Rayleigh surface waves problem in thermoviscoelastic

medium with voids

A. Bucur
Department of Mathematics, Faculty of Mathematics,
Alexandru Ioan Cuza University of Iasi, Iasi, Romania

In this paper we study the propagation of the Rayleigh surface waves in an exponentially functionally
graded half-space filled by a thermoviscoelastic material with voids. We take into consideration the
dissipative character of the porous thermoviscoelastic models upon the propagation waves and study the
damped in time wave solutions. The eigensolutions of the dynamical system are explicitly obtained in
terms of the characteristic solutions.

Key Words: Rayleigh surface waves, Damped in time solutions, Porous thermoviscoelasticity

1 Introduction ponentially varying with depth, proportional to a

common factor exp( x2 ), where is the inverse
The theory of thermoviscoelastic material with
of an inhomogeneous characteristic length. By us-
voids is a recent generalization of the classical the-
ing an idea developed by Destrade [6], we study the
ory of thermoelasticity, by taking into account the
damped in time wave solutions. The propagation
porous and the memory effects. The intended ap-
condition is established in a form of an algebraic
plications to this theory may be found many areas
equation with complex coefficients. The results are
of applied sciences and engineering. Iesan [1], devel-
illustrated for the case of an isotropic and inhomoge-
oped a linear theory of thermoviscoelastic materials
neous thermoviscoelastic with voids half-space. For
with voids in which the time derivative of the strain
the case of an isotropic and homogeneous material,
tensor and the time derivative of the gradient of the
the analytical solutions are explicitly obtained in
volume fraction are included in the set of indepen-
terms of characteristic solutions to the propagation
dent constitutive variables. There are several pa-
equation [8].
pers concerning various problems based on the the-
ory of thermoviscoelastic materials with voids (see,
e.g. Chirita [2], Chirita and Danescu [3], Bucur [4]). 2 Basic equations
Rayleigh surface waves propagation problem
[5] finds its principal application in the interpreta- We consider that a regular region B is filled by a
tion of ground motions due to earthquakes or ex- porous thermoviscoelastic material. We refer the
plosions. Also, many applications have been found motion of the continuum to a fixed system of Carte-
in the industrial world (electronic components, soil sian axes Oxi , (i = 1, 2, 3). We shall employ the
dynamics, filters and sensors, etc.). The propaga- usual summation and differentiation conventions:
tion of surface waves was investigated in a series of Latin subscripts have the range 1, 2, 3, Greek sub-
papers, as, for example, those written by Destrade scripts have the range 1, 2, summation over re-
[6], Bucur et al. [7]. More recently, Chirita and peated subscripts is implied, subscripts preceded by
Danescu [3] studied the propagation of the surface a comma denote partial differentiation with respect
waves for an isotropic and homogeneous thermovis- to the corresponding Cartesian coordinate, and a
coelastic material with voids. They proved that the superposed dot denotes time differentiation.
thermal and viscous dissipation energies influence According to Iesan [1], the fundamental sys-
the attenuation in time and in deep of the half-space tem of field equations consists of the equations of
for the surface waves solutions. motion
The present paper discusses the thermal tji,j + fi = ui ,
(1)
and the memory effects upon the propagation of Hi,i + g + ` = ,
Rayleigh surface waves in a half-space filled by an
exponentially functionally graded thermoviscoelas- in B (0, ), the energy equation
tic material with voids. We consider that the char-
acteristic coefficients and the mass density are ex- T0 = Qi,i + S, (2)

1
in B (0, ), the constitutive equations dissipation energy density, implies
 

tij = Cijrs ers + Bij + Dijk ,k ij + Sij , > 0, > 0, b2 < + 23 , > 0,

Hi = Aij ,j + Drsi ers + di ai + Hi ,  1 2
g = Bij eij di ,i + m + g , > 0, > 0, T0 + < 4 k, (9)
2 T0
= ij eij + a + m + ai ,i , 1 2 
k > 0, b + > + .
Qi = kij ,j + firs ers + bi + aij ,j , 4 3
(3)
3 Surface waves in an anisotropic
with Sij ,Hi , and g given by
thermoviscoelastic porous half-space

Sij = Cijrs ers + Bij + Dijk ,k + Mijk ,k , Throughout this section, we assume B to be the

