Extended Displacement Discontinuity Research Articles

Thermoelastic fracture of two-dimensional hexagonal quasicrystal media weakened by a penny-shaped crack subjected to uniformly antisymmetric heat fluxes

This study focuses on the thermoelastic fracture behavior of quasicrystals, a class of solids that exhibit unique properties between traditional crystals and amorphous materials. The complex atomic structure of quasicrystals poses challenges in understanding their fracture mechanisms under thermoelastic loading conditions. Central to our investigation is an analytical examination of a penny-shaped crack in an infinite three-dimensional body composed of two-dimensional hexagonal quasicrystals, where the upper and lower crack surfaces are applied to a pair of uniformly antisymmetric heat fluxes. This crack problem requires simultaneous consideration of thermal-phason-phonon multiple field coupling. According to the symmetry of field variables with respect to the crack plane, the thermal-phason-phonon coupled crack problem is transformed into a mixed boundary value problem in the upper half space. The extended displacement discontinuities, encompassing both phonon and phason displacement discontinuities, as well as the temperature discontinuity, are chosen as the basic unknown variables to construct the boundary integral-differential equations governing the mixed boundary value problem. Based on these boundary governing equations and Fabrikant's potential theory method, the problem with the crack surface subjected to uniform antisymmetric heat fluxes is solved. The solutions of thermal-phason-phonon fields on the crack plane, and in the full space are given in closed-form. Numerical results are employed to validate the obtained analytical solutions and visually illustrate the spatial distribution of thermal-phonon-phason coupling fields in the vicinity of the crack. The study provides fundamental insights into the behavior of cracks in quasicrystals under thermal loading, with potential implications for the design of new materials and structures.

Fracture Analysis of Planar Cracks in 3D Thermal Piezoelectric Semiconductors

The study of piezoelectric semiconductor (PSC) materials and devices is experiencing rapid growth, giving rise to a burgeoning research domain known as piezotronics. Understanding the fracture behavior of PSC materials is essential for designing robust devices and ensuring their long-term functionality. A planar crack of arbitrary shape in the isotropic plane of a three-dimensional (3D) transversely isotropic thermal PSC body subjected to combined thermal-electrical-mechanical loadings is studied via the extended displacement discontinuity (EDD) boundary integral equation method. Under the framework of linearized basic equations and one-way thermal coupling, the hypersingular boundary integral equations with the volume integrals containing the temperature and carrier concentration are obtained and expressed by the EDDs which include displacement, electric potential, carrier concentration and temperature discontinuities across the crack surfaces and are the basic variables in analyzing crack problems. Based on the finite part of hypersingular integrals, the EDDs near the crack border are proved to have the classical behavior of O(r). Furthermore, the asymptotic solutions of extended stresses including the stress, electric displacement, current density and heat flux at the crack front are presented and display classical singular behavior of O(1/r). The extended stress intensity factors are defined and given in terms of EDDs. A 3D numerical model of a penny-shaped crack in the thermal PSC cylinder is established and used to verify the presented formulae.

Analysis of a penny-shaped crack with semi-permeable boundary conditions across crack face in a 3D thermal piezoelectric semiconductor

In this paper, we study a penny-shaped crack model with electrically and thermally semi-permeable boundary conditions in a three-dimensional transversely isotropic piezoelectric semiconductor. An extended displacement discontinuity boundary element method together with an iterative process is proposed to analyze the penny-shaped crack model. The extended displacement discontinuities across crack face, electric displacement and heat flux along an inner crack cavity, as well as extended stress intensity factors near crack front are obtained via the proposed method. The effects on extended intensity factors near crack front are discussed, including boundary conditions across crack face, applied loads and initial electron concentration. It is shown that boundary conditions across crack face significantly affect extended stress intensity factors near crack front. This implies that a larger initial electron concentration can lead to electrical failure.

Seismic Wave Conversion Caused by Shear Crack Initiation and Growth

In laboratory tests, in rock, elastic wave conversion occurs when shear cracks initiate. In the experiments, the samples are continuously probed with seismic waves during loading that caused shear crack initiation and growth. The initiation of shear cracks is not detected with either transmitted P- or S-waves across the crack. However, shear (S) waves, propagated at normal or oblique incidence on an emerging shear crack, are partially partitioned into compressional (P) waves. The emergence of S–P converted modes is theoretically predicted to arise from wave interaction with an array of similarly oriented microcracks. Wave conversions occur before the initiation of shear cracks can be observed on the rock surface using direct imaging measurements. As shear cracks propagate, until crack coalescence, the converted waves amplitude increases. The trend of changes in amplitude of the S–P waves with respect to load for both normally and obliquely incident shear waves on a shear crack is consistent with the extended displacement discontinuity theory. The consistent observation in all the tests of emergent S–P waves during crack initiation and the detection of peak amplitude prior to crack coalescence provides a basis for the development of seismic monitoring techniques to characterize shear crack initiation, crack growth, and crack coalescence.

