A Linear Theory of Porous Elastic Solids
306PASQUALE GIOVINE considerations.Afterwards we obtain the explicit forms of the constitutive?elds for isotropic materials in terms of the strain measures and apply general hypotheses on macrodisplacements and microstrains to get the linear theory,also formulating a displacement boundary-initial-value problem.
At last,we test our model in applications:(i)we consider deformation processes that are homogeneous and quasi-static and observe properties of porous solids similar to those ascribed to materials with fading memory;(ii)we study the propagation of acoustic waves and?nd solutions in the purely transverse and longitudinal case:in particular,we obtain not only classical elastic waves but also micro-waves propagat-ing without affecting the elastic characteristics of the matrix material.All the results are seen to be consistent with those in[5].
2.Kinematics and Balance Equations
The balance equations for a porous elastic material with empty large voids,that do not diffuse through the skeleton,was obtained in[3],where we supposed that each material element of the body is able to have a microstretch different from and independent of the local af?ne deformation ensuing from the macromotion.This makes the proposal in[1]broader for a continuum with small spherical pores which may contract and expand homogeneously without having a gyratory movement:there we relaxed the former hypothesis by allowing distinct microstrains of the voids along principal axes of microdeformation,but maintained the absence of microrotations of the voids themselves.
The continuous material body B is identi?ed with a?xed region B?of the three-dimensional Euclidean space E,the reference placement,and a generic material element of B?is denoted by x?;thus,the motion of B,as a continuum with ellipsoidal structure,is a pair of smooth mappings on B?× ( is the set of real numbers), x=x(x?,τ)∈E and U=U(x?,τ)∈Sym+,(1) where x gives the spatial position at timeτof the material point which occupied the position x?in the reference placement,U is the microstructural tensor?eld describing the changes in the pore structure and Sym+is the set of second order symmetric tensors with positive determinant.Further,x(·,τ)is a bijection,for each τ,with deformation gradient F in Lin+(Lin+being the collection of all second order tensors with positive determinants):
F(x?,τ)=?x
?x?
(x?,τ)∈Lin+.(2)
The thermal behaviour of the material is described by another smooth mapping on B?× ,the absolute positive temperatureθ=θ(x?,τ)>0.
Through the inverse mapping x?(·,τ)of x(·,τ),we can consider all the relevant ?elds in the theory as de?ned over the current placement Bτ=x(B?,τ)of the body B.The local equations of balance governing an admissible thermo-kinetic process
A LINEAR THEORY OF POROUS ELASTIC SOLIDS 307for porous solids with large voids which are nonconductors of heat are the following (see §4of [3]):
˙ρ+ρdiv ˙x =0
(conservation law of mass),(3)ρ¨x =ρf +div T (balance of linear momentum),(4)
ρκ¨U =ρB ?Z +div (balance for micromomentum),(5)
skw T =skw UZ T +grad U
(balance of angular momentum),(6)and
ρ˙ =T ·grad ˙x +Z ·˙U + ·grad ˙U +ρλ(?rst law of thermodynamics).
(7)
Here the reference placement B ?was chosen so that the reference microinertia tensor ?eld J ?has spherical value:J ?=κI ,where I is the identity tensor and κthe nonnegative microinertia coef?cient depending on the reference geometric features of the pores:
κ=?κ(U ?) 0;(8)moreover,ρis the mass density,f the vector body force,T the Cauchy stress tensor,ρB and ?Z are the resultant symmetric tensor densities per unit volume of external and internal microactions,respectively, the microstress third-order tensor which is symmetric in the ?rst two places, the speci?c internal energy per unit mass and λthe rate of heat generation per unit mass due to radiation;at last,the superposed dot means material time derivative,while skw A denotes the skew (or antisymmetric)part of a second-order tensor A and (grad U )ij :=U ih,k jhk .
Internal microactions include interactive forces between the gross and ?ne struc-tures as well as internal dissipative contributions due to the stir of the surface of pores,while the microstress is normally related to boundary microtractions,even if,in some cases,it could express weakly nonlocal internal effects;?nally,external ones can be interpreted as an externally controlled pore pressure.
