Representation Theory of Three-Dimensional Elliptical Crack Under Dynamic Loading

The relation between the normal displacement on the surface of a dynamical elliptical crack and the normal stress over the crack surface was studied. The three dimensional elastodynamic equations and Fourier Laplace transforms are used. Based on the influence function and the inversion of integral transforms, one can find that if the distribution of normal displacement on the surface of a dynamic elliptical crack is a polynomial of degree n in x 1 and x 2 , then the normal pressure acting over the ellipse is also a polynomial P n(x 1,x 2) of the same degree in x 1 and x 2 .

Analytic model of non-uniform corrosion induced cracking of reinforced concrete structure

In order to perfectly reflect the dynamic corrosion of reinforced concrete (RC) cover in practical engineering,an analytic model of non-uniform corrosion induced cracking was presented based on the elastic-plastic fracture mechanics theory.Comparisons with the published experimental data show that the predictions given by the present model are in good agreement with the results both for natural exposed experiments and short-time indoor tests (the best difference is about 2.7%).Also it obviously provides much better precision than those models under the assumption of uniform corrosion (the maximal improved precision is about 48%).Therefore,it is pointed out that the so-called uniform corrosion models to describe the cover cracking of RC should be adopted cautiously.Finally,the influences of thickness of local rusty layer around the reinforcing steel bar on the critical corrosion-induced crack indexes were investigated.It is found that the thickness of local rusty layer has great effect on the critical mass loss of reinforcing steel,threshold expansion pressure,and time to cover cracking.For local rusty layer thickness with a size of a=0.5 mm,the time to cover cracking will increase by about one times when a/b (a,semi-minor axis;b,semi-major axis) changes from 0.1 to 1 mm.

Coupling of two-dimensional atomistic and continuum models for dynamic crack

A concurrent multiscale method of coupling atomistic and continuum models is presented in the two-dimensional system. The atomistic region is governed by molecular dynamics while the continuum region is represented by construct- ing the mass and stiffness matrix dependent on the coarsening of the grids, which ensures that they merge seamlessly. The low-pass phonon filter embedded in the handshaking region is utilized to effectively eliminate the spurious reflection of high-frequency phonons, while keeping the low-frequency phonons transparent. These schemes are demonstrated by numerically calculating the reflection and transmission coefficient, and by the further application of dynamic crack propa- gation subjected to mode-I tensile loading.

Degradation mechanism of rock under impact loadings by integrated investigation on crack and damage development

Failure of rock under impact loadings involves complex micro-fracturing and progressive damage. Strength increase and splitting failure have been observed during dynamic tests of rock materials. However, the failure mechanism still remains unclear. In this work, based on laboratory tests, numerical simulations with the particle flow code(PFC) were carried out to reproduce the micro-fracturing process of granite specimens. Shear and tensile cracks were both recorded to investigate the failure mode of rocks under different loading conditions. At the same time, a dynamic damage model based on the Weibull distribution was established to predict the deformation and degradation behavior of specimens. It is found that micro-cracks play important roles in controlling the dynamic deformation and failure process of rock under impact loadings. The sharp increase in the number of cracks may be the reason for the strength increase of rock under high strain rates. Tensile cracks tend to be the key reason for splitting failure of specimens. Numerical simulation of crack propagation by PFC can give vivid description of the failure process. However, it is not enough for evaluation of material degradation. The dynamic damage model is able to predict the stress-strain relationship of specimens reasonably well, and can be used to explain the degradation of specimens under impact loadings at macro-scale. Crack and damage can describe material degradation at different scales and can be used together to reveal the failure mechanism of rocks.

MATHEMATICAL PROBLEMS IN THE INTEGRAL-TRANSFORMATION METHOD OF DYNAMIC CRACK

In the investigation on fracture mechanics,the potential function was introduced, and the moving differential equation was constructed. By making Laplace and Fourier transformation as well as sine and cosine transformation to moving differential equations and various responses, the dual equation which is constructed from boundary conditions lastly was solved. This method of investigating dynamic crack has become a more systematic one that is used widely. Some problems are encountered when the dynamic crack is studied. After the large investigation on the problems, it is discovered that during the process of mathematic derivation, the method is short of precision, and the derived results in this method are accidental and have no credibility.A model for example is taken to explain the problems existing in initial deriving process of the integral_transformation method of dynamic crack.

Dual equations and solutions of I-type crack of dynamic problems in piezoelectric materials

This paper firstly works out basic differential equations of piezoelectric materials expressed in terms of potential functions, which are introduced in the very beginning. These equations are primarily solved through Laplace transformation, semiinfinite Fourier sine transformation and cosine transformation. Secondly, dual equations of dynamic cracks problem in 2D piezoelectric materials are established with the help of Fourier reverse transformation and the introduction of boundary conditions. Finally, according to the character of the Bessel function and by making full use of the Abel integral equation and its reverse transform, the dual equations are changed into the second type of Fredholm integral equations. The investigation indicates that the study approach taken is feasible and has potential to be an effective method to do research on issues of this kind.

