Strip-electromechanical model solution for piezoelectric plate cut along two semi-permeable collinear cracks

A strip electric saturation and mechanical yielding model solution is proposed for a piezoelectric plate cut along two equal collinear semi-permeable mode-I cracks with electrical polarization reaching a saturation limit and normal stress reaching a yield stress along a line segment in front of the cracks. By using Stroh formalism and complex variable technique, we derived the analytical solution for the field quantities. Three different situations are investigated when developed electrical saturation zone is bigger/smaller or equal to the developed mechanical yield zone. Numerical results show that the effect of different electric boundary conditions on the crack opening displacement and crack opening potential drop is significant and should not be ignored. The influence of electric load displacement on the energy release rate is also investigated for PZT-4, PZT-5H and BaTiO₃ ceramics, and it may assists for the correct choosing of ceramic for specific job.     

Strip-coalesced interior zone model for two unequal collinear cracks weakening piezoelectric media

In this paper, a mathematical strip-saturation model is proposed for a poled transversely isotropic piezoelectric plate weakened by two impermeable unequal-collinear hairline straight cracks. Remotely applied in-plane unidirectional electromechanical loads open the cracks in mode-I such that the saturation zone developed at the interior tips of cracks gets coalesced. The developed saturation zones are arrested by distributing over their rims in-plane normal cohesive electrical displacement. The problem is solved using the Stroh formalism and the complex variable technique. The expressions are derived for the stress intensity factors （SIFs）, the lengths of the saturation zones developed, the crack opening displacement （COD）, and the energy release rate. An illustrative numerical case study is presented for the poled PZT-5H ceramic to investigate the effect of prescribed electromechanical loads on parameters affecting crack arrest. Also, the effect of different lengths of cracks on the SIFs and the local energy release rate （LERR） has been studied. The results obtained are graphically presented and analyzed.     

Closed-form solution for two collinear cracks in a piezoelectric strip

Under the permeable electric boundary condition, the problem of two collinear anti-plane shear cracks lying at the mid-plane of a piezoelectric strip is investigated. By using the Fourier transform, the associated problem is reduced to a singular integral equation. Solving the resulting equation analytically, the electro-elastic field intensity factors and energy release rates at the inner and outer crack tips can be determined in explicit form. Numerical results for PZT-5H piezoelectric ceramic are also presented graphically. The results reveal that the effect of electric field on crack growth in piezoelectric materials is dependent on applied elastic displacement.     

Closed-form solution for a piezoelectric strip with two collinear cracks normal to the strip boundaries

The problem of two collinear cracks of equal length and normal to the strip boundaries in an infinitely long piezoelectric strip of finite width is analyzed. By using the Fourier series method, the mixed boundary value problem is reduced to triple series equations, which are then transformed to a singular integral equation. For four combined cases of uniform antiplane shear and uniform inplane electric loading at infinity, the solution is obtained in closed-form, and explicit expressions for the electroelastic field are determined. The formulae for calculating the intensity factors of the electroelastic field and the energy release rate at the inner and outer crack tips are given, respectively. Some special cases for the electroelastic field intensity factors and the energy release rate of the present results are discussed.     

Modified strip saturated models for two equal collinear cracks with coalesced zones in piezoelectric media

Two equal collinear cracks with coalesced interior electric saturation zones are analytically studied for two-dimensional(2D) arbitrary polarized semipermeable piezoelectric media based on a modified strip saturated model. The strip saturated model is modified here by varying the strip saturated constant electric displacement condition to the polynomially varying electric displacement conditions. Based on the linear, quadratic,and cubic electric displacement conditions on the inner and outer saturated zones, different modified strip saturated models are proposed and studied for two equal collinear cracks. With the Stroh formalism and the complex variable technique, these fracture problems are reduced into different types of non-homogeneous Riemann Hilbert problems in unknown generalized complex potential functions. These mathematical problems are then solved with the Riemann-Hilbert approach to obtain the stress and electric displacement components at any point of the domain. The explicit expressions for the outer saturated zone length, the crack opening potential(COP), the crack opening displacement(COD), and the local intensity factors(LIFs) are derived. A numerical study is presented for the modified strip saturated model in 2D arbitrary polarized semipermeable PZT-4 material. The obtained results are compared with those of the strip saturated model, and the effects of the polynomially varying saturation condition on the saturated zones and the applied electrical loadings are presented.     