For a material which is a nonconductor of heat,the classical imbalance of entropy comes down to the subsequent one:
ρθ˙η ρλ(Clausius–Duhem inequality),(9)wherein ηis the density of entropy per unit mass;if we introduce the Helmholtz free energy per unit mass ψ:= ?θηand use Equation (7),we obtain a reduced version of this inequality,that is,ρ ˙ψ
+˙θη T ·grad ˙x +Z ·˙U + ·grad ˙U.(10)Remark .The voids theory of Nunziato and Cowin [1]can be recovered by imposing that U is constrained to be spherical (see §5of [3]).
308PASQUALE GIOVINE
3.Constitutive Prescriptions and Thermodynamic Compatibility
Our constitutive equations model a porous elastic body with homogeneous structure.We assume that the overall response of the material depends upon the deformation gradient F ,the microstretch U and its reference gradient ?U ,the temperature θand the time rate of change of the microstretch ˙U :the last one accounts for inelastic surface effects associated with changes in the deformation of the pores in the vicinity of the void boundaries.
Thus,let us call the array S :={F,U,?U,θ}of variables the elastic state of the material and,assuming equipresence,postulate as constitutive relations for constitutive quantities the following:
ψ=?ψ(S ,˙U),η=?η(S ,˙U),T =?T
(S ,˙U),Z =?Z(S ,˙U)and =? (S ,˙U);(11)
all the tilde functions in (11)are assumed to be twice continuously differentiable with respect to all arguments.Also,our constitutive equations have a dependence on an initial state {F,U,θ}={I,U ?,θ?}of the body that we usually suppress in our notations for general considerations,but explicitly express for peculiar applications (if it is necessary).
Now,we have to check the compatibility of the constitutive prescriptions (11)with the Clausius–Duhem inequality in the reduced version (10).Making use of the chain rule of differentiation,we obtain the following equivalent relation that is valid for all values of the constitutive variables S and ˙U :ρψ˙U ·¨U +ρ(η+ψθ)˙θ+ ρψF F T ?T ·grad ˙x ++ ρψ?U F T ? ·grad ˙U
+(ρψU ?Z)·˙U 0,(12)where the operator acts ?lling the near index of two tensors of different dimensions,for example,( A)ijl := ijh A hl .The left-hand member of (12)is linear with respect to ¨U,˙θ,grad ˙x and grad ˙U ,quantities that take up arbitrary values;thus the respective coef?cients in the linear expression must vanish,and hence
ψ=?ψ(S ),η=?ψθ,T =ρψF F T and =ρψ?U F T .(13)This means that the Helmholtz free energy ψ,the entropy η,the Cauchy stress tensor T and the microstress depend upon the elastic state of the material only;moreover,η,T and are determined as soon as the constitutive equation for ψis known.The residual inequality de?nes the dissipation D of the thermo-kinetic process
D :=H ·˙U 0,(14)
where H :=ρ?ψU (S )??Z(S ,˙U)is called the dissipation tensor and it is symmetric.Two invariance principles need to be applied to the theory of constitutive equa-
tions:the objectivity under a rigid motion of the spatial frame of reference and the material symmetry.
A LINEAR THEORY OF POROUS ELASTIC SOLIDS309
With regard to the former,as U transforms like the deformation gradient F,a convenient set of invariant geometric variables could be the following: D:=1
2
(F T F?I),M:=F T U?U?and :=F T ?U(15) (D is the symmetric?nite strain tensor),because they are unaffected by a rigid rotation of the body B and vanish if the displacement of the body from the reference placement B?is rigid.Thus,if we change the elastic state S of the material with the new equivalent and invariant one I:={D,M, ,θ},we can obtain a?rst formal consequence of the principle of material frame-indifference in terms of the constitutive relations(13),namely,
ψ=ˉψ(I),η=?ψθ, =ρF ψ F T and
T=ρFψD F T+ρUψT M F T+ρ?U (F ψ );
(16) moreover,the expression of the symmetric dissipation tensor H changes as it follows: H=ρFψM?Z,(17) with FψM=ψT M F T.
For the latter invariance principle,we consider solids for which the material symmetry is of the type that possesses a center of symmetry and that are isotropic in their dependence of the constitutive?elds upon the elastic state I.
