THREE-DIMENSIONAL ELLIPTIC CRACK UNDER IMPACT LOADING

The dynamic stress intensity factor of a three-dimensionalelliptic crack under impact loading is determined with the finiteelement method. The computation results can take into account theinfluence of time and the ratio of the wave speeds on the stressintensity factor. The present method is suitable not only forthree-dimensional dynamic crack, but also for three-dimensionaldynamic contact.

CONVOLUTION OF THE IMPACT THREE-DIMENSIONAL ELASTO-DYNAMICS AND DYNAMIC STRESS INTENSITY FACTOR FOR AN ELLIPTIC CRACK

This paper presents a formulation for three-dimensional elasto-dynamics with an elliptic crack based on the Laplace and Fourier transforms and the convolution theorem. The dynamic stress intensity factor for the crack is determined by solving a Fredholm integral equation of the first kind. The results of this paper are very close to those given by the two-dimensional dual integral equation method.

Comprehensive investigation of stress intensity factors in rotating disks containing three-dimensional semi-elliptical cracks

Initiation and propagation of cracks in rotating disks may cause catastrophic failures. Therefore, determination of fracture parameters under different working con- ditions is an essential issue. In this paper, a comprehensive study of stress intensity factors （SIFs） in rotating disks containing three-dimensional （3D） semi-elliptical cracks subjected to different working conditions is carried out. The effects of mechanical prop- erties, rotational velocity, and orientation of cracks on SIFs in rotating disks under cen- trifugal loading are investigated. Also, the effects of using composite patches to reduce SIFs in rotating disks are studied. The effects of patching design variables such as mechanical properties, thickness, and ply angle are investigated separately. The modeling and analytical procedure are verified in comparison with previously reported results in the literature.

A FULLY ELLIPTIC CALCULATING PROCEDURE FOR THREE-DIMENSIONAL THERMAL POLLUTION

The use of the mathematical models so far for three-dimensional flow has some limitations because of their simplifications. Many characteristics of the flow field can not be predicted by these models. In this paper the three dimensional elliptic governing equations are solved by finite-volume methods; the buoyancy extensions of the widely tested k-?. model is adapted. The method is first applied to calculate the field of side discharge into open channel flow. The results are in good agreement with those of ref. [7]. Then it is further used to the intake discharge problem which is of a typical layout in cooling-water projects, and the calculated results, which predict in detail the charactreistics of flow field, are reasonable.

Controlling Three-Dimensional Electron-Electron Correlation via Elliptically Polarized Intense Laser Field

The three-dimensional electron-electron correlation in an elliptically polarized laser field is investigated based on a semiclassical model. Asymmetry parameter α of the correlated electron momentum distribution is used to quantitatively describe the electron-electron correlation. The dependence of α on elliptieity e is totally different in three directions. For the z direction （maJor polarization direction）, α first increases and reaches a maximum at ε = 0.275, then it decreases quickly. For the y direction in which the laser field is always absent, the ellipticity has a minor effect, and the asymmetry parameter fluctuates around α = -0.15. However, for the x direction （minor polarization direction）, α increases monotonously with ellipticity though starts from the same value as in the y direction when ε = 0. The behavior of α in the x direction actually indicates a transformation from the Coulomb interaction dominated correlation to the laser field dominated correlation. Therefore, our work provides an efficient way to control the three-dimensional electron electron correlation via an elliptically polarized intense laser field.

Exact analytic solutions for an elliptic hole with asymmetric collinear cracks in a one-dimensional hexagonal quasi-crystal

Using the complex variable function method and the technique of conformal mapping, the anti-plane shear problem of an elliptic hole with asymmetric colfinear cracks in a one-dimensional hexagonal quasi-crystal is solved, and the exact analytic solutions of the stress intensity factors （SIFs） for mode Ⅲ problem are obtained. Under the limiting conditions, the present results reduce to the Griffith crack and many new results obtained as well, such as the circular hole with asymmetric collinear cracks, the elliptic hole with a straight crack, the mode T crack, the cross crack and so on. As far as the phonon field is concerned, these results, which play an important role in many practical and theoretical applications, are shown to be in good agreement with the classical results.

