Green's functions of two-dimensional piezoelectric quasicrystal in half-space and bimaterials

In this paper,we obtain Green’s functions of two-dimensional(2D)piezoelectric quasicrystal(PQC)in half-space and bimaterials.Based on the elastic theory of QCs,the Stroh formalism is used to derive the general solutions of displacements and stresses.Then,we obtain the analytical solutions of half-space and bimaterial Green’s functions.Besides,the interfacial Green’s function for bimaterials is also obtained in the analytical form.Before numerical studies,a comparative study is carried out to validate the present solutions.Typical numerical examples are performed to investigate the effects of multi-physics loadings such as the line force,the line dislocation,the line charge,and the phason line force.As a result,the coupling effect among the phonon field,the phason field,and the electric field is prominent,and the butterfly-shaped contours are characteristic in 2D PQCs.In addition,the changes of material parameters cause variations in physical quantities to a certain degree.

ANALYSIS OF CRACK-TIP SINGULARITIES FOR AN INTERFACIAL PERMEABLE CRACK IN METAL/PIEZOELECTRIC BIMATERIALS

By modeling metal as a special piezoelectric material with extremely small piezoelectricity and extremely large permittivity, we have obtained the analytical solutions for an interracial permeable crack in metal/piezoelectric bimaterials by means of the generalized Stroh formalism. The analysis shows that the stress fields near a permeable interracial crack tip are usually with three types of singularities： r^-1/2±iε and r^-1/2. Further numerical calculation on the oscillatory index ε are given for 28 types of metal/piezoelectric bimaterials combined by seven commercial piezoelectric materials： PZT-4, BaTiO3, PZT-5H, PZT-6B, PZT-7A, P-7 and PZT-PIC 151 and four metals： copper, silver, lead and aluminum, respectively. The explicit expressions of the crack tip energy release rate （ERR） and the crack tip generalized stress intensity factors （GSIF） are obtained. It is found that both the ERR and GSIF are independent of the electric displacement loading, although they seriously depends on the mechanical loadings.

SINGULAR BEHAVIORS OF INTERFACIAL CRACKS IN 2D MAGNETOELECTROELASTIC BIMATERIALS

The singular characteristics of stress, electric displacement and magnetic induction fields near the tip of impermeable interracial cracks in two-dimensional magnetoelectroelastic bimaterials are studied using the generalized Stroh formalism. Two types of singularities are obtained： one is the oscillating singularity 1/2±iε, the other is the non-oscillating singularity 1/2±κ. It is found that the non-zero parameters ε and κ cannot coexist for one transversely isotropic MEE bimaterial, a similar result is obtained for transversely isotropic piezoelectric bimaterials.

Line-integral representations for extended displacements, stresses,and interaction energy of arbitrary dislocation loops in transversely isotropic magneto-electro-elastic bimaterials

In addition to the hexagonal crystals of class 6 mm, many piezoelectric materials （e.g., BaTiO3）, piezomagnetic materials （e.g., CoFe2O4）, and multiferroic com-posite materials （e.g., BaTiO3-CoFe2O4 composites） also exhibit symmetry of transverse isotropy after poling, with the isotropic plane perpendicular to the poling direction. In this paper, simple and elegant line-integral expressions are derived for extended displace-ments, extended stresses, self-energy, and interaction energy of arbitrarily shaped, three-dimensional （3D） dislocation loops with a constant extended Burgers vector in trans-versely isotropic magneto-electro-elastic （MEE） bimaterials （i.e., joined half-spaces）. The derived solutions can also be simply reduced to those expressions for piezoelectric, piezo-magnetic, or purely elastic materials. Several numerical examples are given to show both the multi-field coupling effect and the interface/surface effect in transversely isotropic MEE materials.

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 prop- erty. 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.

