Analysis of a Partially Debonded Conducting Rigid Elliptical Inclusion in a Piezoelectric Matrix

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Debonded arc-shaped interface conducting rigid line inclusions in piezoelectric composites

We consider 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 anti-plane shear stresses and in-plane electric fields. Moreover, one side of the rigid line inclusion has become fully debonded from the matrix or the inhomogeneity leading to the formation of an insulating crack. After the introduction of two sectionally holomorphic vector functions, the problem is reduced to a vector Riemann–Hilbert problem, which can be decoupled sequentially by repeated application of the orthogonality relations between the eigenvectors for two corresponding generalized eigenvalue problems.     

Dynamic Response of a Piezoelectric Material with a Conducting Rigid Inclusion

The problem of a conducting rigid inclusion embedded in an infinite piezoelectric matrix is considered under the action of combined electromechanical impact loads. By using integral transform techniques, the mixed initial-boundary value problem for the case of anti-plane shear load and in-plane electric field is transformed into two systems of dual integral equations, the solutions of which give the singularity coefficients of electroelastic field near the inclusion tips in closed-form in the Laplace transform domain. Numerical results for the stress singularity coefficient in the physical space are presented graphically by numerically solving the resulting Fredholm integral equation and carrying out the numerical inversion of Laplace transform for a PZT-5H material with a conducting rigid line inclusion.     

A conducting arc crack between a circular piezoelectric inclusion and an unbounded matrix

The present paper investigates the problem of a conducting arc crack between a circular piezoelectric inclusion and an unbounded piezoelectric matrix. The original boundary value problem is reduced to a standard Riemann–Hilbert problem of vector form by means of analytical continuation. Explicit solutions for the stress singularities δ=−(1/2)±iε are obtained, closed form solutions for the field potentials are then derived through adopting a decoupling procedure. In addition, explicit expressions for the field component distributions in the whole field and along the circular interface are also obtained. Different from the interface insulating crack, stresses, strains, electric displacements and electric fields at the crack tips all exhibit oscillatory singularities. We also define a complex electro-elastic field concentration vector to characterize the singular fields near the crack tips and derive a simple expression for the energy release rate, which is always positive, in terms of the field concentration vector. The condition for the disappearance of the index ε is also discussed. When the index ε is zero, we obtain conventionally defined electro-elastic intensity factors. The examples demonstrate the physical behavior and the correctness of the obtained solution.     

Singularities of an interface crack in electrostrictive materials

In the present work, the singularities of an interface crack between two dissimilar electrostrictive materials under electric loads are investigated. Within the framework of two-dimensional deformation, the problem is solved using the complex variable method. Three crack models, that is, permeable, impermeable and conducting crack models are considered individually. Complex potentials and intensity factors of total stresses are derived by considering both the Maxwell stresses in the surrounding space at infinity and inside the crack. It is found that, for the above three crack models, the singularities of total stress are the same as those in traditional bi-materials with an interface crack; however, the intensities of the total stress depend on the actual crack model used.     

A piezoelectric screw dislocation interacting with an imperfect piezoelectric bimaterial interface

A general method is presented for the analytical solution of a piezoelectric screw dislocation located within one of two joined piezoelectric half-planes. The bonding along the half-plane is considered to be imperfect with the assumption that the imperfect interface is mechanically compliant and dielectrically weakly (or highly) conducting. For a mechanically compliant interface tractions are continuous but displacements are discontinuous across the imperfect interface. In this context, jumps in the displacement components are assumed to be proportional to their respective interface traction components. Similarly, for a dielectrically weakly conducting interface the normal electric displacement is continuous but the electric potential is discontinuous across the interface. The jump in electric potential is proportional to the normal electric displacement. In contrast, for a dielectrically highly conducting interface the electric potential is continuous across the interface whereas the normal electric displacement has a discontinuity across the interface which is proportional to a certain differential expression of the electric potential. Explicit expressions are derived for the complex field potentials. The results show that there are two dimensionless parameters measuring the interface “rigidity” as compared to one for the purely elastic case. When the imperfect interface is compliant and weakly conducting, the two dimensionless parameters can be positive real values or complex conjugates with positive real parts. When the imperfect interface is compliant and highly conducting the two dimensionless parameters can only be positive real values. An expression for the image force acting on the screw dislocation due to its interaction with a compliant and weakly conducting interface is also given. It is found that the image force is only dependent on two dimensionless generalized Dundurs constants as well as two dimensionless parameters measuring the interface “rigidity”.     

