DSpace at VNU: Homogenized equations of the linear elasticity theory in two-dimensional domains with interfaces highly oscillating between two circles

Pham Chi Vinh · Do Xuan TungHomogenized equations of the linear elasticity theory in two-dimensional domains with interfaces highly oscillating between two circles Received: 11 August 20

Pham Chi Vinh · Do Xuan Tung

Homogenized equations of the linear elasticity theory

in two-dimensional domains with interfaces highly

oscillating between two circles

Received: 11 August 2010 / Revised: 22 November 2010 / Published online: 30 December 2010

© Springer-Verlag 2010

Abstract The main purpose of the present paper is to find homogenized equations in explicit form of the theory

of linear elasticity in a two-dimensional domain with an interface rapidly oscillating between two concentric circles In order to do that, we use the equations of linear elasticity in polar coordinates, and write them and the continuity conditions on the interface in matrix form By standard techniques of the homogenization method,

we have derived the explicit homogenized equations and associate continuity conditions for isotropic and ortho-tropic materials Since the obtained homogenized equations are explicit, i.e their coefficients are expressed explicitly in terms of given material and interface parameters, they are useful in practical applications

1 Introduction

Linear elasticity in domains with rough boundaries or interfaces is closely related to various practical problems such as scattering of elastic waves at rough boundaries and interfaces [1 6], the surface waves in half-spaces with cracked surfaces [7 10], nearly circular holes and inclusions in plane elasticity and thermoelasticity [11–14], and so on When the amplitude (height) of the roughness is very small in comparison to its period, the problems are usually analyzed by the perturbation method When the amplitude of the roughness is much larger than its period, i.e the boundary is very rough, and it oscillates between two parallel surfaces (they are often parallel planes, concentric spheres,… in practical problems), the homogenization method [15,16]

is required, in which the domain containing the very rough boundary is replaced by a new material “strip” whose elastic characteristic has to be determined (see [17]) Mathematically, we have to find the homogenized equations for this “strip” and associated boundary conditions on its boundaries

Nevard and Keller [17] examined the homogenization of a very rough three-dimensional interface that oscillates between two planes and separates two linear anisotropic solids By applying the homogenization method, the authors have derived the homogenized equations, but these equations are still implicit In partic-ular, their coefficients still depend on the solution of a boundary-value problem on the periodicity cell (called

“cell problem”) that includes 27 partial differential equations This problem cannot be solved analytically; it can only be solved numerically, in general Moreover, these equations (Eqs (9.24) in [17]) are incorrect at least in the two-dimensional case (see Remark 1) For two-dimensional very rough boundaries and interfaces, the “cell problem” contains ordinary differential equations, so hopefully it can be solved analytically, then the explicit homogenized equations may be derived In a recent paper [18], the explicit homogenized equations of the linear elasticity in two-dimensional domains with interfaces rapidly oscillating between two straight lines have been obtained To the best of the authors’ knowledge, for the case when the interfaces highly oscillate between two concentric circles, the homogenized equations are not available in the literature, so far Thus, the

P C Vinh (B) · D X Tung

Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science,

334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

E-mail: pcvinh@vnu.edu.vn

main purpose of the present paper is to find them For this end, we use the equations of the linear elasticity in the polar coordinates, then write them and the continuity conditions on the interface in matrix form By using standard techniques of the homogenization method, the explicit homogenized equations and associate continu-ity conditions have been derived for isotropic as well as orthotropic materials Since the obtained homogenized equations are explicit, i.e their coefficient are explicit functions of given material and interface parameters, they are significant in practical applications

2 Formulation of the problem

Consider the plane strain of a linear elastic body that occupies a domain of the plane x1x3 Suppose that

contains+, − whose interface L oscillates between two concentric circles, see Fig.1

In order to study the problem, we use the polar coordinates (r , θ), 0 < r < +∞, 0 < θ ≤ 2π

Sup-pose that, in the polar coordinates, the closed curve L is expressed by the equation r = h(θ/) = h(y) (y = θ/), where h is periodic in y with period 1, and  = 2π/N, N is a positive integer number Assume

that 0<   1, then L is called a very rough interface By a and b we denote the minimum and maximum

values of h (0 < a < b) (see Fig.1) The domain−(+ ) lies inside (outside) the closed curve L We also

assume that, in the domain 0< θ < , i.e 0 < y < 1, any circle r = r0 = const (a < r0 < b) has exactly

two intersections with the curve L.