Interaction between subinterface cracks and interface in metal/piezoelectric ceramic bimaterials

A class of problems concerning subinterface cracks interacting with the interface in metal/piezoelectric ceramic bimaterials are studied. The interaction problem is reduced to a system of integral equations with the aid of the pseudo-traction-electric-displacement method.The equations are solved numerically,and the stress intensity factor,the electric displacement intensity factor and the mechanical strain energy release rate are evaluated.Numerical results of a Cu/PZT-4 bimaterial are given and shown in figures,in which three kinds of remote loading conditions are considered to make comparisons.It is found that the electric loading at infinity plays a quite important role in the present interaction problem.In addition,a conservation law of the first component of the Jk-integral vector is found,which does lead to a consistency check to confirm the effectiveness of the PTEDM as well as the numerical results derived in this paper.

A moving screw dislocation near interfacial rigid lines in two dissimilar anisotropic media

This paper attempts to investigate the problem for the interaction between a uniformly moving screw dislocation and interface rigid lines in two dissimilar.anisotropic. materials. Integrating Riemann-Schwarz＇s symmetry principle with the analysis singularity of complex functions, we present the general elastic solutions of this problem and the closed form solutions for interfaces containing one and two rigid lines. The expressions of stress intensity factors, at the rigid line tips and image force acting on moving dislocation are derived explicitly. The results show that dislocation velocity has an antishielding effect on the rigid line tip and a larger dislocation velocity leads to the equilibrium position of dislocation closing with the rigid line. The presented solutions contain previously known results as the special cases.

SCATTERING OF SH-WAVES BY AN INTERACTING INTERFACE LINEAR CRACK AND A CIRCULAR CAVITY NEAR BIMATERIAL INTERFACE

An analytical method is developed for scattering of SH-waves and dynamic stressconcentration by an interacting interface crack and a circular cavity near bimaterial interface.Asuitable Green’s function is contructed,which is the fundamental solution of the displacement fieldfor an elastic half space with a circular cavity impacted by an out-plane harmonic line source loadingat the horizontal surface.First,the bimaterial media is divided into two parts along the horizontalinterface,one is an elastic half space with a circular cavity and the other is a complete half space.Then the problem is solved according to the procedure of combination and by the Green’s functionmethod.The horizontal surfaces of the two half spaces are loaded with undetermined anti-plane forcesin order to satisfy continuity conditions at the linking section,or with some forces to recover cracks bymeans of crack-division technique.A series of Fredholm integral equations of first kind for determiningthe unknown forces can be set up through continuity conditions as expressed in terms of the Green’sfunction.Moreover,some expressions are given in this paper,such as dynamic stress intensity factor(DSIF)at the tip of the interface crack and dynamic stress concentration factor(DSCF)around thecircular cavity edge.Numerical examples are provided to show the influences of the wave numbers,the geometrical location of the interface crack and the circular cavity,and parameter combinations ofdifferent media upon DSIF and DSCF.

Experimental and numerical study of coal-rock bimaterial composite bodies under triaxial compression

To accurately predict coal burst hazards and estimate the failure of coal pillars in underground coal mining systems,it is of great significance to understand the mechanical behavior of coal-rock bimaterial composite structures.This paper presents experimental and numerical investigations on the response of rock-coal,coal-rock,and rock-coal-rock bimaterial composite structures under triaxial compression.The triaxial compression experiments are conducted under confining pressures in the range of 0-20 MPa.The resulting inside fracture networks are detected using X-ray-based computed tomography(CT).The experimentally observed data indicate that the mechanical parameters of the rock-coalrock composites are superior to those of the rock-coal and coal-rock combinations.After compression failure,the coal-rock combination specimens are analyzed via X-ray CT.The results display that the failure of the coal-rock composite bodies primarily takes place within the coal.Further,the bursting proneness is reduced by increasing confining pressure.Subsequently,the corresponding numerical simulations of the experiments are carried out by using the particle flow code.The numerical results reveal that coal is vulnerable with regard to energy storage and accumulation.