As previously observed,the free energyψdetermines much of the behaviour of the porous material,thus we concentrate on dealing with it in detail?rstly.We assume that the reference placement B?of the body is a homogeneous placement of minimum for the free energy and for which the residual stresses T?,Z?and ?vanish as well asψ?itself.
Also,we are going to obtain the basic equations of the linear theory,hence for the free energyψwe can take the most general homogeneous,quadratic and positive semi-de?nite form of the joint invariants of the elastic state I in the case of a centrosymmetric isotropic material:
ρˉψ(D,M, ,θ)=1
2β1(tr D)2+1
2
β3
tr(sym M)
2
+
+β2tr(D2)+β4tr[(sym M)2]+β5(tr D)
tr(sym M)
+
+β6tr[D(sym M)]+1
2β7tr
(skw M)2
+1
2
·( ? )+
+1
2
γ1(θ?θ?)2+γ2(θ?θ?)tr D+γ3(θ?θ?)tr(sym M)(18) (see the list(2.41)in[6]).
Here,sym A denotes the symmetric part of a second-order tensor A,?is the tensor product,that is,( ? )ijklmn:= ijk lmn,and the sixth-order tensor has the structure
ijklmn=α1(δijδklδmn+δjkδinδlm)+α2δklδinδjm+
+α3δijδknδlm+α4δjkδilδmn+α5(δjkδimδnl+δkiδjlδmn)+
+α6δkiδjmδnl+α7δilδjmδkn+α8(δjlδkmδin+δklδimδjn)+
+α9δilδjnδkm+α10δjlδknδim+α11(δijδkmδnl+δkiδjnδlm)(19)
310PASQUALE GIOVINE (see Equation(4.5)of[7]);the twenty-one coef?cientsβi(i=1,...,7),αj(j= 1,...,11)andγk(k=1,2,3)in(18)and(19)are constants.
Thermal stresses are not of interest in the applications in which we are concerned in the sequel,thus we suppose thatγk=0for k=1,2,3;so,the independent coef?cients in(18)reduce to eighteen.
4.The Linear Theory
Let u be the displacement?eld of the continuum from the reference placement B?and V the change in the microstretch from the reference one(that,for the homogeneity of B?,is constant and can be supposed equal to the identity,that is,U?=I): u(x?,τ):=x(x?,τ)?x?and V(x?,τ):=U(x?,τ)?I,(20) where V is a symmetric tensor as V Sym+.
The deformation gradient F and the?nite strain tensor D are related to the displacement gradient?u by the relations
(?u)T?u,(21) F=I+?u and D=E+1
2
where E:=sym(?u)is called the in?nitesimal strain tensor.Instead the microdeformation characteristics M and are related to V also:
M=V+(?u)T+(?u)T V and =?V+(?u)T ?V.(22) The in?nitesimal theory models physical situations in which the displacement gradient?u,the in?nitesimal microstrain?eld V,the microdeformation gradient ?V and the microvelocity˙V are,in some sense,suf?ciently small that their squares can be neglected.Then,we conclude from(21)2and(22)that,within an error of the second order,the following relations hold:
D=E,M=V+(?u)T and =?V;(23) further,we have sym M=V+E,skw M=W and symmetric in the?rst two places(W:=skw(?u)is the in?nitesimal rotation tensor).
Here,we are interested exclusively with linear porous elastic materials.For such bodies the stresses at any time and point in a process are linear functions of the independent variables?u,V,?V and˙V at the same time and place.In order to obtain their expressions,we substitute the quadratic form(18),withγk=0for each k,into relations(16)3,4and(17)and disregard second order terms in the results;thus T=[(β1+β3+2β5)tr E+(β3+β5)tr V]I+
+2(β2+β4+β6)E+(2β4+β6)V?β7W,
= ?V and
(24)
Z=[β3tr V+(β3+β5)tr E]I+
+2β4V+(2β4+β6)E?β7W?H(I,˙V).
