F.I. Cafarova. 1. Introduction. Journal of Contemporary Applied Mathematics V. 8, No 1, 2018, July ISSN

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1 Jounal of Contempoay Applied Mathematics V. 8, No 1, 2018, July ISSN Enegy Release Rate at the Font of Penny-shaped Inteface Cacks Contained in the PZT/Elastic/PZT Sandwich Cicula Plate unde Action of the Nomal Opening Foces on the Cacks Edges F.I. Cafaova Abstact. This pape studies the Enegy Release Rate (ERR) at the font of the penny-shaped inteface cacks contained in the PZT/Elastic/PZT sandwich cicula plate-disc unde action on the cacks edges opening unifomly distibuted nomal foces. It is assumed that the otationally symmetic stess-stain state in the plate takes place and the investigations ae made by utiliing the exact field equations and elations of electo-elasticity fo pieoelectic mateials.the solution to the coesponding bounday-value poblem is made by utiliing the finite element method (FEM) and the ERR is studied fo vaious pieoelectic (PZT) mateials of the face layes and fo vaious metal-elastic mateials fo the coe laye of the plate. The main attention is focused on the influence of the coupling effect of the mechanical and electical fields on the ERR. At the same time, numeical esults on the effect of the geometical paametes such as face layes thickness, cack s adius and etc. on the ERR ae pesented and discussed. Key Wods and Phases: Enegy Release Rate,pieoelectic mateial, penny-shaped inteface cack, sandwich cicula plate. 1. Intoduction It is known that though Enegy Release Rate (ERR) at cack tips o at a cack font the factue of the mateial o element of constuction contained this cack, is detemined. Fo this pupose it is also used the Stess Intensity Facto (SIF) at the cack tips, howeve, to use the ERR is moe suitable fo the cacks located completely in the pieoelectic mateial o in the inteface between the pieoelectic and elastic mateials. Theefoe, the study of the ERR fo the penny-shaped inteface cacks located between the pieoelectic face and metal-coe layes of the PZT/Elastic/PZT cicula sandwich plate, to which the pesent wok elates also, has a geat significance in the estimation and pognostication of the factue mechanics of the smat layeed systems.note that the detemination of the ERR equies solving the coesponding bounday value poblems fo the PZT/Elastic/PZT layeed systems contained inteface cacks in ode to detemine the stess-stain state 25 c 2011 JCAM All ights eseved.

2 26 F.I. Cafaova in this system and define the ERR though these stesses and stains. Moeove, note that unde fomulation and solution to these poblems one of the main question is the constuction of the electical conditions acoss the cack s edges. Now we conside a bief eview of the elated investigations and the fomulation of the conditions on the penny-shaped cack edges with espect to the electical quantities. Fist, we conside the pape by Kudyatsev et al. (1975) in which a special solution of the stess and displacement fields is obtained fo the penny shaped cack embedded in a pieoelectic mateial.in this pape the so-called pemeable condition on the cack edges is consideed. In othe wod, in this pape it is assumed that the electical potential and the nomal components of the electical displacements ae continuous acoss the cack edge sufaces. The same type of conditions on the cack s edge ae also used in the papes by Paton (1976), Yang (2004) and othe ones listed theein. Note that analyses in the papes by Li, McMeeking and Landis (2008) and Li, Fengand Xu (2009) analye the vaious types of conditions fomulated on the cack edges in the pieoelectic mateials, which is diffe fom the pemeable condition. In the elated investigations besides pemeable conditions, the coesponding impemeable conditions ae also used on the cack edges, accoding to which, it is assumed that the electic displacements on the cack s edge sufaces ae equal to eo. Such conditions, fo instance is used in the pape by Li and Lee (2012) in which an axisymmetic penny-shaped cack poblem fo the infinite pieoelectic laye in the case whee the cack is in the middle plane of the laye is studied. Moeove, the enegetically consistent bounday condition, which was poposed by Landis (2004), is also used unde consideation the cack poblems fo the pieoelectic mateials (see, fo instance, the papes by Zhong (2012), Eskandai et al. (2010) and othes). It should be noted that in all the foegoing woks it is assumed that the penny-shaped cack is embedded completely in a pieoelectic mateial and theefoe fomulation of the pemeable, impemeable, enegetically consistent, semi-consistent and othe types of conditions fo the electical quantities acoss the cack s edge sufaces, becomes necessay. Howeve, in the cases whee the penny-shaped cack is in the inteface between pieoelectic and elastic mediums the necessity fo such conditions disappeas and on the cack s edge face which elate to the pieoelectic medium, the odinay electically-open (o open-cicuit ) and electically-shoted (o shot-cicuit ) conditions ae satisfied. We ecall that the electically-open (o open-cicuit ) condition coincides with the afoementioned impemeable condition. At the same time, we note that the fist attempt to study the poblem elated to the inteface penny-shaped cack between the pieoelectic laye and elastic half-space is made in the pape by Ren et al. (2014). This study is caied out fo the cack s opening mode in the case whee on the cack face, which is in the pieoelectic laye, the open-cicuit condition is satisfied.with this, we complete the consideation the eview of the elated woks caied out duing the last 10 yeas. Note that the eview of the egading woks caied out in ealie yeas can be found in the papes by Kuna (2006, 2010). Analyes of eviewed above woks show that all the investigations caied out theein fo the penny-shaped cacks in pieoelectic mateials and in the inteface between pieo-

