P Leclaire

K V Horoshenkov

A Cummings

TRANSVERSE VIBRATIONS OF A THIN RECTANGULAR POROUS PLATE SATURATED BY A FLUID

A simple model of the transverse vibrations of a thin rectangular porous plate saturated by a #uid is proposed. The model is based on the classical theory of homogeneous plates and on Biot's stress}strain relations in an isotropic porous medium with a uniform porosity. Two coupled dynamic equations of equilibrium relating the plate de#ection and the #uid/solid relative displacement are found and their physical interpretation is given. The energy dissipation by viscous friction is included in the model. An approximate calculation of the natural frequencies of vibration is given for rigid plates with various boundary conditions at the edges. The in#uence of porosity, tortuosity and permeability on the resonances is studied and a condition of maximum damping involving these parameters is given.

INTRODUCTION

The vibration of porous plates is important in the study of #uid}solid interactions with applications in geophysics or in civil engineering, where porous panels are used as sound absorbers. When the material is saturated by air, the approximation of a rigid frame can generally be made. In this approximation, the solid frame is motionless because it is much heavier and more rigid than air. In reality, however, the solid can vibrate in the vicinity of resonance frequencies, resulting in a strong modi"cation of the surface impedance [1,2]. It has been shown [3] that Biot's theory [4,5] can be used to determine the displacement of the solid and its e!ect on the calculation of the surface impedance of rigid porous layers.

Another approach consists in solving the two-dimensional boundary value problem of the vibration of a porous plate. A rigorous and detailed analytical description of this problem including Biot's equations of poroelasticity has been given by Theodorakopoulos and Beskos [6]. In the present article, an additional assumption valid for thin plates is made and an alternative form of the stress}strain relations introduced by Biot [7,8] is used. In this formulation, the equations of equilibrium are derived in a simple way without a signi"cant loss of generality. Two coupled equations describing the dynamic equilibrium of the plate are then obtained by introducing inertia forces. Energy dissipation by viscous friction is included through the use of a dynamic #uid density. The calculated de#ection compares very well with the numerical results of reference [6]. The simplicity of the equations obtained gives a better understanding of the physical phenomena involved in the vibration. These equations make an evaluation of the natural resonance frequencies with a range of boundary conditions at the edges possible.

In this article, the main results of the classical theory of plates are applied to the non-porous case. This can be justi"ed by the fact that at the scale of the wavelengths the porous medium appears as a continuous medium composed of an e!ective solid and an e!ective #uid in Biot's approach of poroelasticity. The plate is treated as a classical homogeneous plate where elastic, inertial and viscous interactions between the solid and the #uid take place within the pores. These e!ects are described by Biot's coupled equations of motion. The continuity conditions on the pressures and velocities at the interface between the porous surfaces and the surrounding #uid were given by Deresiwicz and Skalak [9]. These are fairly simple and involve the porosity factor. From these conditions, it is possible to model the e!ect of #uid loading on the vibration. The #uid behaviour in the pores is at the centre of this article and so it is natural and coherent to include the e!ect of #uid loading at the interfaces. It is thought that the #uid loading will have less e!ect for porous plates than for non-porous plates, though, because porous media are generally permeable. The e!ect of #uid loading is studied in the ba%ed case by the application of the continuity conditions. For simplicity, this e!ect is not implemented numerically. Attention is particularly focussed on the in#uence of permeability, porosity and tortuosity.

STRESS}STRAIN RELATIONS FOR A POROUS PLATE

BIOT'S STRESS}STRAIN RELATIONS IN AN INFINITE POROUS MEDIUM

The stress}strain relations for an unbounded porous medium have been established by Biot. For an isotropic medium with uniform porosity and in the absence of body force, they can be written in the form [8]

GH # P D GH "2 GH # GH , (1)
P D "! M #M , (2)
where

GH " 1 2 du G dx H # du H dx G , (3)
" # # "div u, (4)
"!div w, w"(U!u).

(5, 6)

In these equations, GH is the total stress tensor acting in the porous medium, GH is the Kronecker symbol (the convention of summation on repeated indices being implied), P D is the #uid pressure in the pores, and are the LameH elastic constants of the solid frame de"ned for isotropic media. The coupling coe$cient and the elastic modulus M have been de"ned by Biot [7]. GH is the solid strain tensor expressed as a function of the components of the solid displacement u in the system of co-ordinates (x , x , x ), is the solid dilatation and the #uid content is a function of the #uid displacement U and of the porosity , w being the relative #uid}solid displacement.

THE BENDING OF A POROUS PLATE

The condition of equilibrium of a plate under the action of a load or of a buckling constraint can be written in terms of bending and twisting moments related to the de#ection (see reference [START_REF] Timoshenko | ¹heory of Plates and Shells[END_REF]). The problem of a porous plate is treated as follows: in a "rst step, moments M , M and M are calculated, that are directly related to the de#ection of the solid frame but where the "rst two of these are not the actual bending moments in the #uid-saturated material. Then from a relation obtained for the #uid, it is shown that simple relations can be found between M , M , M and the actual moments. The de#ection w Q "u of the plate in the deformed state is considered "rst (see reference [START_REF] Nowacki | Dynamics of Elastic Systems[END_REF] for the relations between de#ection, curvatures and bending moments). Since the de#ection is undergone by the solid frame, the elements of the strain tensor in the solid can be calculated from the expression for w Q . It is then easily shown that the corresponding stresses are GH # P D , the resultant of these stresses in the plate thickness corresponding to M

, M and M . The advantage of using GH # P D is that most of the results obtained in the theory of thin non-porous homogeneous plates can be applied to the porous case.

The dimensions of the plate are a and b, and its thickness is h. The system of co-ordinates (see Figure 1) is chosen so that the plane de"ned by the axes x and x coincides with the middle surface of the plate before deformation, x and x being parallel to the sides a and b respectively. The axis x is normal to the middle surface. The #uid #ow is assumed to be normal to the plane (x , x ) so that "!*w/*x (see reference [START_REF] Biot | [END_REF]). The component of w along x is written as w"w . The thickness of the plate is taken to be smaller than any acoustic wavelength and the additional assumption is made that the variation of amplitude of the #uid displacement within the plate along x is small. This allows one to make the same assumption as in the non-porous case on the normal component of the total stress tensor: "0, valid for fairly rigid thin plates. The deformations in the thickness of the plate are neglected and only the stresses tangent to the surface are considered.

The de#ection is undergone by the #uid-saturated plate and by the solid frame in particular. The useful elements of the strain tensor in the solid are therefore (see reference [START_REF] Nowacki | Dynamics of Elastic Systems[END_REF])

"!x *w Q *x , "!x *w Q *x , "!x *w Q *x *x . ( 7
)
From the formulae of plane elasticity, the corresponding stresses are

"! Ex 1! *w Q *x # *w Q *x , "! Ex 1! *w Q *x # *w Q *x , "! Ex 1# *w Q *x *x , ( 8
)
where E is Young's modulus and the Poisson ratio of the solid frame. These expressions are a particular form of Hooke's law. The general form is

GH "2 GH # GH . ( 9
)
By identifying each term in this equation with those in equation ( 1) and by the use of the known relations between the elastic constants, it is easily seen that

GH " GH # P D GH , (10)
" "

E (1# )(1!2 ) , (11)
"

" E 2(1# ) . ( 12
)
The stresses GH " GH # P D GH are such that the porous medium behaves as a homogeneous solid of elastic moduli and .

RELATIONS BETWEEN THE BENDING MOMENTS

Two bending moments and one twisting moment may be associated with the stresses GH given by equation (8),

M " F \F x dx "!D *w Q *x # *w Q *x , M " F \F x dx "!D *w Q *x # *w Q *x , (13)
M "

F \F x dx "!D(1! ) *w Q *x *x
, where D"Eh/12(12! ) is the plate #exural rigidity. The actual moments in the #uid-saturated solid are those associated with the total stresses:

M " F \F x dx , M " F \F x dx , M " F \F x dx . (14)
A moment, relative to the #uid, may equally be de"ned as [6] I"

F \F P D x dx . ( 15
)
Equation ( 2) can be written in the form

P D "! M E (1! ) ( # )#M . ( 16
)
Multiplying each term of this equation by x and integrating over the thickness yields

I"! (M #M )!M F \F *w *x x dx (17)
with " M(1! )/E. The second term on the right-hand side could be integrated by parts but is neglected under the assumption on the amplitude of the #uid displacement. It can be seen from equation [START_REF] Timoshenko | ¹heory of Plates and Shells[END_REF] that

M "M # I, M "M # I, M "M (18)
and by using equation ( 17), the following relations are found between the moments:

M "(1# )M # M , M " M #(1# )M , M "M , (19)
I"! (M #M ). ( 20
)

