# Mixed Electrical Conditions on Interface Inclusion in a Piezoelectric Bimaterial under Antiplane Mechanical and Inplane Electric Loadings

Authors: A. Sheveleva; V. Loboda; A. Kryvoruchko
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

An absolutely rigid interface inclusion in a bimaterial piezoelectric space under the action of antiplane mechanical and in-plane electric loadings is analyzed. One zone of the inclusion is electrically insulated while the other part is electrically permeable. This problem is important for engineering application, but it has not been solved earlier in an analytical way. Presenting all electromechanical quantities via sectionally analytic vector functions, the combined Dirichlet-Riemann boundary value problem is formulated. An exact analytical solution of this problem is obtained. Closed form analytical expressions for electromechanical quantities at the interface are derived. Some of these values are also presented graphically along the corresponding parts of the material interface. Singular points of the shear strain and the electric displacement are found and the corresponding intensity factors are determined as well.


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## Body

## 1. Introduction

Piezoelectric materials are used in many engineering applications. These materials are often integrated into sensors, transducers, and actuators, where they can be exposed to substantial mechanical and electrical loading. However, existing microdefects, particularly interface inclusions, can reduce their strength. Therefore, it is very important to study the behavior of piezoelectric ceramics with internal and especially interface defects subjected to the action of mechanical stresses and electrical fields.Many important solutions have been obtained for inclusions in electrically passive materials. For example, the problem of rigid line inclusions in a homogeneous infinite matrix was studied in [1–6] and an interfacial inclusion between two dissimilar media was investigated in [7–9]. Rigid line inclusions in a piezoelectric medium were also actively studied. For instance, Liang et al. [10] and Chen [11] studied the in-plane and antiplane problems, respectively, of a homogeneous infinite piezoelectric medium with such inclusions.Essential progress was made concerning the investigation of rigid inclusions at the interface of piezoelectric materials. Particularly the electroelastic analysis of a conducting rigid line inclusion at the interface of two bonded piezoelectric materials is considered in paper [12]. By combining the analytic function theory and the Stroh formalism the closed-form expressions for the field variables were found. In the paper [13] the generalized two-dimensional problem of a dielectric rigid line inclusion, at the interface between two dissimilar piezoelectric media subjected to piecewise uniform loads at infinity, is studied by means of the Stroh theory. The problem was reduced to a Hilbert problem, and then closed-form expressions were obtained. The special mixed boundary value problem in which a debonded conducting rigid line inclusion is embedded at the interface of two piezoelectric half planes is solved analytically in [14] by employing Stroh formalism. The model based on the assumption that all of the physical quantities, i.e., tractions, displacements, normal component of electric displacements, and electric potential, are discontinuous across the interface defect was used in this paper. The axisymmetric contact problem of a rigid inclusion embedded in the piezoelectric bimaterial frictionless interface subjected to simultaneous far-field compression and electric displacement is studied in [15]. An arc-shaped conducting rigid line inclusion located at the interface between a circular piezoelectric inhomogeneity and an unbounded piezoelectric matrix subjected to remote uniform antiplane shear stresses and in-plane electric fields is considered recently in [16].It should be mentioned that all results concerning the rigid inclusions at the interface between piezoelectric materials are related to the cases of one-type electric conditions at the inclusion. However, in some cases important for engineering applications these conditions can be mixed, i.e., they can change from one part of the inclusion to another. The problem in such case becomes mathematically much more complicated; therefore, the associated analytical solutions are absent to the authors knowledge. Just such solution is suggested in the present paper for a certain case of mixed electrical conditions on the interface inclusion.


