Spectrum of one-dimensional vibrations of a layered medium consisting of a Kelvin-Voigt material and a viscous incompressible fluid Текст научной статьи по специальности «Физика»

УДК 517.958 + 534-18

Spectrum of One-dimensional Vibrations of a Layered Medium Consisting of a Kelvin-Voigt Material and a Viscous Incompressible Fluid

Vladlena V. Shumilova*

Institute for Problems in Mechanics of RAS, Vernadskogo, 101-1, Moscow, 119526,

Russia

Received 24.12.2012, received in revised form 11.02.2013, accepted 11.03.2013 The paper considers a mathematical model for natural vibrations of a periodic layered medium. The medium consists of a viscoelastic Kelvin- Voigt material and a viscous incompressible fluid. For the given model, two homogenized models are derived. They correspond to the cases of transverse and longitudinal vibrations of the layered medium. It is shown that the spectrum of each homogenized model is the union of roots of the corresponding quadratic equations.

Keywords: spectrum, layered medium, homogenized model, viscoelasticity, viscous fluid.

Introduction

In this paper the study initiated in [1] and [2] is continued. The paper is concerned with spectral properties of homogenized models of strongly inhomogeneous layered media. The motivation to study spectral properties of such models is one interesting experimental fact obtained in [3]. It was found that even a small amount of viscous fluid in pores of an elastic solid leads to a qualitative different spectral properties of a continuous elastic solid and an elastic solid saturated with fluid (see [3] for details). Therefore, it would appear natural that media consisting of viscoelastic and fluid components also have some interesting mechanical properties.

In the present paper we consider a mathematical model for natural vibrations of a periodic medium consisting of alternating layers of an isotropic viscoelastic Kelvin-Voigt material and a viscous incompressible fluid. For this medium two homogenized models are derived. They correspond to the cases of transverse and longitudinal vibrations of the layered medium. These homogenized models describe one-dimensional natural vibrations of some viscoelastic KelvinVoigt materials. We also show that the spectrum of each homogenized model is the union of roots of the corresponding quadratic equations. In order to compare results obtained for incompressible and compressible fluid layers, we briefly review the homogenized problems and their spectra given in [1], where fluid was supposed to be compressible.

The paper is organized as follows. In Section 1 we formulate an original mathematical model and derive the corresponding homogenized model. In Sections 2 and 3 we construct homogenized models corresponding to the cases of transverse and longitudinal vibrations, respectively. Then we study the spectral properties of these models.

* v.v.shumilova@mail.ru © Siberian Federal University. All rights reserved

1. Mathematical models

Let Q = (0, l)3 and d is a constant such that 0 < d < 1. Let us denote

Ih = (0, (1 - d)/2) U ((1 + d)/2,1), Is = ((1 - d)/2, (1 + d)/2). Then for a sufficiently small e > 0 we define

Ih = (0, l) n (Ufcez(e/h + ek)) , I| = (0, l) n (UfceZ(e/s + ek)), Qh = Ih x (0, l) x (0, l), QS = IS x (0, l) x (0, l).

Obviously, Q = Qh U QS U Se with Se = dQh n dQS. We assume that the set Qe is occupied by a viscous incompressible fluid whereas the set Qh is occupied by an isotropic viscoelastic Kelvin-Voigt material. In the sequel, the sets Qh and QS are called the viscoelastic and the fluid parts (or layers) of Q, respectively. Note that all viscoelastic and fluid layers of Q are parallel to the x2x3-plane. Denoting Y = (0,1)3 we see that the cube eY is the cell of periodicity of the combined medium Q. In fact, the set Yh = Ih x (0,1) x (0,1) is the viscoelastic part of Y, and the set Ys = Is x (0,1) x (0,1) is the fluid part of Y.