Evaluation of Stress Intensity Factors Subjected to Arbitrarily Distributed Tractions on Crack Surfaces

The stress intensity factors （SIF） considering arbitrarily distributed surface tractions are evaluated based on the sealed boundary finite element method （SBFEM）. The semi-analytical solving process for the stress intensity factors including the effects of surface tractions is presented. Provided are the numerical examples for the evaluation of mode I and Ⅱ stress intensity factors with linear and non-linear distributing forces loaded on the crack surfaces. The crack problems of anisotropy and bimaterial interface are also studied and the stress intensity factors of single-edge-cracked orthotropic material and bi-material interface problems with surface tractions are calculated. Comparisons with the analytical solutions show that the proposed approach is effective and possesses high accuracy.

Finite boundary effects in problem of a crack perpendicular to and terminating at a bimaterial interface

In this paper, the problem of a crack perpendicular to and terminating at an interface in bimaterial structure with finite boundaries is investigated. The dislocation simulation method and boundary collocation approach are used to derive and solve the basic equations. Two kinds of loading form are considered when the crack lies in a softer or a stiffer material, one is an ideal loading and the other one fits to the practical experiment loading. Complete solutions of the stress field including the T stress are obtained as well as the stress intensity factors. Influences of T stress on the stress field ahead of the crack tip are studied. Finite boundary effects on the stress intensity factors are emphasized. Comparisons with the problem presented by Chen et al. (Int. J. Solids and Structure, 2003, 40, 2731–2755) are discussed also.

FURTHER INVESTIGATION OF THE J_2-INTEGRAL IN BIMATERIAL SOLIDS

The J_2-integral induced from the interface of bimaterial solids(J_2^(interface))is stud- ied by numerical method.First,the effect on the J_2-integral induced from the interface is very significant in bimaterial solids,which is inherently related to that induced from the subinterface cracks.Moreover,it can be concluded that either the first or the second component of the J_k- vector is always equal to zero when the contour encloses both the cracks and the whole interface in bimaterial solids.Secondly,it can also be concluded that the interface does produce significant effect on the J_2-integral induced from the subinterface cracks(J_2^(sub))in bimaterial solids.This effect depends on the geometry of the crack arrangement,which is corresponding to the different interaction effect among the cracks and the interface.Moreover,the interface effect on the J_2^(sub) can be neglected when the distance from the crack center to the interface is large enough,which reveals that the bimaterial solids can be regarded as homogenous solids in fracture analysis when the subinterface crack is far enough from the interface.Three examples are given in this paper.

THE ELASTIC FIELD CASED BY TWO INCLUSIONS WITH UNI- FORM DILATATIONAL EIGENSTRAINS IN DISSIMILAR MEDIA

The displacement and the stress states cased by single inclusion are achieved from the fundamental solutions such as nuclei of strain in bimaterals. The elastic field induced by multiple inclusions in dissimilar media could be found from the superstition of that of individual precipitate. In this paper, the effect of the planner interface with parameters of depth from the interface, both pairs of elastic moduli and also shapes of the inclusion are also given, which are of great significance in physical applications.

FURTHER INVESTIGATION OF INTERACTION BETWEEN INTERFACE MACROCRACK AND PARALLEL MICROCRACKS IN BIMATERIAL ANISOTROPIC SOLIDS

In this paper, with the aid of superimposing technique and the Pseudo Traction Method (PTM), the interaction problem between an interface macrocrack and parallel microcracks in the process zone in bimaterial anisotropic solids is reduced to a system of integral equations. After the integral equations are solved numerically, a conservation law among three kinds ofJ-integrals is obtained which are induced from the interface macrocrack tip, the microcrack and the remote field, respectively. This conservation law reveals that the microcrack shielding effect in such materials could be considered as the redistribution of the remoteJ-integral.