A LINEAR THEORY OF POROUS ELASTIC SOLIDS 311
The symmetry in the ?rst two places of the microstress imposes restrictions on the expression (19)of the tensor ;it must be
α1=α2,α4=α5=α6,α7=α10and α8=α9=α11,(25)moreover,Z is symmetric as well as the dissipation tensor H and hence,from (24)3,it results that β7=0.Finally,we need to express H within the same approximation as the other terms in (24)3due to the linear dependence of internal microactions Z upon in?nitesimal variables;by the dissipation inequality (14)2and its smoothness,the dissipation tensor H must vanish whenever ˙V
=0,then we take H =?α(tr ˙V
)I ?2γ˙V ,(26)with αand γas inelastic constants.The imbalance (14)2is satis?ed when
γ 0and 3α+2γ 0.(27)The constitutive equations for the linear theory of a porous elastic material with empty large voids which do not diffuse through the matrix are the following:
T =(λtr E +ω5tr V )I +2μE +ω6V ,
=ω1[I ?Div V +syml (?(tr V )?I)]+ω8I ??(tr V )++2ω2syml (Div V ?I)+2ω7?V +2ω9syml (?V )t ,
Z =(ω3tr V +ω5tr E +αtr ˙V
)I +2ω4V +ω6E +2γ˙V ,(28)where we introduced the new elastic constants,related to previous ones,
λ=β1+β3+2β5,
μ=β2+β4+β6,ω1=2α1,ω2=2α4,ω3=β3,
ω4=β4,
ω5=β3+β5,
ω6=2β4+β6,ω7=α7,
ω8=α3,ω9=2α8.(29)On the third-order tensors in (28)2we used the operators left symmetrization and minor right transposition with the subsequent properties:(syml )ijl :=12( ijl + jil )and ( t )ijl := ilj ,respectively.
We note that in the linear theory the moment of momentum Equation (6)reduces to classical condition of symmetry of the Cauchy stress tensor T and is identically satis?ed by (28)1.
Remark.When one considers thermal phenomena,linear constitutive relations
(28)1,3have to be modi?ed as it follows:the terms ρ(γ2+γ3)(θ?θ?)I and ργ3(θ?θ?)I are added to the right side of (28)1,3,respectively,while,from (16)2,the entropy ηis
η=?γ1(θ?θ?)?(γ2+γ3)tr E ?γ3tr V.(30)
312PASQUALE GIOVINE The eleven elastic constants de?ned in(29),constants that completely describe the microelastic properties of the body,can be further speci?ed if we study the restriction of the free energy densityψto be a positive semi-de?nite form.
Inserting(29)in(18)and neglecting terms of greater order than quadratics in?u, V and?V,we obtain
ρψ=3λ+2μ
6(tr E)2+μdev E·dev E+3ω3
+2ω4
6
(tr V)2+
+ω4dev V·dev V+3ω5+ω6
3
(tr E)(tr V)+ω6dev E·dev V+
+6(ω1+ω7)+9ω8+2(ω2+ω9)
18
?(tr V)·?(tr V)+(31)
+ω7?(dev V)·?(dev V)+2(ω2+ω9)
3
?(tr V)·?(dev V)+
+ω2Div(dev V)·Div(dev V)+ω9?(dev V)·[?(dev V)]t 0, where dev denotes the deviatoric part of a symmetric tensor,that is,dev E:=E?(1/3)(tr E)I.The non-negativeness of the expression in(31)assure us thatω9=0,
while the other coef?cients must resolve the following system of inequalities:μ 0,3λ+2μ 0,(3λ+2μ)(3ω3+2ω4) (3ω5+ω6)2,
4μω4 ω26,ω2 0,ω7 0,3ω8+2(ω1+ω7) 0.
(32)
5.Equations of Motion
The general system of partial differential equations for the motion of linear elastic solids with ellipsoidal structure comes out from the insertion of constitutive relations (28),withω9=0,in the balance equations(3)–(5).Then,for the homogeneous centrosymmetric isotropic materials,the displacement and microstrain equations on u and V are,in components,
ρ=ρ?(1?u i,i),
ρ?¨u j=(λ+μ)u i,ij+μu j,ii+ω5V ii,j+ω6V ji,i+ρ?f,
ρ?κ¨V jk=(ω1V il,il+ω8V ii,ll?ω3V ii?ω5u i,i?α˙V ii)δjk+
+ω1V ii,jk+2ω7V jk,ii+ω2(V ij,ik+V ik,ij)?
