In this study,a trifunctional strategy was developed to prepare a confined Ni-based catalyst(Ni-CeO_(2)@SiO_(2))for dry reforming of methane(DRM)of two main greenhouse gases-CO_(2)and CH_(4).The Ni-CeO_(2)@SiO_(2)cata...In this study,a trifunctional strategy was developed to prepare a confined Ni-based catalyst(Ni-CeO_(2)@SiO_(2))for dry reforming of methane(DRM)of two main greenhouse gases-CO_(2)and CH_(4).The Ni-CeO_(2)@SiO_(2)catalyst was fabricated by utilizing the confinement effect of the SiO_(2)shell and the synergistic interaction between Ni-Ce and the decoking effect of CeO_(2).The catalysts were systematically characterized via X-ray diffraction,N_(2 )adsorption/desorption,transmission electron microscopy,energy dispersive X-ray spectroscopy,hydrogen temperature reduction and desorption set by program,oxygen temperature program desorption,Raman spectroscopy,thermogravimetric analysis,and in situ diffuse reflectance infrared Fourier transform spectroscopy measurements to reveal their physicochemical properties and reaction mechanism.The Ni-CeO_(2)@SiO_(2)catalyst exhibited higher activity and stability than the catalyst synthesized via the traditional impregnation method.In addition,no carbon deposition was detected over Ni-CeO_(2)@SiO_(2)after a 100 h durability test at 800℃,and the average particle size of Ni nanoparticles(NPs)in the catalyst increased from 5.01 to 5.77 nm.Remarkably,Ni-CeO_(2)@SiO_(2)also exhibited superior low-temperature stability;no coke deposition was observed when the catalyst was reacted at 600℃ for 20 h.The high coking and sintering resistance of this confined Ni-based DRM catalyst can be attributed to its trifunctional effect.The trifunctional strategy developed in this study could be used as a guideline to design other high-performance catalysts for CO_(2)and CH4 dry forming and accelerate their industrialization.展开更多
A new technology of functionally graded materials(FGM) mould fabricated by plasma spraying and arc spraying was developed. According to applied characteristic of plastic mould, the reasonable coatings of FGM were desi...A new technology of functionally graded materials(FGM) mould fabricated by plasma spraying and arc spraying was developed. According to applied characteristic of plastic mould, the reasonable coatings of FGM were designed and their microstructures were analyzed. At the same time, some key problems were solved including spray mould fabricating, FGM forming and demoulding, etc. The results show that the service performance of the FGM mould is much more excellent than the one composed of the traditional materials, and the life span can also be greatly increased. The technology will have a significant influence on materials development in mould industry.展开更多
In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinea...In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems （with same stiffness matrix but different right-hand side） on the fine mesh space. The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the Mgorithm produces a numerical solution with the optimal asymptotic H^2-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.展开更多
The forming limit diagram（FLD） is an important tool to be used when characterizing the formability of metallic sheets used in metal forming processes. Experimental measurement and determination of the FLD is timecon...The forming limit diagram（FLD） is an important tool to be used when characterizing the formability of metallic sheets used in metal forming processes. Experimental measurement and determination of the FLD is timeconsuming and therefore the analytical prediction based on theory of plasticity and instability criteria allows a direct and efficient methodology to obtain critical values at different loading paths, thus carrying significant practical importance.However, the accuracy of the plastic instability prediction is strongly dependent on the choice of the material constitutive model [1–3]. Particularly for materials with hexagonal close packed（HCP） crystallographic structure, they have a very limited number of active slip systems at room temperature and demonstrate a strong asymmetry between yielding in tension and compression [4, 5]. Not only the magnitude of the yield locus changes, but also the shape of the yield surface is evolving during the plastic deformation [4]. Conventional phenomenological constitutive models of plasticity fail to capture this unconventional mechanical behavior [4, 6]. Cazacu and Plunkett [6] have proposed generic yield criteria, by using the transformed principal stress, to account for the initial plastic anisotropy and strength differential（SD） effect simultaneously. In this contribution, a generic FLD MATLAB script was developed based on Marciniak–Kuczynski analytical theory and applied to predict the localized necking. The influence of asymmetrical effect on the FLD was evaluated. Several yield functions such as von Mises, Hill, Barlat89, and Cazacu06 were incorporated into analysis. The paper also presents and discusses the influence of different hardening laws on the formability of materials with HCP crystal structures. The findings indicate that the plastic instability theory coupled with Cazacu model can adequately predict the onset of localized necking for HCP materials under different strain paths.展开更多
Long before,the research of form has already entered its heyday,but the study of function is a somewhat new branch of language study.This essay is an attempt to take a brief look at the most significant distinction be...Long before,the research of form has already entered its heyday,but the study of function is a somewhat new branch of language study.This essay is an attempt to take a brief look at the most significant distinction between"form and function",and then aims at making clear the definition of"form and function"in discourse analysis.展开更多
The element-free method is a new numerical technique presented in recent years.It uses the moving least square(MLS) approximation as its shape function,and it is determined by the basic function and weight function.Th...The element-free method is a new numerical technique presented in recent years.It uses the moving least square(MLS) approximation as its shape function,and it is determined by the basic function and weight function.The weight function is the mainly determining factor,so it greatly affects the accuracy of the computational results.The element-free Galerkin method(EFGM) was applied for the solution to plastic large deformation.The simulation of metal rheological forming was successfully done by programming and its results were visualized by using the plotting and data analyses software Tecplot.Then plastic strain under different stages during rheological forming and the three principal stresses at the last deformation were obtained.The example shows the feasibility of EFGM used for metal rheological forming and provides a new method for numerical simulation of rheological forming of complex parts.展开更多
A new form function involving parameters Pi is presented. On the basis of the form function, an initial form of tension structure was Sound by interpolating through the control points on boundary of the structure. The...A new form function involving parameters Pi is presented. On the basis of the form function, an initial form of tension structure was Sound by interpolating through the control points on boundary of the structure. The form function can be controlled by changing beta (1) according to the pre-tension and the boundary of the structure. The final form of a tension structure should be an equilibrium system under the pretension. To examine the nature of the initial form, the FEM was used. Many examples show that the initial form gives a very ideal result for equal or unequal pre-tension in two directions of the structure. In general cases, there is little difference between the initial form and the final one.展开更多
A multi-group pin power reconstruction method that fully exploits nodal information obtained from global coarse mesh solution has been developed.It expands the intra-nodal flux distributions into nonseparable semi-ana...A multi-group pin power reconstruction method that fully exploits nodal information obtained from global coarse mesh solution has been developed.It expands the intra-nodal flux distributions into nonseparable semi-analytic basis functions,and a colorset based form function generating method is proposed,which can accurately model the spectral interaction occurring at assembly interface.To demonstrate its accuracy and applicability to realistic problems, the new method is tested against two benchmark problems,including a mixed-oxide fuel problem.The results show that the new method is comparable in accuracy to fine-mesh methods.展开更多
In the present paper the transformation of symmetric Markov processes by symmetric martingale multiplicative functionals is studied and the corresponding Dirichlet form is formulated.