Interaction between a screw dislocation and an elastic elliptical inhomogeneity with interfacial cracks

The elastic interaction between a screw dislocation and an elliptical inhomogeneity with interfacial cracks is studied. The screw dislocation may be located outside or inside the inhomogeneity. An efficient complex variable method for the complex multiply connected region is developed, and the general solutions to the problem are derived. As illustrative examples, solutions in explicit series form for complex potentials are presented in the case of one or two interfacial cracks. Image forces on the dislocation are calculated by using the Peach-Koehler formula. The influence of crack geometries and material properties on the image forces is evaluated and discussed. It is shown that the interfacial crack has a significant effect on the equilibrium position of the dislocation near an elliptical-arc interface. The main results indicate, when the length of the crack goes up to a critical value, the presence of the interfacial crack can change the interaction mechanism between a screw dislocation and an elliptical inclusion. The present solutions can include a number of previously known results as special cases.

An analytical solution for the stress field and stress intensity factor in an infinite plane containing an elliptical hole with two unequal aligned cracks

The existing analytical solutions are extended to obtain the stress fields and the stress intensity factors(SIFs) of two unequal aligned cracks emanating from an elliptical hole in an infinite isotropic plane. A conformal mapping is proposed and combined with the complex variable method. Due to some difficulties in the calculation of the stress function, the mapping function is approximated and simplified via the applications of the series expansion. To validate the obtained solution, several examples are analyzed with the proposed method, the finite element method, etc. In addition, the effects of the lengths of the cracks and the ratio of the semi-axes of the elliptical hole(a/b) on the SIFs are studied. The results show that the present analytical solution is applicable to the SIFs for small cracks.

Fracture of two three-dimensional parallel internal cracks in brittle solid under ultrasonic fracturing

Similar to hydraulic fracturing(HF), the coalescence and fracture of cracks are induced within a rock under the action of an ultrasonic field, known as ultrasonic fracturing(UF). Investigating UF is important in both hard rock drilling and oil and gas recovery. A three-dimensional internal laser-engraved crack(3D-ILC) method was introduced to prefabricate two parallel internal cracks within the samples without any damage to the surface. The samples were subjected to UF. The mechanism of UF was elucidated by analyzing the characteristics of fracture surfaces. The crack propagation path under different ultrasonic parameters was obtained by numerical simulation based on the Paris fatigue model and compared to the experimental results of UF. The results show that the 3D-ILC method is a powerful tool for UF research.Under the action of an ultrasonic field, the fracture surface shows the characteristics of beach marks and contains powder locally, indicating that the UF mechanism includes high-cycle fatigue fracture, shear and friction, and temperature load. The two internal cracks become close under UF. The numerical result obtained by the Paris fatigue model also shows the attraction of the two cracks, consistent with the test results. The 3D-ILC method provides a new tool for the experimental study of UF. Compared to the conventional numerical methods based on the analysis of stress-strain and plastic zone, numerical simulation can be a good alternative method to obtain the crack path under UF.

Three dimensional K-Tz stress fields around the embedded center elliptical crack front in elastic plates

Through detailed three-dimensional （3D） finite element （FE） calculations, the out-of-plane constraints Tz along embedded center-elliptical cracks in mode I elastic plates are studied. The distributions of Tz are obtained near the crack front with aspect ratios （a/c） of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. Tz decreases from an approximate value of Poisson ratio v at the crack tip to zero with increasing normalized radial distances （r/a） in the normal plane of the crack front line, and increases gradually when the elliptical parameter angle φ changes from 0° to 90°at the same r/a. With a/c rising to 1.0, Tz is getting nearly independent of φ and is only related to r/a. Based on the present FE calculations for Tz, empirical formulas for Tz are obtained to describe the 3D distribution of Tz for embedded center-elliptical cracks using the least squares method in the range of 0.2 ≤ a/c ≤ 1.0. These Tz results together with the corresponding stress intensity factor K are well suitable for the analysis of the 3D embedded centerelliptical crack from field, and a two-parameter K-Tz principle is proposed.