## 2. Basic Equations for a Piezoelectric Material under Out-of-Plane Mechanical Loading and In-Plane Electric Loading

For a piezoelectric material the relationship between the main electromechanical characteristics are defined by the relations (Parton and Kudryavtsev [17])(1)σij=cijksεks-esijEs,Di=eiksεks+αisEs,where σij, εij are the components of stress and strain tensor, Di, Ei are the components of the electric induction and the electric field, cijks, esij are elastic and piezoelectric constants, and αis are dielectric constants.The equilibrium equations in the absence of body forces and free charges are(2)σij,j=0,Di,i=0.The expressions for the deformation and electric field have the form(3)εij=12ui,j+uj,i,Ei=-φ,i,where ui are the components of the displacement vector and φ is the electric potential.For the out-of-plane mechanical loading and in-plane electric loading assuming the material is transversely isotropic with the poling direction parallel to thex3-axis, one has (4)u1=u2=0,u3=u3x1,x2,φ=φx1,x2.Then the constitutive relations take the form(5)σ3iDi=Ru3,iφ,i,where i=1,2 and R=c44e15e15-α11.Introducing the vectors(6)u=u3,φT,t=σ32,D2T,one can write(7)t=Ru,2.The functionsu3 and φ satisfy the equations Δu3=0, Δφ=0; i.e., they are harmonic. Therefore, present them as real parts of some analytic functions (8)u=2ReΦz=Φz+Φ-z-,where Φz=Φ1z,Φ2zT is an arbitrary analytic function of the complex variable z=x1+ix2.Substituting (8) into (7), one gets(9)t=iRΦ′z+iRΦ′z¯.Using the designationQ=iR, we arrive to the equations(10)t=QΦ′z+Q-Φ-′z-.To achieve the purposes of this paper we have to transform further the presentations (8) and (10). Taking into account that(11)u′=Φ′z+Φ-′z-and introducing the vectors(12)v′=σ32,-E1T,P=u3′,D2T,the following relations can be formulated(13)v′=AΦ′z+A-Φ-′z-,(14)P=BΦ′z+B-Φ-′z-,where the matrixes A and B are the following:(15)A=Q11Q1201,B=10Q21Q22.


## 3. Bimaterial Case

Suppose that the planex1,x2 is composed of two half-planes x2>0 and x2<0. The presentations (13) and (14) can be written for regions x2>0 and x2<0 in the form(16)vm=AmΦmz+A-mΦ-mz-,Pm=BmΦ′mz+B-mΦ-′mz-,where m=1 for the region 1 and m=2 for the region 2; A(m) and B(m) are the matrices A and B for the regions 1 and 2, respectively; Φ(m)z are arbitrary vector functions, analytic in the regions 1 and 2, respectively.Next we require that the equalityP(1)=P(2) holds true on the entire axis x1.Then it follows from (16)(17)B1Φ′1x1+i0+B-1Φ-′1x1-i0=B2Φ′2x1-i0+B-2Φ-′2x1+i0.HereFx1±i0 designates the limit value of a function Fz at y→0 from above or below the x1-axis, respectively. Equation (17) can be written as(18)B1Φ′1x1+i0-B-2Φ-′2x1+i0=B2Φ′2x1-i0-B-1Φ-′1x1-i0.The left and right sides of the last equation can be considered as the boundary values of the functions(19)B1Φ′1z-B-2Φ-′2z,B2Φ′2z-B-1Φ-′1z,which are analytic in the upper and lower planes, respectively. But it means that there is a function Mz, which is equal to the mentioned functions in each half-plane and is analytic in the entire plane.Assuming thatMzz→∞→0, on the basis of the Liouville theorem we find that each of the functions in (19) is equal to 0 for any z from the corresponding half-plane. Hence, we obtain(20)Φ-′2z=B-2-1B1Φ′1zforx2>0,Φ-′1z=B-1-1B2Φ′2zforx2<0.Further, we find the jump of the vector function(21)v′x1=v′1x1+i0-v′2x1-i0,when passing through the interface. Determining from the first formula (16) (22)v′mz=AmΦ′mz+A-mΦ-′mz-or(23)v′mx1±i0=AmΦ′mx1±i0+A-mΦ-′mx1∓i0,and substituting in (21), one gets(24)v′x1=A1Φ′1x1+i0+A-1Φ-′1x1-i0-A2Φ′2x1-i0-A-2Φ-′2x1+i0.Finding furtherΦ′2x1-i0=B2-1B-1Φ-′(1)x1-i0 from second equation of (20) and substituting this expression together with the first equation of (20) at x2→+0 in the latest formula lead to(25)v′x1=DΦ′1x1+i0+D-Φ-′1x1-i0,where D=A1-A-2B¯2-1B1. Introducing a new vector function(26)Wz=DΦ′1z,x2>0,-D¯Φ′¯1z,x2<0,the last relation can be written as(27)v′x1=W+x1-W-x1.From the second relations (16) we have(28)P1x1,0=B1Φ′1x1+i0+B-1Φ-′1x1-i0.Taking into account that on the base of (26)(29)Φ′1x1+i0=D-1Wx1+i0,Φ-′1x1-i0=-D--1-1Wx1-i0,and substituting these relations into (28), leads to(30)P1x1,0=SW+x1-S-W-x1,where S=B1D-1. Simple calculations show that(31)S=A1B1-1-A-2B-2-1-1.Representations (27) and (30) are very convenient for solving of antiplane problems for bimaterials with cracks and inclusions at the interface.It is found out that for the considered class of piezoelectric materials the matrixS has the following structure:(32)S=is11s12s21is22,where all skl (k,l=1,2) are real.