Analysis of 3D planar crack problems in one-dimensional hexagonal piezoelectric quasicrystals with thermal effect. part I: Theoretical formulations

Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are performed systematically. Part I of this work derives a series of theoretical formulations that are then used to study the 3D planar crack problems in the QCs. The simple layer potential functions with the extended displacement discontinuities (EDDs) as the unknown variables and the general solution based on quasi-harmonic functions for the QCs under consideration are used to deduce the boundary equations that govern 3D planar crack problems. The hypersingular integral equation method is used to analyze the asymptotic singularities of the coupled thermal-electrical-phonon-phason fields near the crack edge. Expressions are then presented for the extended stress intensity factors (ESIFs) of a mixed model crack in terms of EDDs for arbitrarily-shaped cracks in the QCs, and the basic relationships between the energy release rate and the ESIFs are established. Closed-form solutions for some typical cracks, including an elliptical crack that is subjected to coupled electrical-phonon-phason loadings and a penny-shaped crack that is subjected to antisymmetric thermal loading, are determined via Fabrikant's analysis method. Additionally, both the physical quantities on the crack plane and the corresponding variables in the coupled thermal-electrical-phonon-phason field in the full space are given. The theoretical formulations derived in this paper provide a fundamental basis for development of the numerical approach proposed in Part II of our work, and can also serve as benchmarks for numerical solutions.

Analysis of 3D planar crack problems of one-dimensional hexagonal piezoelectric quasicrystals with thermal effect. part II: Numerical approach

Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are carried out systematically. Part II of the work aims to develop a general numerical approach to study 3D planar crack problems in 1D hexagonal piezoelectric QC media. Based on the theoretical formulations presented in Part I, a direct and effective method is proposed to derive the Green's functions for point extended displacement discontinuities (EDDs). These Green's functions are presented explicitly by a series of potential functions in a compact form. Using the superposition principle, the Green's functions for uniformly distributed EDDs over the crack elements are obtained. Related element solutions are used to construct the numerical approach, known as EDD boundary element method, for 1D hexagonal piezoelectric QCs. The proposed numerical method can be applicative to many complicated planar crack problems, such as multiple cracks, and cracks with non-uniform loadings, for 3D media composed of 1D hexagonal piezoelectric QCs with thermal effect. A comparison of the results obtained from the theoretical solutions given in Part I of the work with those obtained from the numerical method proposed here validates of the present investigation.

Analysis of elastic−plastic fracture in two-dimensional decagonal quasicrystals using the displacement discontinuity method

The generalized Dugdale model of cracks in two-dimensional decagonal quasicrystals is studied by developing the extended displacement discontinuity boundary element method. Expressions for the phonon and phason stress intensity factors and energy release rate are given in terms of the phonon and phason displacement discontinuities near the crack front. The extended displacement discontinuity boundary element method, with an iterative approach of determining the size of the yielding zone, is adopted to numerically simulate the nonlinear fracture problem. The effects of phonon and phason loadings, as well as the yield stress on the size of yielding zone and energy release rate, are discussed.

Fundamental solutions and analysis of the interface crack for two-dimensional decagonal quasicrystal bimaterial via the displacement discontinuity method

The extended displacement discontinuity method is developed to analyze an interface line crack in a two-dimensional decagonal quasicrystal bimaterial. According to the general solutions and Fourier transform, fundamental solutions for phonon and phason displacement discontinuities and phonon and phason stresses are obtained. The delta function in the phonon and phason stresses is replaced by the Gaussian distribution function to eliminate the oscillatory singularity near the crack front. The expressions of phonon and phason stress intensity factors are given in terms of phonon and phason displacement discontinuities. The finite element analysis software COMSOL is used to validate the developed method.