Let the material of the body be isotropic In the polar coordinates (r , θ), the components of the stress tensor

σ rr , σ r θ , σ θθ are related to the displacement components u r , u θ by the following equations [19]:

σ rr = (λ + 2μ)ur ,r +λu r

r +λu θ,θ

σ θθ = (λ + 2μ)u θ,θ

r + λur ,r +(λ + 2μ)u r

σ r θ = μu r ,θ

r + u θ,r−u θ

r



,

where commas indicate differentiation with respect to the spatial variables r , θ, Lame’s constants: λ, μ and

the mass densityρ defined as follows:

λ, μ, ρ =



λ+, μ+, ρ+ for r > h(θ/) λ−, μ−, ρ− for r < h(θ/), (2)

Fig 1 Two-dimensional domains+ and−have a very rough interface L expressed by the equation r = h(θ/) = h(y), where

h (y) is a periodic function with period 1 The curve L oscillates between two concentric circles r = a and r = b (0 < a < b)

whereλ+, μ+, ρ+, λ−, μ−, ρ−are constant The equations of motion have the form [19]

σ rr ,r +1

r σ r θ,θ+1

r (σ rr − σ θθ ) + f r = ρ ¨u r ,

σ r θ,r +1

r σ θθ,θ +2σ r θ

r + f θ = ρ ¨u θ ,

(3)

where f r , f θ are components of the body force, a superposed dot signifies differentiation with respect to time

t Substituting (1) into (3) yields a system of equations for the displacement components whose matrix form is



Ahku,k

where k , h = θ, r, u = [u θ u r]T , F = [ f θ f r]T , the symbol T indicates the transpose of matrices, and

Aθθ = 1

r2



λ + 2μ 0



, A θr = 1

r



μ 0



, A r θ = 1

r



λ 0



, A rr =



μ 0

0 λ + 2μ



,

B= 1

r



μ 0

0 λ + 2μ



, C = 1

r2



0 λ + 4μ

−3μ 0



, D = −1

r2



μ 0

0 λ + 2μ



.

(5)

Assuming that+, − are well welded to each other on L, the continuity for the displacement vector and the

normal traction vector must be satisfied Thus, we have

r (A θθu,θ + Aθru,r + Gu) L n θ + Ar θu,θ+ Arr u,r + Hu L n r = 0 on L, (7)

where n θ , n r are components of the unit normal to L, by the symbol [ϕ] Lwe denote the jump of the quantity

ϕ through L, and

G= 1

r2



0 λ + 2μ

−μ 0



, H= 1

r



−μ 0



Taking into account the fact n θ : n r = −−1h(y)/r : 1 (see also [17]), the continuity condition (7) can be written as

−1h(y) Aθθu,θ+ Aθru,r L− Ar θu,θ+ Arru,r L + −1h(y)[G] Lu − [H]Lu= 0 on L. (9) Our aim is to find the homogenized equation in the explicit form of this problem, i.e the equation for the

leading term V(θ, r, t) in the asymptotic expansion (10) below

3 Explicit homogenized equations for isotropic materials

Following Bensoussan et al [15], Sanchez-Palencia [16] we suppose that u(θ, r, t, ) = U(θ, y, r, t, ), and

U can be expressed by [18]

U= V + N1V + N1θV,θ + N1rV,r

+ 2

N2V + N2θV,θ+ N2rV,r + N2θθV,θθ + N2θrV,θr+ N2rrV,rr

in which V=V(θ, r, t) (being independent of y), N1, N1θ , N 1r , N2, N2θ , N 2r , N2θθ , N2θr , N 2rr , are 2 ×

2-matrix functions of y and r (not depending on θ, t), and they are y-periodic with period 1 In what follows,

byϕ ,ywe denote the derivative ofϕ with respect to the variable y.

The functions N1, N1θ , , N 2rr , are chosen so that the equation (4) and the continuity conditions (6), (9) are satisfied Since y= θ/, we have

Substituting Eq (10) into Eq (4) and taking into account Eq (11) yields an equation which we call Eq (e1) Vanishing of the coefficient of−1of Eq (e

1) leads to



AθθN1,y



,yV+Aθθ



E + N1θ ,y



,yV,θ+AθθN1r ,y+ Aθr



,yV,r = 0, (12)

where E is the identity 2× 2-matrix In order to satisfy Eq (12), N1, N1θ , N 1r are chosen so that



AθθN1,y



,y = 0, Aθθ



E + N1θ ,y



,y = 0, AθθN1r ,y + Aθr



,y = 0. (13) From Eq (10), it is clear that the continuity condition Eq (6) is satisfied if

[N1]L = 0, [N1θ]L = 0, , [N 2rr]L = 0, on L. (14)

Substituting Eq (10) into Eq (9) and taking into account Eq (11) yields an equation which we call Eq (e2) Equating to zero the coefficient of−1of Eq (e2) yields

[AθθN1,y+ G]L V+Aθθ



E + N1θ ,y



LV,θ + [AθθN1r ,y + Aθr]LV,r = 0 on L. (15)

To satisfy Eq (15), we take

[AθθN1,y]L + [G]L = 0, Aθθ



E + N1θ ,y



L = 0, AθθN,y 1r+ Aθr



In view of Eqs (13), (14), (16), N1, N1θ , N 1r are solutions of the following problems:

AθθN1,y

,y = 0, 0 < y < 1, y = y1 , y2;

N1(0) = N1(1),

Aθθ



E + N1θ ,y



,y = 0, 0 < y < 1, y = y1 , y2;

[Aθθ



E + N1θ ,y



N1θ (0) = N1θ (1),



AθθN1r ,y + Aθr



,y = 0, 0 < y < 1, y = y1 , y2;



AθθN1r ,y + Aθr



N1r (0) = N 1r (1),

where y1, y2 (0 < y1 < y2 < 1) are two roots of the equation h(y) = r for y in the interval (0 , 1) in

which r , as a parameter, belongs to the interval (a b) The functions y1(r), y2(r) are two inverse branches of

the function r = h(y) Equating to zero the coefficient of 0of Eq.(e1) provides



AθθN2,y+ AθrN1,r

,y+ CN1

,y+ D V

+

Aθθ



N1+ N2θ ,y



+ AθrN1θ ,r

,y+ AθθN1,y+ CE + N1θ

,y V,θ

+

AθθN2r ,y + Aθr

N1+ N1r

,r

,y+ CN1r

,y + B V,r

+

Aθθ



N1θ+ N2θθ

,y



,y+ Aθθ



E + N1θ ,y V,θθ

+

Aθθ



N1r+ N2θr

,y



+ AθrN1θ

,y+ Aθr+ AθθN1r ,y V,θr

+AθθN2rr ,y + AθrN1r

,yV,rr

×Ar θN1,yV + Arθ



E + N1θ ,y



V,θ+ ArθN1r ,yV,r

,r

+ArrV,r

Setting the coefficient of0of Eq.(e2) zero gives



AθθN2,y+ AθrN,r1

L h −Ar θN1,y

L− [H]L + h[G]L N1

V

+Aθθ



N1+ N2θ ,y



+ AθrN1θ ,r

L h−Ar θ



E + N1θ ,y



L + h[G]L N1θ

V,θ

+AθθN2r ,y+ Aθr

N1+ N1r

,r

L h−Arr + ArθN,y 1r

L + h[G]L N1r

V,r

+Aθθ



N1θ+ N2θθ

,y



L hV,θθ+Aθθ



N1r+ N2θr

,y



+ AθrN1θ

L hV,θr +AθθN2rr ,y + AθrN1r

In order to satisfy (21) we take at y1, y2:

AθθN2,y+ AθrN1,r

L h=Ar θN1,y+ H

L − h[G]L N1,

Aθθ



N1+ N2θ ,y



+ AθrN1θ ,r

L h =Ar θ



E + N1θ ,y



L − h[G]L N1θ ,

AθθN2r ,y+ Aθr

N1+ N1r

,r

L h =Arr+ ArθN1r ,y

Aθθ



N1θ+ N2θθ

,y



L = 0, Aθθ



N1r+ N2θr

,y



+ AθrN1θ

L = 0,

AθθN2rr ,y + AθrN1r

L = 0.