3 Enegy Release Rate at the Font of Penny-shaped Inteface Cacks electic and elastic mateials have been made within the scope of the linea pieoelectic factue mechanics and within the scope of the assumptions that the layes dimensions ae infinite in the plane on which this cack lies. Namely, these infinities allow using the Hankel integal tansfomation method fo the solution to the coesponding bounday value poblems. Howeve, in the cases whee the dimensions of the layes in the planes on which the cacks ae located, ae finite, such as sandwich PZT/Metal/PZT cicula plate-disc the adius of which is commensuable with the adius of the penny-shaped cack, then the methods based on the integal tansfomations, in geneal, is not applicable. As in the pesent pape namely such a case is consideed and theefoe fo a solution to the coesponding bounday value poblem the numeical method, i.e. the finite element method (FEM) is employed. It should be noted that the coesponding buckling delamination poblems wee consideed in the papes by Cafaova et al. (2017), Akbaov et al. (2017) and Cafaova and Rayev (2016). Moeove, note that the coesponding buckling delamination and cack poblems fo the plane-stain state wee consideed in the papes by Aklbaov and Yahnioglu (2013, 2016). 2. Fomulation of the poblem Conside a cicula PZT/Metal/PZT sandwich plate with geomety illustated in Fig. 1a and assume that the thicknesses and pieoelectic mateials of the face layes ae the same, and the mateial of the middle (coe) laye is an elastic one. Also, we suppose that between the coe and face layes thee ae penny-shaped cacks whose locations ae illustated in Fig. 1b.At the same time, Fig. 1b indicates the geometic paametes and the extenal opening foces acting on the cacks edge sufaces. a b Fig. 1. The sketches of the PZT/Metal/PZT plate-disc (a), the geomety of this disc, inteface cacks and extenal opening foces We associate with the lowe face plane of the plate (Fig. 1a) the cylindical coodinate system Oθ, accoding to which, the plate occupies the egion