DYNAMIC EQUILIBRIUM

The case of vibration without energy dissipation is considered "rst. In this case, only elastic and kinetic energies are involved, the elastic energy being responsible for potential forces within the plate while the kinetic energy is the origin of inertia forces. From the expression given by Biot [8] for an unbounded porous medium, the kinetic energy of the plate is expressed as

T" 1 2 4 ( w R Q #2 D w R Q w R #mw R ) d<" h 2 1 ( w R Q #2 D w R Q w R #mw R ) dS, (21)
where , D and m are, respectively, the density of the #uid}solid mixture, the density of the #uid and a mass parameter that can be related to the density of the #uid with the help of the tortuosity . The integration is performed either over the volume < of the plate or over the middle surface S. The elastic energy is given by

U" 1 2 4 ( # # #P D ) d<. ( 22
)
It can easily be seen that

U" 1 2 4 ( # # ) d<. ( 23
)
The equations of dynamic equilibrium can be obtained from Hamilton's principle, that involves minimizing a function of the Lagrangian, the mechanical action, between two instants t and t (see reference [START_REF] Nowacki | Dynamics of Elastic Systems[END_REF]). The method preferred here consists in equating the projection of the potential and inertia forces derived from the elastic and kinetic energies, along the x -, x -and x -axis (see reference [START_REF] Timoshenko | ¹heory of Plates and Shells[END_REF]). The resultant along x of all the potential forces acting on and element dx dx of the middle surface of the #uid-saturated porous plate is

f N " *Q *x dx dx # *Q *x dx dx #q dx dx , ( 24
)
where Q "F \F dx and Q "F \F dx are the total shearing stresses along x , and q is the load applied to the solid as a boundary condition (q has the dimensions of pressure). The inertia forces can be obtained by integrating the time derivative (denoted by a dot over the derived quantity) of the generalized momenta per unit volume of the #uid}solid mixture over the plate thickness. Under the assumption on the amplitude of the #uid displacement, the inertia forces may be expressed as

f G "h( w K Q # D w K ) dx dx . ( 25
)
Since the #uid/solid relative displacement is normal to the middle surface, the equilibrium of the moments acting on the mixture along x and x , respectively, may be written as [START_REF] Timoshenko | ¹heory of Plates and Shells[END_REF] *M *x

! *M *x #Q "0, (26)
*M *x # *M *x !Q "0 (27)
with M "!M . Equating f N and f G , and combining equations ( 24), ( 26) and ( 27) to eliminate Q and Q yields the equation of equilibrium in terms of moments,

*M *x # *M *x !2 *M *x *x " !q#h( w K Q # D w K ) (28)
or, by making use of equations (19),

(1# ) *M *x # *M *x # *M *x # *M *x !2 *M *x *x " !q#h( w K Q # D w K ). (29)
The "rst equation of equilibrium in terms of de#ection is obtained by using expressions (13) for M , M and M . A simple expression is found,

(1# # )D w Q #h( w K Q # D w K )"q (30)
with " ( ) and "*/*x #*/*x . The equation of equilibrium is also referred to as the plate equation. A second relation can be obtained from equation (2). The microscopic forces per unit volume < acting on the #uid along x are given by

*P D *x " M *w Q *x # *w Q *x !M *w *x , (31)
while the inertia forces per unit volume are f G " D w K Q #mw K . Equating the forces (*P D /*x ) d< and f G d< and integrating over the thickness yields the second equation of dynamic equilibrium. Under the assumption on the #uid displacement amplitude, the term containing the space derivative of is neglected throughout the plate and

Mh *w Q *x # *w Q *x !h( D w K Q #mw K )" P, ( 32
)
where the integration constant P"P D (!h/2)!P D (#h/2) is the pressure di!erence corresponding to the boundary conditions applied on the #uid at x "$h/2. Upon inserting the expression for , the coupled equations of dynamic equilibrium are "nally written as

D# Mh 12 w Q #h( w K Q # D w K )"q, ( 33
)
Mh w Q !h( D w K Q #mw K )" P. ( 34
)
The "rst equation may be interpreted as the instantaneous elastic response of the #uid-saturated plate while the second describes the relative motion between the solid and the #uid. The elastic interactions are quanti"ed by the terms where the coupling coe$cient appears. The inertial interactions are given by the terms containing the accelerations.

In a real situation, the relative #uid}solid motion within the pores is responsible for an energy dissipation with a phase shift, corresponding to a delayed after-e!ect [START_REF] Biot | [END_REF]. The e!ects of dissipation by viscous friction should naturally be included in the second equation. They can easily be incorporated by considering the coe$cient m to be frequency dependent,

m( )" ( ) " ( ) D , ( 35
)
being the angular frequency. The frequency-independent coe$cients m and have been de"ned by Biot [8]. The harmonic dependence in time is assumed for the displacements, for q and for P. The dynamic tortuosity ( ) has been introduced by Johnson et al. [13] and can be written as a complex quantity ( j being the imaginary unit complex number)

( )" !j b D D F( ), (36)
where b D is Biot's coe$cient of friction [4] given as a function of the porosity, of the #uid dynamic viscosity and of the permeability by b D " / .

(37)

The viscosity correction function F( ) is close to 1 at low frequencies and describes the e!ects of viscous friction at high frequencies. Various models can be used to evaluate this function [5, 13}16].

The solutions of the plate equations are expressed in the form of double in"nite series,

w Q (x , x )" K L =Q KL K (x ) L (x ), (38)
w(x , x )" K L = KL K (x ) L (x ), (39)
each term of w Q and w satisfying equations ( 33) and (34). =Q KL and = KL are the displacement amplitudes at the position (x , x ). K (x ) and L (x ) are the so-called beam functions chosen to comply with the boundary conditions at the edges. If the plate is simply supported, the products of the beam functions K (x )"sin(m /a) and L (x )"sin(n /b) satisfy exactly the equations of equilibrium. The excitation terms can also be expressed in the form of double in"nite series. Inserting equations ( 38) and (39) into equations ( 33) and (34) leads to a system of two equations and two unknown for each value of the couple (m, n). Solving this system provides the coe$cients =Q KL and = KL . The solutions for w 1 and w are obtained after insertion into equations ( 38) and (39).

FLUID LOADING

The #uid can have an e!ect on the vibration of a plate. This e!ect is accounted for through an additional excitation term or load that must be included in the equations. To evaluate the #uid loading, it is necessary to calculate the surface acoustic pressure:

P(x , x , 0)" ! D 1 w 2 (x , x )G(x , x , 0; x , x , 0) dS. ( 40
)
This integral is often referred to as the Rayleigh integral where G is the Green function associated with the contribution of a surface element dS,

G(x , x , 0; x , x , 0)" e\I0 2 R . ( 41
)
k" /c is the wavenumber and R is given by

R"((x !x )#(x !x ). ( 42
)
In the present study, the term w 2 (x , x ) is the total velocity "eld in the vicinity of the surface. The total velocity "eld is obtained from a study by Deresiewicz and Skalak [9] on the continuity conditions at the interface between two media involving porous materials. The following continuity equation is given on the velocities at the interface separating a porous medium (1) and a #uid (2),

(1!)uR L #; Q L "; Q M L , ( 43
)
where uR L and ; Q L are, respectively, the normal components of the solid velocity u , of the #uid velocity U in the porous medium (1) and ; Q M L the velocity in #uid (2) averaged over the bulk area.

Upon using this condition and the de"nition of the #uid}solid relative displacement [8] w"(U!u),

the total velocity "eld in the vicinity of the surface of the porous plate is given by the simple sum

w R 2 (x , x )"wR Q (x , x )#wR (x , x ), (45)
each component of this "eld being parallel to the x -axis. The additional terms due to #uid loading can be written as [17] f . KL "!

1 P(x , x , 0) K (x ) L (x ) dS, (46)
which can be rewritten in terms of the radiation impedance matrix Z $ KLNO as

f . KL " !j NO Z $ KLNO (w 2 ) NO . ( 47
)
Inserting equations ( 38) and (39) into equations ( 33) and (34) leads to the modal form of the equations of dynamic equilibrium in which the excitation terms q and P appear as q KL and P KL respectively. Equation ( 33) is interpreted as the response of the #uid-saturated plate while equation (34) describes the relative motion between the solid and the #uid and so the #uid loading is accounted for by adding each term f . KL to q KL in the modal form of equation (33).