Analysis solution method for 3D planar crack problems of two-dimensional hexagonal quasicrystals with thermal effects

• An analysis solution method is proposed to analyze 3D planar crack problems for 2D hexagonal quasicrystals. • Corresponding EDD boundary integral equations are transferred to simplified integral-differential forms. • Various kinds of loadings, including normal, tangential and thermal loadings, applied on the crack are considered. • Using the proposed solution, influences of phonon-phason coupling effect on fracture parameters are assessed. An analysis solution method (ASM) is proposed for analyzing arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystal (QC) media. The extended displacement discontinuity (EDD) boundary integral equations governing three-dimensional (3D) crack problems are transferred to simplified integral-differential forms by introducing some complex quantities. The proposed ASM is based on the analogy between these EDD boundary equations for 3D planar cracks problems of 2D hexagonal QCs and those in isotropic thermoelastic materials. Mixed model crack problems under combined normal, tangential and thermal loadings are considered in 2D hexagonal QC media. By virtue of ASM, the solutions to 3D planar crack problems under various types of loadings for 2D hexagonal QCs are formulated through comparison to the corresponding solutions of isotropic thermoelastic materials which have been studied intensively and extensively. As an application, analytical solutions of a penny-shaped crack subjected uniform distributed combined loadings are obtained. Especially, the analytical solutions to a penny-shaped crack subjected to the anti-symmetric uniform thermal loading are first derived for 2D hexagonal QCs. Numerical solutions obtained by EDD boundary element method provide a way to verify the validity of the presented formulation. The influences of phonon-phason coupling effect on fracture parameters of 2D hexagonal QCs are assessed.

Closed-form solutions of an elliptical crack subjected to coupled phonon–phason loadings in two-dimensional hexagonal quasicrystal media

An elliptical crack subjected to coupled phonon–phason loadings in a three-dimensional body of two-dimensional hexagonal quasicrystals is analytically investigated. Owing to the existence of the crack, the phonon and phason displacements are discontinuous along the crack face. The phonon and phason displacement discontinuities serve as the unknown variables in the generalized potential function method which are used to derive the boundary integral equations. These boundary integral equations governing Mode I, II, and III crack problems in two-dimensional hexagonal quasicrystals are expressed in integral differential form and hypersingular integral form, respectively. Closed-form exact solutions to the elliptical crack problems are first derived for two-dimensional hexagonal quasicrystals. The corresponding fracture parameters, including displacement discontinuities along the crack face and stress intensity factors, are presented considering all three crack cases of Modes I, II, and III. Analytical solutions for a penny-shaped crack, as a special case of the elliptical problem, are given. The obtained analytical solutions are graphically presented and numerically verified by the extended displacement discontinuities boundary element method.

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part I: Theoretical solutions

• Planar crack is analyzed for 2D hexagonal quasicrystals with thermal effects. • Extended displacement discontinuity-boundary integral equation method is proposed. • Green's functions for point EDDs are derived by using the Hankel transform method. • The ordinary 1/2 singularity behavior is obtained at the crack edge. • Extended intensity factors are expressed in terms of EDDs across crack surfaces. An extended displacement discontinuity (EDD) boundary integral equation method is proposed for analysis of arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack surface. Green's functions for unit point EDDs in an infinite three-dimensional medium of 2D hexagonal QC are derived using the Hankel transform method. Based on the Green's functions and the superposition theorem, the EDD boundary integral equations for an arbitrarily shaped planar crack in an infinite 2D hexagonal QC body are established. Using the EDD boundary integral equation method, the asymptotic behavior along the crack front is studied and the classical singular index of 1/2 is obtained at the crack edge. The extended stress intensity factors are expressed in terms of the EDDs across crack surfaces. Finally, the energy release rate is obtained using the definitions of the stress intensity factors.

Analysis of arbitrarily shaped planar cracks in three-dimensional isotropic hygrothermoelastic media

ABSTRACTIn the present article, a planar crack of arbitrary shape embedded in three-dimensional isotropic hygrothermoelastic media is investigated. Based on the general solutions and Hankel transform technique, the fundamental solutions for unit-point and extended displacement discontinuities (EDD; including the displacement discontinuities, moisture concentration discontinuity, and the temperature discontinuity) are derived. The EDD boundary integral equations for an arbitrarily shaped, planar crack in the hygrothermoelastic medium are established in terms of the EDD. Utilizing the boundary integral equation method, the singularities of near-crack front fields are analyzed, and the stress, moisture flux, and heat flux intensity factors are all derived in terms of the EDD. As a special case, the analytical solution for a penny-shaped crack under uniform combined loadings is presented. The EDD boundary element method is proposed for numerical simulation. The numerical result for a penny-shaped crack subjected to uniform mechanical–moisture–thermal loading is compared with the analytical solution to verify the correctness of the proposed method. Two coplanar elliptical cracks subjected to combined loadings are simulated as an application, and the influences of applied loadings and the ellipticity ratio are discussed.