4 28 F.I. Cafaova {0 l/2; 0 θ 2π; 0 h}(h = 2h F + h c ) and the penny-shaped cacks occu in { = h F ± 0 ;0 l 0 /2} and in { = h C + h F ± 0 ;0 l 0 /2}. Within these famewok, we suppose that on the cacks edges the unifomly otational symmetic distibuted nomal opening foces with intensity pact and it is equied to detemine the ERR at the inteface cacks font in the PZT/Elastic/PZT sandwich plate caused with this mechanical foces. Fo this pupose, fist, we conside fomulation of the poblems fo detemination of the electomechanical quantities which appea in the plate as a esult of the action of the afoementioned mechanical foces. As we ae consideing the otationally axisymmetic defomation case, theefoe unde the mathematical fomulation of the coesponding poblem we will use the coesponding field equations elated to this case.moeove, below we will denote the values elated to the uppe and lowe face layes by uppe indices (3) and (1), espectively, wheeas the values elated to the coe laye ae denoted by uppe index (2). Assuming that the electo-mechanical state in the sandwich plate unde consideation appeas within the scope of the linea theoy of pieoelecticity fo the face layes and the linea theoy of elasticity fo the coe laye, the coesponding field equations, accoding to the monogaph by Yang (2005), can be witten as follows. Equilibium and electostatic equations: σ (j) + σ(j) + 1 (σ(j) σ (j) σ(j) θθ ) = 0, + σ(j) + 1 σ(j) = 0, j = 1, 2, 3, D (k) + 1 D(k) + D(k) = 0, k = 1, 3 (1) The electo-mechanical constitutive elations fo pieoelectic mateials: σ (k) = c (k) 1111 s(k) + c (k) 1122 s(k) θθ + c(k) 1133 s(k) e (k) 111 E(k) e (k) 311 E(k), σ (k) θθ = c(k) 2211 s(k) + c (k) 2222 s(k) θθ + c(k) 2233 s(k) e (k) 122 E(k) e (k) 322 E(k), σ (k) = c (k) 3311 s(k) + c (k) 3322 s(k) θθ + c(k) 3333 s(k) e (k) 133 E(k) e (k) 333 E(k), σ (k) = c (k) 1311 s(k) e (k) 113 E(k) e (k) 313 E(k), D (k) = e (k) 111 s(k) + e (k) 122 s(k) θθ + e(k) 133 s(k) + ε (k) 11 E(k) + ε (k) 13 E(k), D (k) = e (k) 311 s(k) + e (k) 322 s(k) θθ + e(k) 333 s(k) + ε (k) 31 E(k) + ε (k) 33 E(k), E (k) = ϕ(k), E(k) = ϕ(k). (2)

5 Enegy Release Rate at the Font of Penny-shaped Inteface Cacks Elasticity elations fo the coe laye mateial. σ (2) = λ (2) s (2) + 2µ (2) s (2), σ (2) θθ = λ(2) s (2) + 2µ (2) s (2) θθ, σ(2) = λ (2) s (2) + 2µ (2) s (2), Stain-displacement elations: s (j) = u(j), s(j) σ 2 = 2µ 2 s 2, s 2 = s 2 + s 2 θθ + s2, k = 1, 3. (3) θθ = u(j), s(j) = u(j), s(j) = 1 2 ( u (j) ) + u(j), j = 1, 2, 3. (4) Note that in (1) (4) the following notation is used: σ (j),..., σ (j) and s (j),..., s (j) ae the components of the stess and stain tensos, espectively, u (j) and u (j) ae the components of the displacement vecto, D (k) vecto, E (k) and E (k) and D (k) ae the components of the electical displacement ae the components of the electical field vecto, ϕ (k) is the electic potential, λ (2) and µ (2) ae Lame constants of the coe laye mateial, and c (k) ijkl, e(k) (k = 1, 2, 3) ae the elastic, pieoelectic and dielectic constants, espectively. ε (k) nj nij and Notethat the pieoelectic mateial exhibits the chaacteistics of othotopic mateials with the coesponding elastic symmety axes and becomes electically polaied unde mechanical loads o mechanical defomation placed in an electical field. Accoding to the monogaph by Yang (2005) and othe elated efeences, the polled diection of the pieoelectic mateial will change accoding to the position of the mateial constants in the constitutive elations in (2). In the pesent pape, unde numeical calculations, it is assumed that the O axis diection is the polaied diection. Moeove, in geneal, in the theoy of the pieoelecticity fo simplicity the following notation is used. c (k) 1111 = c(k) 11, c(k) 2211 = c(k) 1122 = c(k) 12, c(k) 3311 = c(k) 1133 = c(k) 13, c(k) 2222 = c(k) 22, c (k) 3322 = c(k) 2233 = c(k) 23, c(k) 3333 = c(k) 33, c(k) 1313 = c(k) 55, e(k) 111 = e(k) 11, e(k) 311 = e(k) 31, e (k) 122 = e(k) 12, e(k) 322 = e(k) 32, e(k) 133 = e(k) 13, e(k) 333 = e(k) 33, e(k) 313 = e(k) 35, e(k) 113 = e(k) 15. (5) Thus, the equations and elations in (1) (6) completes the witing of the field equations. Now we conside mathematical fomulation of the bounday conditions. Bounday conditions on the cacks edges: σ (3) =hf +h C +0 = 0, σ(3) =hf +h C +0 = p, σ(2) =hf +h C 0 = 0, σ(2) = p, =hf +h C 0