?2ω4V jk?1
2ω6(u j,k+u k,j)?2γ˙V jk+ρ?B jk.
(33)
In order to de?ne a displacement boundary-initial-value problem we must assign a time interval[0,ˉτ],ten elastic and two inelastic material constants(λ,μ,ωi,for i=1,...,8,andα,γ,respectively),body forces f and B on B×[0,ˉτ],initial displacement u0and microstrain V0on B,initial velocity v0and microvelocity˙V0on
B and surface displacement?u on?B?[0,ˉτ](?B is the boundary of the body B).
A LINEAR THEORY OF POROUS ELASTIC SOLIDS313 Given the above data,the displacement problem consists of?nding the?elds u and V(and so T,Z and )that correspond to f and
B and satisfy the initial conditions u(·,0)=u0,˙u(·,0)=v0,
V(·,0)=V0and˙V(·,0)=˙V0on B
(34) and the boundary conditions
u=?u and n=0on?B?[0,ˉτ],(35) where n is the outer unit normal to?B.
The last condition on the microstress derives from the request of continuity of n across the boundary;since it is dif?cult to imagine a direct way to act on the pores through the boundary,except for very rare phenomena(see,for example,the case of micro-earthquakes in the basin of the caldera in the Phlegraean?elds in[8]), we assume that is zero outside the continuum,thus condition(35)2follows.
6.Quasi-Static Homogeneous Deformations
The possibility of assigning determined linear elastic constants in a boundary-initial-value problem is connected to experimental situations involving homogeneous deformations;in fact experiments can be used to measure,with our theory,coef?-cients of many real materials,like rock,wood,compact bone,polymer foam,since they are isotropic and have a center of symmetry.
We consider a linear elastic porous material for which Equations(33)apply and that is subjected to a homogeneous deformation process,namely,in which the strain ?eld is independent of position:E≡ˉE(τ);moreover,the deformation process is assumed to be quasi-static and occurs in the absence of body forces,hence the inertia termsρ?¨u andρ?κ¨V can be neglected and both f and B vanish.
Under these hypotheses we search solutions of the micromomentum balance(33)3 in which the microstrain?eld V is independent of position also;(33)3reduces to α˙V iiδjk+2γ˙V jk+ω3V iiδjk+2ω4V jk+ω5ˉE iiδjk+ω6ˉE jk=0,(36) with the initial conditions V jk(0)=0.
For(27),it isγ 0and3α+2γ 0.Now we study the most interesting case,that is,whenγ>0and3α+2γ>0;in the others,that is,if one inequality, or both,are zero,we obtain solutions that are linear combinations of the strain?eld ˉE plus parts of solutions of the strictly positive occurrence.
When j=k,Equation(36)is expressed as
˙V jk+ω4
γV jk+
ω6
2γ
ˉE jk=0,(37)
with the solution given by
V jk(τ)=?ω6
2γ
τ
ˉE jk(σ)e?ω4γ(τ?σ)dσ.(38)
314PASQUALE GIOVINE If j=k,we can linearly combine the three equations obtained from(36)and?nd the solutions for V11and V22
V jj(τ)=V33(τ)?ω6
2γ
τ
(ˉE jj(σ)?ˉE33(σ))e?ω4γ(τ?σ)dσ,(39)
for j=1,2,and an equation for V33only
(3α+2γ)˙V33(τ)+(3ω3+2ω4)V33(τ)+G(τ)=0,(40) where
G(τ):=γω5+ω6(γ+α)ˉ
E33(τ)+2γω5
?αω6
2γ
ˉ
E11(τ)+ˉE22(τ)
+
(41) +
ω6(αω4?γω3)
2γ2
τ
ˉ
E11(σ)+ˉE22(σ)?2ˉE33(σ)
e?