NUMERICAL RESULTS

The plate de#ection w Q computed from the present model is compared to the results of Theodorakopoulos and Beskos [6] with the same numerical data for a water-saturated plate of 4;4;0)2 m. The model of Theodorakopoulos and Beskos can be considered as more rigorous than the present model because the use of the moment I"! F \F P D x dx does not imply any restriction on the #uid. Short wavelengths, i.e. several spatial periods of the #uid displacement, are allowed within the plate whereas the amplitude of the #uid}solid relative displacement is considered as constant throughout the plate thickness in the present model. Figure 2 shows the modulus of the de#ection calculated at the centre for a 4;4;0)2 m sandstone plate saturated by water (see Table 1). The material properties are those of Fatt [6,18]. In the simulation, the plate is uniformly excited by an incident pressure of amplitude P "1400 Pa, constant in the frequency domain. The results for an air-"lled porous panel (&&Y foam'') of 0)5 m;0)5 m;10)70 mm are given in Figure 3. This plate was fabricated at the University of Bradford from recycled car dashboards for noise control application. Its properties are displayed in Table 1. The panel is excited either by a uniform incident pressure of 0)1 Pa amplitude or by a unit point force (1 N) located at x "0)1 m, y "0)15 m. The response is calculated in both cases at x"0)25 m, y"0)25 m. The curves of Figures 2 and3 were calculated for values of (m, n) of up to 40. The results given by the two models are very similar and a di!erence of only a few percent is observed on the amplitude in a wide frequency range. As shown in Figure 3, the discrepancy between these models seems to increase when the frequency reaches values of around 5 kHz. The increasing di!erence is certainly a consequence of the breakdown of the assumption that the plate is much thinner than any acoustic wavelength. Nevertheless, because porous plates will involve smaller Young's moduli and higher loss factors, many applications and experimental observations will be carried out at low frequencies and this model can be considered as useful. It involves simple equations and allows theoretical developments for more complicated con"gurations.

APPROXIMATE CALCULATION OF THE NATURAL FREQUENCIES OF VIBRATION OF A POROUS PLATE

Warburton [19] has provided useful approximate calculations of the natural frequencies of rectangular plates for any combination of the three basic boundary conditions at the edges: i.e., free, simply supported or clamped. Experiments have been carried out in the present investigation to check the accuracy of the prediction on an aluminium plate of 150;50;0)8 mm, with the two a-sides free and the two b-sides clamped. An excellent agreement of less than 3% error has been obtained for the seven "rst resonance frequencies.

A similar evaluation can be made for porous materials if the elastic modulus of the plate is much greater than that of the #uid: i.e., if EM. This condition will be ful"lled in many cases and in particular for rigid porous plates vibrating in air. In this case, only the inertial interactions are considered in the plate equations, which may be written for free vibration as

D w Q #h( w K Q # D w K )"0, D w K Q #m( )w K "0 (48, 49)
and which may be combined into a single equation involving only the de#ection

D w Q #h w K Q "0 (50)
with

" ! D m( ) " ! ( ) D . ( 51
)
The case without attenuation is considered "rst, for the evaluation of the resonance frequencies. In this case, equation (50) indicates that the #uid and the solid vibrate in phase 91)0 8 8 )5 (5,3) or (3,5) 101)2 100)9 (5,4) or (4,5) 126)5 127)5

with an amplitude ratio of (1!/ ). In the restriction stated in this section, the elastic energy is given by

U" 1 2 D 1 *w Q *x # *w Q *x #2 *w Q *x *w Q *x #2 (1! ) *w Q *x *x dS, (52)
the kinetic energy is given by

T" 1 2 h 1 w K Q dS (53)
and, following the method given in reference [19], a frequency factor can be de"ned as

" ! D ha D , "G V #G W a b #2 a b [ H V H W #(1! )J V J W ], (54)
giving a value of the natural frequencies for boundary conditions characterized by the values of the coe

$cients G V , G W , H V , H W , J V and J W .
A useful table of the values taken by these coe$cients for any given set of boundary conditions has been provided by Warburton [19]. For the water-saturated plate of reference [6], the numerical results given by the approximate calculation are close to those provided by the numerical implementation of the model. The eight "rst natural frequencies calculated from the approximate formula (54) are compared to the corresponding &&exact'' frequencies obtained from the full model in Table 2 for Y foam. The agreement is excellent except for the sixth frequency. The reason is that the frequency corresponding to (m, n )"(5, 2) or (2, 5) is 86 Hz. This value is close to that for (m, n)" (4,4). Due to the high loss factor it is not possible to distinguish the two frequencies and only one intermediate value can be read on the curve. As will be shown, the damping by viscous friction can be important at a particular frequency. An evaluation of the resonance frequencies where the viscous friction in the pores are taken into account can be made by using Re( ) instead of ! D / in the expression for . However, the simplicity of the expression for is lost.

INFLUENCE OF POROSITY, TORTUOSITY AND PERMEABILITY ON THE RESONANCES

The in#uence of porosity and tortuosity on the resonance frequencies can be studied for jb D / D "0, this imaginary term mainly a!ecting the quality factor of the resonance. Since the tortuosity is greater than 1 and the porosity is between 0 and 1, the frequency factor increases as the porosity decreases or as the tortuosity increases. As a consequence, the resonance frequency will be higher for higher porosities and for lower tortuosities. This is con"rmed by the present numerical results and by those of reference [6]. The in#uence of permeability is studied by reintroducing the coe$cient b D . The plate equation is then written in a form similar to that of an oscillating system,

D w Q #h ! D #b D / D !j b D / #b D / D w Q "0, (55)
where the expression in parentheses is the dynamic density . By considering the real part of equation ( 55) and from the expression for , it can be observed that the resonance frequency decreases as the ratio b D / increases: i.e., as the permeability decreases. Equation ( 55) also shows that increasing the tortuosity or decreasing the porosity ( also appears in b D

) reduces the imaginary part and therefore increases the quality factor. Starting from 0, an increase of the ratio b D / (i.e. a decrease of the permeability), results in an increase in damping, which reaches a maximum value and then goes back to 0 as b D / tends to in"nity. In the low frequency range: i.e., for F( )K1, the dynamic density or apparent mass of the plate can be written as

" ! D [R(x)#jI(x)], (56)
where the function R(x)#j I(x) gives the change in apparent #uid density due to the motion. This function, given by

R(x)" x 1#x , I(x)" x 1#x , ( 57
)
where x" D /b D is plotted in Figure 4 as a function of the dimensionless frequency x. The maximum of I(x) is and is reached for x"x "1. As a consequence, the frequency for which the damping by viscous friction is maximum is given by

" b D D " D . ( 58
)
The permeability a!ects only while the porosity and the tortuosity a!ect both and the maximum value of the imaginary part of . This frequency is almost identical to Biot's characteristic frequency with the di!erence that it accounts for the tortuosity. It has been introduced by Dunn [20], with a qualitative justi"cation, in the study of the low frequency vibrational modes of a #uid-saturated sandstone cylinder. In some applications, can be chosen to coincide with a plate resonance (given by ) by an appropriate adjustment of the plate dimensions or the clamping conditions.

The porosity, the tortuosity and the permeability which are regarded as macroscopic measurable parameters are closely related to the pore microstructure. The porosity and permeability can have a strong e!ect on the vibration of a porous plate [6].

VISCOUS AND INERTIA FORCES

The in#uence of tortuosity has been little studied and is the subject of this section. Figure 5 shows the modulus of the de#ection calculated from the present model and plotted as a function of frequency for the water-saturated sandstone. This "gure shows that increasing the tortuosity shifts the resonance frequency to lower values and increases the quality factor at the same time, in accordance with the qualitative predictions. As is increased, the resonance frequency and the damping tend to a limit. A physical explanation of this e!ect in this frequency area is that a higher tortuosity, also called the &&drag parameter'', corresponds to higher inertia terms so that the apparent dynamic density of the plate is higher. Since according to the classical theory of plates, the resonance frequencies are inversely proportional to the density, a higher density of the porous plate will result in lower resonance frequencies. At the same time, it is reasonable to think that a higher drag parameter is associated with a lower rate of frictional sliding in the pores so that the viscous losses are reduced and the quality factor is enhanced. In other terms, increasing the tortuosity is equivalent to &&dragging'' or &&pushing'' more #uid. This can be achieved only if the #uid does not slip along the solid, and if the #uid does not slip along the solid no losses by viscous friction occur. The study of the in#uence of tortuosity shows the relative contributions of the inertia forces and the friction forces. The conclusion that can be drawn here is that an increase of the inertia forces corresponds to a decrease of the friction forces and vice versa. These forces determine the resonance frequencies and the damping.

VISCOUS FRICTION IN THE PORES

In the example of Figure 3, the smooth shape of the curve is largely due to the fairly high structural loss factor of the porous frame (see Table 1). The damping is also partly caused by the viscous friction occurring in the pores during the vibration. The in#uence of permeability on the damping is studied numerically in this section in the case of the porous sandstone for which a zero loss factor has been taken (see Table 1). It is shown that a frequency for which the damping by viscous friction is maximum exists and is strongly dependent upon the material parameters.