6 30 F.I. Cafaova σ (2) =hf +0 = 0, σ(2) =hf +0 = p, σ(1) =hf 0 = 0, σ(1) = p, =hf 0 D 3 =hf +h C +0 = 0, D1 =hf 0 = 0, fo 0 l 0/2. (6) Contact conditions between the layes in the aeas which ae out of the cacks: σ (3) = σ (2), σ (3) = σ (2), u (3) = u (2), =hf +h C =hf +h C =hf +h C =hf +h C =hf +h C =hf +h C u (3) = u (2), σ (2) = σ (1), σ (2) = σ (1), =hf +h C =hf +h C =hf =hf =hf =hf u 2 =hf = u 1 =hf, D 3 =hf = 0, D 1 +h C =hf = 0, fo l 0 /2 l/2. (7) Bounday conditions on the face planes of the plate: = 0, σ (3) = 0, σ (1) = 0, =2hF +h C =2hF +h C =0 σ (1) σ (3) = 0, D 3 =0 =2hF = 0, D 1 +h C =0 = 0, fo 0 l/2. (8) Conditions on the lateal bounday of the plate: σ (j) = 0, u (j) = 0, fo j = 1, 2, 3; ϕ (k) =l/2 =l/2 =l/2 = 0 fok = 1, 3, unde0 2h F + h C. (9) This completes the fomulation of all the bounday and contact conditions fo the poblem unde consideation. 3. Method of solution. FEM modeling of the poblem As the analytical o appoximate analytical solution to the poblem unde consideation is impossible theefoe the fomulated poblem is solved numeically by employing FEM. Fo FEM modeling of the poblem, accoding to Yang (2005) and othes, the following functional is intoduced. Π( u (1), u (2), u (3), u (1), u (2), u (3), ϕ (1), ϕ (3) ) = π k=1 Ω (k) [ σ (k) + σ (k) θθ u (k) + σ (k) + σ(k) + σ (k) ] dd

7 Enegy Release Rate at the Font of Penny-shaped Inteface Cacks π whee Ω (1) [ ] E (1) D (1) + E (1) D (1) dd + 1 [ ] 2 2π E (3) D (3) + E (3) D (3) dd Ω (3) 2π 2π l0 /2 0 l0 /2 0 p u (2) p u (1) d 2π =hf d 2π =hf +h C l0 /2 0 l0 /2 0 p u (2) p u (3) d =hf d, (10) =hf +h C Ω (1) = {0 l/2; 0 h F } { = h F 0; 0 l 0 /2} ; Ω (2) = {0 l/2; h F h F + h C } { = h F + 0; 0 l 0 /2} { = h F + h C 0; 0 l 0 /2} ; Ω (3) = {0 l/2; h F + h C 2h F + h C } { = h F + h C + 0; 0 l 0 /2}. (11) Equating to eo the fist vaiation of the functional (10), i.e. fom the elation δπ = 3 k=1 Π δu (k) + 3 k=1 Π δu (k) + Π ϕ (1) δϕ(1) + Π ϕ (3) δϕ(3) = 0 (12) anddoing well-known mathematical manipulations we obtain the equations in (1) and all the coesponding bounday and contact conditions in (7) (9) with espect to the foces and electical displacements. In this way it is poven that the equations in (1) ae the Eule equations fo the functional (10), and the bounday and contact conditions in (7) (9) which ae given with espect to the foces and electical displacements, ae the elated natual bounday and contact conditions. As an usual pocedueof FEM modelling, the solution domains indicated in (11) ae divided into a finite numbe of finite elements. Fo the consideed poblem, each of the finite elements is selected as a standad ectangula Lagange family quadatic finite element with nine nodes and each node has thee degees of feedom, i.e. adial displacement u (j), tansvese displacement u (j) (j = 1, 2, 3) and electic potential ϕ (k) (k = 1, 2). We ecall that unde FEM modelling of the egion containing the cack s tip, as did ou pedecessos, we use odinay (not singula) finite elements. This is because up to now finite elements with oscillating singulaity which appea at the inteface cack tips have not been found. Futhemoe, as shown in the efeences Akbaov (2013), Akbaov and Yahnioglu (2016), Akbaov and Tuan (2009),Henshell and Shaw (1975) and othe ones listed theein, unde calculation of the factue chaacteistics of the element of constuction (such as the

