Acta Mech 218, 333–348 (2011) DOI 10.1007/s00707-010-0426-2 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 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 orthotropic 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 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 particular, 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 334 P C Vinh, D X Tung 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 continuity 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 Formulation of the problem Consider the plane strain of a linear elastic body that occupies a domain of the plane x1 x3 Suppose that contains + , − whose interface L oscillates between two concentric circles, see Fig In order to study the problem, we use the polar coordinates (r, θ ), < r < +∞, < θ ≤ 2π Suppose 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 < 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 < θ < , i.e < 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 σrr , σr θ , σθ θ are related to the displacement components u r , u θ by the following equations [19]: λu r λu θ,θ + , r r (λ + 2μ)u θ,θ (λ + 2μ)u r = + λu r,r + , r r uθ u r,θ =μ + u θ,r − , r r σrr = (λ + 2μ)u r,r + σθ θ σr θ (1) where commas indicate differentiation with respect to the spatial variables r, θ , Lame’s constants: λ, μ and the mass density ρ defined as follows: λ, μ, ρ = λ+ , μ+ , ρ+ for r > h(θ/ ) λ− , μ− , ρ− for r < h(θ/ ), (2) Fig 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 The curve L oscillates between two concentric circles r = a and r = b (0 < a < b) Homogenized equations of the linear elasticity theory 335 where λ+ , μ+ , ρ+ , λ− , μ− , ρ− are constant The equations of motion have the form [19] 1 σrr,r + σr θ,θ + (σrr ) + fr = uă r , r r 2σr θ + f θ = uă , r ,r + , + r r (3) where fr , 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 Ahk u,k ,h + Bu,r + Cu,θ + Du + F = uă in + , (4) where k, h = θ, r , u = [u θ u r ]T , F = [ f θ fr ]T , the symbol T indicates the transpose of matrices, and λ + 2μ , Aθr = μ r r μ 1 μ , C= B= λ + 2μ −3μ r r Aθ θ = μ μ λ , , Ar θ = , Arr = λ + 2μ λ r μ λ + 4μ , D=− 0 λ + 2μ r (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 [u] L = on L , r (Aθ θ u,θ + Aθr u,r + Gu) n + Ar θ u,θ + Arr u,r + Hu L θ n = on L , L r (6) (7) where n θ , nr are components of the unit normal to L, by the symbol [ϕ] L we denote the jump of the quantity ϕ through L, and G= r2 λ + 2μ , −μ Taking into account the fact n θ : nr = − written as −1 −1 h + −μ λ r (8) (y)/r : (see also [17]), the continuity condition (7) can be h (y) Aθ θ u,θ + Aθr u,r −1 H= L − Ar θ u,θ + Arr u,r L h (y)[G] L u − [H] L u = 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 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+ + N1 V + N1θ V,θ + N1r V,r N2 V + N2θ V,θ + N2r V,r + N2θ θ V,θ θ + N2θr V,θr + N2rr V,rr + · · · (10) in which V=V(θ, r, t) (being independent of y), N1 , N1θ , N1r , N2 , N2θ , N2r , N2θ θ , N2θr , N2rr , are × 2-matrix functions of y and r (not depending on θ , t), and they are y-periodic with period In what follows, by ϕ,y we denote the derivative of ϕ with respect to the variable y The functions N1 , N1θ , , N2rr , are chosen so that the equation (4) and the continuity conditions (6), (9) are satisfied Since y = θ/ , we have u,θ = U,θ + −1 U,y (11) 336 P C Vinh, D X Tung 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 −1 of Eq (e1 ) leads to Aθ θ N,y ,y 1θ V + Aθ θ E + N,y ,y 1r V,θ + Aθ θ N,y + Aθr ,y V,r = 0, (12) where E is the identity × 2-matrix In order to satisfy Eq (12), N1 , N1θ , N1r are chosen so that Aθ θ N,y ,y 1θ = 0, Aθ θ E + N,y ,y = 0, 1r Aθ θ N,y + Aθr ,y = (13) From Eq (10), it is clear that the continuity condition Eq (6) is satisfied if [N1 ] L = 0, [N1θ ] L = 0, , [N2rr ] 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 −1 of Eq (e2 ) yields 1θ + G] L V + Aθ θ E + N,y [Aθ θ N,y L 1r V,θ + [Aθ θ N,y + Aθr ] L V,r = on L (15) To satisfy Eq (15), we take 1θ ] L + [G] L = 0, Aθ θ E + N,y [Aθ θ N,y L = 0, 1r Aθ θ N,y + Aθr L = on L (16) In view of Eqs (13), (14), (16), N1 , N1θ , N1r are solutions of the following problems: Aθ θ N,y ,y = 0, < y < 1, y = y1 , y2 ; ] L + [G] L = 0, [N1 ] L = at y1 , y2 ; [Aθ θ N,y (17) N1 (0) = N1 (1), 1θ Aθ θ E + N,y ,y = 0, < y < 1, y = y1 , y2 ; 1θ ] L = 0, [N1θ ] L = at y1 , y2 ; [Aθ θ E + N,y (18) N1θ (0) = N1θ (1), 1r Aθ θ N,y + Aθr ,y 1r Aθ θ N,y + Aθr = 0, < y < 1, y = y1 , y2 ; L = 0, [N1r ] L = 0, at y1 , y2 ; (19) N1r (0) = N1r (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 Homogenized equations of the linear elasticity theory 337 the function r = h(y) Equating to zero the coefficient of Aθ θ N,y + Aθr N,r ,y of Eq.(e1 ) provides + CN,y +D V + 2θ 1θ + Aθr N,r Aθ θ N1 + N,y + 2r 1r Aθ θ N,y + Aθr N1 + N,r + 2θ θ Aθ θ N1θ + N,y + 2θr Aθ θ N1r + N,y + Aθr N1θ 2rr + Aθ θ N,y + Aθr N1r ,y 1θ + Aθ θ N,y + C E + N,y ,y V,θ θ 1r + Aθr + Aθ θ N,y V,θr V,rr 1θ 1r V,θ + Ar θ N,y × Ar θ N,y V + Ar θ E + N,y V,r + Arr V,r Setting the coefficient of ,r V,θ 1r + CN,y + B V,r 1θ + Aθ θ E + N,y ,y ,y ,y ,r ă = + F − ρV (20) of Eq.(e2 ) zero gives Aθ θ N,y + Aθr N,r L 2θ N + N,y h − Ar θ N,y + Aθ θ + 2r 1r Aθ θ N,y + Aθr N1 + N,r 1θ + Aθr N,r 2θ θ + Aθ θ N1θ + N,y L 2rr + Aθ θ N,y + Aθr N1r L L L L − [H] L + h [G] L N1 V 1θ h − Ar θ E + N,y 1r h − Arr + Ar θ N,y + h [G] L N1θ V,θ L + h [G] L N1r V,r L 2θr + Aθr N1θ h V,θ θ + Aθ θ N1r + N,y L h V,θr h V,rr = (21) In order to satisfy (21) we take at y1 , y2 : Aθ θ N,y + Aθr N,r L h = Ar θ N,y +H 2θ 1θ + Aθr N,r Aθ θ N1 + N,y L 2r 1r + Aθr N1 + N,r Aθ θ N,y 2θ θ Aθ θ N1θ + N,y L 2rr Aθ θ N,y + Aθr N1r L L L − h [G] L N1 , 1θ h = Ar θ E + N,y 1r h = Arr + Ar θ N,y L L − h [G] L N1θ , − h [G] L N1r , 2θr + Aθr N1θ Aθ θ N1r + N,y = 0, L (22) = 0, = By integrating Eq (20) along the line r = const, a < r < b from y = to y = 1, and taking into account Eq (22), we obtain 1θ Aθ θ E + N,y 1r V,θ θ + Aθ θ N,y + Aθr V,θr + 1θ Ar θ N,y V + Ar θ E + N,y + 1θ Aθ θ N,y + C E + N,y + CN,y + D V + H/ h 1r V,θ + Ar θ N,y V,r V,θ + y2 y1 1r CN,y + B ,r + Arr V,r ,r V,r V − δG N1 (y2 ) − N1 (y1 ) V ¨ = −δG N1θ (y2 ) − N1θ (y1 ) V,θ − δG N1r (y2 ) − N1r (y1 ) V,r + F − ρ V (23) 338 P C Vinh, D X Tung Here, ϕ = ϕ/ h ϕdy = a1 ϕ+ + a2 ϕ− , a1 = y2 − y1 , a2 = − a1 , y2 y1 (24) = a3 δϕ, a3 = 1/ h (y2 ) − 1/ h (y1 ) , δϕ = ϕ+ − ϕ− ϕ+ and ϕ− 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 θ N,y V ,r Arr V,r 1r Ar θ N,y V,r 1θ Ar θ (E + N,y )V,θ ,r y2 y1 y2 1θ )/ h + Ar θ (E + N,y ,r y1 y2 y1 + Arr / h 1r /h + Ar θ N,y ,r y2 /h + Ar θ N,y y1 V= Ar θ N,y V , ,r V,r = Arr V,r V,r = 1r Ar θ N,y V,r V,θ = 1θ Ar θ (E + N,y ) V,θ ,r , ,r , (25) ,r , which originate all from the equality ϕ,r + ϕ/ h y2 y1 = ϕ (26) ,r that is easily proved by using the relation (24)1 with noting that yk = 1/ h (yk ), k = 1, The application of V, A V , A N1r V , A (E + N1θ )V yields the relations Eq (25) Eq (26) with ϕ being Ar θ N,y rr ,r r θ ,y ,r rθ ,θ ,y It is clear that in order to make Eq (23) explicit, we have to calculate the quantities 1θ q1 = Aθ θ E + N,y 1r 1θ , q2 = Aθ θ N,y + Aθr , q3 = Ar θ E + N,y 1r 1θ q4 = Ar θ N,y , q5 = C E + N,y , 1r , q6 = CN,y , (27) 1 q7 = Ar θ N,y , q8 = Aθ θ N,y , q9 = CN,y , q10 = N1 (y2 ) − N1 (y1 ), q11 = N1θ (y2 ) − N1θ (y1 ), q12 = N1r (y2 ) − N1r (y1 ) Calculating q7 , q8 , q9 : is a y-periodic function with period 1, it is not difficult to verify that From Eq (17) and noting that Aθ θ N,y = Aθ θ N,y ⎧ −1 A ⎪ ⎪ ⎨ θθ ⎪ ⎪ ⎩ −1 A−1 δGa , θθ+ A−1 θθ ≤ y ≤ y1 , −1 A−1 a − E 11+ −1 −1 −1 Aθ θ Aθ θ + δGa1 , δG, y1 ≤ y ≤ y2 , (28) y2 ≤ y ≤ From Eq (28) it follows: = q8 = Aθ θ N,y q9 = CN,y = q7 = Ar θ N,y = A−1 θθ −1 A−1 θ θ + − E δGa1 , −1 CA−1 θ θ Aθ θ −1 −1 A−1 θ θ + − C+ Aθ θ + δGa1 , −1 Ar θ A−1 θ θ Aθ θ −1 −1 A−1 θ θ + − Ar θ + Aθ θ + δGa1 (29) (30) (31) Calculating q1 : From Eqs (18)1,2 it follows: 1θ + E = C∗ ∀ y ∈ [0 1], Aθ θ N,y (32) Homogenized equations of the linear elasticity theory 339 or equivalently: 1θ N,y = A−1 θ θ C∗ − E ∀ y ∈ [0 1], (33) where C∗ = C∗ (r ) Integrating Eq (33) and with the help of the conditions: N1θ (0)=N1θ (1) and [N1θ ] L = at y = y1 we have ⎧ y −1 ⎪ ≤ y ≤ y1 , ⎪ ⎪ Aθ θ − C∗ − E dy + C2 , ⎨ y y1 −1 −1 1θ Aθ θ − C∗ − E dy + C2 , y1 ≤ y ≤ y2 , N = (34) y1 Aθ θ + C∗ − E dy + ⎪ ⎪ ⎪ y −1 ⎩ y2 ≤ y ≤ 1, Aθ θ − C∗ − E dy + C2 where C2 = C2 (r ) From the condition [N1θ ] L = at y = y2 it gives C∗ = A−1 θθ −1 (35) It is clear from Eqs (32) and (35) that = A−1 θθ 1θ q1 = Aθ θ E + N,y −1 (36) 1r Aθ θ N,y + Aθr = C∗ ∀ y ∈ [0 1], (37) −1 1r ∗ = A−1 N,y θ θ C − Aθ θ Aθr ∀ y ∈ [0 1], (38) Determining q2 : From Eqs (19)1,2 it follows: or equivalently: where C∗ = C∗ (r ) Following the same procedure as above we have C ∗ = A−1 θθ −1 A−1 θ θ Aθr , (39) hence, by Eq (37): 1r + Aθr = A−1 q2 = Aθ θ N,y θθ −1 A−1 θ θ Aθr (40) Calculating q3 : From Eq (32) it follows: 1θ = Ar θ A−1 Ar θ E + N,y θ θ C∗ (41) By Eq (35) we have 1θ q3 = Ar θ E + N,y −1 = Ar θ A−1 θ θ Aθ θ −1 (42) Calculating q4 : From Eq (38) it follows: 1r ∗ Ar θ N,y = Ar θ A−1 θ θ C − Aθr (43) On use of Eq (39) yields −1 1r = Ar θ A−1 q4 = Ar θ N,y θ θ Aθ θ −1 −1 A−1 θ θ Aθr − Ar θ Aθ θ Aθr (44) Calculating q5 , q6 : By replacing Ar θ by C in Eqs (42), (44), we obtain the expressions for q5 , q6 , namely 1θ q5 = C E + N,y −1 = CA−1 θ θ Aθ θ −1 1r q6 = CN,y = CA−1 θ θ Aθ θ −1 −1 , −1 A−1 θ θ Aθr − CAθ θ Aθr (45) (46) 340 P C Vinh, D X Tung Calculating q10 , q11 , q12 : From Eq (34), we have −1 q11 = A−1 θ θ + Aθ θ −1 − E a1 (47) Similarly, we obtain −1 q10 = −A−1 θ θ − Aθ θ −1 q12 = A−1 θθ+ −1 A−1 θθ A−1 θ θ + δGa1 a2 , (48) A−1 θ θ Aθr − Aθr + a1 (49) Now we substitute Eqs (29)–(31), (36), (40), (42), (44)–(49) into Eq (23) Finally, the explicit homogenized equation is −1 CA−1 θ θ + δGa2 Aθ θ − + A−1 θθ + −1 CA−1 θ θ Aθ θ −1 −1 A−1 V,θ θ θθ −1 A−1 θ θ + − C+ Aθ θ + δGa1 + D + H/ h + −1 −1 + CA−1 θ θ Aθ θ −1 A−1 θ θ Aθr − δGa1 Aθ θ + −1 −1 A−1 θθ A−1 θθ −1 Ar θ A−1 θ θ Aθ θ V,r + A−1 θ θ Aθr −1 Arr + Ar θ A−1 θ θ Aθ θ + −1 −1 −1 A−1 θ θ + δGa1 − δGa1 Aθ θ + Aθ θ − CA−1 θ θ Aθr + B + A−1 θθ −1 −1 −1 V V,θ A−1 θ θ Aθr − Aθr + −1 A−1 θ θ + − Ar θ + Aθ θ + δGa1 V Ar θ A−1 θθ V,θr + −1 y2 y1 ,r −1 A−1 V,θ θθ ,r −1 A−1 θ θ Aθr − Ar A Ar V,r ,r ă = + F −ρ V (50) This is desired homogenized equation, which defines in the domain a < r < b In the domains r > b and < r < a, the homogenized equations are Ahk+ V,k ,h Ahk− V,k ,h ă r > b, + B+ V,r + C+ V,θ + D+ V + F+ = ρ+ V, (51) ă < r < a + B V,r + C− V,θ + D− V + F− = ρ− V, (52) The continuity conditions on the boundaries r = a and r = b are −1 Ar θ A−1 θ θ Aθ θ + + Arr + −1 −1 A−1 θ θ + − Ar θ + Aθ θ + δGa1 + H Ar θ A−1 θθ −1 Ar θ A−1 θ θ Aθ θ −1 −1 A−1 θθ V,θ L∗ A−1 θ θ Aθr − V Ar θ A−1 θ θ Aθr = 0, and [V] L ∗ = 0, L∗ V,r L∗ L ∗ is lines r = a, r = b (53) Note that the continuity condition (53)1 originates from Ar θ u,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 of Eq.