The modulus of the de#ection in the vicinity of the lowest resonance frequency has been plotted in Figure 6 as a function of frequency for di!erent values of permeability. For low values of permeability ( "10\ m), the resonance curve exhibits a sharp peak close to 21 Hz in this example. The sharpness of the peak implies that the plate and the #uid move practically in phase so that the energy loss by viscous friction in the pores is small. As the permeability is increased, the resonance frequency shifts towards a higher value close to 23 Hz. The peak is also very sharp around this frequency. The interesting point is that an area exists between these two permeabilities where the amplitude of the spectrum decreases dramatically and passes through a minimum. The interpretation of this behaviour is that for each value of permeability, a frequency K exists for which the phase shift between the #uid motion and the solid motion is maximum, the energy loss by viscous friction being maximum at this point. This behaviour occurs only at a single frequency for a given value of permeability. Similar processes can be observed in other resonance areas for other values of permeability. In the example of Figure 6, a value of close to 2;10\ m will make the frequency of maximum damping coincide with the resonance at around 22 Hz and renders the peak much smoother. This result can be predicted with a good precision by using the approximate expression for the frequency of maximum damping given by equation (58). A numerical application for three di!erent values of permeability indicates that the damping by viscous friction reaches a maximum at a frequency close to 44 Hz for "10\ m. 21)7 Hz for "2)2;10\ m and 16 Hz for "3;10\ m. Figure 6 shows the maximum damping corresponding to "2)2;10\. This formula can be used as a tool in some applications to cancel and undesired resonance by making the frequency of maximum damping by viscous friction coincide with a structural resonance of the plate. A physical interpretation of the variation of damping with permeability can be given by a simple argument and by the qualitative conclusions of the preceding section on the relative contributions of the viscous and inertia forces: In the limit of zero permeability, the #uid is totally dragged and no friction occurs so that the energy loss by this mechanism is zero and a sharp peak is predicted. In the limit of in"nite permeability, sliding between the solid and the #uid occurs but without energy loss because of the in"nite permeability. Between these two limits, Biot's attenuation occurs with a maximum in the function of viscous friction losses.

CONCLUSION

A theoretical description of the vibration of a porous plate based on the classical theory of homogeneous plates and on Biot's stress}strain relations in porous media has been proposed. For plates thinner than any acoustic wavelength, the assumption of small variations of the amplitude of the relative #uid/solid displacement throughout the thickness can be made and two simple plate equations are found. The "rst of these characterizes the instantaneous elastic response of the #uid-saturated porous plate while the second describes the relative #uid/solid motion, and the delayed after-e!ect if the complex dissipation term is added. Although it is simpler, the model compares very well with the more rigorous theory of Theodorakopoulos and Beskos [6]. For relatively rigid materials where only the inertial interactions are considered, an approximate calculation of the resonance frequencies has been proposed and the in#uences of porosity, tortuosity and permeability have been studied. It is found that the resonance frequency is an increasing function of porosity and of permeability, and a decreasing function of tortuosity. The damping factor is an increasing function of porosity and a decreasing function of tortuosity, and it reaches a maximum value at a frequency determined by the three parameters.

Figure 1 .

 1 Figure 1. System of co-ordinates in the plate.

Figure 2 .

 2 Figure 2. De#ection at the centre of a simply supported water-saturated sandstone excited by a uniform pressure. . . . . , Theodorakopoulos and Beskos; ** curve, the present model.

Figure 3 .

 3 Figure 3. De#ection at the centre of a simply supported air-"lled porous panel. . . . . . , Theodorakopoulos and Beskos; **, the present model.

Figure 4 .

 4 Figure 4. Real (**) and imaginary (* * *) parts of the function representing the change in apparent #uid density due to the plate motion.

Figure 5 .

 5 Figure 5. First resonance area of a simply supported porous sandstone for di!erent values of tortuosity: **, 1)0; ) ) ) ) ) , 1)2; } } }, 1)5; } ) }, 2)0; * } *, 3)0.

Figure 6 .

 6 Figure 6. First resonance area of a simply supported porous sandstone for di!erent values of permeability.

TABLE 1

 1 Plate properties

	Sandstone	Y foam

TABLE 2

 2 Comparison of the natural frequencies calculated from the approximate formula (54) with those obtained from the full model

	(m, n)	Frequency (approx.) (Hz)	Frequency (theoretical) (Hz)
	(2, 2)	10)1	1 0 )0
	(2, 3) or (3, 2)	25)3	2 4 )9
	(3, 3)	40)5	3 9 )7
	(4, 2) or (2, 4)	50)6	5 0 )1
	(3, 4) or (4, 3)	65)7	6 5 )3
	(4, 4)

ACKNOWLEDGMENTS

The major part of this work has been carried out at the University of Hull under the British Engineering and Physical Sciences Research Council (EP-SRC) grant number GR/L54905.

The authors are grateful to Professor D. C. Hothersall for his encouragement and to M. J. Swift for sharing his knowledge of re-cycled porous materials.


P Leclaire

K V Horoshenkov

M J Swift

D C Hothersall

THE VIBRATIONAL RESPONSE OF A CLAMPED RECTANGULAR POROUS PLATE

The vibrations of a rectangular porous plate can be described by two coupled equations involving the time and space derivatives of the de#ection and of the relative #uid}solid motion. These equations were established previously and are solved in the present article by Galerkin's variational method. As in the classical theory of plates (non-porous), the solutions are approximate and their mathematical form is chosen. The Rayleigh}Ritz decomposition on the basis of orthogonal or quasi-orthogonal eigenfunctions is used and the solutions are inserted in the vibrational equations. This procedure leads to an explicit form for the solid lateral de#ection w Q and for the #uid}solid relative diplacement w within the pores. In this study, the chosen eigenfunctions or trial functions are obtained from the linear combination of sinusoidal and hyperbolic functions. The solutions have a single generic form allowing one to account for any boundary condition at the edges involving simply supported, free or clamped edges. Numerical results are given for three di!erent porous plates with the four edges clamped. Simple experiments have also been carried out in order to check the validity of the method. The results are presented and compared with the predictions.

INTRODUCTION

The vibrations of porous structures are important in problems of sound absorption and in the aeronautical industry. The vibration of porous beams and plates has been scarcely studied analytically to date. A recent and signi"cant result has been given by Theodorakopoulos and Beskos [1], who have extended the classical theory of thin rectangular plates to porous materials including Biot's stress}strain relations in porous media [2]. They have established two coupled governing equations and have given the solutions for a simply supported plate by extending Navier's algebraic solution [START_REF] Szilard | ¹heory and Analysis of Plates: Classical ad Numerical Methods[END_REF] to the porous case. Making an additional assumption about the #uid}solid relative displacement and using an alternate form of Biot's relations [START_REF] Biot | [END_REF], Leclaire et al. [5] have proposed two equations of equilibrium, which are valid in applications where the plate is thinner than any acoustic wavelength. The main advantages of the second model are the simplicity of the equations involved without a signi"cant loss of generality, and the possibility to interpret in a more physical manner the #uid}solid interactions associated with the vibration. While the study of boundary conditions other than simply supported from the equations of Theodorakopoulos and Beskos seems to be a di$cult task, the equations proposed by Leclaire et al. easily lend themselves to the generalization of the analytical and numerical methods usually applied in the classical theory of plates. An overview of the most common methods can be found in the book by Szilard [START_REF] Szilard | ¹heory and Analysis of Plates: Classical ad Numerical Methods[END_REF]. The generalization of one of these methods is the main subject of this article. In the present study, Galerkin's variational method is applied to porous plates where a classical set of trial functions obtained from the linear combination of trigonometric and hyperbolic functions is chosen. The method can be applied for any boundary condition involving simply supported, clamped or free edges. The particular case of all four edges fully clamped is treated and a comparison, in this con"guration, between numerical and experimental results is made for three di!erent porous plates.

Another advantage of the proposed model is that its output parameters are the plate lateral de#ection w Q and the #uid}solid relative displacement w. It can be shown that the simple sum of the time derivatives of w Q and w provides the total velocity "eld in the vicinity of the surface of the plate. The total velocity "eld is needed in the evaluation of the radiated sound power and the global vibroacoustic indicators. In problems of sound radiation by non-porous plates and structures, recent and re"ned models include the surface acoustic pressure generated by the plate motion and the e!ect of the surrounding #uid. The present article is one of the "rst to study the problem of vibration of porous plates including Biot's equations of relative #uid}solid motion in the pores. Although the #uid}solid interaction is studied, the contribution of the radiation impedance on the solid displacement is neglected in a "rst approximation. As we are concerned with problems where the #uid is air, the #uid}solid interaction is not symmetrical in the sense that the radiated sound is entirely generated by the solid motion, while the contribution of the #uid on the solid vibration is neglected. This corresponds to the &&light'' #uid approximation. Obviously, in the case of an acoustical excitation, the plate can be excited acoustically or mechanically, and the #uid does generate the solid vibration. However, this action is included in the excitation term and no other external force due to the radiated acoustic "eld is taken into consideration. Although this assumption is made, a reasonably good agreement can be found on the plate de#ection between the experimental data for di!erent materials saturated by air and the calculated in vacuo response of the plate.