8 32 F.I. Cafaova citical foces, ERR and etc.) the esults obtained by the use of the odinay singula finite elements coincide, with vey high accuacy, with the esults obtained by the use of the odinay finite elements. Table 1The values of the mechanical, pieoelectical and dielectical constants of the selected pieoelectic mateials Mate. c ( 1) 11 (Souce c ( 1) 12 c ( 1) 13 c ( 1) 33 c ( 1) 44 c ( 1) 66 e ( 1) 31 e ( 1) 33 e ( 1) 15 ε ( 1) 11 ε ( 1) 33 Ref.) PZT- 4 (Yang, 2005) PZT- 5H (Yang, 2005) N/m C / m C/V m The algoithm and the pogams to obtain the numeical esults ae coded within the foegoing assumptions by the autho in the FORTRAN pogamming language (FTN77). Employing the standad Rit technique detailed in many efeences, fo instance, in the book by Zienkiewic and Taylo (1989), we detemine the displacements and electical potential at the selected nodes. Afte this detemination, accoding to the elation γ = the enegy elease ate γ (ERR) is detemined, whee U πl 0 l 0, (13) U = π k=1 Ω (k) [ σ (k) + σ (k) θθ u (k) + σ(k) + σ (k) + σ (k) ] dd+ 1 [ ] 2 2π E (1) D (1) + E (1) D (1) dd + 1 [ ] Ω (1) 2 2π E (3) D (3) + E (3) D (3) dd, (14) Ω (3) is the electo-mechanical stain enegy. This completes the consideation the solution method of the poblem though the FEM modeling. 4. Numeical esults and discussions In the pesent pape we conside only the numeical esults elated to the ERR and all the numeical esults ae obtained fo the pieoelectic mateials PZT - 4 and PZT -5H which

9 Enegy Release Rate at the Font of Penny-shaped Inteface Cacks ae selected fo the face layes, howeve the metal mateials - aluminum (Al) and steel (St) ae taken as the coe laye mateials. The values of the elastic, pieoelectic and dielectic constants of the selected pieoelectic mateials and the efeences used fo this pupose ae given in Table 1. Accoding to the monogaph by Gu (2004), the values of Lame s constants of the coe laye mateial ae selected as follows: fo the Al: λ = 48.1 GP a and µ = 27.1 GP a; and fo the St: λ = 92.6 GP a and µ = 77.5 GP a. In ode to analye the coupling effects of the electo-mechanical fields on the ERR, the numeical esults ae obtained fo the following two cases: Case 1. Case 2. e (n) ij e (n) ij = 0, ε (n) ii = 0, (15) 0, ε (n) ii 0. (16) Numeical esults obtained in Case 1 (15) elate to the pue mechanical ERR, howeve the numeical esults obtained in Case 2 (16) elate to the total electo-mechanical ERR and compaison of the esults obtained in Case 2 with the coesponding ones obtained in Case 1 will give the infomation fo estimation of the influence of the coupling electo-mechanical effect on the studied quantities. As noted above, in the pesent pape we conside the numeical esults elated to the dimensionless ERR detemined though the expession γ/(c P 44 ZT l) and the influence of the poblem paametes on this ERR. Unde obtaining these esults, the values of γ ae calculated though the expession (13) and unde this calculation the following appoximate elation is used. γ U πl 0 l 0 ; U = U(l 0 + l 0 ) U(l 0 ), l 0 /l = (17) Note that the numbe 10 8 shown in (17) fo the atio l 0 /l is detemined fom the coesponding convegence equiement which appeas fo the numeical calculation of the deivative U/ l 0. Unde obtaining all the numeical esults illustated in the pesent pape, we assume that the pieoelectic mateials ae polaied along the plate thickness, i.e. the polaied diection of the PZT mateials coincides with the O axis. Moeove, all the numeical esults ae obtained in the case whee h/l = 0.2. Unde FEM modelling we use the symmety of the poblem with espect to the plane = h F +h C /2 and the axial symmety with espect to the O (Fig. 1a) axis and accoding to these symmeties, we conside only the egion {0 l/2; 0 h F + h C /2}unde FEM modelling and divide this egioninto 500 finite elements along the adial diection and 40 finite elements along the plate s thickness diection. Unde fixed numbes of the finite elements, the NDOF depends on the length (o adius) of the penny-shaped cack and the NDOF inceases with inceasing of this length. Fo instance, in the case whee l 0 /l = 0.5 we have NDOF, howeve, in the case whee l 0 /l = 0.3 we have NDOF. All the coesponding PC pogams ae composed by the autho of the pape.

