Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 4, pp. 837-848, Warsaw 2006 PLASTIC INTERFACIAL SLIP OF PERIODIC SYSTEMS OF RIGID THIN INCLUSIONS UNDERGOING LONGITUDINAL SHEAR Vasyl A. Kryven Faculty of Control Measurements and Computer Systems, Electrical Ternopil State Technical University, Ternopil, Ukraine e-mail: mmethod@tu.edu.te.ua Georgiy T. Sulym Faculty of Mechanical Engineering, Bialystok University of Technology, Poland e-mail: sulym@pb.bialystok.pl Myrosłava I. Yavorska Faculty of Control Measurements and Computer Systems, Electrical Ternopil State Technical University, Ternopil, Ukraine e-mail: mmethod@tu.edu.te.ua Plastic interfacial slip at the longitudinal shear of double periodic sys- tems of thin rigid inclusions in elasto-plastic solids is investigated. Pla- stic deformations are considered to be localized in the thin layers on the inclusion-matrix boundary at the inclusion tips. The length of plastic layers and the rupture displacement value at the inclusions tips caused by plastic interfacial slip are determined. Particular cases of uniperio- dical parallel or collinear inclusion systems are analyzed in detail. Key words: longitudinal shear, periodic problem, thin inclusion, elasto- -plastic solid, plastic slip 1. Introduction While investigatingmechanical propertiesof compositematerials, the influence on stress and strain states of inclusions of most various forms and configura- tions should be given into account (Vanin, 1985). Especially, stress-deformed state investigations inbodieswithperiodic systems of inclusions represent con- siderable interest for the strength theory of composites, reinforced materials and for prediction and optimization of their deformation characteristics. For 838 V.A. Kryven et al. linear-elastic bodies, this problem was investigated in a lot of works (Bere- gnytsky et al., 1983; Deliavsky et al., 1998), but for elastic-plastic bodies, has been studied insufficiently. Fig. 1. Geometrical scheme of the problem There are two approaches to plastic deformation analysis near stress con- centrators. The first of them assumes that plastic deformations are located in a region with an unknown boundary (a continuum plastic zone). In the second ome, it is supposed that the plastic deformations are concentrated in some layers with almost vanished thickness (discrete linear plastic zones). Both kinds of zones are observed in experimental works. The localization of plastic deformations in thin layers was found independently with the shape of concentrators: for tension of thin (Leonow et al., 1963) and thick (Kamin- skij et al., 1994) plates with cracks, for tension of plates with circular holes (Rabotnow and Stankiewicz, 1965) and for torsion of circular smooth shafts (Nadai, 1954). Plastic deformations are often located in plastic layers, called Lüders-Czernov’s bands (Nadai, 1954; Sokołowskij, 1969) for materials with sharp transitions between the elastic and plastic states on the stress-strain curve with a well seen plastic flow zone. It should be emphasized that discre- te linear plastic zones are not an absolute alternative for continuum zones. In that case, when the number of discrete plastic zones increases, it can be shown (Kryven, 1983) that in the limit a continuum zone is obtained. The thin layered localizations of plastic zones on interfaces between inclu- sions and a matrix can lead to separate inclusions, which has great influence on the material strength. This phenomenon should be studied in detail. Plastic interfacial slip of periodic systems... 