We now turn to the formulation of mathematical model for the cooperative motion of vis-coelastic and fluid layers of Q. Let us assume that positive constants ph and ps are the densities of the viscoelastic material and the fluid, respectively. Assume also that f (x, t) is the force vector and ue (x,t) is the displacement vector. The equations of motion in the viscoelastic part Qh are as follows

d2ue dae-

ph= ~dX~+fi(x,t), x g Qh, t> 0. (i)

Here <rj are the components of the stress tensor,

du<eA „h

e i

aij = aijkhGkh(ue) + bjfchefch ( ) , x G Q.

and ekh(ue) are the components of the strain tensor,

e (u® ) = 1 (duk + du| 2 \ dxh dxk

Since the viscoelastic material is isotropic the coefficients aijkh and 6ijkh are defined by aijkh = Ae^ij Skh + Me(Sik Sjh + SihSjk ), bjkh = Av Sij Skh + Mv (Sik Sjh + SihSjk),

1 < i, j, k, h < 3,

where Ae and Me are the elastic Lame constants, Av and mv are their viscoelastic counterparts and Sij is the Kronecker symbol.

In the fluid part QS the equations of motion are the Stokes equations

d2ue

= dj + fi(x, t), div ue =0, x G QS, t> 0, (2)

with

'du'

aij = -SijPe + 2MSikSjhekh\ I , x G QS.

Here pe(x,t) is the fluid pressure and ^ is the fluid viscosity.

Besides, at the interface Se between viscoelastic and fluid parts of Q the conditions of continuity of displacement and normal stress are imposed:

Kk = 0, Kk = 0, (3)

where [•]Se denotes the jump across the boundary Se.

Finally, the problem is supplemented by homogeneous initial and Dirichlet boundary conditions: du

U(x, 0) = 0, — (x, 0)=0, x e Q, (4)

we(x,t) = 0, x e dQ, t> 0. (5)

Remark 1.1. In general, the continuity of the normal stress takes the form [afjnj]Se = 0, where nj, j = 1, 2, 3 are the components of the unit normal to S£. Since every layer of Q is parallel to the x2x3-plane, the unit normal to Se is ether n = (1,0, 0) or n = ( — 1,0, 0). This explains the form of the second boundary condition in (3).

To formulate the homogenized problem that corresponds to the original problem (1)-(5) we define the pairs {Zkh(y), Bkh(y)}, {Dkh(y), Akh(y)}, and {Wkh(y, t), Skh(y, t)}. They are solutions of the following auxiliary problems:

da(i)

= 0, y G Y ; div Zkh = y G Ys;

J Zkhdy = 0; [Zfch]s = 0; [a^js = 0;

=0, y G Y ; div Dkh = 0, y G Ys;

dyj

J Dkhdy = 0; [Dkh]s =0; [a^]s = 0;

(6)

(7)

(3)

= 0, w kh(y, 0) = Dkh(y), y G Y ; divy Wkh = 0, y G Y'

dyj

(8)

J Wkhdy = 0, [Wfch]s = 0; [a^]« = 0.

Here Zkh(y), Dkh(y) and Wkh(y,t) are Y-periodic vector functions, Bkh(y), Akh(y) and Skh(y,t) are Y-periodic scalar functions, S = dYh n dYs and

aj = bijimeim(Zkh) + 6ijfch, y e Yh; ajj = 2Meij(Zkh) + M(5ifc¿^ + ¿^¿jfc) — ¿¿jBkh, y e Ys; a^ = bjjimeim(Dfch) + ajj;me;m(Zkh) + ajj^h, y e Yh; aj = 2Mejj(Dkh) — ¿¿jAkh, y e Ys;

a

( ) / kh \

= aijimeim(Wkh) + bj^im , y G Yh;

aj = 2^j (^) - *JSkh, y G Y'.

Then under some additional assumptions on f (x,t) (see [5]) the homogenized problem corresponding to (1)-(5) takes the form

d2ui Oaj^ ,

po^T = dj + fi(x,t), x e Q, t> 0; (9)

du

u(x,t)=0, x e dQ, t> 0; u(x, 0) = — (x, 0) = 0, x e Q; (10)

where p0 = ph\Yh\ + ps\Ys\,

aij = aijkhekh(u) + fajkhekh ^+ 9ijkh(t) * ekh(u), gi(t) * g^(t) = J gi(t - s)g^(s)ds,

aijkh = (aijkh + <Hjimeim(Zkh) + bjimeim(Dkh)) dy + (2peHj(Dkh) - SjAkh)dy, (11) .Jy h Jy s