Hi = Aij ,j + Grsi ers + di + Pij ,j , (4) half-space x2 0 filled by an anisotropic thermo-
g = Fij eij k ,k Rj ,j , viscoelastic porous material that is inhomogeneous
in the sense that the constitutive coefficients and the
and the geometrical relations density mass are exponentially varying with depth,
1
in the form
eij = 2 (ui,j + uj,i ) , (5)
0
Cijrs = Cijrs e x2 , ..., i = i0 e x2 , = %e x2 ,
in B [0, ). Here, ui are the components of the
0
displacement vector, is the void volume fraction, where Cijrs , ...%, and are real constants. If = 0,
is the variation of temperature from the uniform then we have a homogeneous material.
reference absolute temperature T0 > 0, tij are the In what follows, we consider a surface wave
components of stress tensor, Hi are the components that is propagating in the direction of x1 axis, in
of the equilibrated stress vector, g is the intrinsic the half-space x2 0. We assume that the half-
equilibrated force, Qi are the components of the space x2 0 is free of body forces, equilibrated
heat flux vector, is the equilibrated inertia, is body force, and heat supply, and on its surface
the entropy density per unit mass, is the mass x2 = 0 has zero traction, zero equilibrated traction,
density, fi are the components of the body force and zero heat flux. Moreover, we consider that the
vector, ` is the extrinsic equilibrated body force per body is free to exchange heat with the region x2 < 0.
unit mass, and S is the heat supply per unit mass. Thus, we have the following boundary conditions
In what follows, we assume that the internal
t2i (x1 , 0, x3 , t) = 0, H2 (x1 , 0, x3 , t) = 0,
energy density (10)
Q2 (x1 , 0, x3 , t) = 0, x1 , x3 R, t 0.
2W = Cijrs eij ers + Aij ,i ,j + 2 To these boundary conditions we have to adjoin the
(6)
+ Bij eij + Dijk eij ,k + di ,i , asymptotic conditions

and the dissipation energy density lim ui (x, t) = 0, lim (x, t) = 0,

x2 x2

lim (x, t) = 0, lim {tij , hi , g, qi }(x, t) = 0,
= Cijrs eij ers + 2 + Aij ,i ,j + T10 kij ,i ,j x2 x2

+ (Bij + Fij )eij +

(11)
 (Dijk + Gijk )eij ,k for all x1 , x3 R, and t 0.
+ (di + i ),i + Mijk

+ T10 fkij eij ,k The Rayleigh surface wave propagation prob-
   
+ Rj + T10 bj ,j + Pij + T10 aji ,i ,j , lem consists of finding solutions to the boundary
(7) value problem defined by the basic equations (1)-
are positive definite quadratic forms. (5), the boundary conditions (10), and the asymp-
For an isotropic porous thermoviscoelastic ma- totic conditions (11). To solve this problem, we seek
terial, the constitutive equations become solutions U = {u1 , u2 , u3 , , } in the form

U(x1 , x2 , x3 , t) = Vei(x1 vt+px2 ) , (12)

tij = emm ij + 2eij + bij ij
+ emm ij + 2 eij + b ij , where V = {U1 , U2 , U3 , , } is a constant vector,

Hi = ,i + ,i + ,i , is the wave number, p is a complex scalar such that
g = bemm + m emm , (8) Im(p) > 0, (13)
= emm + a + m,
and v is a constant complex parameter so that
Qi = k,i + ,i ,
Re(v) > 0, Im(v) 0. (14)
where , , ..., , and are the constitutive coeffi-
cients and ij are the components of the Kronecker The quantity Re(v) is giving the wave speed, while
delta. In this case, the positive definiteness assump- exp[Im(v)t] is giving the damping in time of the
tion upon the internal energy density and upon the surface wave.

2
  
In view of the relations (12), we can write K=A e11 + pA e12 i de1 + p + i A e21 + pA e22
  
h   i i de2 + i e1 + pe 2 i e %v 2 ,
i x1 vt+ p+ i
x2  
tij = iTij (V, p, v)e h  i , 0
L = P11 + pP120
+ i a01 + p + i 0
P21 + pP220

i x1 vt+ p+ i
  
x2
Hi = ihi (V, p, v)e h

+ i a02 + i R10 + pR20 + i m0 ,
  i, h  
i x1 vt+ p+ i
x2 (15) M = iv a011 + pa012 i b01 + p + i a021
g = iG(V, p, v)e  ,i   i
+ pa022 i b02 + iT0 a01 + pa02 i m0 ,
h 
i x1 vt+ p+ i
x2
= i%N (V, p, v)e h  i,
  
