

_{Js ’(Mi}

_Mi),

as

_{specified by equations}

(2.18).

Then

the main basic

_properties

of the

_composite

can be obtained with the

help

of

equations

(2.19)-(2.23).

Rather than

_expanding

these

_general

_results,

let us illustrate them

by

an

_application

to the vibrations of a fluid and a solid slab sandwiched between two semi-infinite solids.

4. A few

_{applications.}

4.1 THE FLUID SANDWICH VIBRATIONS. - Let

us consider a non-viscous fluid slab of thickness 2a sandwiched between two identical semi-infinite solids. The

_application

of the

localized within the fluid slab. Note first that there are no transverse

_polarized

modes in the fluid slab.

The

_{expression giving}

the

_{symmetrical sagittal}

sandwich modes was found to be

and those for the

_{antisymmetrical sagittal}

ones

where the p o and p are the mass densities of the fluid and of the solid

_{respectively ;}

« 1 and « t are the

outgoing

at

infinity penetration

coefficients of the solid

given by

equations

(3.8)-(3.9)

and ao is the

equivalent

of a

for

the

longitudinal

wave in the

_liquid.

Remark that in the limit a - oo , coth ao a and th a 0 a --+

1,

both

_equations

_(4.1)

and

_(4.2)

reduce to the

_expression

for the localized vibrations at a

_planar

fluid-solid interface

[12].

Figure

1

_represents

the first

_symmetrical

and

_{anti-symmetrical}

modes localized within the fluid slab.

_Defining

the

_speed

of these modes

_by

c, we

_represented

_clct

=

f (kp a)

for the

parameters

characteristic of an acoustic

_Perot-Fabry

interferometer

[13].

In connexion with

this

_{experiment, using equations}

_(2.23)

we also calculated the reflected and transmitted waves

due to an acoustic wave launched in the solid. These results are

_given

in the

_appendix.

Fig.

1. - Localized _waves in _a thin fluid slab of thickness 2 a sandwiched between two identical

semi-infinite solids. The

_figure

_displays

the ratio of the

_speed

of

_propagation

c of these waves to the transverse

speed

of sound c, in the solid as a function of

_akl.

The _parameters used in this calculation are those of an

acoustic

_{interferometer, namely :}

p o = 1010

kglm 3,

, co = 1 900 mis for the

liquid

(N2H4, H20)

and p = 2 200

kg/m 3,

_Ct = 3 750 m/s and _cl = 5 900 m/s for the solid

(Si02).

4.2 THE SOLID SANDWICH VIBRATIONS. - Let

us now consider a solid slab of thickness 2a

sandwiched between two different semi-infinite solids.

_Using

the results

_{(3.14), (3.19)}

and

(2.18),

one obtains a

₍₄

x

_{4 )}

matrix

_expression

for the

_sagital

_part

of the

_{g-1 (MM )}

associated with this

_composite

and

_using

_{equations (3.17), (3.14)}

and

_(2.18)

one obtains a

(2

x

2)

matrix

for the transverse

_part.

Then

_equation

_(2.20)

enables us to calculate the localized vibrations within the 5folid slab. Such localized modes were studied before with another method but

_only

Let us label the two semi-infinite solids

_by

the

_subscripts

1 and 2 and

_keep

for the solid slab the same notation without

_subscript

as in section 3.3. The closed form

_{expression giving}

the localized transverse modes was

_easily

found to be

Figure

2

_represents

the first of the

_infinity

of modes localized within a slab of W sandwiched between Ni and Al. The

_input

_parameters

are

_given

in the table I. The

_speed

c of these

localized modes lies in between the transverse

_speeds

of sound of Al and W. Note that these localized modes exist

_only

when the material sandwiched between the two others has a

transverse

_speed

of sound ct such that Ct Cll ct2. Otherwise one obtains

_only

resonant

modes,

also called

_leaky

waves

[12].

Fig.

2. -

Speed

c of transverse localized waves in a Ni-W-Al sandwich.

Table 1. -

Input

values

_for

the calculation

_of

the velocities

_given

in

_figures

_2,

3 and 4.

where the

_parameters

without index are

_{given by equations}

_(3.20)

and for i = 1 and 2

In order to discuss the different

_possible

localized

_sagittal

modes in such a

_sandwich,

let us

distinguish

four cases

_depending

on the

_respective

values of the transverse

_speeds

of sound of the three different materials and on the existence or non-existence of a localized

_Stoneley

wave

_{having velocity}

_c, at a

_single

interface between the central material and one of the

others.