三相压电复合本构模型中的弧形界面裂纹

深入研究了三相同心圆柱压电复合本构模型中的弧形绝缘界面裂纹问题。采用复势方法获得了该问题的级数形式的解答,并给出了应力、应变、电位移和电场强度等物理量在全场及界面上的分布,同时推导了裂尖处广义强度因子及裂面张开位移和裂面上电势差的表达式。具体计算表明该级数解答收敛迅速,同时显示出第三相混杂区的影响是不能忽略的。由于裂尖处应力奇异性为－1／2,则这种解答不会出现平面应变状态下界面裂纹裂尖处的振荡奇异性,从而不会产生违反物理实际的裂面相互嵌入现象,则该弹性解答也是建立了坚实的物理基础之上。     

Propagation of slip pulse along frictionless contact interface with local separation between two piezoelectric solids

The Stroh formalism of piezoelectric materials,Fourier analysis and singular integral equation technique were used to investigate the existence of a pulse at the fric- tionless interface in presence of local separation between two contact piezoelectric solids. The two solids were combined together by uniaxial tractions and laid in the electric field. The problem was cast into a set of Cauchy singular integral equations,from which the closed-form solutions were derived.The numerical discussion on the existence of such a slip pulse was presented.The results show that such a slip pulse,which has square root singularities at both ends of the local separation zone,can propagate in most material combinations.And the existence of such a slip pulse will not be affected by the applied mechanical and electric fields in some special material combinations.     

ARC INTERFACE CRACK IN A THREEPHASE PIEZOELECTRIC COMPOSITE CONSTITUTIVE MODEL

An in-depth investigation is made on the problem of an arc-shaped interface insulating crack in a three-phase concentric circular cylindrical piezoelectric composite constitutive model. An exact solution in series form is derived by employing the complex variable method. In addition, the distribution of physical quantities such as stresses, strains, electric displacements and electric fields in the whole field and along the interface is also presented. Explicit expressions for crack opening displacement, jump in electric potential on the crack surface and the electro-elastic field intensity factors at the crack tips are obtained. Specific calculations demonstrate that the convergence of the series form solution is satisfactory and that the outer phase (composite phase) will exert a significant effect on the electro-mechanical coupling response of the composite system. Owing to the fact that stresses and electric displacements still possess conventional inverse square root singularities, the oscillating singularities near the crack tip under plane strain conditions will be absent and, as a result, no unphysical interpenetration phenomenon of the two crack surfaces will occur. In conclusion, the elastic solution obtained is also based on a solid physical foundation. Project supported by the National Natural Science Foundation of China (No.59635140), and the Doctorate Foundation of Xi'an Jiaotong University.     

Analytical solutions of uniform extended dislocations and tractions over a circular area in anisotropic magnetoelectroelastic bimaterials

In this paper, we derive the analytical solutions in a three-dimensional anisotropic magnetoelectroelastic bimaterial space subject to uniform extended dislocations and tractions within a horizontal circular area. By virtue of the Stroh formalism and Fourier transformation, the final expression of solutions in the physical domain contains only line integrals over [0,2π] rather than infinite integrals. As the reduced cases, the half-space and homogeneous full-space solutions can be directly derived from the present solutions. Also, in terms of material domains, the present solutions can be reduced to the piezoelectric, piezomagnetic, purely elastic materials with different symmetries of material property. To carry out numerical calculations, Gauss quadrature is adopted. In the numerical examples, the effect of different loading locations on the response at the interface is analyzed. It is shown that, when the magnetic traction or electric dislocation is applied, the physical quantities on the interface may not decrease monotonically as the loading area moves away from the interface. The distributions of different in-plane physical quantities on the upper and lower interfaces under various extended horizontal loadings are compared and the differences are discussed. The work presented in this paper can serve as benchmarks for future numerical studies in related research fields.     