10 34 F.I. Cafaova Table 2.Convegence of the numeical esults with espect to the numbe of FE selected in the adial diection in the case whee l 0 /l = 0.5, h F /l = 0.05, and h C /l = 0.1and the numbe of FE in the O axis diection is 12 fo PZT-5H/Al/PZT-5H P ZT 5H Numbe of FE in NDOF γ/(c44 l) the adial diect. Case 1 Case , Table 3.Convegence of the numeical esults with espect to the numbe of FE selected in the O axis diection in the case whee l 0 /l = 0.5, h F /l = 0.05, and h C /l = 0.1and the numbe of FE in the adial diection is 100 fo PZT-5H/Al/PZT-5H Numbe of the FE in the O axis diec. NDOF P ZT 5H γ/(c44 l) Case 1 Case Now we conside the convegence of the numeical esults with espect to the numbe of finite elements (FE) selected in the adial and O axis diections. Fo this pupose, P ZT 5H conside the numeical esults elated to the dimensionless ERR, i.e. to the γ/(c44 l) fo the PZT-5H/Al/PZT-5H plate. The esults obtained fo vaious values of the FE selected in the adial diection (in the O axis diection) ae given in Table 2 (in Table 3). It follows fom these tables that the convegence of the numeical esults is moe sensitive with espect to the FE numbes selected in the adial diection. These and othe simila esults which ae not given hee allow us to conclude that in the convegence sense of the numeical esults, it is enough to select 500 FE in the adial diection and 40 FE in the O axis diection in ode to obtain esults, the elative eos of which ae less than 0.4%. Note that 40 FE in the O axis diection ae divided in half between the face laye and half thickness of the coe laye.

11 Enegy Release Rate at the Font of Penny-shaped Inteface Cacks The convegence of the numeical esults illustated above gives some confidence on the eliability of the calculation algoithm and PC pogams. Howeve, fo moe detailed veification of the PC pogams and FEM modelling used we conside a compaison of the numeical esults obtained within the scope of the pesent algoithm and PC pogams with the coesponding ones obtained within the scope of the analytical solution method developed in the pape by Li and Lee (2012). We ecall that the pape by Li and Lee (2012) studies an axisymmetic penny-shaped cack poblem fo the infinite pieoelectic laye in the case whee the cack is in the middle plane of the laye and the new analytical method is developed fo detemination of the coesponding fundamental solutions and, by employing this method numeical esults elated to the ERR ae pesented and discussed. Let us employ, in some paticula cases, i.e. in the cases whee on the cack edges the electic displacements ae equal to eo and these edges ae loaded with unifomly distibuted mechanical opening foces with intensityσ 0, ou FEM modelling and the PC pogams fo obtaining the numeical esults consideed in the by Li and Lee (2012). Note that unde FEM modelling of the poblem consideed in the pape by Li and Lee (2012) we assume that l = 1 m, l 0 = m and h = 0.02 m. The values selected fo l 0 and h coincide with the coesponding ones selected in the pape by Li and Lee (2012), howeve, the paamete l does not exist in the pape by Li and Lee (2012) because in that pape it is assumed that the length of the pieoelectic laye in the adial diection is infinite. Thus, within the scope of the foegoing assumptions, we compae the numeical esults obtained with employing of the pesent FEM modelling with the coesponding ones obtained in the pape by Li and Lee (2012) fo the PZT-5H mateial. These esults ae given in Table 4 and it follows fom the coesponding compaison that the FEM modelling and PC pogams developed in the pesent pape ae eliable enough. Table 4.Numeical esults elated to γ (N/m) (i.e. ERR) fo the penny-shaped cack in the middle plane of the infinite PZT-5H pieoelectic laye in the case whee l = 1 m, h = 0.02 m and l 0 = m Souces of σ 0 the esults 10Mpa 20Mpa 30Mpa Pesent esults Results obtained in Li and Lee (2012)

