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DOI: 10.17516/1997-1397-2021-14-3-351-359 УДК 534.18
Effective Acoustic Equations for a Layered Material Described by the Fractional Kelvin-Voigt Model
Alexey S. Shamaev* Vladlena V. Shumilova^
Ishlinsky Institute for Problems in Mechanics RAS Moscow, Russian Federation
Received 10.12.2020, received in revised form 16.01.2021, accepted 05.03.2021 Abstract. The paper is devoted to the construction of effective acoustic equations for a two-phase layered viscoelastic material described by the Kelvin-Voigt model with fractional time derivatives. For this purpose, the theory of two-scale convergence and the Laplace transform with respect to time are used. It is shown that the effective equations are partial integro-differential equations with fractional time derivatives and fractional exponential convolution kernels. In order to find the coefficients and the convolution kernels of these equations, several auxiliary cell problems are formulated and solved. Keywords: homogenization, acoustic equations, viscoelasticity, fractional Kelvin-Voigt model. Citation: A.S. Shamaev, V.V. Shumilova, Effective Acoustic Equations for a Layered Material Described by the Fractional Kelvin-Voigt Model, J. Sib. Fed. Univ. Math. Phys., 2021, 14(3), 351-359. DOI: 10.17516/1997-1397-2021-14-3-351-359.
The study of macroscopic acoustic behavior of heterogeneous viscoelastic materials with periodic microstructure is one of the most significant problems in acoustical engineering when dealing with polymer based composites. The most rigorous and widely accepted mathematical tool for the theoretical part of this study is the theory of homogenization. Using techniques of homogenization, the actual highly inhomogeneous periodic viscoelastic composite can be replaced by the corresponding effective (homogenized) material with the similar acoustic properties.
It is well known that short memory effects in microheterogeneous viscoelastic Kelvin-Voigt materials lead to the appearance of long memory effects in the corresponding effective media (see [1-3]). In other words the acoustic equations for these materials, which are partial differential equations, become partial integro-differential equations after homogenization. The same result was observed for two-phase materials, in which the first phase is an elastic material whilst the second one is a viscoelastic Kelvin-Voigt material [4,5].
In recent years there has been an increasing number of papers devoted to the development of fractional models in viscoelasticity (see, for instance, [6-8] and the reference therein). Such models consist of differential or integro-differential equations with fractional derivatives. The growing popularity of fractional models is explained by their ability of describing the complex behaviour of viscoelastic materials using a small number of parameters.
In this paper, we consider a mathematical model describing small displacements of a two-phase layered viscoelastic material whose behavior is described by the fractional Kelvin-Voigt model. This model consists of a system of partial differential equations with fractional time
*https://orcid.org/0000-0003-2766-6382
tv.v.shumilova@mail.ru https://orcid.org/0000-0003-3830-7924 © Siberian Federal University. All rights reserved
derivatives and rapidly oscillating piecewise constant coefficients, conditions of ideal contact between layers, and homogeneous initial and outer boundary conditions. Using the two-scale convergence method [9,10] and applying the Laplace transform, we show that the corresponding effective model involves a system of partial integro-differential equations with fractional time derivatives and constant coefficients. By solving a number of auxiliary cell problems, we calculate these coefficients and find that the integral parts of the effective equations are of convolution type and their kernels are fractional exponential Rabotnov's functions. Thus, we rigorously establish that long memory effects mentioned above also appear in the effective material that corresponds to the fractional Kelvin-Voigt material.
1. Original acoustic equations
Consider a bounded domain Q = (0, L)3 occupied by two-phase viscoelastic material with a periodic microstructure. Let e C L be a small positive parameter characterizing the heterogeneity period of the viscoelastic material. We suppose that every phase is isotropic and consists of the union of layers that are parallel to the Ox2x3 plane. More precisely, denote
D2e = (0,L) n f Q (e(hi + k),e(h2 + k)U , Dle = (0, L) \ D2e,
\k=0 J
1 — h , 1 + h hi = ——, h2 = ——, 0 <h< 1
and assume that the sets Qie = D1e x (0, L)2 and Q2e = D2e x (0, L)2 are occupied by the first and the second phase, respectively.
Note that the periodicity cell Ye of the above layered material may be extracted in different ways. For our convenience, we will assume that Ye = eY, where Y = (0,1)3 is a unite cube. The cube Y can be decomposed into two parts Y1 and Y2 with a common boundary S as follows:
Yi = ((0, hi) U (h2,1)) x (0,1)2, Y2 = (hi, h2) x (0,1)2, S =({hi}U{h2}) x (0,1)2.
It is obvious that Ye = eYi U eY2 U eS. The part eYi represents the first phase and consists of two layers with the same thickness e(1 — h)/2 while the part eY2 represents the second phase of the layered material and consists of one layer with the thickness eh (see Fig. 1).
