Fn Shiwng,Guo Junhong,*,Yu Jing,b

aSchool of Science,Inner Mongolia University of Technology,Hohhot 010051,China

bCollege of General Education,Inner Mongolia Normal University,Hohhot 010022,China

Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids

Fan Shiwanga,Guo Junhonga,*,Yu Jinga,b

aSchool of Science,Inner Mongolia University of Technology,Hohhot 010051,China

bCollege of General Education,Inner Mongolia Normal University,Hohhot 010022,China

Cracks;Conformal mapping;Hole;Piezoelectric material;Stroh-type formulism

By constructing a new numerical conformal mapping and using the Stroh-type formulism,an anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids is investigated.The explicit expressions of the complex potential function,field intensity factors,energy release rates and mechanical strain energy release rate near the crack tip are obtained under the assumptions that the surfaces of the cracks and hole are electrically permeable and electrically impermeable.Numerical examples are presented to show the influences of the geometrical parameters of defects and applied mechanical/electrical loads on the energy release rate and mechanical strain energy release rate under two electrical boundary conditions.

1.Introduction

Due to the excellent coupling effect between mechanical and electrical fields,the piezoelectric devices have been used in aviation and aerospace industry,such as structural health monitoring,precision positioning and vibration control.The actuators and sensor in smart structures,which scale down favorably in terms of power output and efficiency,have yielded the novel compact piezoelectric hydraulic pumps in the aerospace industry.1However,the brittleness of piezoelectric materials inevitably leads to many kinds of defects(e.g.,cracks and holes)during the processing,manufacturing and in-service periods.Therefore,it is of great significance to understand the fracture behavior of the complicated defects in piezoelectric materials,especially for cracks emanating from holes.

In past decades,many crack problems was considered by researchers.For example,Gao and Yu2addressed the generalized two-dimensional plane problems of a semi-infinite crack in a piezoelectric medium subjected to a line force and a line charge based on the Stroh formalism,and obtained the explicit expressions of the field intensity factors and the Green’s functions.By comparing the electrically impermeable and permeable boundary assumptions,Wang and Mai3pointed out that the electrically impermeable boundary was a reasonable one in engineering applications.Li and Lee4,5analyzed the electroelastic behavior of a piezoelectric ceramic strip containing an anti-plane shear crack.By utilizing the integral transform,Li and his coauthors6–8considered the anti-plane interface cracks in two bonded dissimilar piezoelectric layersunder the electrically permeable and impermeable assumptions.Guo et al.9studied the anti-plane problem of a semiinfinite crack in a piezoelectric strip by using the complex variable function method and the technique of conformal mapping.For the hole problems,Gao and Fan10investigated the two-dimensional problems of an elliptical hole in a piezoelectric material based on the complex potential approach and obtained the explicit solutions in closed form under remotely uniform mechanical and electrical loads.Dai et al.11performed the stress concentration around an elliptic hole in transversely isotropic piezoelectric solids subjected to the uniform mechanical and electrical loads at infinity.The results showed that the electromechanical coupling effect is helpful to reduce the stress concentration.For the cracks emanating from holes,Wang and Gao12analyzed the fracture problem of one and two cracks originating from a circular hole in an infinite piezoelectric solid by using the complex variable method.By constructing a new conformal mapping and using the Stroh-type formulism,Guo et al.13considered the anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material under the electrically impermeable boundary condition.Also,they analyzed the anti-plane fracture behavior of the two non-symmetrical collinear cracks emanating from an elliptical hole14and the multiple cracks emanating from a circular hole15in a piezoelectric solid under differentelectricalboundary conditions.Based on the extended Stroh formalism and the boundary element method,Liang and Sun16successfully identified the hole/crack size,location,and orientation in finite circular piezoelectric plates by using the strain/electrical field data,stress/electrical displacement data,or displacement/electrical charge data under the static loadings.

The key for using the complex variable method to consider the problems of complicated defects such as cracks emanating from holes is to find a suitable conformal mapping.As mentioned above,the previous literature is limited to the relatively simple hole shapes,such as circular or elliptical holes.Thus,the exact solutions of these defects can be derived by the conformal mapping.However,it is very difficult to find an exact conformal mapping for more complicated defects such as cracks emanating from square or triangle holes due to the intricacy of structure and difficulty of mathematics.Considering this,we can effectively solve the complicated defects by constructing a new numerical conformal mapping.Recently,Wang et al.17studied an anti-plane problem of piezoelectric solids containing a regular triangle hole with smooth vertices which emanates an edge crack by constructing a new numerical conformal mapping.Wang et al.18considered the antiplane problems of two cracks emanating from a rhombus hole and a cracked half circular hole at the edge of a half plane in a piezoelectric solid by deriving an approximate mapping function and using the Laurent series.For the classical elasticity,Yan et al.19and Miao et al.20presented the interaction of two collinear cracks emanating from a square hole in a rectangular plate and an infinite plate by using a hybrid displacement discontinuity method and a generalized Bueckner’s principle.The results show that the geometric parameters of defects have a great influence on the failure of materials.