12 36 F.I. Cafaova Fig. 2. The gaphs of the dependence between dimensionless ERR and cack adius fo the PZT-5H/Al/PZT-5H plate Now we conside the esults given in Figs. 2, 3 and 4 which illustate how an incease in the cack adius acts on the ERR. Note that these esults elate to the PZT-5H/Al/PZT- 5H (Fig.2), PZT-4/Al/PZT-4 (Fig. 3) and PZT-5H/St/PZT-5H (Fig. 4) plates and show the gaphs between the dimensionless ERR (denoted as γ / (c P 44 ZT l)) and the dimensionless cack adius (denoted as l 0 /l). Note that in these figues, the dashed lines elate to Case1, howeve the solid lines elate to Case 2. Fig. 3 The gaphs of the dependence between dimensionless ERR and cack adius fo the PZT-4/Al/PZT-4 plate

13 Enegy Release Rate at the Font of Penny-shaped Inteface Cacks Fig. 4. The gaphs of the dependence between dimensionless ERR and cack adius fo the PZT-5H/St/PZT-5H plate Thus, it follows fom Figs. 2, 3 and 4 that fo all the cases unde consideation the pieoelecticity of the face layes causes to decease the ERR in the font of the inteface penny-shaped cack and the magnitude of this decease incease with the cack s adius. Moeove, the analyes of the gaphs given in these figues show that the values of the ERR incease with deceasing of the face layes thickness. 5. Conclusions Thus, in the pesent pape, the ERR at the penny-shape inteface cack contained in the PZT/Elastic/PZT sandwich plate-disc is studied within the scope of the exact equations and elations of the electo-elasticity fo the pieoelectic bodies. The axisymmetic stess-stain state is consideed and the coesponding bounday value poblem is solved numeically by employing FEM. Numeical esults ae pesented and discussed fo the PZT-5H/Al/PZT-5H, PZT-5H/St/PZT-5H and PZT-4/Al/PZT-4 plates. The convegence of the algoithm and PC pogams is tested with espect to the concete cases. Moeove, the validation of the PC pogams and algoithm used in the pesent investigation is examined with espect to the known esults obtained in the pape by Li and Lee (2012). Accoding to analyes of the afoementioned numeical esults obtained fo the ERR it can be dawn the following concete conclusions: 1. The pieoelecticity of the face layes mateials causes to decease the values of the ERR; 2. The values of the ERR incease (decease) with the atio l 0 /l (with the atio h F /l);

