Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 4, pp. 823-836, Warsaw 2015 DOI: 10.15632/jtam-pl.53.4.823 EFFECT OF DAMAGES ON CRACK DEVELOPMENT IN COATING ON ELASTIC FOUNDATION Vagif M. Mirsalimov Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan Republic e-mail: vagif.mirsalimov@imm.az Shahin G. Hasanov Azerbaijan Technical University, Baku, Azerbaijan Republic; e-mail: iske@mail.ru A fracture mechanics problem for a coating linked to the basis made of another elastic material is considered. It is assumed that in the vicinity of technological defect (crack) in the process of loading in the coating, there will arise prefracture zones (damages) that are modeled as the areas of weakened interparticle bonds of the material. In the prefracture zones (interlayers of the overstressedmaterial), the coatingmaterial is deformed beyond the limit of elasticity. It is considered that during loading in the cross section of the coating in the vicinity of the crack, there is an arbitrary number of rectilinear prefracture zones. The condition determining the limiting value of the external load at which the crack growth happens is obtained. Keywords: coating, elastic foundation, crack with plastic end zones, prefracture zones 1. Introduction Analysis of the present state of coatings revealed that the coating materials have crack-visible discontinuities. In the cross sections of the coating, there arise transition zones where physico- -mechanical features of the material differ from the features of the basic coating. The indicated damages in the coating cross section may have both natural origin (lamination, inclusions, po- res) or be caused by technological processes. In spite of importance of the enumerated factors on coating strength, up today, these issues have not found due consideration in calculation me- thods. Development of calculation models of investigation of damage of a coating on an elastic foundation is a very urgent problem. A problem on interactions of damages on crack growth (Mirsalimov and Rustamov, 2012a,b) is an important problem of strength theory. Wide litera- ture, e.g. Kulchytsky-Zhyhailo and Rogowski (2007), Haj-Ali (2009), Tukashev and Adilhanova (2010), Ameri et al. (2011), Hasanov (2010, 2013), Hasanov and Mirsalimov (2014) and others has been devoted to investigation of the stress strain state and fracture of the coating on an elastic foundation. 2. Formulation of problem Consider with respect to Cartesian coordinates x,y a double-layer body consisting of a coating of thickness h with the elastic characteristics G1 (shear modulus) and µ1 (Poisson ratio) linked with an elastic half-plane with characteristics G2 and µ2 (Fig. 1). Consider a fracture mechanics problem for a double layer body when the normal load P is applied to theexternal surfaceof thecoating.Theremainingpartof thecoating isnot loaded. It is accepted that the coatingmaterial has a crackwith endplastic zones (Leonov-Panasyuk-Dugdale 824 V.M.Mirsalimov, Sh.G. Hasanov Fig. 1. A scheme of the problem of interaction of prefracture zones and cracks with end zones in the coating crackmodel), seeLeonov andPanasyuk (1959),Dugdale (1960). It is assumed that in the coating material in the vicinity of the crack, after repeated loading, there appear damages (prefracture zones) that are modeled as areas of interparticle bonds of the material. At the