 0
N = k11 0
+ pk12 + p + i k210 0
+ pk22 ivT 0 0
a ,
i x1 vt+ p+ i
x2
Qi = iqi (V, p, v)e , (20)
and
where
   Qij = C
e1ij1 , Rij = C
e1ij2 , Sij = C
e2ij2 . (21)
Tij = C eijr1 + pC eijr2 Ur + D e ij1 + pD e ij2
   Since we seek a non-trivial solution V =
i B eij + M 0 + pM 0 + i 0 , {U1 , U2 , U3 , , } of the homogeneous system of al-
ij1 ij2 ij
   
ei2 i dei gebraic equations (18), we must impose that
hi = G e r1i + pG e r2i Ur + A ei1 + pA

det H = 0,
 
+ P1i 0
+ pPi20
+ i a0i , (22)
h  
G = (Fer1 + pFer2 )Ur + e1 + pe 2 i e where
  i
+ R10 + pR20 + i m0 ,

H11 H12 H13 G1 T1
  H21 H22 H23 G2 T2
%N = (r1 0 0
+ pr2 )Ur + a01 + pa02 i m0
H =

i a0 , H31 H32 H33 G3 T3
. (23)
 I1 I2 I3 K L
0 0
qi = iv(fir1 + fir2 )Ur iv a0i1 + pa0i2 J1 J2 J3 M N

i b0i + (ki1
0 0
+ pki2 ), Further, we will denote by pn , n = 1, ..., 5, the eigen-
(16) values with Im(pn ) > 0, so that the asymptotic
C 0
emnpq = Cmnpq 0
ivCmnpq , dem = d0m ivd0 conditions (11) are satisfied. This assumption can
m,
0 0 0 0 always be fulfilled by means of appropriate con-
Bmn = Bmn ivBmn , Amn = Amn ivAmn ,
e e
e mnp = D0 ivD0 , 0 0
straints upon the constitutive coefficients and by
D mnp mnp em = dm ivm , taking some appropriate restrictions upon the wave
0
Femn = Bmn0
ivFmn , e = 0 iv 0 , speed and on the damping rate.
Gmnp = Dmnp ivG0
e 0
mnp .
 (n) (n) (n)
Let V (n) = U1 , U2 , U3 , (n) , (n) be

(17) the eigensolution corresponding to the eigenvalue
According to the relations (12), (15), and (16), we pn , n = 1, ..., 5. Following [3], we seek solutions of
can write the evolution equations (1) and (2) in the the Rayleigh surface waves propagation problem, in
form the form
Hir Ur + Gi + Ti = 0,
5
Ir Ur + K + L = 0, (18) X
U(x1 , x2 , t) = n V (n) ei(x1 vt+pn x2 ) , (24)
Jr Ur + M + N = 0,
n=1

where with = (1 , 2 , 3 , 4 , 5 ) is a constant vector to


i
 be determined such that the boundary conditions
Hij = Qij + pRij + p + (Rji + pSij ) %v 2 ij , (10) hold true. In view of Eqs.(10,24), we obtain
 
Gi = D
e 1i1 + pD e 1i2 i B e1i + p + i D e 2i1 the following secular equation for the parameter v


+ pD e 2i2 i B T21 T (2) T (3) T (4) T (5) 

 (1)
2i ,
e 
   21 21 21 21 
0 0 0 T (1) T (2) T (3) T (4) T (5) 
Ti = M1i1 + pM1i2 + i 1i0
+ p + i M2i1  22 22 22 22 22 
 (1) (2) (3) (4) (5) 
T23 T23 T23 T23 T23  = 0, (25)

0
+ pM2i2 + i 21
0
,  (1) (2) (3) (4) (5) 
   h h2 h2 h2 h2 
e 2j1 + p + i G  2
Ij = G
e 1j1 + pG

e 1j2 + pG e 2j2  q (1) q (2) q (3) q (4) q (5) 
  2 2 2 2 2
+ i Fe1j + pFe2j ,
(n) (n)
h    where Tij = Tij (V (n) , pn , v), hi = hi (V (n) , pn , v),
Jj = iv f1j1 0
+ pf1j20
+ p + i f 0
2j1 + pf 0
2j2 (n)
 i G(n) = G(V (n) , pn , v), qi = qi (V (n) , pn , v), and
+ iT0 j1 0
+ pj20
, %N (n) = %N (V (n) , pn , v). We have to select the so-
(19) lutions of the Eq.(25) that satisfy the conditions