14 38 F.I. Cafaova 3. The magnitude of the ERR depends not only on the electo-mechanical popeties of the face layes mateials, but also on the mechanical popeties of the elastic coe laye. Fo instance, the values of the ERR obtained fo the plate with the St coe laye ae significantly less than the coesponding ones obtained fo the same plate with the Al coe laye. Refeences [1] Akbaov, S.D. (2013), Stability Loss and Buckling Delamination: Thee-Dimensional Lineaied Appoach fo Elastic and Viscoelastic Composites, Spinge, Heidelbeg, New Yok, USA. [2] Akbaov, S.D. and Yahnioglu, N. (2013), Buckling delamination of a sandwich platestip with pieoelectic face and elastic coe layes, Appl. Math. Model.,37, [3] Akbaov, S.D. and Yahnioglu, N. (2016), On the total electo-mechanical potential enegy and enegy elease ate at the inteface cack tips in an initially stessed sandwich plate-stip with pieoelectic face and elastic coe layes, Int. J. Solids Stuct., 88-89, [4] Akbaov, S.D. and Tuan, A. (2009), Mathematical modelling and the study of the influence of initial stesses on the SIF and ERR at the cack tips in a plate-stip of othotopic mateial, Appl. Math. Model.,33, [5] Akbaov, S.D., Cafaova,F.I. and Yahnioglu, N.(2017) Buckling delamination of the cicula sandwich plate with pieoelectic face and elastic coe layes unde otationally symmetic extenal pessue. AIP Confeence Poceedings 1815, (2017); doi: / View online: Published by the Ameican Institute of Physics: pp [6] Cafaova, F.I., Akbaov, S.D. and Yahnioglu, N. (2017), Buckling delamination of the PZT/Metal/PZT sandwich cicula plate-disc with penny-shaped inteface cacks, Smat Stuct. Syst., 19(2), [7] Cafaova F.I. and Rayev, O.A. (2016). Stability loss of the PZT/Metal/PZT sandwich ciculsa plate-discunde open-cicuit condition. Tansactionsof NAS of Aebaijan, Issue Mechanics, 36 (4), pp [8] Eskandai, M., Moeini-Adakani, S.S. and Shodja, H.M. (2010), An enegetically consistent annula cack in a pieoelectic medium, Eng. Fact. Mech., 77, [9] Gu, A.N. (1999), Fundamentals of the Thee-Dimensional Theoy of Stability of Defomable Bodies, Spinge-Velag, Belin, Heidelbeg, Gemany. [10] Gu, A.N. (2004), Elastic Waves in Bodies With Initial (Residual) Stesses, A.C.K., Kiev, Ukaine.

15 Enegy Release Rate at the Font of Penny-shaped Inteface Cacks [11] Henshell, R.D. and Shaw, K.G. (1975), Cack tip finite elements ae unnecessay, Int. J. Nume. Meth. Eng., 9, [12] Kuna, M. (2006), Finite element analysis of cacks in pieoelectic stuctues: a suvey, Ach. Appl. Mech., 76, [13] Kuna, M. (2010), Factue mechanics of pieoelectic mateials whee ae we ight now?,eng. Fact. Mech., 77, [14] Kudyatsev, B.A., Paton, V.Z. and Rakitin, V.I. (1975), Beakdown mechanics of pieoelectic mateials axisymmetic cack on bounday with conducto, Pikl. Math. Mekh.,39, [15] Landis, C.M. (2004), Enegetically consistent bounday conditions fo electomechanical factue, Int. J. Solids Stuct., 41, [16] Li, Y.D. and Lee, K.Y. (2012), Thee dimensional axisymmetic poblems in pieoelectic media: Revisited by a eal fundamental solutions based new method, Appl. Math. Model.,36, [17] Li, Y.S., Feng, W.J. and Xu, Z.H. (2009), A penny-shaped inteface cack between a functionally gaded pieoelectic laye and a homogeneous pieoelectic laye, Mecanica, 44(4) [18] Li, W., McMeeking, R,M. and Landis, C.M. (2008), On the cack face bounday conditions in electo-mechanical factue and an expeimental potocol fo detemining enegy elease ates, Eu. J. Mech. A/Solids, 27, [19] Paton, V.Z. (1976), Factue mechanics of pieoelectic mateials, Acta Astonaut, 3(9-10), [20] Ren, J.N., Li, Y.S. and Wang, W. (2014), A penny-shaped intefacial cack between pieoelectic laye and elastic half-space, Stuct. Eng. Mech., 62(1), 1-17 [21] Yang, F. (2004), Geneal solutions of a penny-shaped cack in a pieoelectic mateial unde opening mode loading, Q. J. Mech. Appl. Math., 57(4), [22] Yang, J. (2005), An Intoduction to the Theoy of Pieoelecticity, Spinge, New Yok, USA. [23] Zienkiewic, O.C. and Taylo, R.L. (1989), The Finite Element Method: Basic Fomulation and Linea Poblems. Vol. 1., Fouth Ed., McGaw-Hill Book Company, Oxfod, UK. [24] Zhong, X.C. (2012), Factue analysis of a pieoelectic laye with a penny-shaped and enegetically consistent cack, Acta Mech., 223,

16 40 F.I. Cafaova Faile I. Cafaova Genje State Univesity, Genje, Aebaijan