Some basic problems in anisotropic bimaterials with a viscoelastic interface

This research is devoted to the study of anisotropic bimaterials with Kelvin-type viscoelastic interface under antiplane deformations. First we derive the Green’s function for a bimaterial with a Kelvin-type viscoelastic interface subjected to an antiplane force and a screw dislocation by means of the complex variable method. Explicit expressions are derived for the time-dependent stress field induced by the antiplane force and screw dislocation. Also presented is the time-dependent image force acting on the screw dislocation due to its interaction with the Kelvin-type viscoelastic interface. Second we investigate a rectangular inclusion with uniform antiplane eigenstrains embedded in one of the two bonded anisotropic half-planes by virtue of the derived Green’s function for a line force. The explicit expressions for the time-dependent stress field induced by the rectangular inclusion are obtained in terms of the simple logarithmic and exponential integral functions. It is observed that in general the stresses exhibit the logarithmic singularity at the four corners of the rectangular inclusion. Our results also show that when one side of the rectangular inclusion lies on the viscoelastic interface, the interfacial tractions are still regular at the two corners of the inclusion which are located on the interface. Last we address a finite Griffith crack normal to the viscoelastic interface by means of the obtained Green’s function for a screw dislocation. The crack problem is formulated in terms of a resulting singular integral equation which is solved numerically. The time-dependent stress intensity factors at the two crack tips are obtained and some interesting features are discussed.     

Electro-elastic interaction between a piezoelectric screw dislocation and circular interfacial rigid lines

The electro-elastic interaction between a piezoelectric screw dislocation located either outside or inside inhomogeneity and circular interfacial rigid lines under anti-plane mechanical and in-plane electrical loads in linear piezoelectric materials is dealt with in the framework of linear elastic theory. Using Riemann–Schwarz’s symmetry principle integrated with the analysis of singularity of complex functions, the general solution of this problem is presented in this paper. For a special example, the closed form solutions for electro-elastic fields in matrix and inhomogeneity regions are derived explicitly when interface containing single rigid line. Applying perturbation technique, perturbation stress and electric displacement fields are obtained. The image force acting on piezoelectric screw dislocation is calculated by using the generalized Peach–Koehler formula. As a result, numerical analysis and discussion show that soft inhomogeneity can repel screw dislocation in piezoelectric material due to their intrinsic electro-mechanical coupling behavior and the influence of interfacial rigid line upon the image force is profound. When the radian of circular rigid line reaches extensive magnitude, the presence of interfacial rigid line can change the interaction mechanism.     

Coulomb traction on a penny-shaped crack in a three dimensional piezoelectric body

The axisymmetric problem of a penny-shaped crack embedded in an infinite three-dimensional (3D) piezoelectric body is considered. A general formulation of Coulomb traction on the crack surfaces can be obtained based on thermodynamical considerations of electromechanical systems. Three-dimensional electroelastic solutions are derived by the classical complex potential theory when Coulomb traction is taken into account and the poling direction of piezoelectric body is perpendicular to the crack surfaces. Numerical results show that the magnitude of Coulomb tractions can be large, especially when a large electric field in connection with a small mechanical load is applied. Unlike the traditional traction-free crack model, Coulomb tractions induced by an applied electric field influence the Mode I stress intensity factor for a penny-shaped crack in 3D piezoelectric body. Moreover, compared to the current model, the traditional traction-free crack model always overestimates the effect of the applied electric load on the field intensity factors and energy release rates, which has consequences for 3D piezoelectric fracture mechanics.     

Non-local theory solution for a Mode I crack in piezoelectric materials

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in the piezoelectric materials subjected to a uniform tension loading. The permittivity of the air in the crack is considered. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the crack length, the materials constants, the electric boundary conditions and the lattice parameter on the stress and the electric displacement fields near the crack tips. It can be obtained that the effects of the electric boundary conditions on the electric displacement fields are large. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present at the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allowing us to use the maximum stress as a fracture criterion.     

Analysis of bi-material interface cracks with complex weighting functions and non-standard quadrature

A boundary element formulation is developed to determine the complex stress intensity factors associated with cracks on the interface between dissimilar materials. This represents an extension of the methodology developed previously by the authors for determination of free-edge generalized stress intensity factors on bi-material interfaces, which employs displacements and weighted tractions as primary variables. However, in the present work, the characteristic oscillating stress singularity is addressed through the introduction of complex weighting functions for both displacements and tractions, along with corresponding non-standard numerical quadrature formulas. As a result, this boundary-only approach provides extremely accurate mesh-insensitive solutions for a range of two-dimensional interface crack problems. A number of computational examples are considered to assess the performance of the method in comparison with analytical solutions and previous work on the subject. As a final application, the method is applied to study the scaling behavior of epoxy–metal butt joints.     