The equations proposed by Leclaire et al. [5] are brie#y recalled in section 2. In section 3, these equations are rewritten in an integral form, as required in Galerkin's variational method, the mathematical form of the solutions is chosen, the coe$cients involved in these solutions are calculated from the boundary conditions at the edges and explicit forms are given for w Q and w. The evaluation of the di!erent elastic coe$cients involved in the model is discussed in section 4. Finally, an experimental validation of the theoretical results is proposed in section 5.

EQUATIONS OF EQUILIBRIUM FOR A POROUS PLATE

Two simple equations describing the dynamic equilibrium of thin rectangular porous plates have been proposed by Leclaire et al. [5] for a plate of thickness h subjected to a load q and with a pressure di!erence of P"P(!h/2)!P(#h/2) in the #uid between the two surfaces:

D# Mh 12 w Q #h( w K Q # D w K )"q, ( 1
)
Mh w Q !h( D w K Q #mw K )" P. ( 2
)
Here w Q is the plate de#ection, w is the #uid}solid relative displacement, D is the #exural rigidity, and M are Biot's elastic parameters [6], and D are, respectively, the densities of the #uid}solid mixture and of the #uid, m is the mass parameter introduced by Biot [START_REF] Biot | [END_REF],

given by D /, where is the tortuosity and is the porosity. The space derivatives are written with the help of the operators " ( ) and "*/*x#*/*y of the system of co-ordinates (x, y) while the double dots denote the second time derivative. In the "rst equation of equilibrium (or plate equation), (D# Mh/12) w Q represents the internal potential force (per unit surface) within the #uid-saturated plate, while the inertia terms h w K Q and h D w K , and the load q will be considered as external forces. Similarly, an internal force associated with the #uid}solid relative displacement may be de"ned and is given by Mh w Q , while the external forces can be taken to be hmw K , h D w K Q and P.

VARIATIONAL METHOD FOR SOLVING THE PLATE EQUATIONS

VARIATIONAL EQUATIONS

According to the virtual work principle, the condition of equilibrium of a system is that the total work performed by the internal and external forces during a small virtual displacement is zero:

W G # W C "0. (3)
The virtual displacement and the virtual work are de"ned at constant time. They must be distinguished from the real displacement and from the corresponding work. The virtual work principle can be seen as a particular form of a variational principle formulated as

X X F dz"0. ( 4
)
According to this principle, a function F characterizing the evolution of a system from a perturbed state towards a stable equilibrium state is minimum (extremum in a more general manner) between z and z . In Galerkin's variational method, the function to be minimized is the total virtual work between the boundaries of the middle surface (S):

1 (W G #W C ) dS"0. (5)
The virtual work is calculated from a virtual displacement of the system and from forces obtained directly from the governing di!erential equation which is known a priori and not from the calculation of energies. Upon applying this method to a porous plate and referring to the forces de"ned in section 2, the internal work in the plate is given by the product of the potential force (D# Mh/12) w Q and the virtual displacement w Q :

W G " D# Mh 12 w Q w Q . ( 6
)
The external work is given by

W C "[h( w K Q # D w K )!q] w Q . ( 7
)
Inserting the expressions for W G and W C into Equation ( 5) yields the variational equation of the plate:

1 D# Mh 12 w Q #h( w K Q # D w K )!q w Q dS"0. ( 8
)
For a virtual displacement w, a second variational equation can be obtained for the #uid}solid relative displacement by making use of equations ( 2) and ( 5):

1 Mh w Q !h D w K Q # D w K ! P w dS"0. (9)
In the following, sinusoidal dependence in time is assumed for all quantities so that the terms containing the time can be discarded, the second derivative introducing the squared angular frequency .

CHOICE OF THE MATHEMATICAL FORM OF THE SOLUTIONS

In the linear domain, the displacements are small and are assumed to be normal to the surface (a, b) de"ned by the lateral dimensions of the plate which coincide with the system of co-ordinates (x, y). The problem consists in "nding suitable expressions for w Q and w verifying the system of integral equations ( 8) and ( 9) as well as satisfying the boundary conditions at the edges of the plate. Various approximate functions have been tried in the case of non-porous plates where only one governing equation is considered, such as a linear combination of trigonometric and hyperbolic functions [7,8], polynomials (see reference [9] for example) or trigonometric functions [10]. The following approach is used in the present study: a possible solution for the solid de#ection is sought "rst. The general form of this solution is also applied to the other unknown w and to the excitation terms. All these terms are then inserted in equations ( 8) and ( 9) leading to new equations. Finding solutions to these is a veri"cation of their compatibility and provides the amplitude coe$cients of w Q and w.

The form of the trial functions chosen is the combination of trigonometric and hyperbolic functions. As in the Rayleigh}Ritz method, the classical decomposition of the solutions on the basis of orthogonal eigenfunctions is used. The solid de#ection is written as

w Q (x, y)" P L =Q PL PL (x, y). ( 10
)
Since the plate is rectangular, it is convenient to write each function PL (x, y) as the product of two independent functions called the beam function P (x) and L (y) of one variable only:

PL (x, y)" P (x) L (y). ( 11
)
The chosen form for the beam function is

P (x)"C P sin P x a #C P cos P x a #C P sinh P x a #C P cosh P x a , ( 12
)
where P is a frequency parameter corresponding to the rth root of a characteristic equation. The mathematical form of this equation and the coe$cients C P , C P , C P , C P are imposed by the boundary conditions.

BOUNDARY CONDITIONS

The standard boundary conditions are applied to the porous solid which is considered as a homogeneous e!ective medium in Biot's approach of porous media. For a clamped beam of length a, the conditions at the edges are that the solid displacement and its "rst derivative are zero: w Q (0)"w Q (a)"0 and w Q (0)"w Q (a)"0, corresponding to edges without motion of translation or rotation. The determinant of the homogeneous boundary conditions is then obtained from the following equations:

0 #C P #0 #C P "
) 13
Numerical values for P can be found in reference [7]. The values of C P , C P , C P , C P and P are then inserted into the expression for the beam function P (x). The beam function in the y direction L (y) is deduced from the expression for P (x), where x, a and r are replaced by y, b and n.

If the beam is simply supported, C P "1, C P "C P "C P "0 and the frequency equation is sin( P )"0. The same generic expression can be used for P (x) in the case of other boundary conditions involving a combination of simply supported, free, clamped or guided edges. Each case is described by a di!erent set of coe$cients C P , C P , C P , C P and a frequency equation leading to di!erent values of P .

SOLUTIONS

In this study, one seeks solutions for w(x, y), w Q , w, q and P with the same mathematical form as w Q :

w(x, y)"

P L = PL PL (x, y), w Q (x, y)" P L ( w Q ) PL P (x) L (y),
w(x, y)"

P L w PL P (x) L (y),
q(x, y)"

P L Q PL P (x) L (y),
P(x, y)"

P L P PL P (x) L (y). (14)
Here, the coe$cients =Q PL and = PL are the unknowns of the problem. After insertion into equations ( 8) and ( 9), the solutions, if they exist, must satisfy the following equations:

P L ? @ =Q PL D# Mh 12 ( '4 P L G I #2 '' P L G I # P L G I ) !h P L G I != PL D h P L G I !Q PL P L G I dx dy"0, (15)
P L ? @ (=Q PL [ Mh( '' P L G I # P '' L G I )#h D P L G I ] != PL h D P L G I ! P PL P L G I ) dx dy"0. ( 16
)
The fourth and second order derivatives are, respectively, denoted by the exponents (IV) and (II). The eigenfunctions and their derivatives satisfy the orthogonality requirement (depending on the choice of eigenfunctions, this condition can be satis"ed approximately).

The terms with non-identical subscripts r, i and n, k are therefore zero or negligible and equations ( 15) and ( 16) can be rewritten as Solving system (17, 18) for a couple (r, n) yields the coe$cients =Q PL , = PL and the responses are given by equations (10) and (11). For a simply supported plate, the integrals take a simple form and a solution similar to Navier's algebraic solution [START_REF] Szilard | ¹heory and Analysis of Plates: Classical ad Numerical Methods[END_REF] for a porous plate [1] is obtained.