The viscoelastic material we propose to study is described by the fractional Kelvin-Voigt model. Its constitutive equations between the components of the stress and strain tensors have the form
= aijkH(x)ekh(ue) + bijkh(x)ehh (D?ue), 0 < a < ^ (1)
where ue(x,t) is the displacement vector, ae(x) = a(e-ix) and be(x) = b(e-ix) are Y^-periodic tensors describing the elastic and viscous properties of the material, ae and e(ue) are the stress and strain tensors, and Df is the Caputo fractional time derivative of order a,
«<"•)=1 (g+dh ),D>- ^ jr.—, £
aijkh(y) = AsSijSkh + l^s(SikSjh + SihSjk), y € Ys, bijkh(y) = CsSijSkh + Vs(SikSjh + SihSjk), y € Ys, y = e-ix, s = 1, 2, 1 ^ i,j, k,h ^ 3.
eY,
eY
eYi
eh
Fig. 1. The first and the second phases of the layered material
Here As and ns are the Lame parameters of Qs£, Zs and ns are parameters describing the viscous behavior of Qs£, r(a) is Euler's gamma function, and Sj is Kroneker's delta. Note that in (1) and everywhere below we assume summation with respect to repeated indices.
The motion of the viscoelastic material in the phase Qs£ is described by the system of partial differential equations with fractional time derivative
d\
Ps
dt2
M
dxn
+ fi(x,t) in Qse X (0,T), S = 1, 2,
(2)
where ps = const > 0 is the density of the material in Qse and f(x,t) are the components of the volume external force vector.
On the boundaries between the layers we assume the condition of ideal contact. It means the continuity of displacements and normal stresses at each layer interface and is written as
Mk
o, kei]k =o,
(3)
where the square brackets denote the jump in the enclosed quantity across the boundary
Se = dQle n dQ2£.
Finally, we accept that the boundary conditions on dQ for displacements as well as the initial conditions for displacements and velocities are homogeneous, i.e.
s. _. due
ue\ao, =0, uE\t=0 =0,
dt
0.
t=0
(4)
Problem (2)-(4) is a mathematical model describing the general motion of the two-phase viscoelastic material. Our aim now is to deduce the corresponding effective (homogenized) model that describes the limit dynamic behavior of the original two-phase viscoelastic material as e ^ 0.
x
x
s
2. Effective acoustic equations
To construct the homogenized problem, we will use the method proposed in [5,11,12]. This method was developed for the homogenization of acoustics equations in two-phase dissipative media with periodic microstructure. Its main tools are the Laplace transform and the concept of two-scale convergence introduced by G. Nguetseng [9].
First, applying the Laplace transform ue(x,t) ^ uex(x) and f (x,t) ^ fx(x), we convert the evolutionary problem (2)-(4) into the stationary one. As a result, we obtain the following
boundary value problem for Laplace transforms:
2
PsA2u\i = —+ fxi(x) in Qse, s = 1, 2,
dxj (5)
ui = 0, [ueJSe =0, [a^ie]|Se = 0,
where
vj = ia'ijkh(x) + A°'bejkh(x)) ekh(uV).
Next, using the basic properties of two-scale convergence and repeating the same arguments as in [5,11,12], we can show that the homogenized problem that corresponds to problem (5) and which is constructed for e ^ 0 has the form
2 daii
PoA un = d^T + fii(x) in Q, uA|an = 0, (6)
where
P0 = Pi(1 — h) + P2h, = dijkhekh(u\),
dijkh = Y (cijkh(y) + c}jlm(y)eL(.Qkxh» dy, (7)
cjkh(y) = aijkh(y) + Aa bijkh(y), e«m(Qkxh) = 1 ( ^Qm + ^
Here the vector-valued functions Qih(y) are Y-periodic solutions to the following cell problems: d
d i
JT (cijkh(y) + cijlm(y)eylm(Qf)) =0 in Y, Qkxhdy = 0, ayj Jy (8)
[Qih]|y1=hs =0, [cijkh(y)+ cijlm(y)elm,(Q\h)] \y1=hs =0, S = 1, 2.
Now we apply the inverse Laplace transform to the homogenized stationary problem (6). We have
P0S = P1 + fi(x,t) in Q x (0,T), (9)
dt2 dxj
du
u|dn = 0, u|t=o = 0, —
=0
t=o
with
Vij = dijkh(t) * ekh(u), (10)
where the symbol * denotes the operation of convolution with respect to time t.