Nevertheless,no investigation on four edge cracks emanating from a square hole has been reported up to now in any material including piezoelectric material.In fact,an increase of the number of cracks shows a different and interesting fracture behavior,which can provide an important reference in engineering practice.Therefore,this paper focuses on the anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids by constructing a new numerical conformal mapping,and the explicit solutions of the electroelastic fields are derived finally.

2.Basic equations

For the transversely isotropic piezoelectric solids,the poling direction is along the positivex3-direction and the isotropic plane is in thex1ox2-plane.The anti-plane deformation is determined by the out-of-plane displacement and the inplane electrical potential,which are the functions ofx1andx2.The generalized Hooke’s law for a two-dimensional antiplane problem of piezoelectric solids is

Under the assumption of small deformation,the gradient equation can be expressed as

The elastic and electrostatic equilibrium equations in the absence of the body force are

In Eqs.(1)–(3),iis from 1 to 2,the repeated indices denote summation,and a comma in the subscripts denotes a partial differentiation;c44,e15and ε11are the elastic constants,piezoelectric constants and dielectric permittivity,respectively;σ3i,γ3i,u3,Di,Eiand φ are the stress,strain,elastic displacement,electrical displacement,electrical field and electrical potential,respectively.

Substituting Eqs.(1)and(2)into Eq.(3)leads to the following governing equation for the two-dimensional anti-plane problem of piezoelectric solids:

where ?2= ?2/?x21+ ?2/?x22istwo-dimensionalLaplace operator,u=[u3，φ]Tis the generalized displacement vector,and the material matrix B0can be expressed as

By introducing a generalized stress function vector φ,the general solution21to Eqs.(4)and(1)can be written in the Stroh formalism as

wherez=x1+ix2,f(z)is an analytic function to be determined by the boundary conditions,B=iB0,and A=I is a 2×2 unit matrix.

It is found that once the analytic function vector f(z)is determined,the stress,strain,displacement,elastic displacement,electrical field and electrical potential for the antiplane problem of piezoelectric solids with defects can be obtained from Eqs.(1),(2)and(6).

3.Description of problem and conformal mapping

3.1.Description of problem

The mechanical/electrical boundary conditions along the surface of the cracks and hole can be expressed as

whereSis the boundary along the surface of defect,t3andDnrepresent the anti-plane traction and electrical displacement of normaldirection along the surface ofthe boundary,respectively.

3.2.Conformal mapping

To solve the above boundary value problem(i.e.,Eqs.(6)and(7))efficiently,we construct a new numerical conformal mapping which transforms the boundary of complicated defect(Fig.1)in thez-plane into a unit circle in the ζ-plane.Firstly,Savin22proposed a conformal mapping:

which maps the exterior of the square hole inz-plane onto the exterior of a circular hole in ζ-plane.Ris a constant related to the side length of square hole,i.e.,R=0.59136a.It is noted that the square hole is not an exact one,but it is a regular one with smooth vertices,and the radius of curvaturerof its corners isr=0.014a.The deviation becomes smaller if more items are taken in Eq.(8).For the current problem,the former four items have a high accuracy.

Inspired by the conformal mapping Eq.(8),we propose a new mapping(Fig.2)which maps the exterior region of four cracks emanating from a square hole inz-plane onto the interior region of a unit circle in ζ-plane as follows:

Fig.1 Four edge cracks emanating from a square hole in piezoelectric solids.

in which the mapping function μ(ζ)that maps outer region of four cracks emanating from a circular hole inz1-plane onto the interior of a unit circle in ζ-plane was proposed firstly by Guo et al.23,i.e.,

The real parametersci(i=1，2，3)in Eq.(10)can be determined by

whereliare the lengths of cracks inz1-plane(Fig.2),which can be determined by the following relationship:

As shown in Fig.2,Eq.(9)maps the pointsA～Hinzplane onto the corresponding pointsA1～H1inz1-plane,respectively,and then Eq.(10)maps the pointsA1～H1inz1-plane onto the corresponding pointsA′±～H′

Fig.2 Schematic diagram of conformal mapping.