loading with external loading in the interlayer of the overstressed material a plastic flow is formed. Let, for definiteness, the power loadings change so that plastic deformation is realized in the area of weakened interparicle bonds of the material. Interaction of the faces of prefracture zones is modeled by the lines of plastic flow between them (degenerated plastic deformation zones). Under constant stress the sizes of plastic flow zones (prefracture zones) depend on the form of the material. General tendency to formation of areas with a broken structure of the material at early stages of fracture in the form of narrow layers occupying slight volume of the body compared with its elastic zone (Panasyuk, 1991; Mirsalimov, 1987; Rusinko andRusinko, 2011) is knownwell from practice. The sizes of prefracture zones at the end plastic deformation zones at the crack tips are unknown beforehand and should be defined. Interaction of the prefracture zones in the vicinity of technological defect (crack) may reduce to the loss of crack stability, appearance of new cracks. It is assumed that the prefracture zones are oriented in the direction of action of maximal tensile stresses appearing in the coating. Since the indicated zones are small compared with the remaining elastic part of the coating, one can mentally remove them changing it by cuttings whose surfaces interact between them- selves by some law corresponding to the action of the removed material. In the coating cross section the crack with end zones is of length 2l1 along the axis x1. Let in the coating in the vicinity of the crack there will be (N − 1) prefracture zones of length 2lk (k = 1,2, . . . ,N) (Fig. 1). In the centers of prefracture zones and the crack with end zones, we locate the origin of a local system of coordinates xkOkyk whose axes xk coincide with the prefracture zones and the crack, andmake the angles ak with the axis x (Fig. 1). Under the action of the external power load P on the coating surface in bonds connecting the prefracture zone faces and the cracks in end zones, there will arise normal qyk = σS and tangential qxkyk = τS stresses (k = 1,2, . . . ,N), where σS is the yield point of the coating material for tension; τS is the yield point of the material for shear. The boundary conditions of the problem are written in the form (the upper index 1 corre- sponds to the coating, the upper index 2 to the half-plane): — for y =0 σ(1)y =−Pδ(x) τ(1)xy =0 (2.1) — for y =−h u(1)+iv(1) = u(2)+iv(2) σ(1)y +iτ (1) xy = σ (2) y +iτ (2) xy (2.2) — for y1 =0 λ11 < x < λ21 σ (1) y =0 τ (1) xy =0 (2.3) Effect of damages on crack development in coating on elastic foundation 825 — for y1 =0,−ℓ1 ¬ x1 ¬ λ11 and λ21 ¬ x1 ¬ ℓ1 σ(1)y1 = σs τ (1) x1y1 = τs (2.4) and σ(1)yk = σs τ (1) xkyk = τs on Lk (k =1,2, . . . ,N) (2.5) where Lk are the faces of the k-th prefracture zone; δ(x) is Dirac’s impulse function, σx,σy,τxy are stress tensor components; u,v are displacement vector components: as y →−∞ the displa- cements and stresses disappear. 