Generalized 2D problem of piezoelectric media containing collinear cracks

The generalized 2D problem in piezoelectric media with collinear cracks is addressed based on Stroh's formulation and the exact electric boundary conditions on the crack faces. Exact solutions are obtained, respectively, for two special cases: one is that a piezoelectric solid withN collinear cracks is subjected to uniform loads at infinity, and the other is that a piezoelectric solid containing a single crack is subjected to a line load at an arbitrary point. It is shown when uniform loads are applied at infinity or on the crack faces that, the stress intensity factors are the same as those of isotropic materials, while the intensity factor of electric displacement is dependent on the material constants and the applied mechanical loads, but not on the applied electric loads. Moreover, it is found that the electric field inside any crack is not equal to zero, which is related to the material properties and applied mechanical-electric loads. The project supported by the National Natural Science Foundation of China (19772004)     

Exact solutions of two semi-infinite collinear cracks in piezoelectric strip

Using the complex variable function method and the conformal mapping technique,the fracture problem of two semi-infinite collinear cracks in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load on the partial crack surface.Analytic solutions of the field intensity factors and the mechanical strain energy release rate are derived under the assumption that the surfaces of the crack are electrically impermeable.The results can be reduced to the well-known solutio...     

DYNAMIC BEHAVIOR OF TWO COLLINEAR PERMEABLE CRACKS IN A PIEZOELECTRIC LAYER BONDED TO TWO HALF SPACES

IntroductionIn the fracture mechanics studies for piezoelectric materials,differently electricboundary conditions at the crack surfaces have been proposed by many researchers.Forexample,for the sake of analytical simplification,the assumption that the cra…     

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…     

Solutions to a limited-permeable crack or two limited-permeable collinear cracks in piezoelectric/piezomagnetic materials

The solutions of a limited-permeable crack (case I) or two collinear limited-permeable cracks (case II) in piezoelectric/piezomagnetic materials subjected to a uniform tension loading were investigated in this paper using the generalized Almansi’s theorem. At the same time, the electric permittivity and the magnetic permeability of air in crack were firstly considered. Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables were jumps of displacements across crack surfaces, not the dislocation density functions or the complex variable functions. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials to obtain the relations among electric displacement intensity factors, magnetic flux intensity factors and stress intensity factors at crack tips.     

一维六方压电准晶对称条形体中共线双半无限快速传播裂纹的解析解

利用复变函数方法, 通过构造保角映射, 分析了一维六方压电准晶中对称条形体中共线双半无限快速传播裂纹的反平面问题, 给出了在电非渗透型与电渗透型两种情况下动态应力强度因子的解析解. 当速度趋于0时可退化为已有的对应的静力问题的解. 得到的解析解对准晶材料的工程应用中有一定的理论价值.     

Nonlocal theory solution of two collinear cracks in the functionally graded materials

In this paper, the interaction of two collinear cracks in functionally graded materials subjected to a uniform anti-plane shear loading is investigated by means of nonlocal theory. The traditional concepts of the nonlocal theory are extended to solve the fracture problem of functionally graded materials. To make the analysis tractable, it is assumed that the shear modulus varies exponentially with the coordinate vertical to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near the crack tips. The nonlocal elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials. The magnitude of the finite stress field depends on the crack length, the distance between two cracks, the parameter describing the functionally graded materials and the lattice parameter of the materials.     

Asymptotic elastic solution for two straight cracks of arbitrary length and location

An infinite elastic plane containing two straight cracks of arbitrary length and location is analyzed within the framework of elastostatics. The mathematical formulation is based on the stress solution for a single crack and leads to a system of singular integral equations that govern the crack surface displacement densities. The solution series in terms of the reciprocal of the crack centre distance is not suitable for cracks that are spaced too closely. It is shown by way of examples that the method of asymptotic solution is convenient for developing approximation expressions of the stress and displacement field with certain characteristics. The formulas for the stress intensity factors and crack opening are given for the case of a constant tensile load. Graphical results are given for the variations of the stress intensity factors with parameters depending on the relative positions of the cracks.     