DYNAMICALLY ASYMPTOTIC SOLUTIONS NEAR MODE II CRACK-TIPS

Description on dynamic behaviours of crack-tips is one of the important foundations to develop a reasonable dynamic fracture criterion. In order to describe the dynamic behaviours of the crack-tip in a material with low viscosity-number, the displacement potential function is assumed as a mathematical expression with exponential singularity. The asymptotic linear differential equations determining plane crack-tip field are established based on the mechanical constitutive model for elastic-viscoplastic materials. According to the conditions of determining solutions for dynamic cracks of mode II, the crack-tip stress fields are numerically simulated based on the asymptotic linear differential equations. Results show the asymptotic linear equations can well describe the crack-tip fields of plane dynamic cracks in the elastic-viscoplastic material with low viscosity-number.

Field structure at mode Ⅲ dynamically propagating crack tip in elastic-viscoplastic materials

An elastic-viscoplastic mechanics model is used to investigate asymptotically the mode Ⅲ dynamically propagating crack tip field in elastic-viscoplastic materials. The stress and strain fields at the crack tip possess the same power-law singularity under a linear-hardening condition. The singularity exponent is uniquely determined by the viscosity coefficient of the material. Numerical results indicate that the motion parameter of the crack propagating speed has little effect on the zone structure at the crack tip. The hardening coefficient dominates the structure of the crack-tip field. However, the secondary plastic zone has little influence on the field. The viscosity of the material dominates the strength of stress and strain fields at the crack tip while it does have certain influence on the crack-tip field structure. The dynamic crack-tip field degenerates into the relevant quasi-static solution when the crack moving speed is zero. The corresponding perfectly-plastic solution is recovered from the linear-hardening solution when the hardening coefficient becomes zero.

THE THREE-DIMENSIONAL STRESS INTENSITY FACTOR UNDER MOVING LOADS ON THE CRACK FACES

The dynamic stress intensity factor history for a half plane crack in an otherwise unbounded elastic body,with the crack faces subjected to a traction distribution consisting of two pairs of combined mode point loads that move in a direction perpendicular to the crack edge is considered.The analytic expression for the combined mode stress intensity factors as a function of time for any point along the crack edge is obtained.The method of solution is based on the application of integral transform together with the Wiener-Hopf technique and the Cagniard-de Hoop method. Some features of the solution are discussed and graphical results for various point load speeds are presented.

Numerical study on static and dynamic fracture evolution around rock cavities

In this paper,a numerical code,RFPA2D（rock failure process analysis）,was used to simulate the initiation and propagation of fractures around a pre-existing single cavity and multiple cavities in brittle rocks.Both static and dynamic loads were applied to the rock specimens to investigate the mechanism of fracture evolution around the cavities for different lateral pressure coefficients.In addition,characteristics of acoustic emission（AE） associated with fracture evolution were simulated.Finally,the evolution and interaction of fractures between multiple cavities were investigated with consideration of stress redistribution and transference in compressive and tensile stress fields.The numerically simulated results reproduced primary tensile,remote,and shear crack fractures,which are in agreement with the experimental results.Moreover,numerical results suggested that both compressive and tensile waves could influence the propagation of tensile cracks;in particular,the reflected tensile wave accelerated the propagation of tensile cracks.

Analysis of dynamic stress intensity factors of thick-walled cylinder under internal impulsive pressure

Dynamic stress intensity factors are evaluated for thick-walled cylinder with a radial edge crack under internal impulsive pressure. Firstly, the equation for stress intensity factors under static uniform pressure is used as the reference case, and then the weight function for a thick-walled cylinder containing a radial edge crack can be worked out. Secondly, the dynamic stresses in uncracked thick-walled cylinders are solved under internal impulsive pressure by using mode shape function method. The solution consists of a quasi-static solution satisfying inhomogeneous boundary conditions and a dynamic solution satisfying homogeneous boundary condi- tions, and the history and distribution of dynamic stresses in thick-walled cylinders are derived in terms of Fourier-Bessel series. Finally, the dynamic stress intensity factor equations for thick-walled cylinder containing a radial edge crack sub- jected to internal impulsive pressure are given by dynamic weight function method. The finite element method is utilized to verify the results of numerical examples, showing the validity and feasibility of the proposed method.

Incubation time fracture criterion for FEM simulations

The paper is discussing problems connected with embedment of the incubation time criterion for brittle fracture into finite element computational schemes. Incubation time fracture criterion is reviewed; practical questions of its numerical implementation are extensively discussed. Several examples of how the incubation time fracture criterion can be used as fracture condition in finite element computations are given. The examples include simulations of dynamic crack propagation and arrest, impact crater formation （i.e. fracture in initially intact media）, spall fracture in plates, propagation of cracks in pipelines. Applicability of the approach to model initiation, development and arrest of dynamic fracture is claimed.