3. The method of the boundary-value problem solution For the solution of the problemunder considerationwe use the superposition principle. Thenwe can represent the state of a double-layer body in the form of the sumof two stress-strain states: 1) adhesive connection of materials without a crack and prefracture zones under the action of the external normal load P on the external surface of the coating; 2) stress-strain state of a coating with a crack and prefracture zones on the faces of which the stresses equal in value and opposite in sign, defined by the first stress-strain state for y1 =0 and on Lk are additionally applied. The boundary conditions for the first stress state are of the form (2.1)-(2.2). For the solution of boundary value problem (2.1), (2.2), we use Papkovich-Neiber’s four functions Fmn (x,y) (n,m = 1,2). Two of them are for the coating (upper index 1) and two for the half-plane (upper index 2). The stresses and strains are expressed by the Parkovich-Neiber function by the known for- mulae (Uflyand, 1967) σ (m) y 2Gm =2(1−µm) ∂Fm2 ∂y − ∂ 2Fm1 ∂y2 −y∂ 2Fm2 ∂y2 τ (m) xy 2Gm = ∂ ∂x [ (1−2µm)Fm2 − ∂Fm1 ∂y −y ∂Fm2 ∂y ] u(m) =− ∂Fm1 ∂x −y ∂Fm2 ∂x v(m) =(3−4µm)Fm2 − ∂Fm1 ∂y −y ∂Fm2 ∂y (3.1) Taking into account the symmetryof theproblem inx,weuse theFourier cos-transformation. Accept that F11 = ∞∫ 0 (Asinhαy +Bcoshαy)cosαx dα F12 = ∞∫ 0 (C sinhαy +Dcoshαy)αcosαx dα F21 = ∞∫ 0 Eeαy cosαx dα F22 = ∞∫ 0 Feαyαcosαx dα (3.2) 826 V.M.Mirsalimov, Sh.G. Hasanov Satisfying by functions (3.1), (3.2) boundary conditions (2.1), (2.2), we get a system of six linear algebraic equationswith respect to six unknown functions A(α), B(α), C(α), D(α), E(α), F(α) 2(1−µ1)(C coshαh+Dsinhαh)−Asinhαh−Bcoshαh −αh(C sinhαh+Dcoshαh)=− P 2πG1α2 (1−2µ1)(C sinhαh+Dcoshαh)−Acoshαh−B sinhαh −αh(C coshαh+Dsinhαh)= 0 B = E (3−4µ1)D−A =(3−4µ2)F −E G1[2(1−µ1)C −B] = G2[2(1−µ2)F −E] G1[(1−2µ1)D−A] = G2[(1−2µ2)F −E] (3.3) Solving algebraic system of equations (3.3) by themethod of successive exclusion of unknowns, we find the coefficients A(α), B(α), C(α), D(α), E(α), F(α) D = ∆1 ∆ F = ∆2 ∆ E = B A = 1 a11 (B1−a12B−a13C −a14D−a15F) B = 1 c11 ( B2− a21 a11 B1− c12C −c13D− c14F ) C = 1 A∗11 ( b∗2− c21 c11 b∗1−A∗12D−A∗13F ) Here a11 =−sinhαh a12 =−coshαh a13 =2(1−µ1)coshαh−αhsinhαh a14 =2(1−µ1)sinhαh−αhcoshαh a15 =0 a21 =−coshαh a22 =−sinhαh a23 =(1−2µ1)sinhαh−αhcoshαh a24 =(1−2µ1)coshαh−αhsinhαh a25 =0 a31 =−1 a32 =1 a33 =0 a34 =3−4µ1 a35 =−(3−4µ1) a41 =−G1 a42 =1 a43 =0 a44 = G1(1−2µ1) a45 =−G2(1−2µ2) a51 =0 a52 =1−2G1(1−µ1) a53 =2G1(1−µ1) a54 =0 a55 =−2G2(1−2µ2) B1 =− P 2πG1α2 B2 =0 B3 =0 B4 =0 B5 =0 c11 = a22−a12 a21 a11 c12 = a23−a13 a21 a11 c13 = a24−a14 a21 a11 c14 = a25−a15 a21 a11 c21 = a32−a12 a31 a11 c22 = a33−a13 a31 a11 c23 = a34−a14 a31 a11 c24 = a35−a15 a31 a11 c31 = a42−a12 a41 a11 c32 = a43−a13 a41 a11 c33 = a44−a14 a41 a11 c34 = a45−a15 a41 a11 c41 = a52−a12 a51 a11 c42 = a53−a13 a51 a11 c43 = a54−a14 a51 a11 c44 = a55−a15 a51 a11 A∗11 = c22− c12 c21 c11 A∗12 = c23− c13 c21 c11 Effect of damages on crack development in coating on elastic foundation 827 A∗13 = c24− c14 c21 c11 A∗21 = c32− c12 c31 c11 A∗22 = c33−c13 c31 c11 A∗23 = c34− c14 c31 c11 A∗31 = c42− c12 c41 c11 A∗32 = c43−c13 c41 c11 A∗33 = c44− c14 c41 c11 b∗1 =− a21 a11 B1 b ∗ 2 =− a31 a11 B1 b ∗ 3 =− a41 a11 B1 b ∗ 4 =− a51 a11 B1 ∆ = ( A∗22−A∗12 A∗21 A∗11 )( A∗33−A∗13 A∗31 A∗11 ) − ( A∗23−A∗13 A∗21 A∗11 )( A∗32−A∗12 A∗31 A∗11 ) ∆1 = M1 ( A∗33−A∗13 A∗31 A∗11 ) −M2 ( A∗23−A∗13 A∗21 A∗11 ) ∆2 = M2 ( A∗22−A∗12 A∗21 A∗11 ) −M1 ( A∗32−A∗12 A∗31 A∗11 ) M1 = b ∗ 3− c31 c11 b∗1− ( b∗2− c21 c11 b∗1 )A∗21 A∗11 M2 = b ∗ 4− c41 c11 b∗1− ( b∗2− c21 c11 b∗1 )A∗31 A∗11 By means of formulae (3.1), (3.2), we find the stress