Pijkh = p(SikSjh + SihSjk)\Ys\ + (bijkh + bijimeim(Zkh)) dy+

JYh (12)

+ / (2peij (Zkh) - SijBkh) dy, JY 3

/( ( dW kh\\

f a,ijimeim(Wkh) + biji„iei„A gt j j dy+

r ( (dW kh\ \ (13)

+ JYs -dr J - Sij n dy'

Remark 1.2. To obtain the homogenized problem (9) and (10) we modify the results given in [4]. Namely, the auxiliary problems (6) and (8) have the same form as in [4], but we change auxiliary problems which define the initial conditions for Wkh(y,t). Nevertheless, setting Pkh(y,t) = Bkh(y)S'(t) + Akh(y)S(t) + Skh(y,t) in formula (5.3) from [4], we can easily derive problems (7).

In what follows we suppose that f(x,t) = 0. Then the homogenized problem (9), (10) describes natural vibrations of the homogeneous viscoelastic medium. In order to define the spectrum of the homogenized problem we apply the Laplace transform to equations (9), (10). We have g / ggk \

X2poiti = dj [(aijkh + Xpijkh + gijkh(A)) dxxh) , x e Q, (14)

u(x,A)=0, x e dQ, (15)

where u(x, A) and gijkh(A) are the Laplace transforms of u(x,t) and gijkh(t), respectively. Taking A for a spectral parameter, the spectrum of the homogenized problem (9), (10) is the set S = {A e C : u(x, A) ^ 0}, where u(x, A) is a solution of (14), (15).

It should be noted that if the set Q® is occupied by a viscous compressible fluid, then the condition divu® = 0 in (2) is replaced by the condition p® = —7divu®, where 7 = c2ps (here c is the speed of sound in the fluid). However, in this case the corresponding homogenized model is also described by system (9), (10) (see [1]) while the periodic auxiliary problems differ from (6)-(8). It is clear that incompressible fluid models can be considered as a limiting case of compressible fluid models when the acoustic speed c goes to infinity.

2. The case of transverse vibrations

In this section we consider the displacement vectors u®(x,t) and u(x,t) such that u®(x,t) = (u\(xl,t), 0,0) and u(x,t) = (ui(xi,t), 0,0). Then it is easy to see that the homogenized system (9) contains only one integro-differential equation:

d2u1 " du" "

= aiv'i + pi~gf + gi(t) * ui.

Hereinafter, the following notation is used: ai = ana, pi = pan, gi(t) = g^^t), i = 1, 2. To determine the constants ai, p1 and the kernel of convolution gi(t) we solve the auxiliary problems (6)-(8) for k = h =1 and find

Z 11(y) = (z(yi), 0,0), D11(y) = (0,0,0), W 11(y,t) = (0,0,0), y G Y;

B11(y) = -, A11(y) = -, S11(y,t) = 0, y G Ys, 1 - d 1 - d

where a1 = a1111 = Ae + 2^e, b1 = 61111 = Av + 2^,v, and

y1d

1 - d'

for y! G (0, (1 - d)/2],

z(y1) ={ - y1 + -, for y1 G /s,

, for y1 G [(1 + d)/2,1). Using formulas (11)-(13) we obtain

a1 = , p1 = t-^, g1(t)=0-1 — d 1 — d

Finally, we find that the homogenized problem takes the form

= a1u'i + P^^' X1 G (0,1), t> 0; (16)

du

u1(0,t)= u1(1,t) = 0, t> 0; u1(x1,0) = -df- (x1,0) = 0, x1 G (0,1). (17)

It follows from (16), (17) that in the case of transverse vibrations the homogenized problem does not contain long-term memory and describes one-dimensional vibrations of the viscoelastic Kelvin-Voigt material.