By integrating Eq (20) along the line r = const, a < r < b from y = 0 to y = 1, and taking into account

Eq (22), we obtain



Aθθ



E + N1θ ,y



V,θθ + AθθN1r ,y+ Aθr V,θr

+ArθN1,y V + Arθ



E + N1θ ,y



V,θ + ArθN1r ,y V,r

,r+ Arr V,r ,r

+AθθN1,y +C



E + N1θ ,y



V,θ+CN1r

,y + B V,r

+CN1

,y + D V+H/hy2

y1V− δGN1(y2) − N1(y1)V

−δGN1θ (y2) − N1θ (y1)V,θ − δGN1r (y2) − N 1r (y1)V,r + F − ρ ¨V = 0. (23)

ϕ =

1

 0

ϕdy = a1ϕ+ + a2 ϕ−, a1= y2 − y1 , a2= 1 − a1 ,



ϕ/hy2

y1= a3 δϕ, a3=1/h(y2) − 1/h(y1), δϕ = ϕ+ − ϕ−.

(24)

ϕ+andϕ− are independent of y, they are the values of ϕ in +and−, respectively

In order to derive Eq (23), the following relations have been employed:



Ar θN1,yV



,r

 +Ar θN1,y /hy2

y1V=Ar θN1,y V,r ,



ArrV,r

,r

 +Arr /hy2

y1V,r =Arr V,r

,r ,



Ar θN,y 1rV,r



,r

 +Ar θN1r ,y /hy2

y1V,r =Ar θN1r ,y V,r



,r , (25)



Ar θ (E + N1θ

,y )V ,θ



,r

 +Ar θ (E + N1θ

,y )/hy2

y1

V,θ =Arθ (E + N1θ

,y ) V ,θ



,r ,

which originate all from the equality

ϕ ,r +ϕ/hy2

that is easily proved by using the relation (24)1with noting that y

k = 1/h(y k ), k = 1, 2 The application of

Eq (26) withϕ being A r θN1,yV, A rrV,r , A r θN1r ,yV,r , A r θ (E + N1θ

,y )V ,θ yields the relations Eq (25)

It is clear that in order to make Eq (23) explicit, we have to calculate the quantities

q1=Aθθ



E + N1θ ,y



, q2= AθθN1r ,y+ Aθr , q3=Ar θ



E + N1θ ,y



,

q4= Ar θN1r ,y , q5=C

E + N1θ ,y



, q6= CN1r

,y ,

q7= ArθN1,y , q8= AθθN1,y , q9= CN1

,y ,

q10= N1(y2) − N1(y1), q11= N1θ (y2) − N1θ (y1), q12= N1r (y2) − N 1r (y1).

(27)

Calculating q7, q8, q9:

From Eq (17) and noting that AθθN1,y is a y-periodic function with period 1, it is not difficult to verify that

AθθN1,y =

⎧

⎪

⎨

⎪

A−1θθ −1A−1

θθ+ δGa1, 0≤ y ≤ y1 ,

A−1θθ −1A−1

11 +a1− EδG, y1≤ y ≤ y2 ,

A−1θθ −1A−1

θθ+ δGa1, y2≤ y ≤ 1.

(28)

From Eq (28) it follows:

q8= AθθN1,y =A−1θθ −1A−1

θθ+− EδGa1, (29)

q9= CN1

,y =CA−1θθ A−1θθ −1A−1

θθ+− C+A−1

θθ+

q7= ArθN1,y =ArθA−1

θθ A−1θθ −1A−1

θθ+− Arθ+A−1

θθ+

Calculating q1: From Eqs (18)1,2it follows:

Aθθ



N1θ

or equivalently:

N1θ

where C∗ = C∗(r) Integrating Eq (33) and with the help of the conditions: N1θ (0)=N1θ (1) and [N1θ]L = 0

at y = y1we have

N1θ =

⎧

⎪

⎨

⎪

⎪

y

0



A−1

y

y1



A−1

θθ+C∗− Ed y+y1

0



A−1

θθ−C∗− Ed y+ C2, y1≤ y ≤ y2 ,

y

1



A−1

(34)

where C2= C2(r) From the condition [N1θ]L = 0 at y = y2it gives

It is clear from Eqs (32) and (35) that

q1=Aθθ



E + N1θ ,y



Determining q2: From Eqs (19)1,2it follows:

or equivalently:

where C∗= C∗(r) Following the same procedure as above we have

hence, by Eq (37):

q2= AθθN1r ,y+ Aθr = A−1θθ −1A−1θθAθr (40)