ω4
γ
(τ?σ)dσ;
at last,the solution of Equation(40)is
V33(τ)=?1
3α+2γ τ
G(σ)e?3ω3+2ω4
3α+2γ
(τ?σ)dσ.(42)
Now,we can use previous results to give expression to the Cauchy stress ten-sor T during an homogeneous deformation;by inserting(38),(39)and(42)in the constitutive relation(28)1,we have
T jk=2μˉE jk(τ)?ω26
2γ
τ
ˉE jk(σ)e?ω4γ(τ?σ)dσ,if j=k,and
T jj=(λtrˉE(τ)+2μˉE jj(τ))?3ω5+ω6
3α+2γ τ
G(σ)e?3ω3+2ω4
3α+2γ
(τ?σ)dσ+
+ω6
2γ
τ
(3ω5+ω6)ˉE33(σ)?ω5trˉE(σ)?ω6ˉE jj(σ)
e?
ω4
γ
(τ?σ)dσ.
(43)
The viscoelastic behaviour in the linear range of a continuum with voids during
quasi-static homogeneous motions was observed in[5].Our relation(43)for the
stress T of a porous elastic solid with ellipsoidal structure generalizes the expression
(4.2)of[5],thus,when3ω3+2ω4=0,we recover the similarity with the stress arising in linear viscoelasticity,while,in general,our material is,in the in?nitesimal
approximation,an isotropic simple material with fading memory of order2(see§40
and41of[9]).
7.Small-Amplitude Acoustic Waves
In this application we investigate the propagation of small-amplitude acoustic waves
for a material characterized by the system(33)in absence of body forces(i.e.,f=0
and B=0).
A LINEAR THEORY OF POROUS ELASTIC SOLIDS 315
We introduce the microdisplacement vector y and then consider wave solutions of (33)of the form
u(x,τ)=ζh φ(x,τ)and y(x,τ)=ξg φ(x,τ),(44)where φ(x,τ)=Re[e i bτ?(a +i b/c)m ·x ];ζand ξare the wave amplitudes;h ,g and m are unit vectors representing the directions of macro-and microdisplacement and of propagation,respectively;b is the frequency;a and c are the wave attenuation and the wave speed,respectively.
The in?nitesimal microstrain ?eld V is related to y by the equality V =sym (?y)(=grad y in the linear theory),thus,by substituting (44)in the system of linear Equations (33),we are led to the following:
ρ?b 2+μ a +i b 2 h +(λ+μ) a +i b 2(m ·h)m ζ++ a +i b 2 ω62g + ω5+ω62 (m ·g)m ξ=0and a +i b c ω3?(ω1+ω8) a +i b c 2+i αb I ?(45)?(ω1+ω2) a +i b 2
m ?m (m ·g)++ ω4? ω22+ω7 a +i b c 2
+iγb ?ρ?κ2b 2 2sym (g ?m) ξ++ ω5(m ·h)I +ω6sym (h ?m) ζ =0.We say that the waves are purely transverse if h is equal to g and perpendicular to the propagation m .With these positions,the system (45)is equivalent to
ρ?b 2+μ a +i b c
2 ζ+ω62 a +i b c 2ξ=0and (46) a +i b c 2 ω4? ω2
2+ω7 a +i b
c 2+i γb ?ρ?κ2b 2 ξ+ω62
ζ =0.Apart from trivial ones,we have:(i)a static solution with b =0and attenuation
a = (4μω4?ω26
)/[2μ(ω2+2ω7)](the radicand is nonnegative for (32)1,4,5,6)in which the amplitudes are related by ω6ξ=2μζ;(ii)if ω6=0,a transverse macro-wave (i.e.,with ξ=0)propagating without attenuation (a =0)at a constant speed c 2=μ/ρ?and which does not affect the porosity of the material;(iii)if
316PASQUALE GIOVINE ω6=0,a transverse micro-wave(i.e.,withζ=0)that propagates with attenuation a=γc/(ω2+2ω7)and speed given by
c2=(ω2+2ω7)
2ω4?ρ?κb2+
(2ω4?ρ?κb2)2+4γ2b2
2γ2
:(47)
in the expressions of a and c only coef?cients associated with the properties of the pores appear,thus the micro-wave propagates without modifying the elastic features of the matrix material,but is both dispersed and attenuated for the variations in the porosity which accompany the wave.