## 4. Formulation of the Problem for an Absolutely Rigid Inclusion with Mixed Electrical Condition at the Face

Consider an absolutely rigid thin inclusionc≤x1≤b at the interface x2=0. It is assumed that this inclusion is electrically insulated for c≤x1≤a and is electrically permeable at a<x1<b (a<b). Such situation can take place, e.g., for a rigid electrically permeable interface layer having a part covered by electrical insulator (Figure 1).Figure 1
An absolutely rigid inclusion at the interface of two piezoelectric materials with electrically insulated partc,a and electrically permeable part a,b.Then the boundary conditions at the interface are of the form(33)ε131=ε132=0,D21=D22=0forc<x1<a,(34)ε131=ε132=0,E1=0D2=0fora<x1<b,(35)σ23=0,D2=0,ε31=0,E1=0forx1∉c,b.We also assume that a vectorP∞=ε13∞,D2∞T is prescribed at infinity.Consider (30) in the following expanded form: (36)ε131x1,0=is11W1+x1+s12W2+x1+is11W1-x1-s12W2-x1,D21x1,0=s21W1+x1+is22W2+x1-s21W1-x1+is22W2-x1,in which (32) was taken into account. Combining (36) one arrives at the presentations(37)ε131x1,0-imjD21x1,0=tjFj+x1+γjFj-x1,where(38)Fjz=W2z+isjW1z,and tj=s12-mjs22, γj=-s12+mjs22/tj, sj=s11+mjs21/tj, m1,2=∓-s11s12/s21s22.It follows from the last equations thats1,2=-m1,2, γ2=1/γ1, and the values m1,2 are real.Because according to (38) Fj+x1-Fj-x1=W2+x1-W2-x1+isjW1+x1-W1-x1, then with use of (27) one gets(39)-E1x1,0+isjσ23x1,0=Fj+x1-Fj-x1.It is sufficient to apply relations (37) and (39) in the following analysis only for j=1; therefore, assuming j=1, (37) and (39) can be presented in the form(40)ε131x1,0-im1D21x1,0=t1F1+x1+γ1F1-x1,(41)-E1x1,0+is1σ23x1,0=F1+x1-F1-x1,where m1=--s11s12/s21s22, s1=-m1.Taking into account that forx1∉c,b the relationships F1+x1=F1-x1=F1x1 are valid, it follows from (40) that(42)1+γ1t1F1x1=ε131x1,0-im1D21x1,0forx1⟶∞.Using the fact that the functionF1z is analytic in the whole plane cut along c,b and applying the conditions at infinity, one gets from the last equation(43)F1zz→∞=ε~13-iD~2,where(44)ε~13=ε13∞r1,D~2=m1D2∞r1,r1=1+γ1t1.