components |x1| ¬ ℓ1, y1 = 0 and Lk (|xk| ¬ ℓk, yk =0, k =1,2, . . . ,N). The boundary conditions of the problem for the second stress-strain state take the form: — for y =0 σ(1)y =0 τ (1) xy =0 (3.4) — for y =−h σ(1)y =0 τ (1) xy =0 (3.5) — for y1 =0 σy1 − iτx1y1 = { −(σ1y1 − iτ 1 x1y1 ) on the crack faces σs − iτs − (σ1y1 − iτ 1 x1y1 ) on the end zone faces of the crack (3.6) — for yk =0, |xk| ¬ ℓk σyk − iτxkyk = σs − iτs − (σ 1 yk − iτ1xkyk) (k =1,2, . . . ,N) (3.7) By means of the Kolosov-Muskhelishvili formulae (Muskhelishvili, 1977), we represent bo- undary conditions (3.5)-(3.7) in the form of the boundary value problem for finding the two analytic functions Φ(z) and Ψ(z) y =0 Φ(z)+Φ(z)+zΦ′(z)+Ψ(z)= 0 y =−h Φ(z)+Φ(z)+zΦ′(z)+Ψ(z)= 0 yk =0 Φ(xk)+Φ(xk)+xkΦ′(xk)+Ψ(xk)= Fk (3.8) where F1 = { −(σ1y1 − iτ 1 x1y1 ) on the crack faces σS − iτS − (σ1y1 − iτ 1 x1y1 ) on the end zone faces of the crack Fk = σS − iτS − (σ1yk − iτ 1 xkyk ) (k =1,2, . . . ,N) 828 V.M.Mirsalimov, Sh.G. Hasanov We look for the complex potentials Φ(z) and Ψ(z) (Panasyuk et al., 1977) in the form Φ(z)= 1 2π N+1∑ k=0 ℓk∫ −ℓk gk(t) t−zk dt Ψ(z)= 1 2π N+1∑ k=0 e−2iαk ℓk∫ −ℓk [ gk(t) t−zk − Tke iαk (t−zk)2 gk(t) ] dt (3.9) where Tk = te iαk +z0k zk =e −iαk(z−z0k) Using transformation formulae (Muskhelishvili, 1977) in transfoming into the new system of coordinates Φk(zk)= Φ ( zke iαk +z0k ) Ψk(zk)= e 2iαk [ Ψ ( zke iαk +z0k ) +z0kΦ ′ ( zke iαk +z0k )] (3.10) we write complex potentials Φn(zn) and Ψn(zn) for the considered problem in the system of coordinates xnOnyn Φn(zn)= 1 2π N+1∑ k=0 ℓk∫ −ℓk gk(t) t−zk dt Ψn(zn)= 1 2π N+1∑ k=0 e2iαnk ℓk∫ −ℓk [ gk(t) t−zk − (Tk −zn)eiαk (t−zk)2 g0k(t) ] dt (3.11) where zk =e −iαk ( zne iαn +z0n −z0k ) αnk = αn −αk Having defined by theKolosov-Muskhelishvili formula (Muskhelishvili, 1977) the stresses on the axis xn and substituting into boundary condition (3.8), after some transformations, we get a system of N +2 integral equations: — for |x| < ∞ ∞∫ −∞ [g00(t) t−x +g0N+1(t)K0,N+1(t−x)+g0N+1(t)L0,N+1(t−x) ] dt =− N∑ k=1 ℓk∫ −ℓk [g0k(t)K0,k(t,x)+g 0 k(t)L0,k(t,x)] dt ∞∫ −∞ [g0N+1(t) t−x +g00(t)KN+1,0(t−x)+g00(t)LN+1,0(t−x) ] dt =− N∑ k=1 ℓk∫ −ℓk [g0k(t)KN+1,k(t,x)+g 0 k(t)LN+1,k(t,x)] dt (3.12) Effect of damages on crack development in coating on elastic foundation 829 — for |x| < ℓn (n =1,2 . . . ,N) ℓk∫ −ℓk g0n(t) t−x + ∑ k 6=n [ ℓk∫ −ℓk g0k(t)Knk(t,x)+g 0 k(t)Lnk(t,x) ] dt + ∞∫ −∞ [g00(t)Kn,0(t,x)+g 0 0(t)Ln,0(t,x)] dt + ∞∫ −∞ [g0N+1(t)Kn,N+1(t,x)+g 0 N+1(t)Ln,N+1(t,x)] dt = πFn(x) (3.13) Here K0,N+1(x)= KN+1,0(x)= x x2+h2 L0,N+1(x)= LN+1,0(x)= ih (x+ih)2 K0,k(t,x)= eiαk 2 ( 1 Tk −x− ih/2 + 1 Tk −x+ih/2 ) L0,k(t,x)= e−iαk 2 Tk −Tk +ih (Tk −x+ih/2)2 Kn,0(t,x) = 1 2 ( 1 t+ih/2−Xn + e−2iαn t− ih/2−Xn ) Ln,0(t,x)= 1 2 ( 1 t− ih/2−Xn − t+ih/2−Xn (t− ih/2−Xn)2 e−2iαn ) KN+1,k(t,x)= eiαk 2 ( 1 Tk −x+ih/2 + 1 Tk −x− ih/2 ) LN+1,k(t,x)= e−iαk 2 Tk −Tk − ih (Tk −x− ih/2 )2 Kn,N+1(t,x)= 1 2 ( 1 t− ih/2−Xn + e−2iαn t+ih/2−Xn ) Ln,N+1(t,x)= 1 2 ( 1 t+ih/2−Xn − t− ih/2−Xn (t+ih/2−Xn )2e −2iαn ) Knk(t,x)= eiαk 2 ( 1 Tk −Xn + e−2iαn (Tk −Xn)2 ) Lnk(t,x)= e−iαk 2 ( 1 Tk −Xn − Tk −Xn (Tk −Xn)2 e−2iαn ) Xn = xe iαn +z0n (3.14) For convenience, in (3.12) and (3.13) and further, we omit the index in xn. From the system of N +2 singular integral equations (3.12) and (3.13) we exclude two unknown functions g00(t) and g0N+1(t). We can write