By definition, the spectrum of problem (16), (17) is the union of all A G C so that the corresponding spectral problem

poA2U1 = (a1 + P1A)u/1', X1 G (0,-), (18)

u1(0, A) = u1(1, A) = 0 (19)

has a non-trivial solution u1(x1,A). In order to define the values of A we seek a solution of problem (18) and (19) in the form

^ k

u1(x1,A) = ^^ vk (A)sin—X1. (20)

fc=1

Substituting (20) into (18) gives

^ k J3(A2 + P1CfcA + azCfc) vfc (A) sin =0

with Ck = n2k2/(p0-2). The spectrum of problem (16), (17) is the union of roots of the quadratic equations

A2 + P1Cfc A + azCfc =0 (21)

for all k G N. It is clear that for every fixed value of k G N the roots of equation (21) lie in the left half-plane (A : ReA < 0}. Let us denote

ki = max! k : k G N U {0}, k< -21- VPo"ll •

I n^i J

Since defining the spectrum of problem (16), (17) is reduced to finding the roots of the quadratic equations (21), the following statement is valid.

Theorem 2.1. The spectrum S1 of problem (16), (17) has the form

S1 = (A1fc}fc=1 U (A2fc}fc=1,

where A1Mfc = 1 (-&Cfc ± , k=l,2,....

In particular, A1k, A2k £ R for k = 1,..., k1, and A1k, A2k G R for all k > k1. Moreover, the

following asymptotic relations are valid:

A1k+, Aik^f-^+as k

Therefore, in the case of transverse vibrations the spectrum of the homogenized model contains k1 pairs of complex conjugate eigenvalues and infinite number of real eigenvalues. In particular, if / < n61/(^Y/(1 — d)p0a1) then the spectrum S1 contains only real eigenvalues.

Note that if we change a1 and in Theorem 2.1 for a1 and b1, respectively, then this theorem gives the spectral properties of the problem that describes one-dimensional vibrations (along the x1-axes) of the original Kelvin-Voigt material. Moreover, the equality a1/^1 = a1/b1 means that eigenvalues A1k of the latter problem and of problem (16), (17) have identical asymptotic behavior as k ^ ro.

To conclude this section we suppose that the original fluid is compressible with the large enough value of 7. Our aim now is to study the behavior of the spectrum of the corresponding homogenized problem as 7 ^ ro. It is known (see [1]) that this homogenized problem has the form

Po= A1U' + B1 dU" + G1(t) * <, X1 G (0,Z), t> 0; (22)

u1 (0,t)= u1(/, t) = 0, t> 0; u (x1,0) = -dt-(x1,0)=0, x1 G (0,/), (23)

where

A1 = p!(4^2a1(1 - d)+ 7&2d), B1 = 2^b1p1, G1(t) = -Q1e-?i,

Q1 = P?d(1 - d)(Yb1 - 2M«1 ^ C = P1P2, P1 = 2^(1 -d) + b1d, P2 = Y(1 - d) + a1d

We see that problem (22), (23) describes one-dimensional vibrations of the viscoelastic material with long-term memory. Furthermore, it was shown in [1] that the spectrum S2 of problem (22), (23) takes the form

S2 = (A1fc }feC=1 U (A2k }fc=1 U (A3fc}1,

where Aik, i = 1,2,3 are the roots of the cubic equation

A3 + (£ + BC)A2 + (B1C + A1)CfcA + (A1£ - Q1)Cfc = 0. (24)

Now we divide the left-hand side of (24) by y and consider the limit as 7 ^ ro. Since

^(e+BiCfc) ^ (1 - d)pi, ^(Bie+Ai) ^ bipi, ^(Aie - ?i) ^ aipi,

Y Y Y

two roots of (24) approach the roots of the quadratic equation (21) as y ^ ro. Furthermore, as it follows from Vieta's theorem the last root of (24) approaches —ro because other two roots of (24) are bounded as y ^ ro. Therefore, we observe the following interesting fact: there is a qualitative difference in the form of problems (16), (17) and (22), (23) and we cannot obtain the first problem as a limiting case of the second problem as y ^ ro. However, if we study eigenvalues of these two problems then eigenvalues of problem (16), (17) are the finite limits of eigenvalues of problem (22), (23) as y ^ ro.