P L =Q PL D# Mh 12 (I I #2I I #I I )!h I I != PL h D I I " P L Q PL I I . ( 17
The coe$cients Q PL and P PL correspond to the external exciting forces applied to the porous plate and are determined by the way the plate is excited. Rigorously, the external forces are responsible for the motion of the plate, which in turn, contributes to the loading of the plate but this contribution is neglected in this study in a "rst approximation. The coe$cients Q PL and P PL can be written [START_REF] Szilard | ¹heory and Analysis of Plates: Classical ad Numerical Methods[END_REF] as 17) and (18) are equal to F P (x ) L (x ) and 0 respectively. If the plate is excited acoustically by a plane wave of amplitude P G , the pressure applied on the #uid}solid mixture and the pressure di!erence in the #uid are uniformly distributed. In this case,

Q PL " ? @ q(x,
Q PL "16P G / rn (20)
with (r, n) odd [START_REF] Szilard | ¹heory and Analysis of Plates: Classical ad Numerical Methods[END_REF]. From reference [11], the values taken for P PL are Q PL . From previous theoretical works, di!erent mechanisms of energy dissipation can easily be included in the calculations. Forces of viscous frictions in the pores can be introduced in equations (1, 2) and (8,9). These forces are derived from a dissipation potential and their e!ect is accounted for by considering the tortuosity to be complex and frequency dependent,

( )" !j D F( ). ( 21
)
where is the #uid dynamic viscosity and is the solid permeability. The function F( ) is the viscosity correction function and can be evaluated from a model [START_REF] Allard | Sound Propagation in Porous Media: Modelling Sound Absorbing Materials[END_REF]. This function is close to 1 in the low-frequency domain where this study is carried out. The dynamic tortuosity ( ) is inserted instead of into equations ( 9) and (18). Structural damping is included in the model by considering complex elastic moduli. For air-saturated materials, the energy loss by thermal exchanges between the solid structure and the #uid can be included by considering a dynamic compressibility [START_REF] Lafarge | [END_REF]. The #uid compressibility appears in Biot's elastic coe$cients.

A NOTE ON BIOT'S ELASTIC COEFFICIENTS

Biot's coe$cients of interest in this study are and M and are given by [6]

"(1#Q/R), M"R/, (22, 23)
where the elastic coe$cients Q and R can be written as

Q" (1!c) Q (1!c) Q /K Q #/K D , R" (1!c) Q /K Q #/K D with c"K/ Q K Q and Q "1!.
In these expressions, K Q , K D and K are, respectively, the bulk moduli of the solid constituting the frame, of the #uid and of the frame at constant #uid pressure. The coe$cient c can be seen as a coe$cient characterizing the consolidation state of the heterogeneous material. Since K is less than the Hashin}Strikman upper bound

Q K Q [14]
, c is between 0 (non-consolidated materials) and 1 (consolidated materials). The following relation is further obtained:

"1!c Q (24)
and the bounds for are (consolidated materials) and 1 (non-consolidated materials). It is interesting to note the similarity for c"0 between the expression for Q and R and the Hashin}Strikman lower bound. Biot's elastic coe$cients can be determined from hydrostatic experiments [15]. Since the #uid is air, the approximation K D K Q , K can be made. The materials are fairly rigid and so we shall also consider that c"1 and " in the numerical computation. As a result, the coe$cients of interest are approximated by

QK0, RKK D , MKK D /.
It may be noticed that the #exural rigidity or bending sti!ness

D" Eh 12(1! ) " K(1!2 )h 4(1! ) (25)
is a key parameter in the problems of plate vibration. It involves Young's modulus E and the Poisson ratio of the solid frame. The coe$cient K given as a function of E and by K"E/3(1!2 ) must therefore be evaluated in any case. As a consequence, the parameters c, and M can be determined without making approximations if the bulk modulus K Q of the solid constituting the porous frame is known.

NUMERICAL AND EXPERIMENTAL RESULTS

PRELIMINARY TESTS

Three plates were studied. Their properties are summarized in Table 1. The plates were obtained by consolidating the particles of a material with a binder. The CoustoneR plate contains #int particles with a mean grain size of about 1 mm and an epoxy rubber binder. The other two plates designated as G foam and Y foam are fabricated from particles of plastic foam obtained from recycled car dashboards. Their grain size varies from a few tens of m to 5 mm and the binder is manufactured by Hyperlast LTD.S For each material, the permeability was measured from #ow resistivity tests. The tortuosity has not been measured but estimated from Berryman's formula [16] "(1#1/)/2 as a value between 1)2 and 1)8 depending on the porosity. It was noticed that this coe$cient does not have a strong in#uence on the response of the plates studied. The numerical value of Young's modulus used for the Coustone is the one provided by the manufacturer. For the dashboard foams, E was evaluated experimentally from a simple experiment of bending of a cantilever beam submitted to a known force at the free end and from the corresponding de#ection formula.

TABLE 1

Information necessary for the computation of the plate response

NUMERICAL RESULTS

The plate response to a uniform acoustical excitation was calculated from the variational method described in section 3. The Gauss integration method with 40 terms of Legendre polynomial was used in order to obtain a fast and accurate evaluation of the integrals (I )}(I ). The response was calculated at the centre for three plates. The values for the density and the dynamic viscosity of air are D "1)2 kg/m and "1)839;10\ kg/m/s respectively. A typical value of 0)35 is taken for the Poisson ratio for all materials. In a "rst step, the computation included only the energy loss by viscous friction between the solid and the #uid. From the experimental results, it appeared that the observed damping could not be explained by this mechanism alone. The viscoelastic damping in the solid structure plays an important role and complex Young's modulus and the Poisson ratio had to be considered. An imaginary part between 10 and 15% of the real part has been taken for E and for (Table 1). Since a detailed analysis of structural damping is not the subject of this article and for the sake of simplicity, these values were chosen arbitrarily and independent of frequency, in order to match the experimental results on the damping.

EXPERIMENTS AND COMPARISON

The experimental set-up is shown in Figure 1. A porous panel clamped at the four edges was excited by the low-frequency noise produced by a loudspeaker of 40 cm diameter. E!ective clamping was achieved by using two heavy steel frames of about 25 kg each such that the mass of the sample was only a few per cent of the mass of the frames. The plate de#ection was measured by using an accelerometer of negligible mass (a few grams). With the help of a microphone at 1 cm from the surface, the amplitude of the incident pressure P G at (x"a/2, y"b/2) was evaluated to a value of about 0)1 Pa. The theoretical and experimental plate de#ections are plotted in Figures 2}4 versus frequency for three di!erent porous plates. Since the absolute value of the experimental de#ection was not available, an arbitrary reference value was taken for the theoretical calculation chosen to enable the comparison. The experimental response was measured at (x"a/2, y"b/2). Since the plate is excited by a uniform pressure, only the odd modes with an even number of nodal lines are

TABLE 2

First theoretical and experimental resonance frequencies (in Hz) for the G foam plate excited. A reasonable good agreement is found when comparing the experimental and theoretical frequencies for these modes (see Tables 2}4), with a maximum error around 10% in the range of samples studied. The resonance frequencies are strongly dependent on Young's modulus E, the Poisson ratio and the porosity. In the case of the G foam, the "rst resonance, located at about 10 Hz, was in the low-frequency limit sensitivity of the microphone so that the experimental results are not reliable below this frequency.

The general shape of the experimental curves compares well with the theoretical curves. The discrepancies can be due to the fact that frequency-independent elastic moduli have been considered in the model in this simple analysis. In addition, it has been assumed for the calculations that the incident pressure "eld was uniformly distributed over the surface of the plate in the whole frequency range, while it is not perfectly constant in the real experimental con"guration.

CONCLUSION

The de#ection of a porous plate and the relative #uid}solid displacement are important in the study of the e!ect of resonances on the surface impedance of porous layers. In this article, a variational method for solving the plate equations has been used and the solutions have been given for di!erent porous materials with the four edges clamped. The e!ect of #uid loading has been neglected and no external force other than the excitation terms has been taken into consideration. This e!ect is closely related to the calculation of global vibroacoustic indicators: i.e., the mean square velocity, the radiated sound power and radiation e$ciency. Nevertheless, the "rst comparison between experimental and theoretical results on the plate defection is reasonably good if the viscoelastic damping in the solid is accounted for.

R

  CEP Acoustics Ltd, Unit C Falcon Court, Clayton Business Park, Clayton Le Moors, Accrington BB5 5JD, U.K. S Hyperlast Ltd, Station Road, Birch Vale, High Peak SK22 1BR, U.K.

Figure 1 .

 1 Figure 1. Experimental set-up.

Figure 2 .

 2 Figure 2. Calculated and measured de#ection of the G foam: **, theory; ))))))))), experiment.

Figure 3 .

 3 Figure 3. Calculated and measured de#ection of the Y foam: **, theory; ))))))))), experiment.

Figure 4 .

 4 Figure 4. Calculated and measured de#ection of the Coustone: **, theory; ))))))))), experiment.

TABLE 3

 3 First theoretical and experimental resonance frequencies (in Hz) for the > foam plate

	Theory	18)5	6 6 )9	111)5	159)7
	Experiment	18)8	6 9 )8	115	168

TABLE 4

 4 First theoretical and experimental resonance frequencies (in Hz) for the Coustone plate

	Theory	41)7	147)0
	Experiment	38)8	K133

ACKNOWLEDGMENTS

We are grateful to Prof. A. Cummings for his encouragement and his comment on this article and to E. Chadwick for producing coustone plates. This work has been carried out with the support of the British Engineering and Physical Sciences Research Council (EPSRC) grant number GR/L54905.