Calculating q3: From Eq (32) it follows:

Ar θ



E + N1θ ,y



= Ar θA−1

By Eq (35) we have

q3=Ar θ



E + N1θ ,y



= ArθA−1

θθ A−1θθ −1. (42)

Calculating q4: From Eq (38) it follows:

Ar θN1r ,y = Ar θA−1

θθ



C∗− Aθr

On use of Eq (39) yields

q4= ArθN1r ,y = ArθA−1

θθ A−1θθ −1A−1θθAθr − ArθA−1

θθAθr (44)

Calculating q5, q6:

By replacing Ar θ by C in Eqs (42), (44), we obtain the expressions for q5, q6, namely

q5=C



E + N1θ ,y



q6= CN1r

,y = CA−1θθ A−1θθ −1A−1θθAθr − CA−1θθAθr (46)

Calculating q10, q11, q12:

From Eq (34), we have

q11=A−1

θθ+A−1θθ −1− Ea1. (47) Similarly, we obtain

q10= −A−1θθ−A−1θθ −1A−1

θθ+ δGa1a2, (48)

q12= A−1θθ+A−1θθ −1A−1θθAθr − Aθr+



Now we substitute Eqs (29)–(31), (36), (40), (42), (44)–(49) into Eq (23) Finally, the explicit homogenized equation is



CA−1θθ + δGa2A−1

θθ−



A−1θθ −1A−1

θθ+− C+A−1

θθ+

δGa1+ D +H/hy2

y1



V

+A−1θθ −1A−1

θθ+ δGa1− δGa1A−1

θθ+A−1θθ −1+ CA−1θθ A−1θθ −1

V,θ

+CA−1θθ A−1θθ −1A−1θθAθr − δGa1A−1

θθ+



A−1θθ −1A−1θθAθr − Aθr+



− CA−1θθAθr + B V,r +ArθA−1

θθ A−1θθ −1A−1

θθ+− Arθ+A−1

θθ+



δGa1V

,r

+ A−1θθ −1V,θθ+ A−1θθ −1A−1θθAθr V,θr +Ar θA−1

θθ A−1θθ −1V,θ

,r

+Arr + ArθA−1

θθ A−1θθ −1A−1θθAθr − ArθA−1

θθAθr V,r



,r + F − ρ ¨V = 0. (50)

This is desired homogenized equation, which defines in the domain a < r < b In the domains r > b and

0< r < a, the homogenized equations are



Ahk+V,k

,h+ B+V,r + C+V,θ+ D+V + F+ = ρ+¨V, r > b, (51)



Ahk−V,k

,h+ B−V,r + C−V,θ+ D−V + F− = ρ−¨V, 0 < r < a. (52)

The continuity conditions on the boundaries r = a and r = b are



ArθA−1

θθ A−1θθ −1A−1

θθ+− Arθ+A−1

θθ+



δGa1+ H V

L∗

+Arr + ArθA−1

θθ A−1θθ −1A−1θθAθr − ArθA−1

θθAθr V,r

L∗

+ArθA−1

θθ A−1θθ −1V,θ

L∗= 0, and [V]L∗= 0, L∗is lines r = a, r = b. (53) Note that the continuity condition (53)1originates from

Ar θu,θ+ Arr u,r + Hu L∗= 0, L∗is lines r = a, r = b. (54) Substituting Eq (10) into (54) yields an equation denoted by Eq.(e3) By equating to zero the coefficient of

0of Eq.(e3) we have



Ar θN,y1 + HV + Arθ



E + N1θ ,y



V,θ+Arr + ArθN,y 1r



V,r

Integrating Eq (55) along L∗from y = 0 to y = 1 and using the results obtained above we obtain Eq (53.1).

We summarize our results in the following theorem

Theorem 1 Let u (θ, r, , t) satisfy Eqs (4), (6), (9) with Ahk , B, C, D, G, H, ρ defined by Eqs (2), (5), (8),

F is given, the curve L: r = h(y) is a very rough interface which oscillates between two concentric circles

r = a and r = b (a < b) and h(y) is a differentiable y-periodic function with period 1 In addition,

sup-pose u=U (θ, y, r, , t) and U(θ, y, r, , t) has asymptotic form Eq ( 10) Then, V (θ, r, t) is a solution of Eqs.