Discrete Element Modeling of Asphalt Concrete Cracking Using a User-def ined Three-dimensional Micromechanical Approach

We established a user-defined micromechanical model using discrete element method （DEM） to investigate the cracking behavior of asphalt concrete （AC）. Using the ＂Fish＂ language provided in the particle flow code in 3-Demensions （PFC3D）, the air voids and mastics in asphalt concrete were realistically built as two distinct phases. With the irregular shape of individual aggregate particles modeled using a clump of spheres of different sizes, the three-dimensional （3D） discrete element model was able to account for aggregate gradation and fraction. Laboratory uniaxial complex modulus test and indirect tensile strength test were performed to obtain input material parameters for the numerical simulation. A set of the indirect tensile test were simulated to study the cracking behavior of AC at two levels of temperature, i e, -10 ℃ and 15 ℃. The predicted results of the numerical simulation were compared with laboratory experimental measurements. Results show that the 3D DEM model is able to predict accurately the fracture pattern of different asphalt mixtures. Based on the DEM model, the effects of air void content and aggregate volumetric fraction on the cracking behavior of asphalt concrete were evaluated.

Exact Solutions for Piezoelectric Materials with an Elliptic Hole or a Crack under Uniform Internal Pressure

The existing investigations on piezoelectric materials containing an elliptic hole or a crack mainly focus on remote uniform tensile loads.In order to have a better understanding for the fracture behavior of piezoelectric materials under different loading conditions,theoretical and numerical solutions are presented for an elliptic hole or a crack in transversely isotropic piezoelectric materials subjected to uniform internal pressure and remote electro-mechanical loads.On the basis of the complex variable approach,analytical solutions of the elastic and electric fields inside and outside the defect are derived by satisfying permeable electric boundary condition at the surface of the elliptical hole.As an example of PZT-4 ceramics,numerical results of electro-elastic fields inside and outside the crack under various electric boundary conditions and electro-mechanical loads are given,and graphs of the electro-elastic fields in the vicinity of the crack tip are presented.The non-singular term is compared to the asymptotic one in the figures.It is shown that the dielectric constant of the air in the crack has no effect on the electric displacement component perpendicular to the crack,and the stresses in the piezoelectric material depend on the material properties and the mechanical loads on the crack surface and at infinity,but not on the electric loads at infinity.The figures obtained are strikingly similar to the available results.Unlike the existing work,the existence of electric fields inside an elliptic hole or a crack is considered,and the piezoelectric solid is subjected to complicated electro-mechanical loads.

THE FUNDAMENTAL SOLUTIONS FOR THE PLANE PROBLEM IN PIEZOELECTRIC MEDIA WITH AN ELLIPTIC HOLE OR A CRACK

Based on the complex potential method, the Greed’s functions of the plane problem in transversely isotropic piezoelectric media with an elliptic hole are obtained in terms of exact electric boundary conditions at the rim of the hole. When foe elliptic hole degenerates into a crack, the fundamental solutions for the field intensity factors arc given. The general solutions for concentrated and distributed loads applied on the surface of the hole or crack are produced through the superposition of fundamental solutions With the aid of these solutions , some erroneous results provided previously in other works are pointed out More important is that these solutions can be used as the fundamental solutions of boundary element method to solve more practical problems in piezoelectric media.

Interactions of penny-shaped cracks in three-dimensional solids

The interaction of arbitrarily distributed penny-shaped cracks in three-dimensional solids is analyzed in this paper. Using oblate spheroidal coordinates and displacement functions, an analytic method is devel- oped in which the opening and the sliding displacements on each crack surface are taken as the basic unknown functions. The basic unknown functions can be expanded in series of Legendre polynomials with unknown coefficients. Based on superposition technique, a set of governing equations for the unknown coefficients are formulated from the traction free conditions on each crack surface. The boundary collocation procedure and the average method for crack-surface tractions are used for solving the governing equations. The solution can be obtained for quite closely located cracks. Numerical examples are given for several crack problems. By comparing the present results with other existing results, one can conclude that the present method provides a direct and efficient approach to deal with three-dimensional solids containing multiple cracks.

Closing law and stress intensity factor of elliptical crack under compressive loading

The solution of surface displacement of an elliptical crack under compressive-shear loading was obtained by using the complex function method. The closing mode was established by analyzing the geometrical condition of closing crack, and the corresponding critical stress was solved. The result corrects the traditional viewpoint, in which there exist only open or close states for an elliptical crack, and points out that the local closing is also one of crack states. Based on them, the effect of the closed crack on stress intensity factor was discussed in detail, and its rational formulae are put forward.