the solutions to equations (3.12) in the following way g00(t)= ∞∫ −∞ [D0(t)W1(x− t)+D0(t)W2(x− t)+DN+1(t)W3(x− t) +DN+1(t)W4(x− t)] dt (3.15) 830 V.M.Mirsalimov, Sh.G. Hasanov g0N+1(t)= ∞∫ −∞ [D0(t)W3(x− t)+D0(t)W4(x− t)+DN+1(t)W1(x− t) +DN+1(t)W2(x− t)] dt Here D0(x)=− 1 π N∑ k=1 ℓk∫ −ℓk [g0k(t)K0,k(t,x)+g 0 k(t)L0,k(t,x)] dt DN+1(x)=− 1 π N∑ k=1 ℓk∫ −ℓk [g0k(t)KN+1,k(t,x)+g 0 k(t)LN+1,k(t,x)] dt W1(x)= 1 2 [M1(x)+N1(x)] W2(x)= 1 2 [M2(x)+N2(x)] W3(x)= 1 2 [M2(x)−N2(x)] W4(x)= 1 2 [M1(x)−N1(x)] M1(x)=− i 4π ∞∫ −∞ sgns[eh|s|+h(|s|+s)] sinh |s|h+ |s|h eisx ds M2(x)= i 4π ∞∫ −∞ sgns sinh |s|h+ |s|h eisx ds N1(x)=− i 4π ∞∫ −∞ sgns{[eh|s|−h(|s|+s)]eisx − (1+ isx)+hs} sinh |s|h−|s|h ds N2(x)=− i 4π ∞∫ −∞ sgns(eisx −1− isx) sinh |s|h−|s|h ds (3.16) Now substituting into (3.15) the expressions for D0(x) and DN+1(x) from (3.16), we find g00(x)=− N∑ k=1 ℓk∫ −ℓk [g0k(u)M0,k(u,x)+g 0 k(u)N0,k(u,x)] du g0N+1(x)=− N∑ k=1 ℓk∫ −ℓk [g0k(u)MN+1,k(u,x)+g 0 k(u)NN+1,k(u,x)] du (3.17) Here M0,k(u,x)= 1 π ∞∫ −∞ [W1(x− t)K0,k(u,t)+W2(x− t)L0,k(u,t) +W3(x− t)KN+1,k(u,t)+W4(x− t)LN+1,k(u,t)] dt N0,k(u,x)= 1 π ∞∫ −∞ [W1(x− t)L0,k(u,t)+W2(x− t)K0,k(u,t) +W3(x− t)LN+1,k(u,t)+W4(x− t)KN+1,k(u,t)] dt (3.18) MN+1,k(u,x)= 1 π ∞∫ −∞ [W3(x− t)K0,k(u,t)+W4(x− t)L0,k(u,t) +W1(x− t)KN+1,k(u,t)+W2(x− t)LN+1,k(u,t)] dt Effect of damages on crack development in coating on elastic foundation 831 NN+1,k(u,x)= 1 π ∞∫ −∞ [W3(x− t)L0,k(u,t)+W4(x− t)K0,k(u,t) +W1(x− t)LN+1,k(u,t)+W2(x− t)KN+1,k(u,t)] dt Now substituting these formulas into (3.13), after some transformations, we get a system of N singular integral equations of the considered problem for |x| ¬ ℓn (n =1,2, . . . ,N) ℓk∫ −ℓk g0k(t) t−x dt+ N∑ k=1 ℓk∫ −ℓk [g0k(t)Rnk(t,x)+g 0 k(t)Snk(t,x)] dt = πFn(xn) (3.19) where Rnk(t,x)= (1−δnk)Knk(t,x)+rnk(t,x) Snk(t,x)= (1−δnk)Lnk(t,x)+snk(t,x) (3.20) and rnk(t,x)= ∞∫ −∞ [Kn,0(τ,x)M0,k(t,τ)+Ln,0(τ,x)N0,k(t,τ) +Kn,N+1(τ,x)MN+1,k(t,τ)+Ln,N+1(τ,x)NN+1,k(t,τ)] dτ snk(t,x) = ∞∫ −∞ [Kn,0(τ,x)N0,k(t,τ)+Ln,0(τ,x)M0,k(t,τ) +Kn,N+1(τ,x)NN+1,k(t,τ)+Ln,N+1(τ,x)MN+1,k(t,τ)] dτ (3.21) After substituting into (3.21) integrals (3.18), the kernels rnk(t,x) and snk(t,x)will be repre- sented by three-fold iterated integrals. After integration, these expressions may be represented by single integrals. Omitting very bulky calculations, for the kernels rnk(t,x) and snk(t,x) we finally find rnk(t,x)= ∞∫ 0 [( 1 sinhhs+hs + 1 sinhhs−hs ) Hnk(Xn,Tk,s,αn,αk) + ( 1 sinhhs+hs − 1 sinhhs−hs ) Gnk(Xn,Tk,s,αn,αk) ] ds snk(t,x) = ∞∫ 0 [( 1 sinhhs+hs − 1 sinhhs−hs ) Hnk(Xn,Tk,s,αn,−αk) + ( 1 sinhhs+hs + 1 sinhhs−hs ) Gnk(Xn,Tk,s,αn,−αk) ] ds (3.22) where Hnk(Xn,Tk,s,αn,−αk)= eiαk 4 {sin(Xn −Tk)s − sin(Tk −Xn)s〈hs+e−2iαn[1−hs+s2(Tk −Tk)(Xn −Xn)+h2s2]〉 + 〈s(Tk −Tk)− e−2iαn[(Tk −Tk)+hs2(Xn −Xn −Tk +Tk)]〉cos(Tk −Xn)s +e−hs[sin(Tk −Xn)s+e−2iαn sin(Tk −Xn)s]} Gnk(Xn,Tk,s,αn,αk)= eiαk 4 {−[1+e−2iαn(−1+hs)]sin(Tk −Xn)s−hssin(Tk −Xn)s −s(Tk −Tk)cos(Tk −Xn)s− e−2iαn(Xn −Xn)scos(Tk −Xn)s+e−hs[sin(Tk −Xn)s − e−2iαn sin(Tk −Xn)s+e−2iαn sin(Xn −Xn −Tk +Tk)scos(Tk −Xn)s]} (3.23) 832 V.M.Mirsalimov, Sh.G. Hasanov Note that the functions rnk(t,x) and snk(t,x) are regular. They define the effect of faces of the band on the stress state near the crack tips. To the system of singular equations (3.19) for internal cracks, we should add the additional conditions ℓk∫ −ℓk g0k(t) dt =0 (k =1,2, . . . ,N) (3.24) Using the procedure