(e ) we have 1θ 1r Ar θ N,y V,θ + Arr + Ar θ N,y V,r + H V + Ar θ E + N,y L∗ = (55) Integrating Eq (55) along L ∗ from y = to y = and using the results obtained above we obtain Eq (53.1) We summarize our results in the following theorem Theorem 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 In addition, suppose 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) Homogenized equations of the linear elasticity theory 341 Remark For the two-dimensional case, from (9.12) and (9.13) in [17], one can show that ci jkδ ∂ yδ χkmn (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 Mimn ≡ (the right-hand side of Eq (9.24) in [17] must be zero) Remark It is easy to see that −1 CA−1 θ θ Aθ θ A−1 θθ a1 A−1 θθ+ −1 −1 −1 A−1 θ θ + − C+ Aθ θ + a1 = −1 −1 A−1 θ θ + − Aθ θ + Aθ θ A−1 θθ −1 Ar θ A−1 θ θ Aθ θ −1 −1 −1 a1 = −1 CA−1 θ θ Aθ θ A−1 θθ −1 −1 A−1 θ θ − − C− Aθ θ − (−a2 ), −1 −1 A−1 θ θ − − Aθ θ − Aθ θ −1 A−1 θ θ Aθr − Aθr + = (−a2 )Aθ θ − −1 A−1 θ θ + − Ar θ + Aθ θ + a1 = −1 A−1 θθ −1 Ar θ A−1 θ θ Aθ θ (−a2 ), (56) A−1 θ θ Aθr − Aθr − , −1 −1 −1 −1 A−1 θ θ − − Ar θ + Aθ θ − (−a2 ) Therefore, Eqs (50) and (53)1 can be written as −1 ∗ CA−1 θ θ + δG a1 Aθ θ + + H/ h y2 y1 −1 CA−1 θ θ Aθ θ + A−1 θθ V+ −1 + + −1 −1 Ar θ A−1 θ θ Aθ θ + + Arr + A−1 θ θ Aθr −1 −1 −1 Ar θ A−1 θθ V,θr + −1 −1 A−1 θθ V,θ L∗ = 0, − −1 + CA−1 θ θ Aθ θ −1 V,θ A−1 θ θ Aθr − Aθr − ,r −1 A−1 V,θ θθ ,r −1 A−1 θ θ Aθr − Ar θ Aθ θ Aθr A−1 θ θ Aθr −1 −1 ∗ A−1 θ θ − − Ar θ − Aθ θ − δG a2 V −1 ∗ A−1 θ θ − − Ar θ − Aθ θ − δG a2 + H Ar θ A−1 θθ −1 Ar θ A−1 θ θ Aθ θ A−1 θθ −1 Ar θ A−1 θ θ Aθ θ V,r + −1 A−1 θθ −1 −1 ∗ A−1 θ θ − − C− Aθ θ − δG a2 + D −1 −1 ∗ ∗ A−1 θ θ − δG a2 − δG a2 Aθ θ − Aθ θ −1 Arr + Ar θ A−1 θ θ Aθ θ + −1 −1 ∗ A−1 θ θ Aθr − δG a2 Aθ θ − − CA−1 θ θ Aθr + B −1 A−1 V,θ θ θθ A−1 θθ V Ar A1 Ar V,r ă = 0, + F −ρ V ,r (57) L∗ V,r L∗ L ∗ is lines r = a, r = b, (58) where δG∗ = G− − G+ and noting that Aθ θ being symmetry Thus, the relations Eq (56) ensures the homogenized equations and the continuity conditions treat + , − equally In component form, Eqs (51)–(53) are written as ⎧ 1 1 ⎪ ⎪ − r 1(λ + 2μ)+ Vr − r (λ + 3μ)+ Vθ,θ + r (λ + 2μ)+ Vr,r + r μ+ Vr,θ θ ⎨ + r (λ + μ)+ Vθ,r θ + ( + 2)+ Vr,rr + fr + = + Văr , r > b 1 1 ⎪ ⎪ − r μ+ Vθ + r (λ + 3μ)+ Vr,θ + r μ+ Vθ,r + r (λ + 2μ)+ Vθ,θ θ ⎩ + r (λ + μ)+ Vr,r θ + μ+ Vθ,rr + f θ + = + Vă , r > b r2 ( + 2μ)a1 (λ + 2μ)+ 3μ+ − − λ + 2μ + ra3 δλ] Vr + λ + 2μ −1 λ + 2μ 3μ λ + 2μ −1 − δμa2 (λ + 2μ)− λ + 2μ 1 δμ r2 (λ + 2μ)+ λ + 2μ − 3μ λ + 2μ + 3μ 3μλ + λ + 2μ − λ + 2μ λ + 2μ Vθ,θ + r δμa1 (λ + 2μ)+ λ + 2μ −1 −1 λ λ + 2μ λ λ + 2μ − μ −1 λ + 2μ Vr,r (59) −1 μ+ −1 a1 − λ+ 342 P C Vinh, D X Tung + 1 r2 μ −1 δμa1 μ+ δ(λ + 2μ)a1 r2 + r + + λ + 4μ μ δμa1 r μ λ + 2μ −1 μ 1− μ −1 −1 + Vθ,r ,r λ r λ + 2μ 4μ(λ + μ) λ + 2μ + −1 μ −1 − λ+ Vr ,r Vθ,θr + λ + 4μ μ (λ + 4μ)+ − + −1 λ + 2μ −1 λ + 2μ −1 1 r μ Vr,θ θ + λ λ + 2μ + r2 λ λ + 2μ δ(λ + 2μ)a1 r (λ + 2μ)+ + − μ+ 1 r λ + 2μ 1 r μ −1 −1 δ(λ + 2μ)a2 μ− μ δ(λ + 2μ)a1 μ+ ,r ,r −1 μ+ μ+ − −1 λ + 4μ μ + μ Vθ,θ θ + − μ − ra3 δμ Vθ μ −1 Vθ,r λ r λ + 2μ λ + 2μ λ λ + 2μ λ λ + 2μ + + λ + 2μ δμa1 r μ + 1 r μ −1 −1 − λ+ Vr + 4μ(λ + μ) λ + 2μ = 0, Vθ,θ Vr,θr (61) λ Vr r (62) L∗ Vr,r L∗ L ∗ is lines r = a, r = b, (63) L∗ , Vr,θ + −1 μ+ −1 −1 λ + 2μ λ + 2μ λ r λ + 2μ 1− −1 ,r ⎧ 1 1 ⎪ ⎪ − r 1(λ + 2μ)− Vr − r (λ + 3μ)− Vθ,θ + r (λ + 2μ)− Vr,r + r μ− Vr,θ θ ⎨ + r (λ + μ)− Vθ,r θ + (λ + 2μ)− Vr,rr + fr = Văr , < r < a 1 1 ⎪ ⎪ − r μ− Vθ + r (λ + 3μ)− Vr,θ + r μ− Vθ,r + r (λ + 2μ)− Vθ,θ θ ⎩ + r (λ + μ)− Vr,r θ + μ+ Vθ,rr + f θ − = Vă , < r < a, ( + 2μ)a1 r (λ + 2μ)+ Vr,θ −1 + f θ = Vă , a < r < b, Vr, (60) ,r − λ + 3μ + + Vθ,θ + fr = Văr , a < r < b, Vr,r 1 − (λ + 2μ) + μ Vθ −1 λ + 2μ Vθ − μ Vθ r + L∗ μ −1 Vθ,r L ∗ is lines r = a, r = b, L∗ (64) L∗ [Vr ] L ∗ = 0, [Vθ ] L ∗ = 0, L ∗ is lines r = a, r = b (65) Remark Consider the case of plane stress Then, the corresponding constitutive equations have the form Eq (1) in which λ 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 which λ is replaced by λ , and V does not depend on the time t Homogenized equations of the linear elasticity theory 343 As an example, we consider the case when the interface L is of “comb-type” as illustrated in Figure For this case, y1 = α, ≤ α ≤ 1, y2 = Since y1 , y2 not depend on r , all in Eqs (60), (61) so, consequently, Eqs (60), (61) are simplified to ⎧ − α1 Vr − r12 α2 Vθ,θ + r1 α3 Vr,r + r12 α4 Vr,θ θ ⎪ ⎪ ⎨ r + r α5 Vθ,r θ + α6 Vr,rr + fr = Văr , < r < a β V + β V + 1β V + β V ⎪ ⎩ r 1 θ r 2 r,θ r θ,r r θ,θ θ + r β5 Vr,r θ + β6 V,rr + f = Vă , < r < a (66) where α1 = μ α2 = δμ α3 = λ + 3μ λ + 2μ δ(λ + 2μ)a1 (λ + 2μ)+ −1 λ λ + 2μ 3μ λ + 2μ −1 α4 = μ α6 = λ λ + 2μ β1 = λ λ + 2μ −1 λ + 2μ δμa1 μ+ − (λ + 3μ)+ + + λ + 4μ μ Fig The interface of comb-type μ a1 + λ + 3μ λ + 2μ λ + 2μ −1 −1 + λ + 2μ , , 3μλ + λ + 2μ λ + 2μ − λ+ + (67) −1 μ + −1 , 4μ(λ + μ) , λ + 2μ μ −1 λ + 2μ −1 −1 −1 δμa2 (λ + 2μ)− λ + 2μ λ , λ + 2μ λ + 2μ λ + 3μ μ β2 = δ(λ + 2μ)a1 β3 = λ + 2μ −1 λ + 2μ , α5 = −1 1 − μ+ (λ + 2μ)+ λ + 2μ δ(λ + 3μ)a1 (λ + 2μ)+ − λ + 2μ + −1 δ(λ + 2μ)a2 μ− μ 1 − (λ + 2μ) + μ − λ + 3μ + δ(λ + 3μ)a1 μ+ −1 −1 − (λ + 3μ)+ + μ , μ+ + μ+ − μ λ + 3μ μ −1 , μ −1 , 344 P C Vinh, D X Tung 30 α αk (1011 dyne/cm2 ) 25 α1, α3, α6 20 15 α 10 α 0 0.1 0.2 0.3 0.4 0.5 α 0.6 0.7 0.8 0.9 Fig The dependence of the coefficient αk , (k = 1, 2, , 6) on the parameter α Here, we take λ+ = 7.583 × 1011 dyne/cm2 , μ+ = 6.334 × 1011 dyne/cm2 , λ− = 3.653 × 1011 dyne/cm2 , μ− = 2.823 × 1011 dyne/cm2 30 β2 β 20 15 βk (10 11 dyne/cm ) 25 β5 10 β1, β3, β6 0 0.1 0.2 0.3 0.4 0.5 α 0.6 0.7 0.8 0.9 Fig The dependence of the coefficient βk , (k = 1, 2, , 6) on the parameter α Here, we take λ+ = 7.583 × 1011 dyne/cm2 , μ+ = 6.334 × 1011 dyne/cm2 , λ− = 3.653 × 1011 dyne/cm2 , μ− = 2.823 × 1011 dyne/cm2 −1 , β5 = α5 , β6 = α4 , λ + 2μ a1 = − α, a2 = α, ϕ = (1 − α)ϕ+ + αϕ− β4 = (68) One can see, from Eqs (67), that the effective coefficients αk , βk of Eq (66) are functions of the parameter α They are the corresponding coefficients of Eqs (59), (62) when α = 0, α = 1, respectively Figures 3, show the dependence of the effective