Dynamic behavior of two unequal parallel permeable interface cracks in a piezoelectric layer bonded to two half piezoelectric materials planes

IntroductionDuetotheintrinsicelectro_mechanicalcouplingbehavior,piezoelectricmaterialsareveryusefulinelectronicdevices.However,mostpiezoelectricmaterialsarebrittlesuchasceramicsandcrystals.Therefore ,piezoelectricmaterialshaveatendencytodevelopcriticalcracksduringthemanufacturingandthepolingprocesses.So ,itisimportanttostudytheelectro_elasticinteractionandfracturebehaviorsofpiezoelectricmaterials.Theincreasingattentiontothestudyofcrackproblemsinpiezoelectricmaterialshasledtoalotofsignificantw…     

Analysis of a partially debonded elliptic inhomogeneity in piezoelectric materials

A generalized solution was obtained for the partially debonded elliptic inhomogeneity problem in piezoelectric materials under antiplane shear and inplane electric loading using the complex variable method. It was assumed that the interfacial debonding induced an electrically impermeable crack at the interface. The principle of conformal transformation and analytical continuation were employed to reduce the formulation into two Riemann-Hilbert problems. This enabled the determination of the complex potentials in the inhomogeneity and the matrix by means of series of expressions. The resulting solution was then used to obtain the electroeiastic fields and the energy release rate involving the debonding at the inhomogeneity-matrix interface. The validity and versatility of the current general solution have been demonstrated through some specific examples such as the problems of perfectly bonded elliptic inhomogeneity , totally debonded elliptic inhomogeneity, partially debonded rigid and conducting elliptic inhomogeneity, and partially debonded circular inhomogeneity.     

POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS(Ⅰ)--SEVERAL EXACT SOLUTIONS

For the orthotropic piezoelectric plane problem, a series of piezoelectric beams is solved and the corresponding exact solutions are obtained with the trial-anderror method on the basis of the general solution in the case of three distinct eigenvalues, in which all displacements, electrical potential, stresses and electrical displacements are expressed by three displacement functions in terms of harmonic polynomials. These problems are rectangular beams having rigid body displacements and identical electrical potential, rectangular beams under uniform tension and electric displacement as well as pure shearing and pure bending, beams of two free ends subjected to uniform electrical potential on the upper and lower surfaces.     

2D dynamic analysis of cracks and interface cracks in piezoelectric composites using the SBFEM

The dynamic stress and electric displacement intensity factors of impermeable cracks in homogeneous piezoelectric materials and interface cracks in piezoelectric bimaterials are evaluated by extending the scaled boundary finite element method (SBFEM). In this method, a piezoelectric plate is divided into polygons. Each polygon is treated as a scaled boundary finite element subdomain. Only the boundaries of the subdomains need to be discretized with line elements. The dynamic properties of a subdomain are represented by the high order stiffness and mass matrices obtained from a continued fraction solution, which is able to represent the high frequency response with only 3–4 terms per wavelength. The semi-analytical solutions model singular stress and electric displacement fields in the vicinity of crack tips accurately and efficiently. The dynamic stress and electric displacement intensity factors are evaluated directly from the scaled boundary finite element solutions. No asymptotic solution, local mesh refinement or other special treatments around a crack tip are required. Numerical examples are presented to verify the proposed technique with the analytical solutions and the results from the literature. The present results highlight the accuracy, simplicity and efficiency of the proposed technique.     

Interaction between a screw dislocation and a viscoelastic piezoelectric bimaterial interface

The complex variable method is employed to derive analytical solutions for the interaction between a piezoelectric screw dislocation and a Kelvin-type viscoelastic piezoelectric bimaterial interface. Through analytical continuation, the original boundary value problem can be reduced to an inhomogeneous first-order partial differential equation for a single function of location z = x + iy and time t defined in the lower half-plane, which is free of the screw dislocation. Once the initial, steady-state and far-field conditions are known, the solution to the first order differential equation can be obtained. From the solved function, explicit expressions are then derived for the stresses, strains, electric fields and electric displacements induced by the piezoelectric screw dislocation. Also presented is the image force acting on the screw dislocation due to its interaction with the Kelvin-type viscoelastic interface. The derived solutions are verified by comparing with existing solutions for the simplified cases, and various interesting features are observed, particularly for those associated with the image force.