for converting (Panasyuk et al., 1977; Mirsalimov, 1987) at conditions (3.24), the system of complex singular integral equations (3.19) is reduced to a system of N × M algebraic equations for determining the N × M unknowns g0k(tm) (k = 1,2, . . . ,N; m =1,2, . . . ,M) 1 M M∑ m=1 N∑ k=1 ℓk[g 0 k(tm)Rnk(ℓktm,ℓnxr)+g 0 k(tm)Snk(ℓktm,ℓnxr)] = F 0 n(x 0 r) M∑ m=1 g0n(tm)= 0 (n =1,2, . . . ,N) (3.25) If in (3.25) we pass to complex conjugate values, we get onemore N ×M algebraic equations. For completeness of algebraic equations, we need 2×N complex equations determining the sizes of prefracture zones. The solution of the system of integral equations is sought in the class of everywhere bounded functions (stresses). Consequently, it is necessary to add to system (3.25) the conditions of stress boundedness at the ends of the crack and prefracture zones xk =±lk (k = 1,2, . . . ,N). These conditions are of the form M∑ m=1 (−1)mg0n(tm)cot 2m−1 4M π =0 (n =1,2, . . . ,N) M∑ m=1 (−1)M+mg0n(tm)tan 2m−1 4M π =0 (3.26) The obtained resolving systems of equations can be determines under the given external load the stress-strain state of the coating linked with elastic foundation in the availability of a crack and arbitrary number of prefracture zones in the coating. The united resolving system of equations becomesnonlinear because of theunknownvalues lk (k =1,2, . . . ,N). For its solution, we use the method of successive approximations the essence of which is the following. We solve system (3.25) at some definite values l∗k (k = 1,2, . . . ,N) of the sizes of prefracture zones and the crack end zones with respect to the remaining unknowns. The remaining unknowns enter into the system linearly. The values of l∗k and the found quantities gk(tm) are substituted into (3.26), i.e. into the unused equations of the system. The taken values of the parameters l∗k and the appropriate values gk(tm) will not, generally speaking, satisfy equations (3.26). Therefore, by selecting the values of the parameters l∗k, wewill repeat the calculations until equations (3.26) of system (3.25) and (3.26) are satisfied with the given accuracy. At each approximation, the algebraic system is solved by the Gauss method with choosing the principal element. Using the solution of the problem, calculate the opening on the faces of the crack and prefracture zones −1+κ 2G1 πℓk M M1∑ m=1 gk(tm)= vk(x0k,0)− iuk(xok,0) (k =1,2, . . . ,N) Here, M1k is the number of nodal points contained in the interval (−lk,x0k). Effect of damages on crack development in coating on elastic foundation 833 For the displacement vector modulus on the faces of the crack and prefracture zone for x = x0k, we have V0k = √ u2k +v 2 k = 1+κ 2G1 πℓk M √ A2k +B 2 k (3.27) where Ak = M1k∑ m=1 vk(tm) Bk = M1k∑ m=1 uk(tm) (k =1,2, . . . ,N) To determine the external load at which the crack propagation occurs, we use the criterium of critical opening of crack faces at the foundation of the plastic deformations zone. Then the condition determining the limiting value of the external load will be the equality V01(λ11)= δc V01(λ21)= δc (3.28) where δc is a characteristic of the fracture toughness of the coating material defined experimen- tally. The obtained solution of the problems allows one to predict the appearance of new cracks in the coating material. Tp achieve that, the problem statement should be