±in ζ-plane,respectively,in which the plus sign underneath indices of the alphabet denotes above or right plane,and the minus sign stands for below or left plane.

4.Explicit solution to problem

4.1.Complex potentials

For the current problem,the complex potential vector can be taken as24

where c∞stands for a complex constant vector to be determined by the remote loading conditions,c∞zis the complex potential vector related to the external loads without defect,f0(z)is an unknown complex vector which stands for the potential vector disturbed by defect,and f0(∞)=0.Then,we solve the unknown complex potential vector f0(z)under the boundary conditions.

Differentiating Eq.(6)with respect tox1and defining F(z)=df(z)/dz,one has

Inserting Eq.(13)into Eq.(14)and then takingz→∞r(nóng)esult in

Thus,the constant vector c∞can be obtained from Eq.(15)as

If the electrically impermeable boundary condition is adopted and the surfaces of the crack and hole are free of mechanical loads,Eq.(7)can be written as

Substituting Eq.(13)into Eq.(17),we have

Combined with conformal mappingz= ω(ζ),in the ζplane,Eq.(18)can be transformed into

where f0(ζ)is an analytic function inside the unit circle and ω(ζ)is analytic inside the unit circle except for the point ζ=0.According to the Cauchy integral formula for an arbitrary point within|ζ|＜ 1,Eq.(21)becomes

Using the residue theorem in complex variable function,we derive the right integral of Eq.(22)as

Substituting Eq.(23)into Eq.(22),we have

Differentiating Eq.(24)with respect to ζ and defining F0(ζ)=df0(ζ)/dζ,we find

in whichR=0.59136aand ω′(ζ)can be obtained from Eq.(9)as

Also,μ′(ζ)in Eq.(26)can be determined from Eq.(10),that is,

For the electrically permeable case,the square hole and the cracks are free of traction on their surfaces and are filled with air of a dielectric permittivity.Therefore,the boundary condition17on the surfaces of hole and cracks can be expressed as

whereD02is unknown constant to be determined from the condition

Similar to the treatment of Guo et al.23,theD02can be obtained as

Therefore,we obtain the complex potential vector f(z)with defect by inserting Eqs.(16)and(24)into Eq.(13).Furthermore,all the electroelastic quantities of the anti-plane problem for four edge cracks emanating from a square hole in piezoelectric solids can be derived.Due to their long and complex expressions,the electroelastic quantities,such as stress and electrical displacement,are no longer given here.Next,we will give the explicit solution of the field intensity factors and the energy release rate at the crack tip.

4.2.Field intensity factors

For the electrically impermeable boundary,the vector of field intensity factors is defined as

From Eq.(27),we have μ′(1)=0 as ζ→ 1.Besides,we note that ω′(1)=0.By using the L’Hospital rule,Eq.(31)becomes

Inserting Eq.(25)into Eq.(32)and noting Eq.(15),we obtain the expressions of the field intensity factors near the crack tip under the electrically impermeable boundary as follows:

For the electrically permeable boundary,the field intensity factors can be defined similar to Eq.(36)as

in whichkpσandkpDare the stress and electrical displacement intensity factors under the electrically permeable boundary at the crack tip,respectively.Repeating the process of Eqs.(13)–(25),we have

Substituting Eq.(30)into Eq.(39),we can rewrite the expression as

Eq.(40)shows that the stress intensity factor only relates to the geometrical parameters of defects and the mechanical load,but the electrical displacement intensity factor is also dependent on the material constants.The electrical load has no effect on the electrical displacement intensity factors for the electrically permeable boundary,which is different from those for the electrically impermeable one.The similar conclusion can be found in previous work.12,13

4.3.Energy release rate and mechanical strain energy release rate

For the electrically impermeable crack,the energy release rate is equivalent to theJ-integral,i.e.,

whereksandkEare the strain and electrical field intensity factors at the crack tip,respectively,which have the following relationship with the stress and electrical displacement intensity factors

Inserting Eqs.(33)and(42)into Eq.(41),and noting Eq.(34),we can obtain the energy release rate at the right crack tip as

In order to compare the results under different fracture criterions,we also present the mechanical strain energy release rates:

However,for the electrically permeable case,the energy release rate is equivalent to the mechanical strain energy release rate,i.e.,

It can be found from Eq.(33)that the stress and electrical displacement intensity factors decouple each other.However,the energy release rate and mechanical strain energy release rate can show the interplay between the mechanical and electrical fields.