3
 h  
(5) iT0 b b)5 + b2 + w2
(14). It is not easy to see that the secular equa- U1 = b ) (b +

b(+2b
tion has such a solution for characterizing Rayleigh mbb
i
(5) (5) (5)
waves. However, for particular materials, the sec- 2 , U2 = p5 U1 , U3 = 0,
n h
ular equation can be solved by means of numerical = T
(5) 0
( )52 + w2 + b
b + 2b
2

b(+2b
b )
methods. i o
m2 (b + 2b ) 5 m2 w2 ,
4 Application for an isotropic h h
(5) = bT0 b (
b + 2b)52 + bw2 + (
b + 2b
)
thermoviscoelastic porous half-space
(+2b
b )  b i
2 + w2 bb2 5
b
In this section, we illustrate the above results for the  b o
class of isotropic and homogeneous porous thermo- + w2 2 + w2 ,
viscoelastic media ( = 0). According to Eqs.(6,14), (31)
we can write the characteristic equation (23) in the with
form w = iv, (32)
(p2 + 1) v 2 ]2 (p, v) = 0,
[b (26) and
b = + w ,
b = + w , bb = b + wb ,
where
h  in 
b = + w , b = b + w , b = + w .
(p, v) =
b + 2b (p2 + 1) v 2 bk + iv
The implicit and explicit expression for the secular
h   
(p2 + 1)2 + k 12 b v 2 iv bc + m equation was obtained in [8]. Also, some numerical
i h   simulations were presented for a specific material.
mT0 (p2 + 1) iv c 12 b v 2
2
io n h Acknowledgment
+ m2T0 + 12 kbbb + iv b T0 (bb
io h This work was supported by a grant of the Roma-
+ b) (p2 + 1)2 + iv3 mT0 (bb + b) + bbbc nian National Authority for Scientific Research and
i
2 2 2 2 Innovation, CNCS-UEFISCDI, project number PN-
T0 ( v ) (p + 1).
b
(27) II-RU-TE-2014-4-0320.
We will denote by pi , i = 1, ..., 5, the solutions of the
Eq.(26) with Im(pi ) > 0. Like in [3], for p = p2 and References
for p = p3 we obtain the following eigensolutions
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n p 1 1 materials with voids, J. Elast., Vol. 104, 369
o n o
2
V (2) = , , 0, 0, 0 , V (3) = 0, 0, , 0, 0 , (28)
384, 2011.
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() () ()
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by viscoelastic porous materials, J. Therm.
n h  b Stresses, Vol. 38(1), 96109, 2015.
(1)
U1 = (b k
1
(b
k
w) 2
1 + k 2 (3) Chirita, S. and Danescu, A., Surface waves
 w) i
+ w2 + w c mT0 + m) 1 + cw
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 b
m 2 T0
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2
2 + w + 3 ,
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i1
(1)
= 2 (b k w) i(b k T0 w)1 + w Stresses, Vol. 38(2), 229249, 2015.
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c + mT0 ) , (5) Lord Rayleigh, On waves propagating along
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(1) = (b iw1
b)1 + mT
T0
(b 0b the plane surface of an elastic solid, Int. J.
k
 b w) 2
i Eng. Sci., Vol. 25(2), 205211, 1987.
T0 2 + w2 ,
(6) Destrade, M., Seismic Rayleigh waves on an
(29)
(4) i
h
w b
exponentially graded, orthotropic half-space,
U1 = b ) (bk + w)4 + (bc + mT0 )],
b
Proc. R. Soc. A, Vol. 463, 495-502, 2007.
k(+2b
(4) (4) (4)
U2 = p4 U1 , n U3 = 0, h
(7) Bucur, A.V., Passarella, F., and Tibullo, V.,
= b1
(4)
k(b + 2b )42 + kw2 + cw ( + 2b
b ) Rayleigh surface waves in the theory of ther-
k(+2b ) i o
2 3
+ T0 w cw
4 + , moelastic materials with voids, Meccanica,
n h Vol. 49(9), 20692078, 2014.
(4) = w (
b + 2b )4
2
+ mT0 b
) + w2
2 ( + 2b
k(+2b
b )
i o (8) Bucur, A., Rayleigh surface waves problem in
bT2 0 4 + mT 2 w
2
,
b 0
linear thermoviscoelasticity with voids, Acta.
(30) Mech., DOI:10.1007/s00707-015-1527-8, 2016.