complemented with the condition (criterion) of the crack appearance (discontinuity of interparticle bonds of the material). In place of such a condition, we accept the criterium of critical opening of prefracture zone faces |(v+k −v − k )− i(u + k −u − k )|= δcr (k =2, . . . ,N) (3.29) where δcr is the characteristics of resistance of the material to cracking. Using the obtained solution, we can write the limit condition in the form V0k(x ∗ k)= δcr (k =2, . . . ,N) (3.30) where x∗k is the coordinate of the point of the prefracture zone at which discontinuity of the material interparticle bonds occurs. These additional conditions enables finding the coating parameters at which new cracks appear in the coating cross section. Dependences of the length of the crack-tip zone d1 = (l1 −λ11)/l1 on the value of the load p∗ = P/hσs for different values of the crack length l∗ =(λ21−λ11)/l1 forα1 =45◦ and z0 =(0.05h−i0.25h) are depicted inFig. 2.Thedependences of the prefracture zone length l2/h on the dimensionless value of the external load P/hσs under different orientation angles of the prefracture zone location in the case l∗ = 0.75 are depicted in Fig. 3. The dependences of the opening of prefracture zone faces δ/δ0 along the prefracture zone x2/l2 at different orientation angles of the prefracture zone location in the case l∗ = 0.75 are depicted in Fig. 4. Here δ0 = πE1δc/8σs. Figure 5 represents the dependence of the critical load pc = P/hσs on the dimensionless length of the crack λ = l∗/h for α1 =45◦. 4. Conclusions Experimental data from operational practice of the pair “coating-elastic foundation” convin- cingly show that at the design stage it is necessary to take into attention the cases when the coating may have damages and cracks. The existing methods of strength analysis of the pair “coating-elastic foundation” ignore this case. Sucha situationmakes it impossible todesign apa- ir “coating-foundation” withminimal specific consumption ofmaterials at guaranteed reliability and durability. 834 V.M.Mirsalimov, Sh.G. Hasanov Fig. 2. Dependence of the length of the left end zone d1 =(ℓ1−λ11)/R1 on the dimensionless external load p∗ = P/hσs for different values of the crack length ℓ∗ =(λ21−λ11)/ℓ1 for α1 =45◦ and z0 =0.05h− i0.25h Fig. 3. Dependence of the prefracture zone length ℓ2/h on the dimensionless length of the external load P/hσs at different orientation angles of the prefracture zone location for the case ℓ∗ =0.75 Fig. 4. Dependence of the prefracture zone faces opening δ/δ0 along the prefracture zone at different orientation angles of the prefracture zone location for the case x2/ℓ2. Here δ0 = πE1δc/8σs In this connection, it is necessary to realize the limit analysis of thepair “coating-foundation” in order to establish ultimate loads at which cracking and crack growth in the coating occurs. The size of the limitingminimal prefracture zone at which a crack appears should be considered as a design characteristic of the coating material. Based on the suggested designed model that takes into account the availability of damages (zones ofweakened interparticle bonds of thematerial) and crackswith end zones in the coating, we developed amethod of calculation of coating parameters at which cracking and crack growth occurs. Knowing the basic values of critical parameters of cracking and the effect of materials properties on