P Leclaire

O Umnova

K V Horoshenkov

L Maillet

Porosity measurement by comparison of air volumes

A method for measuring the volume of a solid with arbitrary shape and the open porosity ͑volume fraction of the open pores͒ of air-saturated porous materials is proposed. The experimental principle is inspired from Beranek's method ͓J. Acoust. Soc. Am. 13, 248 ͑1942͔͒ based on the application of the ideal gas law to the gas surrounding the solid. The originality of the proposed method lies in the comparison of the volume of a measurement chamber containing the sample with that of an empty reference chamber. During a measurement, the reduction of volume in the measurement chamber due to the introduction of the solid is compensated by increasing the volume of a piston connected to the measurement chamber. An important aspect of this experimental system is that the influence of temperature can be considerably reduced as nearly the same variations of temperature apply simultaneously to the measurement and the reference chambers. A preliminary calibration experiment without sample in the measurement chamber is carried out in order to equalize the volumes of the two chambers. The calibration can be performed once and for all, thus minimizing the number of operations in the measurement process. The porosities of several materials are measured and compared to reference values. The precision and possible future improvements are discussed.

I. INTRODUCTION

The porosity of a porous material corresponds to the volume proportion of fluid contained in the pores, i.e., the total volume of the fluid in the pores divided by the volume of the sample. It is an essential parameter in the physical and acoustical modeling of porous media. In this article, a method for measuring the porosity of air-saturated materials with reduced thermal effect is proposed. The applications also concern chemistry, the pharmaceutical industry, and powder technology. Determining the porosity is equivalent to measuring the volume of a solid with arbitrary shape. The most common methods involve the saturation of the sample with water. For example, the volume of solid can be measured from the rise of the water level after having plunged the sample in water in a measuring cylinder. Alternatively for a porous solid, the density of the dry sample and the density of the sample when saturated with water provide an evaluation of the porosity. However, water-saturation methods are not always appropriate as these can damage the sample or modify its properties. The description of various methods for measuring porosity such as mercury porosimetry, buoyancy, or methods based on density measurements can be found in Ref. 1. Different types of porosities are defined and open, trapped, or total porosity can be determined depending on the methods selected. The method proposed in this article is based on the compressibility of the air in the connected pores and therefore this method yields the open porosity.

II. PREVIOUS WORKS

Beranek [START_REF] Beranek | [END_REF] proposed a simple method for the determination of porosity without involving saturation of the sample with water. This method is based on the great difference in compressibility between a solid and a gas. In Beranek's method, a porous sample is introduced in an air-tight chamber that is connected to a U-tube manometer as shown in Fig. 1.

The situation shown in Fig. 1 can be obtained by pouring water from the left branch of the manometer if the U-tube is initially empty. The difference of pressure (aϪb) corresponds to the gauge pressure in the chamber. The pressures can be related to volumes with the help of Boyle's law ͑ideal gas law for a constant temperature process͒ and if this experiment is done with and then without sample, it is possible to infer the volume proportion of solid and the porosity. Boyle's law assumes that the processes are isothermal. Therefore the experiment must be carried out very slowly and in a thermostatic container, for example, in a tank containing a mixture of liquid water and ice.

Based on this principle, Champoux et al. 3 designed a much more elaborate system providing results with an excellent precision in a wide range of porosities. Their system involves a micrometer drive for precise variations of volume in the chamber, a sensitive differential pressure transducer to replace the U-tube manometer, and a large air reservoir to isolate the apparatus from atmospheric pressure fluctuation. Champoux et al. also gave solutions to achieve a good temperature stabilization.

III. NEW METHOD BASED ON VOLUME COMPARISON BETWEEN A REFERENCE CHAMBER AND A MEASUREMENT CHAMBER WITH VARIABLE VOLUME

A. Experimental setup

The experimental principle of the proposed method is shown in Fig. 2. Two air-tight chambers of approximately 60 ml, a reference and a measurement chamber, are connected to the branches of a U-tube manometer containing water. The volume of the reference chamber must be slightly higher than that of the measurement chamber for a reason that will become apparent later. The manometer is made up of glass tubes with an inner diameter around 5 mm, large enough to neglect capillary effects. Water can be removed from and reinjected in the manometer with the help of a water-drawing piston. If valves A and B are closed, drawing water will have the effect of increasing both the reference volume and the volume in which the sample is contained. A measurement piston of 20 ml capacity is connected to the measurement chamber. If smaller volumes are involved, a smaller piston of 10 ml capacity can be used instead.

B. Calibration

A calibration experiment is carried out first in order to equalize the volumes of the reference chamber and the measurement chamber in the absence of sample and with the measurement piston to zero. The two chambers are first set at atmospheric pressure ͑valves A and B opened͒ and starting from a high level of water in both branches, A and B are closed and water is drawn. If the volumes of the two chambers were identical, the pressure difference (aϪb) would be close to zero. Equalizing the two volumes can be achieved by introducing volume absorbers, e.g., pieces of nonporous solid in the reference chamber and by repeating the procedure described in this section several times until (aϪb) approaches zero. The need to introduce volume absorbers in the reference chamber in order to equalize the volumes explains why, initially, the reference chamber must have a greater volume than that of the measurement chamber.

C. Principle of a measurement with reduced thermal influence

The proposed technique involves a number of operations that should be carried out in a certain order and is explained in this section. After the calibration has been completed, a porous sample is placed in the measurement chamber. This is done after the two chambers are set at atmospheric pressure ͑A and B opened͒. The reduction in the volume of air due to the introduction of the porous solid is compensated by varying the volume of the measurement piston ͑starting from the zero position͒. The measurement piston should be moved only when B is open. After each movement of the waterdrawing piston, the chambers are returned to atmospheric pressure by opening A and B. As water is drawn in the successive tries ͑with A and B closed͒, the difference in head levels (aϪb) between the two branches of the manometer approaches 0 when the volume in the measurement piston corresponds to the volume of solid of the porous sample. The proportion of solid ͑and hence the porosity͒ is obtained by dividing the volume of solid by the volume of the sample.

The originality of this method lies in the fact that the volumes of the two chambers are compared and equalized so that temperature variations or any atmospheric pressure fluctuation act simultaneously on both chambers. Consequently, the thermal effects nearly ''cancel out'' and this method has the advantage that the influence of temperature is considerably reduced. Also, the calibration experiment needs to be done only once, so that measurement on any new sample will involve only one measurement after placing the sample in the measurement chamber, thus minimizing the number of operations in the experimental procedure. Although minimized, the temperature still has an influence as the experimental system is not perfectly symmetrical. The movements of all the pistons involved in this experiment should be smooth and gentle. In particular, one should make sure that the passage of the air through the narrow constriction be- tween the measurement piston and the chamber is slow. After some experience, it takes about 20 min to calibrate the device and about 15 min per sample to determining the porosity of materials described in the next section.

IV. MEASUREMENTS, RESULTS, AND PRECISION

The porosities of seven samples were measured: two sands, expanded clay granulates, a porous concrete, spherical particles, and two sugar granulates. The results and useful information about the samples are listed in Table I. The porosities measured by the method proposed in this article were compared to reference values for three samples: the spherical particles and the two sands. It was not possible to compare the results with reference values for the expanded clay granulates, the porous concrete, or the sugar crystals.

A. Spherical particles

The particles studied were sugar beads with a particle size around 1 mm used to decorate cakes in bakeries. These beads represent an excellent model of uniform spherical particles with a very narrow particle size distribution. A sample volume of 28 ml of beads was measured in a measuring cylinder. The error on the determination of the sample volume was taken to be Ϯ0.5 ml ͑Ϯ1.8%͒. The experiment was carried out and the final volume read on the large measurement piston was 17 ml with an error estimated to Ϯ0.2 ml. Hence, the total error on the reading on the piston is estimated to be Ϯ0.4 ml ͑Ϯ2.4%͒ if the error in the calibration experiment is assumed to be the same. The overall precision is therefore close to Ϯ4.2% and the porosity is 1Ϫ17/28 ϭ0.393Ϯ0.017. The reference values displayed in Table I correspond to theoretical values for a random packing of spheres. The value of porosity generally accepted for a random dense packing of uniform spheres is 0.36 ͑Refs. 1 and 4͒. For a random loose packing of uniform spheres, the value of 0.45 can be taken as a reference. 4 In the experiment, no particular action was taken ͑e.g., rolling, shaking, or com-pressing͒ to pack the sample. Apart from eventual electrostatic forces resulting from friction between particles, no interparticle interactions that can occur in water-saturated particles were expected and a measurement of porosity performed in air should give values between these two limits. The value of 0.393Ϯ0.017 we obtained was in the expected range. The experiment was repeated several weeks later for this sample. The sample volume selected was the same as in the first experiment ͑28 ml͒. In the second experiment, the calibration procedure was carried out first and the final value of the difference (aϪb) read was ϩ0.5 mm. Then the sample was placed in the measurement chamber and the data recorded. Table II shows the sequence of data recorded in the measurement process. Starting from 18 ml, the volume set on the graduated piston was varied in successive tries: 18, 14, 17 ml, . . . and the corresponding values of (aϪb) were read: ϩ3.5, Ϫ3, ϩ2 mm, . . . . This difference decreased until it reached a value close to the same difference of ϩ0.5 mm obtained in the calibration process. The volume displayed on the piston was then considered to be the final value and the actual volume of solid. The value obtained for the porosity was 0.414Ϯ0.017. This greater value seems to indicate that the sample was less compacted than in the first experiment.