## 5. Solution of the Problem

Consider now the problem formulated by the interface conditions (33)–(35) and illustrated by Figure 1. Relations (27) and (30) and, consequently, (40) and (41) ensure satisfying equation P(1)x1,0=P(2)x1,0 for the whole interface and, accordingly, satisfying the second and third interface conditions (35). Further satisfaction of first and forth conditions (35) provides the analyticity of the function F1z for the whole plane with a cut along the segment c,b of the interface. Satisfying the remaining boundary conditions (33) and (34) with use of (40) and (41), one gets the following equations:(45)F1+x1+γ1F1-x1=0forc<x1<a,ReF1+x1+γ1F1-x1=0,ReF1+x1-F1-x1=0fora<x1<b.The last two relations lead to the equation(46)ReF1±x1=0fora<x1<b.Introducing further the substitution(47)F1z=iΦ1z,system (34) and (35) can be written in the form(48)Φ1+x1+γ1Φ1-x1=0forc<x1<a,(49)ImΦ1±x1=0fora<x1<b.Taking into account that the functionΦ1z is analytic outside the segment c,b and using the prescribed values of the stress and the electric fields at infinity, one gets, from (43) and (44), the following condition for Φ1z at infinity(50)Φ1zz→∞=E1∗-iσ32∗,where E1∗=-D~2, σ32∗=ε~13, r1=t1(1+γ1).Relations (48) and (49) present the combined Dirichlet-Riemann boundary value problem. The solution of such problem concerning a rigid stamp was found by Nahnein and Nuller [18] and, concerning an in-plane interface crack, it was developed by Loboda [19] and Kozinov et al. [20]. Using these results, an exact solution of problem (37) and (38) is the following. (51)Φ1z=PzX1z+QzX2z,where(52)Pz=C1z+C2,Qz=D1z+D2,X1z=ieiϕzz-cz-b,X2z=eiϕzz-cz-a,φz=2εln⁡b-az-clz-a+a-cz-b,ε=12πln⁡γ1,l=b-c,C1=-σ32∗cos⁡β-E1∗sin⁡β,D1=E1∗cos⁡β-σ32∗sin⁡β,C2=-c+b2C1-β1D1,D2=β1C1-c+a2D1,β=εln⁡1-1-λ1+1-λ,β1=εa-cb-c,λ=b-al.This solution satisfies the condition at infinity (50) and the equation ∫cbF1+x1-F1-x1dx1=0 (Knysh et al. [21]). The last equation is obtained using (41) from the conditions of the electric potential uniqueness for overcoming the inclusion and the requirement of its equilibrium.Using solution (51) together with formula (40), one gets(53)m1D21x1,0+iε131x1,0=-Qx1x1-a+iPx1x1-br1exp⁡iφx1x1-cforx1>b,(54)m1D21x1,0=-t1Px1x1-cb-x11-γ1cosh⁡φ0x1+1+γ1sinh⁡φ0x1+-t1Qx1x1-cx1-a1+γ1cosh⁡φ0x1+1-γ1sinh⁡φ0x1fora<x1<b,where φ0x1=2εtan-1⁡a-cb-x1/b-cx1-a.Substituting solution (51) into (41) gives the following formulas:(55)s1σ23x1,0+iE1x1,0=2αPx1b-x1-iQx1a-x1exp⁡iφ∗x1x1-cforc<x1<a,(56)σ23x1,0=2s1x1-cPx1b-x1cosh⁡φ0x1+Qx1x1-asinh⁡φ0x1fora<x1<b,where φ∗x1=2εln⁡b-ax1-c/la-x1+a-cb-x1, α=(γ1+1)2/4γ1.It is very important that the obtained solution is not oscillating in this case at the right crack tip and, therefore, commonly used intensity factors can be introduced. Thus, we introduce further the following mechanical strain and electrical displacement intensity factors (IFs):(57)KD=limx1→a+02πx1-aD21x1,0,Kε=limx1→b+02πx1-bε131x1,0.Using (54) and taking into account that φ0a=ln⁡γ1, one can find (58)KD=r1Qam1a-c2πα.The intensity factorKε can be found from formula (53) and can be written in the form (59)Kε=r12πlPb.Substituting the expressions for Pb and Qa one gets (60)KD=1m1πl2α1-λε13∞cos⁡β-m1D2∞sin⁡β-2εε13∞sin⁡β+m1D2∞cos⁡β,(61)Kε=πl2ε13∞sin⁡β+m1D2∞cos⁡β+2ε1-λε13∞cos⁡β-m1D2∞sin⁡β.Use of (55) for x1→a-0 permits obtaining the following expressions of σ23x1,0 via the stress intensity factor Kε:(62)σ23x1,0=-2αr1s1Kε2πa-x1forx1⟶a-0.