839 2. Common case of the double-periodic problem 2.1. Problem formulation Within the framework of antiplane deformation we consider a double- periodic problem connected with the plastic interfacial slip of thin rigid plate inclusions forming a rectangular lattice in an ideal elastic-plastic body with the Treska-St. Venant or Huber-Mises-Hencky yield condition τ2yz(x,y)+ τ 2 xz(x,y)= k 2 k is the yield limit of thematerial subject to shear. The inclusions occupy the sectors x =2na, |y+2mh| ¬ l (n,m ∈ Z),where 2a, 2h are distances between the inclusion centers in horizontal and vertical directions respectively. Fig. 2. Scheme of the conformal mapping Consider nowthe case,when thenon-zero component w(x,y) of thedispla- cement vector is symmetric with respect to straight-lines directed along the inclusions axis and asymmetric with respect to straight-lines passing across the inclusion centers and directed perpendicular to the inclusions. We assume also that conditions of ideal soldering with the matrix are satisfied before loading, and the plasticity zones arising as a consequence of the loading are very thin layers adjoined to the inclusions and they initiate from the points of maximum stress concentrations. It means that we suppose that the borders of plasticity zones are thin layers and areas of inclusions x =2na±0, L ¬ |y+2mh| ¬ l, where the length of the plasticity strips l−L must be determined (2L is the length of the inclusion ideal contact zone). The stress field is determined by two non-zero stress tensor compo- nents τxz(x,y) and τyz(x,y), and according with Hooke’s law we have τxz(x,y) = µ∂w/∂x, τyz(x,y) = µ∂w/∂y (µ – shear modulus of the mate- rial). Taking into account geometrical symmetry of the problem, we assume 840 V.A. Kryven et al. that the body loading is such that on the boundary of a periodic rectangle ((2n−1)a; (2m−1)h), the shear stress τxz =0. Let us denote the stress τyz in the tips of the periodic rectangle by τ0 and determine the composite stress- deformed state considering τ0 as given. A constant τ0 determine the loading of the body. According with the equilibrium equations andHooke’s law, the stress ten- sor components τxz, τyz in the elastic part of the body are described by an analytical function τ(ζ) = τyz + iτxz of the complex variable ζ = x + iy. Due to periodicity of the problem, it is sufficient to determine them in the rectangle −a ¬ x ¬ a, −h ¬ y ¬ h. The function τ(ζ) is analytical in this rectangle with a notch along the segment x =0, |y| ¬ l. Due to symmetry of the problem instead of the periodic rectangle, it is sufficient to consider the rectangle ABCD: 0¬x ¬ a, 0¬ y ¬ h. The following conditions must be satisfied on its boundary: |τ(ζ)|= k (ζ = iy, L ¬ |y| ¬ l) Reτ(ζ)= 0 (ζ = iy, |y| < L) Imτ(ζ)= 0 ((ζ = iy, l < |y| ¬ h)∨ (ζ = x± ih, 0¬ x ¬ a)∨ ∨(ζ = a+iy, |y| ¬ h)) (2.1) The first one is a plasticity condition τ2yz(ξ)+ τ 2 xz(ξ) = k 2 on the region ζ = iy, L ¬ |y| ¬ l. The second condition is the ideal soldering condition w(x,y) = 0 on the inclusion surface ζ = iy,−L ¬ y ¬ L without plastic slip. The condition w = 0 on the given interval is equivalent to ∂w(x,y)/∂y =0. Due to Hook’s law, it can be reduced to τyz(0,y) = 0 (|y| ¬ L). The last equation in (2.1) describes the condition τxz = 0 on the remained boundary of the region ABCD. Equation (2.1) together with the condition τ(a,h) = τ0 (2.2) formulae the boundary value problem for the function τ(ζ). 2.2. Solution to the problem Boundary problem (2.1), (2.2) in the rectangular ABCD for the function τ(ζ) can be easily reduced to a problem in a half-plane for a new function τ1(η) by mapping η(ζ) the rectangular to the half-plane Reη ­ 0. We search a solution to problem (2.1), (2.2) as a composition of functions τ(ζ)= τ1 [ −isn (iKζ h ,c )] (2.3) Plastic interfacial slip of periodic systems... 