Purely elastic waves occur when the inelastic coef?cientγvanishes;in this case we have no attenuation and obtain:(iv)a transverse micro-wave(ζ=0)that propagates with speed c2=(ω2+2ω7)b2/(ρ?κb2?2ω4),whenρ?κb2>2ω4 andω6=0;(v)two distinct types of complete transverse waves,if the determi-
nant of the coef?cient matrix of the system of amplitudesζandξvanish,that is, (ρ??μ/c2)(2ω4?ρ?κb2+(ω2+2ω7)b2/c2)+ω26/(2ρ?)=0.In all elastic cases the wave speeds are functions of the frequency b.
When we supposed in(ii),above,that the coef?cientω6of the nonspherical part of the microstrain in the stress tensor T is null,we set ourselves in the hypotheses of Cowin and Nunziato in[5]and,in fact,we found their transverse wave solution;the others are new.Also,we observe that our micro-wave solutions in(iii),above,can be recognized in some developments of[10].
Purely longitudinal waves succeed if h,g and m are equal.Now the system(45) reduces to
ρ?b2+(λ+2μ)
a+i
b
c
2
ζ+(ω5+ω6)
a+i
b
c
2
ξ=0,
a+i b
c
ω3?(ω1+ω8)
a+i
b
c
2
+iαb
ξ+ω5ζ
=0and
a+i b
c
2ω4?ω
a+i
b
c
2
+2iγb?ρ?κb2
ξ+ω6ζ
=0,
(48)
whereω=ω1+2ω2+2ω7.
If the attenuation is null and the coef?cients of the microstrain in the stress tensor vanish,we have the classical longitudinal macro-wave(ξ=0)found in[5]propa-gating at a constant speed c2=(λ+2μ)/ρ?and which does not affect the porosity of the material.Instead,a purely elastic wave occurs,when the inelastic coef?cients αandγare zero:it is a complete longitudinal wave with the same speed as that of the previous one that exists if the determinant of the system of the amplitudesζand ξvanish,that is,(λ+2μ)(ω3+2ω4?ρ?κb2)+ρ?b2[2(ω1+ω2+ω7)+ω8]=0. Some other solutions of the type acoustic waves could be found with a more accurate study of our system,but this will be reported in a forthcoming work.
A LINEAR THEORY OF POROUS ELASTIC SOLIDS317
Finally,we want to mention a mixed case in which longitudinal and transverse features are present.We suppose that the direction of macrodisplacement h is parallel to the propagation m,but it is perpendicular to the microdisplacement g.Thus, Equations(45)are reduced to
ρ?b2+(λ+2μ)
a+i
b
2
ζ=0,
a+i b
ω4?
ω
2
2
+ω7
a+i
b
2
+iγb?ρ?
κ
2
b2
ξ=0,
(49)
when the coef?cients of the microstrain in the stress tensor are null.A complete
elastic solution without attenuation and with amplitudesζandξdifferent from zero
exists with speed c2=(ω2+2ω7)b2/(ρ?κb2?2ω4)(ρ?κb2>2ω4),if this quantity is constant and also equal to the classical one(λ+2μ)/ρ?.
So,we have a solution where the macrodisplacement is orthogonal to the microdis-
placement,if the microelastic properties of the pores are,in some sense,constrained
to the elastic characteristics of the matrix material.
8.Concluding Remarks
In[3]the theory of continua with microstructure was successfully applied to describe
porous solids with very large vacuous pores viewed as materials with ellipsoidal
microstructure.Hence,in this paper,we have developed in some detail the related
linear theory of a nonconducting homogeneous centrosymmetric isotropic porous
material by proposing the appropriate constitutive features and by specializing them
with thermodynamic restrictions and invariance principles.
The voids theory in[1]is contained in the present model and the yielded results in
our applications to quasi-static homogeneous deformations and to small-amplitude
acoustic waves are in accordance with those previously derived in[5].In particular,we
observed that,during quasi-static homogeneous motion,the porous solid behaves like
an isotropic simple material with fading memory in the linear range and it reduces to a
viscoelastic medium when the microstructural variable remains spherical.Moreover,
analyzing the propagation of waves,we arrived at some new solutions like a purely
transverse micro-wave spreading without perturbing the elastic properties of the
matrix material and a mixed complete elastic one which comes out only if the macro-
and microelastic properties of the material are related.
Acknowledgements
This research was supported by the Italian M.U.R.S.T.through‘Fondi per la ricerca
scienti?ca40%’.
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