B. Redhill and Olivine sands

For the Redhill sand, a sample volume of 29Ϯ0.5 ml ͑error Ϯ1.7%͒ was selected. The porosity experiment was carried out and the final volume read on the large measurement piston was 16Ϯ0.4 ml ͑error of 0.2 ml for the measurement added to 0.2 ml for the calibration, i.e., Ϯ2.5%͒. The porosity was found to be 0.448Ϯ0.019 ͑total error of Ϯ4.2%͒. The usual method by water saturation was used, providing a reference porosity of 0.41Ϯ0.06. This value was obtained by pouring 22Ϯ0.5 ml of sand in 34Ϯ0.5 ml of water. The water level rose to a level of 47Ϯ0.5 ml so that the volume of the solid was 13Ϯ1 ml. Although the total precision of Ϯ10% was not excellent, this measurement provided a reference value to which to compare the value obtained with the new method.

The same experiments ͑air-based method and water satu-ration͒ were performed for the Olivine sand. A sample volume of 13.5Ϯ0.5 ml and a solid volume measured with the smaller piston of 7.5Ϯ0.2 ml were measured. This error of Ϯ0.2 ml corresponds to Ϯ0.1 ml for the measurement added to Ϯ0.1 ml for the calibration. For the water saturation experiment ͑sample volume of 40Ϯ0.5 ml, solid volume of 23Ϯ1 ml͒, the precision was slightly better than for the Redhill sand because it was possible to select a greater sample volume.

C. Expanded clay granulates

This material is composed of particles with a range of sizes between 2 and 3 mm. The particularly of this material is that each particle itself is porous. The expected high value of porosity was confirmed by the experimental result. It may be noticed that this method leads to the measurement of the open porosity only. The trapped porosity corresponding to air inclusions inside the porous particles is not determined. There was no alternative method to which to compare the result as saturating the grains with water posed the problems of damage to the sample and modification of its porosities. Two main reasons explain why the precision was not as good as for the other samples. The greater size of the grains was responsible for a greater uncertainty in the evaluation of the sample volume ͑the value was 20Ϯ1 ml͒. The high porosity of the materials implied that the measured solid volume with the air-based system was small so that the relative error was greater. The reading on the small measurement piston was 5.5Ϯ0.2 ml and the total error was Ϯ8.6%.

D. Porous concrete

The sample volume was 24Ϯ1 ml and the measured solid volume was 16.5Ϯ0.4 ml.

E. Sugar crystals

The porosities of an ordinary sugar and of a cocrystalline sugar were measured and compared. Cocrystalline sugar is made up of spherical shaped agglomerates which incorporate some air. 5 Because these agglomerates are porous, the bulk density of cocrystallized sugars ͑0.62-0.64 g/cm 3 ) is lower than that of conventional sugars ͑0.81-0.84 g/cm 3 ) ͑Ref. 6͒. The measured densities of the ordinary and the cocrystallized sugars in this study were, respectively, 0.95 and 0.61 g/cm 3 . The porosities were measured and the expected higher value for the cocrystalline sugar was confirmed by the experimental results. A porosity of 0.377Ϯ0.014 was obtained for ordinary sugar crystals while the porosity of cocrystalline sugar was found to be 0.583Ϯ0.029 ͑see Table I͒.

V. ALTERNATIVE DESIGN AND FUTURE IMPROVEMENTS

Discarding the error on the measurement of the sample volume which does not depend on the proposed measurement system, the error on the reading of the volume of solid in the measurement chamber was in the order of 2.5% for the beads, the two sands, and the porous concrete. For the clay granulates this error was 3.6%. This greater error can be explained by the fact that the clay granulates have a significantly higher porosity than the other materials resulting in a smaller volume of solid measured in the chamber. Since the sizes of the measurement chamber and of the porous samples are limited, improving the precision requires increasing the precision of the value read on the measurement piston as well as the precision of the calibration. A possible solution to achieve this is depicted in Fig. 3 where two measurement pistons are connected to the measurement chamber. This configuration corresponds to a double scale precision reading. The idea is that the larger piston provides a gross value of the volume while the smaller piston can be used for fine tuning. The capacity of the small piston should correspond to a convenient integer number of the unit graduation on the large piston, for example, 1 ml capacity for 1 ml unit graduation.

The improvement of the precision on the reading must be accompanied by an improvement in the calibration. A possible solution is to design the reference chamber slightly smaller that the measurement chamber and to connect a small piston to it. This solution avoids the repeated opening and closing of the reference chamber to introduce or remove the ''volume absorbers.'' The equalization of the volumes is achieved by varying the volume of the calibration piston until (aϪb) is small. A greater flexibility as well as symmetry are gained with this modified system.

Although attractive, these solutions were not attempted in the present study. The reason for this is that the best solution to achieve excellent precision is to redesign the entire system in a similar way as Champoux et al. improved Beranek's design. The improvement of the present setup will require precise systems for varying the volumes ͑these improvements are more costly͒. A possible solution for this purpose is the use of precision motor driven syringe volume variators. A sensitive differential pressure piezometer will also be required as well as an improved thermal insulation for the chambers ͑the thermal effects are already greatly reduced in comparison to Beranek's method but can be reduced further͒. Because the implementation is far more expensive than the present device, it was not attempted in the present study. However, after these improvements are carried out, it is expected that our system will perform as well as the system of Champoux et al. but with less dependence upon temperature variations. Furthermore, this refined system can incorporate the new features described in Fig. 3, for even more flexibility and ease of use.

VI. DISCUSSION

It is thought that the technique proposed in this article is the best possible to reduce the thermal effects on the measurements, the reason being that the volumes of the reference and measurement chambers are equal in the end so that the temperature variations ͑acting simultaneously on both cham-bers͒ ''cancel out.'' If the proper modus operandi is followed, the measurement is fast, simple, and does not require the application of a complicated mathematical formula. The porosities of several materials were measured with a typical precision of the order 5% for porosities of the order of 0.4. It was also found that in the present experimental configuration, the error increases with porosity and the experimental setup presented in this article is adapted to the measurement of low porosities. More costly future improvements should increase the precision and permit the measurement of higher porosities for materials such as fibrous materials and highly porous foams with open cells ͑reticulated foams͒. The method yields the open porosity and this limitation must be borne in mind with respect to foams containing closed pores.

FIG. 1 .

 1 FIG. 1. Experimental setup based on Beranek's method.

FIG. 3 .

 3 FIG.3. Improvement in the design of the air volume comparison system for the measurement of porosity.

TABLE I .

 I Porosity measurements results.

			Measured	Reference
	Sample Name	Information on sample	porosity	value
	Spherical	Ϸ1 mm particles with narrow size	0.393Ϯ0.017	0.36 -0.45
	particle	distribution
	Redhill sand-	Coarse sand 0.59Ϯ0.16 mm	0.448Ϯ0.019	0.41Ϯ0.04
	Grade 14/25
	Olivine sand	Finely dispersed sand	0.444Ϯ0.028	0.425Ϯ0.024
	Expanded clay	Double porosity sample ͑collection of	0.725Ϯ0.063
	granulates	porous particles͒. 2 to 3 mm
	Laterlite LECA
	Porous concrete	Mix of limestone 5Ϯ1 mm, Cauldon	0.312Ϯ0.021
	mix	RHP cement, Limestone dust, Sika
		Plastocrete and water.
	Sucrose	Ordinary commercial sugar crystals	0.377Ϯ0.014
	Cocrystalline	Commercial cocrystalline honey	0.583Ϯ0.029
	sugar	granules

TABLE II .

 II Example of recorded data for evaluation of porosity ͑data for the spherical particles͒.

			Approximate time before
	Volume set on measurement	Pressure difference (aϪb)	taking the reading
	piston in ml	͑mm͒	͑min͒
	18	ϩ3.5	1
	14	Ϫ3	1
	17	ϩ2	1
	15	Ϫ2.5	1
	15.7	Ϫ1	2
	16	Ϫ0.5	3
	16.4 ͑final value͒	Ϸ ϩ0.5	5

ACKNOWLEDGMENT

This work was supported by the Engineering and Physical Sciences Research Council ͑EPSRC͒ Grant No. GR/ R43761/01.