841 where sn is a Jacobian elliptic function reversed to one given by the integral equality ζ = h K η∫ 0 dt √ (1+ t2)(1+ c2t2) (2.4) where K is the full elliptic integral of the first mode K = 1∫ 0 dt √ (t2−1)(1− c2t2) and c (0 < c < 1) is themodular of the elliptic integral-solution of the equation aK = hK′ and K′ – full elliptic integral of the second mode K′ = 1/c∫ 1 dt √ (t2−1)(1− c2t2) This gives a possibility of changing the boundary value problem for the func- tion τ(ζ) in the rectangle to the problem for the new unknown function τ1(ζ) in the half-plane (simpler for further calculations). The function ζ(η) is a conformal mapping of the right half-plane Reη > 0 into the rectangle 0 < Reζ < a,−h < Imζ < h. The points 0,−i,−i/c, i/c, i correspond to −ih, a− ih, a+ih, ih, respectively. The infinitely remote point (E′) of the plane corresponds to the point ζ = a (E). Therefore, it is easy to verify that the function τ1(η) will satisfy the following boundary conditions in the half-plane Reη ­ 0: |τ1(η)|= k ( Reη =0, sn iKL h ¬ |Imη| ¬ sn iKl h ) Reτ1(η)= 0 ( Reη =0, |Imη| ¬ sn iKL h ) Imτ1(η) = 0 ( Reη =0, |Imη| > sn iKl h ) τ1 ( i c ) = τ0 (2.5) For further simplification we will omit the modular at the argument of the Jacobian elliptic function sn. The solution to boundary problem (2.5) can be received in the way given by Kryven (1979), by taking into account that according to conditions (2.5) the conformalmapping τ1(ζ) is knowna prioribecause thehalf-plane Reη ­ 0 is transformed into the half-circle Reτ1 > 0, |τ1| < k with the cut along the 842 V.A. Kryven et al. segment τ0 < Reτ1 < k, Imτ1 =0. After direct construction of the mapping, we find the function τ(ζ) by substituting τ1(η) in (2.3) τ(ζ)= k snKl h √ sn2 iKζ h + sn2KL h − snKL h √ sn2 iKζ h + sn2Kl h sn iKζ h √ sn2Kl h − sn2KL h (2.6) Let us find the length of the plastic strips by taking into account the third relation of condition (2.5). From the limit case in (2.6) at ζ → a+ih, after some calculations we receive sn KL h = k2− τ201 k2+ τ201 sn Kl h Here we have τ01 = √ p+k2q− √ p−k2q√ 2q p = k4+ τ40 −2k2c2τ20 sn2 Kl h q =2τ20 ( 1−c2 sn2 Kl h ) The quantity τ01 corresponds to the stress component τyz in the point (a,0) of the rectangular boundary. To obtain the plasticity strip length d = l−L we determine L from the function reversed to sn by integral (2.4). Themaximal displacement jump [[w]] = g is achieved in the inclusion tips, and it is given by the following formula g = k µ l∫ L τyz(+0,y) dy (2.7) where µ denotes the shear modulus of the body material. We can find the component τyz from formula (2.6) τyz(0,y) = k snKl h √ sn2Ky h − sn2KL h snKy h √ sn2Kl h − sn2KL h L < |y| < l By substituting the variable sn2(Ky/h)= t into integral (2.7), we have g = kh 2µK snKl h√ sn2Kl h − sn2KL h sn2Kl h∫ sn2KL h √ t− sn2KL h t(1− t)(1− c2t2) dt (2.8) Let us look at particular cases of the slip zone evolution problem for pe- riodic systems of inclusions. Plastic interfacial slip of periodic systems... 843 3. The case of a collinear system of rigid thin equidistant inclusions in the same plane Acollinear inclusion system is a consequence of the double-periodic systemone when distance between the centers of inclusions in the horizontal direction trends to infinity: a →∞. In this case, we have sn iKζ h → sinh πζ 2h sn Kl h → sin πl 2h sn KL h → sin πL 2h Therefore, at a →∞we receive from (2.6) τ(ζ)= k sin πl 2a √ sinh2 πζ 2h − sin2 πL 2h − sin πL 2a √ sinh2 πζ 2h − sin2 πl 2h sinh πζ 2h √ sin2 πl 2h − sin2 πL 2h Due to the conditions τ∞yz = τ0, τ ∞ xz =0, we have at infinity τ0 = k sin πl 2h − sin πL 2a√ sin2 πl 2h − sin2 πL 2h The dependence of the plastic strips length on the load is given by d = l− 2h π arcsin (k2− τ20 k2+ τ20 sin πl 2h ) (3.1) Displacement jumps in the inclusion tips can also be expressed in a closed form as a function of the load g = 2kh πµ sin πl 2h√ sin2 πl 2h − sin2 πL 2h [f2(l)−f2(L)] (3.2) f2(y)= sin πL 2h arctan [ sin πL 2h tan ( arcsin cos πy 2h cos πL 2h )] −arcsin cos πy 2h cos πL 2h 4. The case of a complanar system of rigid thin equidistant inclusions Aperiodic systemof parallel inclusions is a consequence of the double-periodic systemwhenthedistancebetween the inclusioncenters in thevertical direction trends to