coefficients αk , βk on α Homogenized equations for orthotropic elastic materials In this section, we derive explicit homogenized equations for the orthotropic elastic materials in twodimensional domains with very rough interfaces of circular type Homogenized equations of the linear elasticity theory 345 Consider the plane strain of a orthotropic linear elastic body that occupies a domain of the plane x1 x3 Suppose that is made of from + , − whose interface L rapidly oscillates between two concentric circles (Fig 1) In the polar coordinates (r, θ ), the components of the stress tensor σrr , σr θ , σθ θ are related to the displacement components u r , u θ by the following relations (see [20,21]): ur u θ,θ + , r r u θ,θ ur = c12 u r,r + c22 + , r r uθ u r,θ = c66 + u θ,r − , r r σrr = c11 u r,r + c12 σθ θ σr θ (69) where the material constants ci j and the mass density ρ defined as follows: ci j , ρ = ci j+ , ρ+ for r > h(r/ ), ci j− , ρ− for r < h(r/ ), (70) where ci j+ , ρ+ , ci j− , ρ− are constant The equations of motion have the form [19] 1 σrr,r + σr θ,θ + (σrr − σθ θ ) + fr = uă r , r r 2r + f = uă , σr θ,r + σθ θ,θ + r r (71) where fr , f θ are components of the body force Substituting Eq (69) into equations of motion Eq (71) yields a system of equations whose matrix form is Ahk u,k ,h + Bu,r + Cu,θ + Du + F = uă in + , (72) k, h = θ, r , u = [u θ u r ]T , F = [ f θ fr ]T and c22 c12 , Ar θ = , Aθr = c c 66 66 r r 1 c66 α1 , D=− , C= B= c −α 11 r r r Aθ θ = c c66 , Arr = 66 , c11 c 12 r c66 , (73) c22 where α1 = c22 + 2c66 , α2 = c22 + c66 − c12 We also assume that L, then the continuity conditions are [u] L = on L , r (Aθ θ u,θ + Aθr u,r + Gu) L +, − are well welded to each other on n θ + Ar θ u,θ + Arr u,r + Hu L nr = on L , (74) (75) where G= Since n θ : nr = - −1 h(y)/r r2 −c66 0 c22 , H= c12 −c66 r (76) : 1, the continuity condition Eq (75) can be written as −1 h (y) Aθ θ u,θ + Aθr u,r + −1 L − Ar θ u,θ + Arr u,r h (y)[G] L u − [H] L u = on L L (77) Analogously as above, we have the following theorem Theorem Let u(θ, r, , t) satisfy Eqs (72), (74), (77) with Ahk , B, C, D, G, H, ρ defined by Eqs (70), (73), (76), 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 In addition, suppose 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 the problem Eqs (50)–(53), in which Ahk , B, C, D, G, H, ρ are defined by Eqs (70), (73), (76) 346 P C Vinh, D X Tung By using Eqs (70), (73), (76), we can write equations Eqs (50)–(53) in component form as follows: ⎧ 1 1 ⎪ ⎪ − r 1c22+ Vr − r (c22 + c66 )+ Vθ,θ + r c11+ Vr,r + r c66+ Vr,θ θ ⎨ + r (c12 + c66 )+ Vθ,r θ + c11+ Vr,rr + fr + = + Văr , r > b (78) 1 1 ⎪ ⎪ − r c66+ Vθ + r (c22 + c66 )+ Vr,θ + r c66+ Vθ,r + r c22+ Vθ,θ θ ⎩ + r1 (c12 + c66 )+ Vr,r θ + c66+ Vθ,rr + f θ + = + Vă , r > b a1 c22 c22+ r2 + a1 δc66 r2 + r − + r2 α2 c22 a1 