5.Numerical examples

To discuss the effects of the geometrical parameters of complicated defect considered in this paper and the combined mechanical/electrical loads on the normalized energy release rate and the mechanical strain energy release rate under two electrically boundary conditions,we take PZT-7 as a model material with the following material constantsc44=2.50×1010N/m2,e15=13.50 C/m2,ε11=171×10-10C/Vm andJcr=5.0 N/m,whereJcrstands for the critical energy release rate.25

Fig.3 Variation of normalized energy release rate J/Jcrwith ratio of crack length to side length of square hole.

Fig.4 Effect of applied electromechanical loads on normalized energy release rate.

Fig.4 shows the effects of the applied electromechanical loads on the normalized energy release rates for given geometrical parameters of defect.For the electrically impermeable boundary condition,when the positive electrical load(or no electrical load)is applied on the solid,the energy release rate increases with increasing mechanical load,which indicates that the mechanical loads always promote the crack growth(Fig.4(a)).When the negative electrical load is applied on it,the energy release rate decreases slowly first and then increases quickly with increasing mechanical load.For the electrically permeable boundary condition,however,the energy release rate only depends on the magnitude of mechanical load.In other words,the energy release rate is almost independent of the electrical load,which is also clear in Fig.4(b).Fig.4(b)shows that the energy release rate decreases with the increasing absolute value of negative electrical field,and it increases first and then decreases with the positive electrical field.It indicates that the negative electrical load always retards the crack growth,but the positive electrical load may either promote or retard the crack growth for the electrically impermeable boundary condition,which is greatly dependent on the mechanical load,while the energy release rate is a constant which is independent of the electrical load under the electrically permeable boundary condition.

Fig.5 shows the effects of the applied electromechanical loads on the normalized mechanical strain energy release rates under two electrical boundary conditions, whereGcr=5.0 N/m denotes the critical mechanical strain energy release rate.When the geometrical parameters of defect and the electrical load are fixed,the mechanical strain energy release rate increases with increasing mechanical load.Moreover,the mechanical strain energy release rate under the electrically impermeable boundary condition is greater than that under the electrically permeable one when σ∞32≤2.5 MPa.However,the mechanical strain energy release rate of the electrically permeable cracks will be wholly larger than that of the electrically impermeable ones when σ∞32≥5.5 MPa.Thus,we can control the crack growth by adjusting the applied mechanical loads.We can observe from Fig.5(b)that the mechanical strain energy release rate increases linearly with increasing electrical load for the electrically impermeable boundary condition.The result shows that the positive electrical load always promotes the crack growth and the negative electrical load retards the crack growth,which is in agreement with the experimental result.26

Comparing Fig.4(b)with Fig.5(b),we can see that the effect of the electrical load on the crack growth is different for the electrically impermeable boundary condition by using the different fracture criterions.

Fig.5 Effect of applied electromechanical loads on normalized mechanical strain energy release rate.

6.Conclusion

In this paper,by constructing a new numerical conformal mapping and using the Stroh-type formalism,the explicit expressions of the complex potential function,the field intensity factorsandenergyreleaseratesandthemechanicalstrainenergy release rate at the crack tip are derived for the anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids.Numerical examples are provided to show the effects of the geometrical parameters of complicated defect and applied mechanical/electrical loads on the energy release rate and mechanical strain energy release rate under the electricallypermeableandimpermeableboundaryconditions.Insummary,some useful conclusions are drawn as follows:

(1)The increases of the length of horizontal cracks(left and right cracks)and size of the square hole always promote thecrack growth undertwo electricalboundary conditions.

(2)The appearance of vertical cracks greatly affects the stress concentration of the horizontal cracks.The increase of the length of vertical cracks can promote or retard the horizontal crack propagation,which is strongly dependent on the lengths of the two horizontal cracks.

(3)The mechanical load always promotes the crack growth.The electrical load has no effect on the crack growth under the electrically permeable boundary condition,while the effect of the electrical load on the crack growth depends on the fracture criterion under the electrically impermeable boundary condition.

Acknowledgements

This work was supported by the National Natural Science Foundation of China(Nos.11262012,11502123,11462020 and 11262017)and the Inner Mongolia Natural Science Foundation(Nos.2015JQ01 and 2015MS0129)of China.

Appendix A.Supplementary material

Supplementary data associated with this article can be found,in the online version,at http://dx.doi.org/10.1016/j.cja.2016.08.018.

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15 January 2016;revised 9 March 2016;accepted 3 May 2016

Available online 15 October 2016

?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.Tel.:+86 471 6576143.

E-mail addresses:fsxwang@163.com(S.Fan),jhguo@imut.edu.cn,guojunhong114015@163.com(J.Guo),yujing3622@163.com(J.Yu).Peer review under responsibility of Editorial Committee of CJA.