infinity: h →∞. Here we have sn iKζ h → tan πζ 2a sn Kl h → tanh πl 2a sn KL h → tanh πL 2a 844 V.A. Kryven et al. Thus from (2.6) we obtain: τ(ζ)= k tanh πl 2a √ tan2 πζ 2a +tanh2 πl 2a − tanh πL 2a √ tan2 πζ 2a +tanh2 πl 2a tan πζ 2a √ tanh2 πl 2a − tanh2 πL 2a Since limζ→∞τ(ζ) = τ0, limy→∞ tan(iπy/2a) = i, and due to the last formula, we find τ0 = k tanh πl 2a √ 1− tanh2 πL 2a − tanh πL 2a √ 1− tanh2 πl 2a√ tanh2 πl 2a − tanh2 πL 2a τ∞yz = τ0 τ∞xz =0 and d = l− 2a π arcsinh (k2− τ20 k2+ τ20 sinh πl 2a ) (4.1) In this case, the displacement jump in the inclusion tips can be expressed in a closed form from integral (2.8) g = ka µπ tanh πl 2a tanh πL 2a cosh2 πL 2a √ tanh2 πl 2a − tanh2 πL 2a [f1(l)−f1(L)] (4.2) f1(y)= 1 2sinh πL 2a ln ∣∣∣∣∣∣ 1− cosh πL 2a √ tanh2 πy 2a − tanh2 πL 2a 1+cosh πL 2a √ tanh2 πy 2a − tanh2 πL 2a ∣∣∣∣∣∣ −arcsin tanh πL 2a tanh πy 2a 5. Results and discussion The dependence of the plastic strips length and the displacement jump on the load τ0/k is shown in Fig.3. In the case of collinear inclusions (lines 1-3) this length decreases as the distance 2h between the inclusions increases. The jump value of the displacement g in the inclusion tips changes in the same way. If h → ∞, we arrive at the case of a single inclusion (Vytvytsky and Kryven, 1979). Thus, from (3.1), we have d = 2τ20 l k2+ τ20 g = kl µ ( 1− k2− τ20 k2+ τ20 arccos k2− τ20 k2+ τ20 ) (5.1) That is a dependence of d on τ0 for one inclusion. As a matter of fact, it does not differ from the dependence described by curve 3 in Fig.3. It means that in the case of inclusions periodically situated on the same plane, the Plastic interfacial slip of periodic systems... 845 Fig. 3. Length of plastic strips versus load; 1, 2, 3 – l/h =0.9, 07, 02; 3, 4, 5, 6, 7 – l/a=0.1, 1, 2, 3, 5 solution for l/h ¬ 0.2 does not differ from the solution corresponding to a single inclusion any longer. In the case of a periodic system of coplanar inclusions for a → ∞, we also obtain expression (5.1) from (3.1) for a single inclusion. The plastic strips length decreases again together with the decrease of distances 2a between the inclusions (lines 4-7 in Fig.3). Moreover, the obtained results for the coplanar inclusions system, as a mater of fact, do not differ from those obtained from formula (5.1) for a single already inclusion at l/a ¬ 0.1. Solution (2.6) to the two-periodic problem can be treated as an approxi- mate one for the case of plastic interfacial slip of a rigid thin inclusion x =0, |y| ¬ l in the rectangular |x| ¬ a, |y| ¬ h deformed by a constant shear stress τyz = τ0 along the sides y =±h, |x| ¬ a. Accuracy of such an approximation can be determined as ameasure of nonhomogeneity of the stress τyz on hori- zontal sides of the periodic rectangular (in both cases, the stresses τxz =0). From formula (2.6), it follows that the stress component τyz monotoni- cally decrease on the periodic rectangular side from the point (0,h) to the point (a,h), while the second stress component equals zero. The stress com- ponent τyz attains its maximum value on this rectangle side τ01 = k snKl h cnKL h − snKL h cnKl h√ sn2Kl h − sn2KL h cn2x =1− sn2x at the point (0,h), and the minimum value τ0 at the point (a,h). Thus, the measure of nonhomogeneity of the stress τyz on the horizontal sides of theperiodical rectangle for thedouble-periodic problemcanbedefined as Θ = max τ0∈(0,k) τ01− τ0 τ01 = max L∈(0,l) τ01− τ0 τ01 (5.2) 846 V.A. Kryven et al. The regions (between lines and the axis l/h = 0) in the space of geome- trical parameters for which, according to the solution to the double-periodic problem, the stress τyz on the horizontal sides of the periodical rectangle can be treated as constant with an accuracy Θ are given in Fig.4. Fig. 4. Regions of geometric parameters in which the solution to the double-periodic problem gives an approximation of the solution corresponding