3. The case of longitudinal vibrations

In this section we assume that we(x,t) = (0, w2(x2,t), 0) and u(x,t) = (0, «2(x2, t), 0). In order to obtain the corresponding homogenized problem, we need to determine the constants a2 and #2, and the kernel of convolution g2(t). To do this, we solve the auxiliary problems (6)-(8) for k = h = 2 and find

Z22(y) = (z(yi), 0,0), D22(y) = (0,0,0), W22(y,t) = (0,0,0), y G Y;

22

22

bi d

aid

B (y) = —2M — bi2 — ^, A (y) = —ai2 — —4, S(y,t) = 0, y G Ys,

1d

1d

where ai2 = aii22 = Ae, b12 = b1122 = Av. Using (11)-(13) we obtain

a d2 b d2

«2 = ai (1 — d) + 2a 12 d + , #2 = bi (1 — d) + 4^d + 2bi2d + , g2(t)=0.

1 — d 1 — d

Therefore, the homogenized problem takes the form

Po-

d 2«2 dt2

«2^2' + #2-^, x2 G (0,1), t> 0; d«2

«2(0,t) = «2 (1,t) = 0, t> 0; «2(x2, 0) = —2 (x2, 0) = 0, x2 G (0,1).

dt

(25)

(26)

We see that problem (25), (26) has the same form as problem (16), (17). Hence, to describe the spectral properties of problem (25), (26) one needs to use Theorem 2.1 and change ai and # i for a2 and #2, respectively. However, since a2222 = a i, b2222 = b i and a2/#2 = a i/b i, eigenvalues A ik of problem (25), (26) and eigenvalues of the problem describing one-dimensional vibrations (along the x2-axes) of the original Kelvin-Voigt material have different asymptotic behavior as k ^ ro.

It should be noted that if original fluid is assumed to be compressible with large enough value of Y then the corresponding homogenized problem, as in the case of transverse vibrations, describes one-dimensional vibrations of the viscoelastic material with long-term memory (see [1]). Moreover, the spectrum of this problem is the union of roots of the cubic equations (24) with the subscript 1 changed for 2 and constants in the equations are

A2 = a i (1 — d) + d(Y + (a 12 — y)c3 + b 12C4), B2 = b i (1 — d) + d(2^ + b 12C3), Q2 = P id(1 — d)(Y — a 12 + b 12P 1P2)2,

where c3 = —6i2Pi(1 — d), c4 = pi(1 — d)(Y — ai2 + bi2PiP2)- Finally, assuming 7 ^ то and repeating the above-mentioned arguments we can easily obtain results that are similar to the results obtained in the previous section.

This work was supported by the Russian Foundation for Basic Research, grant 13-01-00384

a.

References

[1] A.S.Shamaev, V.V.Shumilova, Spectrum of one-dimensional oscillations in a combined stratified medium consisting of a viscoelastic material and a viscous compressible fluid, Izvestiya Ros. Akad. Nauk, Mekhanika Zhidkosti i Gaza, (2013), no. 1, 17-25 (in Russian).

[2] A.S.Shamaev, V.V.Shumilova, On the spectrum of one-dimensional oscillations of a laminated composite with components of elastic and viscoelastic materials, Sib. Zh. Ind. Mat., 15(2012), no. 4(52), 124-134 (in Russian).

[3] L.D.Akulenko, S.V.Nesterov, Inertial and dissipative properties of a porous medium saturated with viscous fluid, Izvestiya Ros. Akad. Nauk, Mekhanika Tverdogo Tela, 40(2005), no. 1, 109-119 (in Russian).

[4] V.V.Shumilova, Homogenization of acoustic equations for a partially perforated viscoelastic material with channels filled with a fluid, Sovremennaya Matematika. Fundamentalnye Napravleniya, 39(2011), 185-198 (in Russian).

[5] G.Nguetseng, Asimptotic analysis for a stiff variational problem arising in mechanics, SIAM J. Math. Anal., 21(1990), no. 6, 1394-1414.

Спектр одномерных колебаний слоистой среды, состоящей из материала Кельвина-Фойгта и вязкой несжимаемой жидкости

Владлена В. Шумилова

Рассмотрена математическая модель, описывающая собственные колебания периодической слоистой среды, составленной из вязкоупругого материала Кельвина-Фойгта и вязкой несжимаемой жидкости. Для данной модели построены две усредненные модели, соответствующие поперечным и продольным колебаниям слоистой среды. Показано, что спектр каждой усредненной модели есть объединение корней соответствующих квадратных уравнений.

Ключевые слова: спектр, слоистая среда, усредненная модель, вязкоупругость, вязкая жидкость.