Extended displacement discontinuity method for analysis of penny-shaped cracks in three-dimensional thermal piezoelectric semiconductors

Abstract In this paper, the extended displacement discontinuity (EDD) boundary element method is developed to analyze a penny-shaped crack in the isotropic plane of a three-dimensional (3D) transversely isotropic thermal piezoelectric semiconductor (PSC). The generalized Almansi's theorem and the operator theory are used to obtain the general solutions under generalized loading. The fundamental solutions for extended displacement discontinuities (EDDs) applied on the penny-shaped crack surface, which include the displacement, electric potential, carrier density, and temperature discontinuities, are derived. By using the EDD boundary element method, the EDD under mechanical, electrical and heat loading near the edge of a penny-shaped crack is calculated. The stress and heat flux intensity factors of the crack are obtained. The influence of uniform and non-uniform heat loadings on the fracture of 3D transversely isotropic thermal PSCs is investigated.

Effects of thermal and electric boundary conditions on fracture of three-dimensional thermopiezoelectric media

An arbitrarily shaped planar crack under different thermal and electric boundary conditions on the crack surfaces is studied in three-dimensional transversely isotropic thermopiezoelectric media subjected to thermal–mechanical–electric coupling fields. Using Hankel transformations, Green functions are derived for unit point extended displacement discontinuities in three-dimensional transversely isotropic thermopiezoelectric media, where the extended displacement discontinuities include the conventional displacement discontinuities, electric potential discontinuity, as well as the temperature discontinuity. On the basis of these Green functions, the extended displacement discontinuity boundary integral equations for arbitrarily shaped planar cracks in the isotropic plane of three-dimensional transversely isotropic thermopiezoelectric media are established under different thermal and electric boundary conditions on the crack surfaces, namely, the thermally and electrically impermeable, permeable, and semi-permeable boundary conditions. The singularities of near-crack border fields are analyzed and the extended stress intensity factors are expressed in terms of the extended displacement discontinuities. The effect of different thermal and electric boundary conditions on the extended stress intensity factors is studied via the extended displacement discontinuity boundary element method. Subsequent numerical results of elliptical cracks subjected to combined thermal–mechanical–electric loadings are obtained.

Analysis of arbitrarily shaped, planar cracks in a three-dimensional transversely isotropic thermoporoelastic medium

The displacement discontinuity boundary integral equation method is extended to analyze planar cracks of arbitrary shape embedded in a three-dimensional, transversely isotropic, thermoporoelastic medium. Based on the general solutions and Hankel transform technique, the fundamental solutions for unit-point, extended displacement discontinuities (including the displacement discontinuities, pore pressure discontinuity, and the temperature discontinuity) are derived. The extended displacement discontinuity boundary integral equations are established for an arbitrarily shaped, planar crack in the isotropic plane of the thermoporoelastic medium in terms of the extended displacement discontinuities. Using the boundary integral equations method, the singularities of near-crack front fields are analyzed, and the stress, fluid flux and heat flux intensity factors are derived in terms of the extended displacement discontinuities. To validate the analytical solution, the EDD boundary element method is proposed. The numerical simulation of a penny-shaped crack under combined uniform mechanical-pore pressure–thermal loadings is compared with the analytical solution to validate the correctness of the proposed method. As an application, two coplanar elliptical cracks are numerically simulated. The influences of the applied, combined mechanical-pore-pressure-thermal loadings, the crack distance, the ellipticity ratio as well as the size of cracks are all studied.

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part II: Numerical solutions

• 2D hexagonal quasicrystals planar crack with thermal effects analyzed numerically. • Green's functions are derived for uniform triangular and rectangular elements. • The EDD-BEM is proposed for planar crack analysis in infinite 2D hexagonal QC body. • Elliptical crack with different loadings are investigated and presented graphically. The extended displacement discontinuity (EDD) boundary element method is developed to analyze an arbitrarily shaped planar crack in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack face. Green's functions for uniformly distributed EDDs over triangular and rectangular elements for 2D hexagonal QCs are derived. Employing the proposed EDD boundary element method, a rectangular crack is analyzed to verify the Green's functions by discretizing the crack with rectangular and triangular elements. Furthermore, the elliptical crack problem for 2D hexagonal QCs is investigated. Normal, tangential, and thermal loads are applied on the crack face, and the numerical results are presented graphically.