Application of scaled boundary finite element method in static and dynamic fracture problems

The scaled boundary finite element method （SBFEM） is a recently developed numerical method combining advantages of both finite element methods （FEM） and boundary element methods （BEM） and with its own special features as well. One of the most prominent advantages is its capability of calculating stress intensity factors （SIFs） directly from the stress solutions whose singularities at crack tips are analytically represented. This advantage is taken in this study to model static and dynamic fracture problems. For static problems, a remeshing algorithm as simple as used in the BEM is developed while retaining the generality and flexibility of the FEM. Fully-automatic modelling of the mixed-mode crack propagation is then realised by combining the remeshing algorithm with a propagation criterion. For dynamic fracture problems, a newly developed series-increasing solution to the SBFEM governing equations in the frequency domain is applied to calculate dynamic SIFs. Three plane problems are modelled. The numerical results show that the SBFEM can accurately predict static and dynamic SIFs, cracking paths and load-displacement curves, using only a fraction of degrees of freedom generally needed by the traditional finite element methods.

CFD study： Effect of pulsating flow on gas-solid hydrodynamics in FCC riser

Gas–solid flow in a fluid catalytic cracking （FCC） riser exhibits poor mixing in the form of a core–annulus flow pattern and a dense bottom/dilute top distribution of solids. To enhance gas–solid mixing, studies on dense fluidized beds have suggested using a pulsating flow of gas. The present study investigates the effect of pulsating flow on gas–solid hydrodynamics inside the FCC riser employing computational fluid dynamics. Two flow conditions are investigated： a cold flow of air-FCC catalyst in a pilot-scale riser and a reactive flow in an industrial-scale FCC riser. In the cold-flow riser, pulsating flows cause the slug flow of solids and thus increase the average solid accumulation in the flow domain and solid segregation towards the wall. In the industrial FCC riser, pulsating flows produce radial profiles that are more homogeneous. Pulsating flows further improve the conversion and yield in the initial few metres of height. At 7 m, the conversion from pulsating flow is 59%, compared with 44% in without pulsating flow. The results and analysis presented here will help optimize flow conditions in the circulating fluidized bed riser, in not only FCC but also applications such as fast pyrolysis and combustion.

Numerical comparison of two modes of gas-solid riser operation： Fluid catalytic cracking vs CFB combustor

Two modes of gas-solid riser operation, i.e., fluid catalytic cracking （FCC） and circulating fluidized bed combustor （CFBC）, have been recognized in literature; particularly in the understanding of choking phenomena. This work compares these two modes of operation through computational fluid dynamics （CFD） simulation. In CFD simulations, the different operations are represented by fixing appropriate boundary conditions： solids flux or solids inventory. It is found that the FCC and CFBC modes generally have the same dependence of solids flux on the mean solids volume fraction or solids inventory. However, during the choking transition, the FCC mode of operation needs more time to reach a steady state; thus the FCC system may have insufficient time to respond to valve adjustments or flow state change, leading to the choking. The difference between FCC and CFBC systems is more pronounced for the systems with longer risers. A more detailed investigation of these two modes of riser operation may require a three-dimensional full loop simulation with dynamic valve adjustment.

Asymptotic Solution of Steady Dynamic Crack Growth in a Power-law Elastic-plastic Solid

Within the infinitesimal deformation and plastic incremental theory, an asymptotic solution of near tip fields is presented in this paper for mode Ⅰ steady dynamic crack growth in a power law incompressible elastic plastic solid under plane strain conditions. This solution shows that the near tip stresses and strains are fully continuous and bounded, and depend only on one free parameter that must be determined from the far field.

A novel pore-fracture dual network modeling method considering dynamic cracking and its applications

Unconventional reservoirs are normally characterized by dual porous media, which has both multi-scalepore and fracture structures, such as low permeability or tight oil reservoirs. The seepage characteristicsof such reservoirs is mainly determined by micro-fractures, but conventional laboratory experimentalmethods are difficult to measure it, which is attribute to the dynamic cracking of these micro-fractures.The emerging digital core technology in recent years can solve this problem by developing an accuratepore network model and a rational simulation approach. In this study, a novel pore-fracture dualnetwork model was established based on percolation theory. Fluid flow in the pore of two scales, microfracture and matrix pore, were considered, also with the impact of micro-fracture opening and closingduring flow. Some seepage characteristic parameters, such as fluid saturations, capillary pressure, relative permeabilities, displacement efficiency in different flow stage, can be predicted by proposedcalculating method. Through these work, seepage characteristics of dual porous media can be achieved.