Eqs (50)–(52) and the continuity conditions Eq (53)

Remark 1 For the two-dimensional case, from (9.12) and (9.13) in [17], one can show that c i j k δ ∂ y δ χ kmn (in Nevard and Keller’s notations) are constant in domains x3 > h(y) and x3 < h(y) From this fact, it can

be shown that the correct homogenized equation for the two-dimensional case is Eq (9.24) in [17], in which

M i mn ≡ 0 (the right-hand side of Eq (9.24) in [17] must be zero)

Remark 2 It is easy to see that



CA−1θθ A−1θθ −1A−1

θθ+− C+A−1

θθ+



a1=CA−1θθ A−1θθ −1A−1

θθ−− C−A−1

θθ−



(−a2),



A−1θθ −1A−1

θθ+− A−1θθ+A−1θθ −1a1=A−1θθ −1A−1

θθ−− A−1θθ−A−1θθ −1(−a2),

a1A−1

θθ+



A−1θθ −1A−1θθAθr − Aθr+



= (−a2 )A−1θθ−A−1θθ −1A−1θθAθr − Aθr−



,



ArθA−1

θθ A−1θθ −1A−1

θθ+− Arθ+A−1

θθ+



a1=ArθA−1

θθ A−1θθ −1A−1

θθ−− Arθ+A−1

θθ−



(−a2).

(56)

Therefore, Eqs (50) and (53)1can be written as



CA−1θθ + δG∗a1A−1

θθ+



A−1θθ −1A−1

θθ−− C−A−1

θθ−

δG∗a2+ D

+H/hy2

y1



V+A−1θθ −1A−1

θθ− δG∗a2− δG∗a2A−1

θθ−A−1θθ −1+ CA−1θθ A−1θθ −1V,θ

+CA−1θθ A−1θθ −1A−1θθAθr − δG∗a

2A−1

θθ−



A−1θθ −1A−1θθAθr − Aθr−



− CA−1θθAθr + B V,r+Ar θA−1

θθ A−1θθ −1A−1

θθ−− Ar θ−A−1

θθ−



δG∗a2V

,r

+ A−1θθ −1V,θθ + A−1θθ −1A−1θθAθr V,θr+ArθA−1

θθ A−1θθ −1V,θ

,r

+Arr + ArθA−1

θθ A−1θθ −1A−1θθAθr − ArθA−1

θθAθr V,r



,r + F − ρ ¨V = 0, (57) 

Ar θA−1

θθ A−1θθ −1A−1

θθ−− Ar θ−A−1

θθ−



δG∗a2+ H V

L∗

+Arr + Ar θA−1

θθ A−1θθ −1A−1θθAθr − Ar θA−1

θθAθr V,r

L∗

+Ar θA−1

θθ A−1θθ −1V,θ

whereδG∗= G−− G+and noting that Aθθbeing symmetry Thus, the relations Eq (56) ensures the homog-enized equations and the continuity conditions treat+, −equally

In component form, Eqs (51)–(53) are written as

⎧

⎪

⎪

−1

r2(λ + 2μ)+ V r− 1

r2(λ + 3μ)+ V θ,θ+1

r (λ + 2μ)+ V r ,r+ 1

r2μ+ V r ,θθ

+1

r (λ + μ)+ V θ,rθ + (λ + 2μ)+V r ,rr + f r+= ρ+¨Vr , r > b

−1

r2μ+ V θ + 1

r2(λ + 3μ)+ V r ,θ+1

r μ+ V θ,r + 1

r2(λ + 2μ)+ V θ,θθ

+1

r (λ + μ)+ V r ,rθ + μ+V θ,rr + f θ+ = ρ+¨V θ , r > b

(59)