Closed-form solutions for two anti-plane collinear cracks in a magnetoelectroelastic layer

In this paper we develop closed form solutions for anti-plane mechanical and in-plane electric and magnetic fields for two collinear cracks in magneto-electro-elastic layer of finite thickness under the conditions of permeable crack faces using integral transform method. The anti-plane mechanical shear or displacement and in-plane electrical and magnetic loading are applied to the top and bottom surfaces of the layer for the two cases considered. Expressions for shear stresses, electric displacements and magnetic inductions in the vicinity of the cracks are derived as well as intensity factors for two cracks in magneto-electro-elastic layer. Numerical results for stress intensity factors and energy release rate are shown graphically.     

Interaction between collinear periodic cracks in an infinite piezoelectric body

The problem of collinear periodic cracks in an infinite piezoelectric body is studied. Effect of saturation strips at the crack-tips is taken into account. By means of the Stroh formalism and the conformal mapping technique, the general periodic solutions for collinear cracks are obtained. The stress intensity factors and the size of saturation strips are derived analytically, and their dependencies on the ratio of the periodicity on the half-length of the crack are analyzed in detail. Numerical results show the following two facts. (1) When h/l>4.0, the stress intensity factors become almost identical to those of a single crack in an infinite piezoelectric body. This indicates that the interaction between cracks can be ignored in establishing the criterion for the crack initiation in this case. (2) The speed of the saturation strip size of periodic cracks approaching that of a single crack depends on the electric load applied at infinity. In general, a large electric load at infinity is associated with a slow approaching speed.     

Closed-form solution for two collinear mode-III cracks in an orthotropic elastic strip of finite width

The problem of an orthotropic strip containing two collinear cracks normal to the strip boundaries is considered. The Fourier series method is used to reduce the associated boundary value problem to triple series equations, then to a singular integral equation, which can be solved analytically. Under remote uniform antiplane shear loading, the stress field and the crack sliding displacement are determined analytically and stress intensity factors are also given in a closed form.     

Crack bifurcation predicted for dynamic anti-plane collinear cracks in piezoelectric materials using a non-local theory

A non-local theory of elasticity is applied to obtain the dynamic interaction between two collinear cracks in the piezoelectric materials plane under anti-plane shear waves for the permeable crack surface boundary conditions. Unlike the classical elasticity solution, a lattice parameter enters into the problem that make the stresses and the electric displacements finite at the crack tip. A one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and electric displacement near the crack tips. By means of the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations in which the unknown variable is the jump of the displacement across the crack surface. The solutions are obtained by means of the Schmidt method. Crack bifurcation is predicted using the strain energy density criterion. Minimum values of the strain energy density functions are assumed to coincide with the possible locations of fracture initiation. Bifurcation angles of ±5° and ±175° are found. The result of possible crack bifurcation was not expected before hand.     

Basic solution of two parallel non-symmetric permeable cracks in piezoelectric materials

The behavior of two parallel non-symmetric cracks in piezoelectric materials subjected to the anti-plane shear loading was studied by the Schmidt method for the permeable crack electric boundary conditions. Through the Fourier transform, the present problem can be solved with two pairs of dual integral equations ip which the unknown variables are the jumps of displacements across crack surfaces. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials. Finally, the relations between electric displacement intensity factors and stress intensity factors at crack tips can be obtained. Numerical examples are provided to show the effect of the distance between two cracks upon stress and electric displacement intensity factors at crack tips. Contrary to the impermeable crack surface condition solution, it is found that electric displacement intensity factors for the permeable crack surface conditions are much smaller than those for the impermeable crack surface conditions. At the same time, it can be found that the crack shielding effect is also present in the piezoelectric materials.     

Dynamic behavior of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips

The dynamic interaction of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips subjected to the anti-plane shear harmonic stress waves was investigated. By using the Fourier transform, the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is jump of displacement across the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter, the circular frequency of the incident waves and the thickness of the strip upon stress, electric displacement and magnetic flux intensity factors of cracks.