3. Solutions of auxiliary cell problems
Passing to the inverse Laplace transforms in (8) we see that Qkh(y,t) depends on the Dirac function S(t) and cannot be expressed in explicit form without some additional explanations. In order to do this and at the same time derive direct formula for calculation of components of
the tensor d(t), we will proceed in the following way. Let us represent the solutions Qkh(y) to problems (8) in the form
Vkh(y)
Qf(y)= Z kh(y)+ Aa _ M ih , M kh = const, (11)
where vector-valued functions Zkh(y), Vkh(y) and parameters Mkh are to be specified.
In a first step, let us define the vector-valued functions Zkh(y) as Y-periodic solutions to the cell problems
d t
-7- (bijkh(y) + bijtm(y)eylm(Zkh)) =0 in Y, Zkhdy = 0,
dyj Jy (12)
[Z k%!=hs =0, [bijkh(y) + bijlm(y)e«m (Z kh)]\yi=ha =0, * =1 2
In a second step, using the solutions Zkh(y) to problems (12), we define the vector-valued functions Vkh(y) as Y-periodic solutions to the cell problems
d
Tj (aijkh(y) + aijim(y)eym(Zkh) + hjlm(y)eym(Vkh)) =0 in Y, [aijkh(y) + OijiMelm(Zkh) + bijim(y)e«m(Vkh)] ^ = 0, s = 1, 2, (13)
/ Vkhdy = 0, [Vkh]\ =h =0.
Jy iyi=hs
To write out solutions to problems (12) and (13), we introduce 1-periodic piecewise linear function z(yi) defined by
^ yi e (CU,),
z(yi)={ _y, + o, yi e (hi,h2),
2
(y, _ l)h
1 - h
, yi e (h2,1).
It is easy to check that Zkh(y) = Zhk (y) and Vkh(y) = Vhk (y), so that we need only to find Zkh(y) and Vkh(y) for k ^ h. Solving problems (12) for k ^ h, we obtain
Zii(y) = (ciz(yi), 0,0), Z22(y) = Z33(y) = (c2z(yi), 0,0),
Zi2(y) = (0, czz(yi), 0), Zi3(y) = (0, 0, C3z(y,)), Z23(y) = (0, 0, 0),
where
CI = J^(1 _ h)(b2 _ bi), C2 = 7^(1 _ h)(Z2 _ Zi), C3 = —(1 _ h)(V2 _ ni),
b,2 bi2 ni2
bi2 = bih + b2(1 _ h), ni2 = nih + n2(1 _ h), bs = Zs + 2ns. Substituting Zkh(y) into problems (13) and solving them for k < h, we derive
Vii(y) = (cAz(yi), 0,0), V22(y) = V33(y) = (c5z(yi), 0,0),
Vi2(y) = (0, cez(yi), 0), Vi3(y) = (0,0, c6z(yi)), V23(y) = (0, 0,0),
where
1 1 — h
C4 = vr(1 - h)(bia2 - b2ai), C5 = —2— ((A2 - Ai)bi2 - (C2 - Ci)ai2),
bi2 bi2
C6 = -2-(1 - h)(niM2 - V2Pi), ai2 = aih + a2(1 - h), as = As + 2^.
Now, after defining Zkh(y) and Vkh(y) in (11), we can find parameters Mkh. It follows from (8), (12), and (13) that Mkh satisfies the system d
-J- (aijlm(y)eim(Vkh) + Mkhbijim(y)elm(Vkh» =0 in Y, (14)
yj
[aijim(y)eim(Vkh) + Mkhbijim(y)efm(Vkh)]\yi=hs =0, S = 1, 2. (15)
Substitute Vkh(y) found above into (14) and (15). It is easy to check that equations (14) are always fulfilled for any parameters Mkh. Further, from the boundary conditions (15) we calculate the required values of Mkh:
mii=m22=m33=- a2, b12
Mi2 = M2i = Mi3 = M3i = Mi2
ni2
Applying the inverse Laplace transform to (11), we get
Qkh(y, t) = S(t)Zkh (y) + Ra-i(M kh,t)V kh(y), where Rv(fi,t) denotes fractional exponential Rabotnov's function [13]:
Rv(M = tV n=nr[(l + n)(1 + v)] •
=0
Next we substitute the decomposition (11) into (7) to obtain
jA
dijkh = Aijkh + AaBjkh + Gijkh(A),
where the components of the tensors A, B, and G(A) are given by the formulas
Aijkh = I (aijkh(y) + aijim(y)eym(Zkh) + bijUy^Vkh» dy, (16)
'Y
Bijkh = Y (bijkh(y) + bijim(y)eym(Zkh)) dy, (17)
GiMX) = xa -Mkh fY (*ijim(y)eym(Vkh) + Mkhbijim(y)eym(Vkh)) dy.