to the periodical rectangle with the relative error not exceeding Θ The closed-form solutions to problems under consideration enable one to determine the effective shearmodulus in composites reinforced by rigid bands in the sameway as it was done for the elastic problemby Ju andChenTsung- Muh (1994a,b), Porohovsky et al. (1998), and finally to formulate conditions of composite rupture on the basis of known deformation or energetic criteria. References 1. Beregnytsky L.T., Panasiuk V.V., Stashchuk M.G., 1983, Interaction of Rigid Linear Inclusions and Cracks in the Deformable Solid, Kyiv, Naukova dumka (in Russian) 2. DeliavskyM.V.,OnyshkoL.J.,OnyshkoO.E., 1998,Analyticalmechanic of composite fibroid distraction.Review,Fizyko-KhimichnaMekanikaMateria- liv, 34, 6, 45-55 (in Ukrainian) 3. Ju J.W., Chen Tsung-Muh, 1994a, Effective elastic moduli of two dimen- sional brittle solids with interacting microcracks. Part 1. Basic formulations, Trans. ASME. J. Appl. Mech., 61, 2, 349-357 4. Ju J.W., Chen Tsung-Muh, 1994b, Effective elastic moduli of two dimen- sional brittle solids with interactingmicrocracks. Part 2. Evolutionary damage models,Trans. ASME. J. Appl. Mech., 61, 2, 358-366 Plastic interfacial slip of periodic systems... 847 5. Kaminskij A.A., Usikowa G.I., Dmitriewa E.A., 1994, Experimental in- vestigations of plastic deformation distributions near a crack tip during static loads,Applied Mechanics, 30, 11, 71-76 (in Russian) 6. Kryven V.A., 1979, Antiplane deformation of the elastic-plastic body weake- ned by double- periodic system of cracks, Fizyko-Khimichna Mekanika Mate- rialiv, 15, 1, 31-33 (in Russian) 7. Kryven V.A., 1983, The generalization of indications for plastic zone during antiplane deformation of ideal elastic-plastic bodies with sharp stress concen- trators,Dopovidi AN URSR, Academy of Science of Ukrainian Soviet Socialist Republic, Ser. A, 19, 2, 31-34 (in Ukrainian) 8. Leonow M.Ja., Vytvytsky P.M., Jarema S.Ja., 1963, The plastic layers of plateswithcrack-likedconcentrators,DokladyANSSSR,Academy of Science of Union of Soviet Socialist Republics, 148, 3, 541-544 (in Russian) 9. Nadai A., 1954, Plasticity and Fracture of Solids, Moskwa, Izdatelstwo Ino- strannoj Literatury (in Russian) 10. Porohovsky V.V., Opanasovych B.K., Deliavsky M.V., 1998, Effective elastic characteristics of a body with the system of rectilinear thin inclusions in the conditions of longitudinal shear, Math. Metody i Fiz.-Mech. Polia, 41, 3, 109-116 (in Ukrainian) 11. Rabotnow Ju.N., Stankiewicz O.F., 1965,Experimental indication of pla- stic zones on themodels with titanium alloys, Izwestija AN SSSR. Academy of Science of Union of Soviet Socialist Republics, Mekanika, 2, 108-109 (in Rus- sian) 12. Sokołowskij W.W., 1969, Theory of Plasticity, Moskwa, Wysszaja Szkola (in Russian) 13. Vanin G.A., 1985, Micromechanics of Composite Materials, Kyiv, Naukova Dumka (in Russian) 14. Vytvytsky P.M., Kryven V.A., 1979, Antiplane deformation of the body with rigid thin inclusion,Dopovidi ANURSR,Academy of Science ofUkrainian Soviet Socialist Republic. Ser. A, 15, 2, 104-108 (in Ukrainian) Plastyczny międzyfazowy poślizg okresowego układu sztywnych cienkich inkluzji podczas wzdłużnego ścinania Streszczenie W ramach antypłaskiego stanu odkształcenia przeanalizowano plastyczny poślizg na granice kontaktu dwuokresowego układu cienkich sztywnych inkluzji z ośrodkiem 848 V.A. Kryven et al. sprężysto-plastycznympodczas ścinania. Założono, że odkształcenia plastyczne znaj- dują sięw cienkichwarstwachna granicy inkluzji w otoczeniu ich końców.Wyznaczo- no długość warstw plastycznych oraz wartość skoku przemieszczenia spowodowane- go plastycznymprześlizganiem.Dokładnie rozpatrzono również szczególne przypadki jednookresowych zagadnień dla inkluzji w jednej i w równoległych płaszczyznach. Manuscript received June 17, 2004; accepted for print August 23, 2006