## 6. Numerical Illustration

The results for the shear stress jumpσ23x1,0 for c=-10 mm, b=10 mm, D2∞=0.005 C/m2, and different values of ε13∞ are shown in Figure 2. Lines I, II, and III in this figure correspond to ε13∞=0.0001, 0.00015, 0.0002, respectively. The results for the electric displacement D2(1)x1,0 for x1∈(a,b) and x1>b are shown in Figures 3 and 4, respectively. The variation of the strain ε13(1)x1,0 on the inclusion continuation is shown in Figure 5. The same values of ε13∞ as in Figure 2 were chosen in Figures 3–5. The materials with the characteristics c44(1)=43,7·109 Pa, e15(1)=17 C/m2, α11(1)=15,1·10-9 C/(V·m), c44(2)= 42,47·109 Pa, e15(2)=-0,48 C/m2, and α11(2)= 0,0757·10-9 C/(V·m) [22] were used for these calculations.Figure 2
The shear stressσ23x1,0 jump distribution over the insulated region of the inclusion for different values of ε13∞.Figure 3
The electric displacementD2(1)x1,0 for x1∈(a,b).Figure 4
The electric displacementD2(1)x1,0 for x1>b.Figure 5
The variation of the strainε13(1)x1,0 on the inclusion continuation.It can be seen from the results in Figure2 that the stress jump is sufficiently small at the pointa, but it grows with tending x1 to the left tip of the inclusion. On the other hand the electric displacement D2(1)x1,0 essentially grows at the right neighborhood of the pointa (Figure 3) and keeps moderate values at the left vicinity of the right crack tip. It is interesting to note that D2(1)x1,0 remains limited at the right vicinity of the point b and, moreover, it modestly decreases at a distance from this point (Figure 4). At the same time as it follows from Figure 5 the shear deformation ε13(1)x1,0 is singular at the right neighborhood of the point b.Therefore, it grows very fast for x1 tending to this point and it promptly decreases a distance from it.


## 7. Conclusion

A rigid inclusion between two semi-infinite piezoelectric spaces with electrically insulated and electrically permeable parts of the crack faces under the action of antiplane mechanical and in-plane electric loadings has been analyzed. This problem is important for engineering application, because the presence of inclusions disturbs the field variables in the matrixes and as a result affects the load-carrying capacity of engineering structure. Although such situation can take place for any engineering device containing piezoelectric element, it has not been investigated earlier in an analytical way to the authors’ knowledge. Firstly the electromechanical quantities are presented by using of sectionally analytic vector functions. A combined Dirichlet-Riemann boundary value problem (48) and (49) is formulated and its exact analytical solution is derived. Analytical expressions (53)–(56) for the main electromechanical characteristics along the interface are presented in a closed form. Singular points of these values and the stress and electric intensity factors are determined. The variation of the shear stress jump σ23x1,0, electric displacement D2(1)x1,0, and the strain ε13(1)x1,0 along different sections of the inclusion and its continuation are presented in a graphical form for certain positions of the point a dividing the electrically insulated and electrically permeable zones of the inclusion.



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