A modified maximum tangential tensile stress criterion for three-dimensional crack propagation

The three-dimensional （3D） crack propagation is a hot issue in rock mechanics. To properly simulate 3D crack propagation, a modified maximum tangential tensile stress criterion is proposed. In this modified criterion, it is supposed that cracks propagate only at crack front in the principal normal plane. The tangential tensile stress at crack front in the principal normal plane in local coordinates is employed to determine crack propagation, which is calculated through coordinate transformation from global to local coordinates. New cracks will propagate when the maximum tangential tensile stress at crack front in the principal normal plane reaches the tensile strength of rock-like materials. Compared with the previous crack propagation criteria, the modified crack propagation criterion is helpful in calculating 3D crack stress intensity factor, and can overcome the limitations of propagation step determined by individual experiences in previous studies. Finally, the 3D crack propagation process is traced by element-free Galerkin method. The numerical results agree well with the experimental ones for a frozen resin sample with prefabricated 3D cracks.

Analytic solutions to problem of elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals

By means of the complex variable function method and the technique of conformal mapping, the anti-plane shear problem of an elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals is investigated. The solution of the stress intensity factor （SIF） for mode III problem has been found. Under the condition of limitation, both the known results and the SIF solution at the crack tip of a circular hole with two straight cracks and cross crack in one-dimensional hexagonal quasicrystals can be obtained.

Screw dislocations interacting with two asymmetrical interfacial cracks emanating from an elliptical hole

The interaction between screw dislocations and two asymmetrical interfacial cracks emanating from an elliptic hole under loads at infinity is studied. The closed-form solution is derived for complex potentials. The stress intensity factor and the critical applied stress for the dislocation emission are also calculated. In the limiting cases, well-known results can be obtained from the present solutions. Moreover, new exact solutions for a screw dislocation interacting with some complicated cracks are derived. The results show that the shielding effect increases with the increase in the length of the other cracks and the minor semi axis, but decreases with the increase of dislocation azimuth. The repulsion acting on the dislocation from the other phase and the other crack extend in the horizontal direction, which makes the dislocation emission at the crack tip take place more easily, but the minor semi axis of the elliptical hole extending in the vertical direction makes it more difficult.

SCATTERING OF SH-WAVE BY CRACKS ORIGINATING AT AN ELLIPTIC HOLE AND DYNAMIC STRESS INTENSITY FACTOR

The method of complex function and the method of Green's function are used to investigate the problem of SH-wave scattering by radial cracks of any limited length along the radius originating at the boundary of an elliptical hole, and the solution of dynamic stress intensity factor at the crack tip was given. A Green's function was constructed for the problem, which is a basic solution of displacement field for an elastic half space containing a half elliptical gap impacted by anti-plane harmonic linear source force at any point of its horizontal boundary. With division of a crack technique, a series of integral equations can be established on the conditions of continuity and the solution of dynamic stress intensity factor can be obtained. The influence of an elliptical hole on the dynamic stress intensity factor at the crack tip was discussed.

A NOVEL METHOD FOR EXACT SOLUTIONS OF ELLIPTICAL CRACKS IN PIEZOELECTRICS

According to the constitutive relationship in linear piezoceramics, elliptical crack problems in the impermeable case are reconsidered with the hypersingular integral equation method. Unknown displacement and electric potential jumps in the integral equations are approximated with a product of the fundamental density function and polynomials, in which the fundamental density function reflects the singular behavior of electroelastic fields near the crack front and the polynomials can be reduced to a real constant under uniform loading. Ellipsoidal coordinates are cleverly introduced to solve the unknown displacement and electric potential jumps in the integral equations under uniform loading. With the help of these solutions and definitions of electroelastic field intensity factors, exact expressions for mode Ⅰ, mode Ⅱ and mode Ⅲ stress intensity factors as well as the mode Ⅳ electric displacement intensity factor are obtained. The present results under uniform normal loading are the same as the available exact solutions, but those under uniform shear loading have not been found in the literature as yet.