Extended displacement discontinuity method for analysis of cracks in 2D thermal piezoelectric semiconductors

The extended displacement discontinuities method has previously been used for crack analysis of elastic materials, piezoelectric media, magneto-electro-elastic media and piezoelectric semiconductors. Here, this method is extended to study cracks in two-dimensional n-type thermal piezoelectric semiconductors. The extended displacement discontinuities include the conventional displacement discontinuity, electric potential discontinuity, carrier density discontinuity, as well as temperature discontinuity across crack faces; correspondingly, the extended stresses represent conventional stress, electric displacement, electric current, and heat flux. Employing a Fourier transform, the fundamental solutions for a line crack under uniformly distributed extended displacement discontinuities on the crack faces are derived under mechanical, electrical, and heat loading. Based on the obtained fundamental solutions, an extended displacement discontinuity boundary element method is developed. The stress and heat flux intensity factors at the crack tip are calculated under different combined loadings.

Analysis of a three-dimensional arbitrarily shaped interface crack in a one-dimensional hexagonal thermo-electro-elastic quasicrystal bi-material. Part 2: Numerical method

The extended displacement discontinuity boundary integral equation and boundary element method are extended to analyze a three-dimensional (3D) arbitrarily shaped interface crack in a one-dimensional hexagonal, quasicrystal bi-material with both electric and thermal effects under combined phonon-phason-electric-thermal loadings. Based on the analogy with the analysis method for three-dimensional, transversely isotropic magnetoelectrothermoelastic bi-materials (Dang et al., in press) the numerical method is proposed for one-dimensional quasicrystal bi-materials. Firstly, the fundamental solutions for uniformly distributed, extended displacement discontinuities applied over a constant triangular element are obtained by integrating the fundamental solutions for unit-point extended displacement discontinuities given in Part 1 over the triangular area (Zhao et al., 2017). Secondly, in order to eliminate the oscillatory singularity near the crack front, the Delta function in the integral–differential equations is approximated by the Gaussian distribution function, and the Heaviside step function is replaced by the Error function accordingly. Thirdly, the extended stress intensity factors without oscillatory singularities and the energy release rate are all obtained in terms of the extended displacement discontinuities. At last, the extended displacement discontinuity boundary element method is proposed to validate the analytical solution. In the numerical simulation, the multi-physical behavior of an elliptical, interface crack is numerically simulated. The correctness of the proposed numerical method, the influence of the applied, combined phonon-phason-electric-thermal loadings, the material-mismatch, and the ellipticity ratio are all studied.

Analysis of cracks in one-dimensional hexagonal quasicrystals with the heat effect

The extended displacement discontinuity (EDD) method is proposed to analyze cracks in the periodical plane of one-dimensional (1D) hexagonal quasicrystals with the heat effect. Based on the operator theory and the Fourier transform, the fundamental solutions for EDDs are derived, where the EDDs include phonon and phason displacement discontinuities and the temperature discontinuity. The EDD boundary integral equation method is used to analyze the singularities of the near-crack tip fields, and the extended stress intensity factor (ESIF) expressions are obtained in terms of the EDDs across the crack faces. The EDD boundary element method is proposed to calculate the ESIFs of cracks in 1D hexagonal quasicrystals. COMSOL software is used to validate the developed method. The influences of applied mechanical and heat loads on cracks in a finite plate are investigated.

Analysis of a three-dimensional arbitrarily shaped interface crack in a one-dimensional hexagonal thermo-electro-elastic quasicrystal bi-material. Part 1: Theoretical solution

The extended displacement discontinuity boundary integral-differential equation method is adapted to analyze a three-dimensional interface crack of arbitrary shape in a one-dimensional, hexagonal thermo-electro-elastic quasicrystals bi-material. The extended displacement discontinuities include phonon and phason displacement discontinuities, electric potential discontinuity, as well as temperature discontinuity across the interface crack; while the extended stresses represent phonon and phason stresses, electric displacement and heat flux, respectively. An analysis method is proposed based on the analogy between the governing equations for one-dimensional hexagonal electro-thermo-elastic quasicrystals and three-dimensional transversely isotropic magnetoelectrothermoelastic media. By using the analogy method, the fundamental solutions for unit-point extended displacement discontinuities on the interface are obtained. Using the superposition principal, the extended displacement discontinuity boundary integral-differential equations are established. The singular indices and the singular behaviors of the near crack border fields are studied, and the combined extended stress intensity factors (SIFs) are derived in terms of the EDDs.