Therefore, the constitutive equations (10) take the form
aij = Aijkhekh(u) + Bijkhekh(D<a'u) + Gjkh(t) * ekh(u), (18)
where Gijkh(t) are the inverse Laplace transforms of Gijkh(X):
Gijkh(t) = Ra-1 (Mkh ,t) j (aijim(y)eim(Vkh) + M khbijim(y)elm(Vkh)) dy. (19)
From (18) we see that the effective acoustic equations (9) are partial integro-differential equations with fractional time derivative and constant coefficients. It is interesting to note that their kernels are expressed via two different Rabotnov's functions Ra-1 (—a12/b12,t) and Ra-l(-Ml2/ni2,t).
4. Components of the tensors A, B, and G(t)
Before proceeding to the calculation of the tensors A, B, and G(t), let us note that
Ajkh = Ajikh = Akhij, Aijkh = 0 whenever SjSkh + SikSjh + SihSjk = 0,
A2222 = A3333, A1122 = A1133, A1212 = A1313, A2222 — A2233 = 2A2323 and similarly for the tensors B and G(t). Moreover, it is easy to see that
A2323 = Mi(1 — h) + Mih, B2323 = ni(1 — h)+ nih, G2323(t)=0.
Therefore, it is sufficient to find the components of A, B, and G(t) with indexes {1111}, {2222}, {1122}, and {1212}. To do this, we first substitute the found solutions to problems (12) and (13) into formulas (16) and (17). This yeilds
A1111 = ai(1 — h) + a2h + ci h(ai — 0,2) + cAh(bi — b2), A2222 = ai(1 — h) + a2h + C2h(Xi — X2) + c5h((i — (2), A1122 = Ai(1 — h) + X2 h + C2h(ai — a2) + c5h(bi — bi), A1212 = Mi(1 — h) + Mih + c3h(^i — m2) + c6h(ni — n2), B1111 = bi(1 — h) + b2h + cih(bi — b2), B2222 = bi(1 — h)+ b2h + cih((i — (2), B1212 = ni(1 — h) + n2h + c3h(ni — m), B1122 = (1 (1 — h) + (2h + c2h(bi — b2).
Taking into account the above values of constants c and using trivial transformations, we obtain
A1111 = j2r (aib2h + aib2(1 — h)) , A1212 = ~2r (m2Vih + Minl(1 — h)) , b12 V12
A2222 = ai(1 — h)+ a2h + h(1 — ^ — (2] (an((1 — (2) — 2bu(Xi — X2)),
b12
1h
A1122 = (biX2h + b2Xi(1 — h)) + -t-(1 — h)((2 — (1 )(aib2 — a2bi), b12 b12
B1111 = ^, B2222 = bi(1 — h) + bih — h (1 — h)((i — (i)2, b12 b12
B1122 = (bi(ih + bi(i(1 — h)), B1212 = —. bi2 ^12
In order to find the components of G(t), we substitute the solutions Vkh(y) to problems (13) and the parameters Mkh into formulas (19). As a result, we get
h(1 — h) 2 a12
Giiii(t) =---3-(aib2 — aibi) Ra-i — — ,t
b12 V b12
h(1 — h) 2 a12
G2222 (t) = — yb32 ' ((Xi — X2 )bi2 — ((1 — (2)aiiY Ra-1 y — , t J ,
h(1 — h) a12
Gii22(t) =--bi-(aib2 — aibi) ((Xi — Xi)bi2 — ((1 — (2)au) Ra-i ^ — — ,t
Gi212 (t) = — h(1 — h (Mini — M2Vi)2Ra-i( — — .
mi V nii J
To conclude, we note that our results can be considered as a generalization of those obtained in the case of two-phase layered viscoelastic material described by a standard Kelvin-Voigt model (a = l). Indeed, the effective acoustic equations for the last material also have form (9) with the constitutive equations (18), where Aijkh and Bijkh are defined by the same formulas as above. Moreover, the components of G(t) are found by using the formulas presented here, in which we should put a = l and take into account that
This work was accomplished within the Russian State Assignment under contract no. АААА-
А20-120011690138-6.
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Эффективные уравнения акустики для слоистого материала, описываемого дробной моделью Кельвина-Фойгта
Алексей С. Шамаев Владлена В. Шумилова
Институт проблем механики им. А. Ю. Ишлинского РАН Москва, Российская Федерация
Аннотация. Статья посвящена построению эффективных уравнений акустики для двухфазного слоистого вязкоупругого материала, описываемого моделью Кельвина-Фойгта с дробными производными по времени. Для этой цели используется теория двухмасштабной сходимости и преобразование Лапласа по времени. Показано, что эффективные уравнения являются интегро-дифференциальными уравнениями в частных производных с дробными производными по времени и дробно-экспоненциальными ядрами свертки. Для того чтобы найти коэффициенты и ядра сверток этих уравнений, сформулированы и решены несколько вспомогательных задач.
Ключевые слова: усреднение, уравнения акустики, вязкоупругость, дробная модель Кельвина-Фойгта.