Analysis on non-oscillatory singularity behaviors of mode Ⅱ interface crack tip in orthotropic bimaterial

The fracture behaviors near the mode II interface crack tip for orthotropic bimaterial are studied. The non-oscillatory field, where the stress singularity exponent is a real number, is discussed by the complex function method and the undetermined coefficient method. From the research fracture problems, the stress functions with ten undetermined coefficients and an unknown singularity exponent are introduced when △1 〉 0 and △2 〉 0. By the existence theorem of non-trival solutions for the system of eight homogeneous linear equations, the characteristic equation, the stress singularity exponent, and the discriminating condition of the non-oscillatory singularity are found. By the uniqueness theorem of the solutions for the system of twelve non-homogeneous linear equations with ten unknowns, the ten undermined coefficients in the stress functions are uniquely determined. The definitions of the stress intensity factors are given with the help of one-sided limit, and their theoretical formulae are deduced. The analytic solutions of the stresses near the mode II interface crack tip are derived. The classical results for orthotropic material are obtained.

Modeling core-spreading of interface dislocation and its elastic response in anisotropic bimaterial

Interracial dislocation may have a spreading core corresponding to a weak shear resistance of interfaces. In this paper, a conic model is proposed to mimic the spreading core of interfacial dislocation in anisotropic bimaterials. By the Stroh formalism and Green＇s function, the analytical expressions of the elastic fields are deduced for such a dislocation. Taking Cu/Nb bimaterial as an example, it is demonstrated that the accuracy and efficiency of the method are well validated by the interface conditions, a spreading core can greatly reduce the stress intensity near the interfacial dislocation compared with the compact core, and the elastic fields near the spreading core region are significantly different from the condensed core, while they are less sensitive to a field point that is 1.5 times the core width away from the center of the spreading core.

AN INVESTIGATION ON STRESS INTENSITY FACTORS OF INTERFACE CRACK FOR THREE TYPES OF SPECIMENS

The problems of finite bimaterial plates, hearing uniform tension, compact: tension and three point bending, are studied by using the eigenfunction expansion variation method (EEVM). And interfacial stress intensity factors (SIFs) are determined. The SIFs varying with shear modulus mu and Poisson's ratios nu of both materials are discussed.

AN EXPLICIT TENSOR EXPRESSION FOR THE FUNDAMENTAL SOLUTIONS OF A BIMATERIAL SPACE PROBLEM

In this paper, by using the method of tensor operation, the fundamental solutions, given in the references listed, for a concentrated force in a three-dimensional biphase-infinite solid were expressed in the tensor form, which enables them to be directly applied to the boundary integral equation and the boundary element method for solving elastic mechanics problems of the bimaterial space. The fundamental solutions for Mindlin's problem, Lorentz's problem and homogeneous space problem are involved in the present results.

CRACK KINKING IN PIEZOELECTRIC MATERIALS

The solution for a class of two-dimensional electro- elastic branched crack problems was presented. Explicit Green's function for an interface crack subjected to an edge dislocation was developed using the extended Stroh formulation, which allows the branched crack problem to be expressed in terms of coupled singular integral equations. The kink was modelled as the continuous distribution of edge dislocations and the dislocation density function was defined on the line of the branch crack only. Competition between crack extension along the interface and kinking into the substrate was investigated using the integral equations and the maximum energy release rate criterion. The numerical results have shown the effect of electric field on the path of crack extension. (Edited author abstract) 24 Refs.

Calculation of transient dynamic stress intensity factors at bimaterial interface cracks using a SBFEM-based frequency-domain approach

A frequency-domain approach based on the semi-analytical scaled boundary finite element method(SBFEM) was developed to calculate dynamic stress intensity factors(DSIFs) at bimaterial interface cracks subjected to transient loading.Be-cause the stress solutions of the SBFEM in the frequency domain are analytical in the radial direction,and the complex stress singularity at the bimaterial interface crack tip is explicitly represented in the stress solutions,the mixed-mode DSIFs were calculated directly by definition.The complex frequency-response functions of DSIFs were then used by the fast Fourier transform(FFT) and the inverse FFT to calculate time histories of DSIFs.A benchmark example was modelled.Good re-sults were obtained by modelling the example with a small number of degrees of freedom due to the semi-analytical nature of the SBFEM.