1

r2



δ(λ + 2μ)a1

(λ + 2μ)+



3μ+−



3μ

λ + 2μ

  1

λ + 2μ

−1

(λ + 2μ)−

 1

λ + 2μ

−1

−λ + 2μ + ra3 δλ] V r+ 1

r2



δμ

 1

(λ + 2μ)+

 1

λ + 2μ

−1

−

 1

μ

−1 1

μ+

a1

−



3μ

λ + 2μ

  1

λ + 2μ

−1

V θ,θ+1

r



δμa1

(λ + 2μ)+



λ

λ + 2μ

  1

λ + 2μ

−1

− λ+

+



3μλ

λ + 2μ



+ λ + 2μ −



3μ

λ + 2μ

  1

λ + 2μ

−1

λ

λ + 2μ



V r ,r



δ(λ + 2μ)a1

r (λ + 2μ)+



λ

λ + 2μ

  1

λ + 2μ

−1

− λ+

V r

,r

r2

 1

μ

−1

V r ,θθ +1

r

 1

μ

−1

V θ,θr +

 1

r



λ

λ + 2μ

  1

λ + 2μ

−1

V θ,θ

,r

+



λ

λ + 2μ

2 1

λ + 2μ

−1 +



4μ(λ + μ)

λ + 2μ



V r ,r

,r

+  f r = ρ ¨V r , a < r < b, (60)

1

r2



δμa1

μ+



(λ + 4μ)+−

λ + 4μ

μ

  1

μ

−1

−δ(λ + 2μ)a2

μ−

 1

μ

−1

− μ − ra3 δμ

V θ

r2



δ(λ + 2μ)a1



1

λ + 2μ

−1

1

(λ + 2μ)+−

 1

μ

−1 1

μ+

 +



λ + 4μ μ

  1

μ

−1

V r ,θ

+1

r



λ + 4μ μ

  1

μ

−1

− λ + 3μ + δ(λ + 2μ)a μ+ 1



μ+−

 1

μ

−1

V θ,r

+



δμa1

r



1−

 1

μ

−1 1

μ+

V θ

,r

r2

 1

λ + 2μ

−1

V θ,θθ+1

r



λ

λ + 2μ

  1

λ + 2μ

−1

V r ,θr

+



1

μ

−1

V θ,r

,r

+

 1

r

 1

μ

−1

V r ,θ

,r

+  f θ = ρ ¨V θ , a < r < b, (61)

⎧

⎪

⎪

−1

r2(λ + 2μ)− V r− 1

r2(λ + 3μ)− V θ,θ+1

r (λ + 2μ)− V r ,r+ 1

r2μ− V r ,θθ

+1

r (λ + μ)− V θ,rθ + (λ + 2μ)−V r ,rr + fr−= ρ−¨Vr , 0 < r < a

−1

r2μ− V θ + 1

r2(λ + 3μ)− V r ,θ+1

r μ− V θ,r + 1

r2(λ + 2μ)− V θ,θθ

+1

r (λ + μ)− V r ,rθ + μ+V θ,rr + f θ− = ρ−¨V θ , 0 < r < a,

(62)



δ(λ + 2μ)a1

r (λ + 2μ)+



λ

λ + 2μ

  1

λ + 2μ

−1

− λ+

V r+λ

r V r

L∗

+



λ

λ + 2μ

2 1

λ + 2μ

−1 +



4μ(λ + μ)

λ + 2μ



V r ,r

L∗

+

 1

r

λ + 2μ

  1

λ + 2μ

−1

V θ,θ

L∗

= 0, L∗is lines r = a, r = b, (63)



δμa1

r



1−

 1

μ

−1 1

μ+

V θ− μ

r V θ

L∗

+



1

μ

−1

V θ,r

L∗

+

 1

r

 1

μ

−1

V r ,θ

L∗

Remark 3 Consider the case of plane stress Then, the corresponding constitutive equations have the form

Eq (1) in whichλ is replaced by λ = λ(1 − 2ν)/(1 − ν), ν is Poisson’s ratio, and the equations of motion

Eq (3) are replaced by the corresponding equilibrium equations Consequently, all obtained results are still valid for the plane stress in whichλ is replaced by λ, and V does not depend on the time t.

This is desired homogenized equation, which defines in the domain a < r < b In the domains r > b and

0< r < a, the homogenized equations are

... y2 < 1) are two roots of the equation h(y) = r for y in the interval (0 , 1) in< /i>

which r , as a parameter, belongs to the interval (a b) The functions y1(r),... are two inverse branches of< /i>