δc66 c66+ r δc22 c66+ a1 δc66 r c66 c22 1− −1 −1 c22 − c22 + ra3 δc12 Vr α2 c22 − c66+ c22+ c66 −1 − c22 −1 Vθ,θ c66+ 1 r c66 −1 c22 − c12+ Vr ,r −1 Vθ,θ ,r + fr = Văr , a < r < b, 1 c66 + ,r −1 (79) −1 c66 −1 Vr,θ − c22 + c66 1 r c22 −1 − c66 − ra3 δc66 Vθ α1 c66 + c66+ Vθ −1 ,r a2 δc22 c66− c66 α1 c66 + c12 r c22 Vr,r c22 c66 − −1 c12 c22 c12 c22 Vθ,θr + − −1 c66 −1 c66 −1 a1 δc22 r c22+ −1 1 r c66 c66+ − −1 a2 δc66 c22− c22 c66 − Vr,r + α1 c66 α1+ − + c12 c22 c12 c22 −1 − c12 α2 c12 − c12+ + + c11 c22 c22 Vr,θ θ + c11 + a1 δc22 r2 + −1 −1 1 r c66 −1 c22 c22 + + c22+ −1 c22 c22 δc66 c22+ + α2 c22 α2+ − −1 Vθ,θ θ + Vθ,r c12 r c22 c22 −1 Vr,θr + f θ = Vă , a < r < b, (80) − c22− Vr − r12 (c22 + c66 )− Vθ,θ + r1 c11− Vr,r + r12 c66− Vr,θ θ ⎪ ⎪ ⎨ r + r (c12 + c66 )− Vθ,r θ + c11− Vr,rr + fr − = Văr , < r < a − r12 c66− Vθ + r12 (c22 + c66 )− Vr,θ + r1 c66− Vθ,r + r12 c22− Vθ,θ θ ⎪ ⎩ + r1 (c12 + c66 )− Vr,r θ + c66 V,rr + f = Vă , < r < a (81) + Vθ,r ,r a1 δc22 r c22+ + + c12 c22 c11 + c12 r c22 Vr,θ ,r c22 c12 c22 c22 −1 − c12+ Vr + c22 −1 −1 − c12 c22 c12 Vr r L∗ Vr,r L∗ , L ∗ is lines r = a, r = b, Vθ,θ L∗ (82) Homogenized equations of the linear elasticity theory a1 δc66 r + 1− c66 c66 −1 c66+ −1 + Vθ,r L∗ 347 c66 Vθ r Vθ − 1 r c66 [Vr ] L ∗ = 0, [Vθ ] L ∗ = 0, −1 L∗ , Vr,θ L ∗ is lines r = a, r = b, (83) L∗ L ∗ is lines r = a, r = b (84) It is clear that when the materials of + and − are the same, i.e ci j+ = ci j− , ρ+ = ρ− , then Eqs (78)1 , (79), (81)1 coincide with each other The same conclusion is true for Eqs (78)2 , (80), (81)2 It is also not difficult to verify that when the materials are isotropic, i.e c11 = c22 = λ + 2μ, c12 = λ, c66 = μ, then Eqs (59), (60), (61), (62), (63), (64) coincide with Eqs (78), (79), (80), (81), (82), (83), respectively Remark All obtained results in this section also hold for the plane stress in which the constants c11 , c22 , c12 /c , c = c − c2 /c , c = c − c c /c , and V does are replaced, respectively, by c11 = c11 − c13 33 22 22 12 13 23 33 23 33 12 not depend on the time t Conclusions In this paper, we consider the homogenization of the plane strain problem of the theory of linear elasticity in two-dimensional domains with interfaces rapidly oscillating between two concentric circles By an appropriate asymptotic expansion of the solution, and using the homogenization method, we have derived the explicit homogenized equations and associate continuity conditions in explicit form for the isotropic and orthotropic elastic materials Since these equations are explicit, they are convenient tools to investigate various practical problems of the theory of linear elasticity in two-dimensional domains with rough interfaces rapidly oscillating between two concentric circles Acknowledgments The work was 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