them, cracking and crack growth phenomenon may be controlled by means of design-technological decisions at the design stage. Effect of damages on crack development in coating on elastic foundation 835 Fig. 5. Dependence of the critical load pc = P/hσs on the dimensionless length of the crack λ = ℓ1/h for α1 =45◦ Numerical realization of the obtained equations enables solution of the following practically important design problems: 1) to estimate the guaranteed resource of the pair “coating-elastic foundation” with regard to expected defects and loading conditions; 2) to establish the admissible deficiency level and maximum values of workload ensuring sufficient reliability reserve; 3) to choosematerials with necessary complex of static and cyclic fracture toughness charac- teristics. References 1. Ameri M., Mansourian A., Heidary Khavas M., AlihaM.R.M., Ayatollahi M.R., 2011, Crackedasphalt pavementunder traffic loading– a3Dfinite element analysis,Engineering Fracture Mechanics, 78, 8, 1817-1826 2. DugdaleD.S., 1960,Yieldingof steel sheets containing slits,Journal of theMechanics andPhysics of Solids, 8, 100-108 3. Haj-Ali R., 2009, Cohesivemicromechanics: a new approach for progressive damagemodeling in laminated composites, International Journal of Damage Mechanics, 18, 8, 691-720 4. Hasanov Sh.H., 2010, Calculated method of research of fatigue fracture of the road covering, Structural Mechanics of Engineering Constructions and Buildings, 2, 14-20 5. Hasanov Sh.H., 2013, Solution of fracturemechanics for the transverse crack in the cross section of road cover, Journal of Mechanics of Machines, Mechanisms and Materials, 2, 23, 35-40 6. Hasanov Sh.H., Mirsalimov V.M., 2014,Modeling of stress-strain state of road covering with cracks,Acta Polytechnica Hungarica, 11, 215-234 7. Leonov M.Ya., Panasyuk V.V., 1959, Propagation of fine cracks in solid body (in Russian), Prikladnaya Mekhanika, 5, 391-401 8. Kulchytsky-ZhyhailoR., Rogowski G., 2007, Stresses of hard coating under sliding contact, Journal of Theoretical and Applied Mechanics, 45, 4, 753-771 9. Mirsalimov V.M., Rustamov B.E., 2012, Effect of damages on crack-visible of the cavity ope- ning displacement in burning solid fuel, International Journal of Damage Mechanics, 21, 373-384 10. Mirsalimov V.M., Rustamov B.E., 2012, Interaction of prefracture zones and crack-visible cavity in a burning solid with mixed boundary conditions,Acta Mechanica, 223, 627-643 11. Mirsalimov V.M., 1987, Non-One Dimensional Elastoplastic Problems (in Russian), Nauka, Moscow 836 V.M.Mirsalimov, Sh.G. Hasanov 12. Muskhelishvili N.I., 1977, Some Basic Problem of Mathematical Theory of Elasticity, Kluwer, Amsterdam 13. Panasyuk V.V., 1991, Mechanics of Quasibrittle Fracture of Materials (in Russian), Naukova Dumka, Kiev 14. Panasyuk V.V, Savruk M.P., Datsyshyn A.P., 1977, A general method of solution of two- dimensional problems in the theory of cracks,Engineering Fracture Mechanics, 9, 481-497 15. Rusinko A., Rusinko K., 2011,Plasticity and Creep of Metals, Springer, Berlin 16. Tukashev J.B., Adilhanova L.A., 2010, Investigation of stress-strain state of road covering, Geology, Geography and Global Energy, 37, 2, 163-165 17. Uflyand Ya.S., 1967, Integral Transforms in the Theory of Elasticity (in Russian), Nauka, Leningrad Manuscript received October 20, 2014; accepted for print April 6, 2015