An

_infinity

of localized sandwich modes exist. Their

_speed

_{is ct. The} variation of the

_speed

c as a function of 2

_akll

of the first of these modes is

_given

in

_figure

3 for the Ni-W-Al sandwich. When 2

_akll

--> _oo, the

velocity

of this sandwich mode goes to the

velocity

of the

Stoneley

wave

_existing

at the

_single

W-Al interface. The other modes

_(not

shown in the

figure)

come out of the bulk band of Al for

_higher

values of 2

_akll

and when 2

_akll

--+ _oo, their

velocity

tends

_{asymptotically}

to the

_velocity

_{ct of W.}

ii)

Ct ctl ct2 with no existence of a

_Stoneley

wave

One

_expects

similar results as

_above,

_apart

that all localized mode velocities c will tend to ct for 2

akll

--+ oo.

Only

one localized mode exists below ct1 and its

_velocity

c - _{cs for 2}

_akll

--+ oo. The

_velocity

of such a mode is

_represented

in

_figure

4 for a W-Al-Ni sandwich.

iv)

ctl Ct ct2 and ctl ct2 ct

There is no

_truly

localized mode within the sandwich.

5. Discussion.

Let us first mention here two other

_problems

which were solved

_recently

with the same

formalism.

(i)

Closed form

_expressions

for bulk and surface transverse elastic waves in

_{superlattices}

made of 3 and 4 different

_layers

were obtained

_[16].

(ii)

Analytic expressions

for the elastic energy of interaction of a

_point

defect with a

_planar

defect,

a

_plane

of dilatation and half a

_plane

of dilatation in a

_plane

defect have been

calculated

_[17].

The

_implications

for

_segregation

of

_impurities

near

_{high angle grain}

boundaries are also discussed in this work.

Many

other

_problems

related in

_particular

to the static and vibrational

_properties

of

_layered

Fig. 3.

Fig.

4.

Fig.

3. -

Speed

c of the first

_sagittal

localized wave in a Ni-W-Al sandwich.

Fig.

4. -

Speed

c of the

_sagittal

localized wave in a W-Al-Ni sandwich.

Explicit expressions

of the elastic response functions were obtained

_only

for a few

_layered

systems :

free surfaces and a

_planar

interface between two static solids

_[19]

with conventional solutions of the differential

_equations.

The formalism described _{in this paper enables} one to consider more

_{complex layered composites mostly}

because the double

_boundary

conditions

(continuity

of stresses and

_{displacements}

at each

_interface)

are combined

_(see

_Eqs.

_(2.18))

into a

_{single boundary}

condition.

Let us

_finally

stress that the

_theory

outlined in this _paper and illustrated

_by

two

_simple

examples

of

_{layered composites applies}

to _{any elastic}

_composite

_{without any restriction} as to the number and the form of the constituents. Of course in such cases and

_especially

if one

deals with

_{non-isotropic}

_{constituents,}

the whole theoretical

_procedure

described in Section 2 becomes numerical and should be

_compared

with the finite element method.

_Advantages

were

_[20]

found in the use of the interface response

method,

mostly

because many

physical

Appendix

The fluid sandwich :

localized,

reflected and transmitted waves.

Consider a

_sagittal

bulk

_propagating

wave

_coming

_{from X3}

= _{+ oo .} The

corresponding

eigenvector

has as

_components

where

Using

this incident wave

_as

1 U(D»

in

_equation

(2.23),

one obtains from the second and

third term of this

_equation

the reflected and transmitted waves. Let us recall that the a i and a, to be used in the response function

appearing

in this

equation

were defined as

and

_provide

then the reflected and transmitted waves

_{corresponding}

to the incident wave

(Al).

In what follows we

_give

the final

_expressions

of these waves, where the ci, and cit take the values

given by equation

(A2).

Let us define also

and

With these

notations,

the reflected

_(in

the solid at _X3 «--

a)

wave has the

_following

_components

The wave transmitted in the fluid slab

_{(- a «}

_X3

_{a )}

is

The wave transmitted in the solid at _{X3 «-- - a} was found to be

The above results

_(A.7)-(A.9)

also contain the waves

_{corresponding}

to the localized

symmetrical

and

_{antisymmetrical}

modes whose

_dispersion

relation is

_{given by Dl}

= 0

(Eq. (4.1))

and

_D2

= 0

(Eq. (4.2))

respectively.

The unnormalized wave

_{corresponding}

to the

symmetrical

modes

_{(Dl = 0)}

and the

_{antisymmetrical}

ones

_{(D2 = 0)}

is obtained

_by

multiplication

of the

_expressions

_(A.7)-(A.9)

_{by Dl}

and

_{D2, respectively,}

and

_taking

respectively

the limit

_{Dl -->}

0. Note that

B

is in this case a common normalization factor and

that for these modes localized within the fluid

slab,

Remerciements.

L’un d’entre nous

(L. D.)

voudrait

_exprimer

ici sa

_profonde

reconnaissance au Professeur

Jacques

Friedel pour

l’inspiration

de

_départ

et son intérêt constant et souriant. En effet le

présent

travail et

_{plus particulièrement}

la théorie

_générale

de

_réponse

des interfaces

_peut

être considéré comme une

_{généralisation logique}

des travaux initiaux du Professeur

_Jacques

Friedel sur les défauts

_ponctuels.

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