A non-incremental time-space algorithm is proposed for numerical. analysis of forming process with the inclusion of geometrical, material, contact-frictional nonlinearities. Unlike the widely used Newton-Raphso...A non-incremental time-space algorithm is proposed for numerical. analysis of forming process with the inclusion of geometrical, material, contact-frictional nonlinearities. Unlike the widely used Newton-Raphson counterpart, the present scheme features an iterative solution procedure on entire time and space domain. Validity and feasibility of foe present scheme are further justiced by the numerical investigation herewith presented.展开更多
In order to solve fuzzy mathematical programming with soft constraints,the initial models must first be converted into crisp models.Membership functions are employed to describe the fuzzy right-hand side parameters ne...In order to solve fuzzy mathematical programming with soft constraints,the initial models must first be converted into crisp models.Membership functions are employed to describe the fuzzy right-hand side parameters needed to achieve this conversion.In some cases,echelon form membership functions(EFMFs)are required to depict the actual fuzzy situation.However,due to their discrete properties,fuzzy programming problems with such membership functions cannot be modeled by traditional methods.Motivated by these challenges,this paper introduces a novel absolute value representation modeling approach to formulate fuzzy programming using EFMFs.This approach can translate a discrete model to a continuous one which can then be easily solved.Finally,by means of a numerical example,the effectiveness of our new approach is demonstrated.展开更多
A new and simple method is developed to establish the pseudo orthogonal properties (POP) of the eigenfunction expansion form (EEF) of crack-tip stress complex potential functions for cracked anisotropic an...A new and simple method is developed to establish the pseudo orthogonal properties (POP) of the eigenfunction expansion form (EEF) of crack-tip stress complex potential functions for cracked anisotropic and piezoelectric materials, respectively. Di?erent from previous research, the complex argument separation technique is not required so that cumbersome manipulations are avoided. Moreover, it is shown, di?erent from the previous research too, that the orthogonal properties of the material characteristic matrices A and B are no longer necessary in obtaining the POP of EEF in cracked piezoelectric materials. Of the greatest signi?cance is that the method presented in this paper can be widely extended to treat many kinds of problems concerning path- independent integrals with multi-variables.展开更多
Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level met...Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.展开更多
In this paper a zero-density estimate of the large sieve type is given for the automorphic L-function L f (s,χ),where f is a holomorphic cusp form and χ a Dirichlet character of mod q.
Adaptive fuzzy neural inference systems are used to illustrate the primary nodal number of plant life-forms. Categorization of two candidate areas is carried out using the water-energy dynamic (for Ecuador, South Amer...Adaptive fuzzy neural inference systems are used to illustrate the primary nodal number of plant life-forms. Categorization of two candidate areas is carried out using the water-energy dynamic (for Ecuador, South America) and Macedonia, Southern Europe), within which the life-form spectra are distributed. Genetic optimization methods are used to expand the primary nodal number to the complete number of life-form categories. The distribution of the elements exhibits a stochastic, binomial distribution and the utopia line and curve are summarized which enhance accuracy of the climatic data and of the consequent numbers of plant species occurrences. Expansion of the distribution of each life-form category is approximated within the Z utopia hyperplane with use of the functional approximation algorithm. This process gives additional structure and informative value to the Z plane, enhancing our ability to make informed policy decisions concerning species and ecosystem conservation.展开更多
Most material distribution-based topology optimization methods work on a relaxed form of the optimization problem and then push the solution toward the binary limits.However,when benchmarking these methods,researchers...Most material distribution-based topology optimization methods work on a relaxed form of the optimization problem and then push the solution toward the binary limits.However,when benchmarking these methods,researchers use known solutions to only a single form of benchmark problem.This paper proposes a comparison platform for systematic benchmarking of topology optimization methods using both binary and relaxed forms.A greyness measure is implemented to evaluate how far a solution is from the desired binary form.The well-known ZhouRozvany(ZR)problem is selected as the benchmarking problem here,making use of available global solutions for both its relaxed and binary forms.The recently developed non-penalization Smooth-edged Material Distribution for Optimizing Topology(SEMDOT),well-established Solid Isotropic Material with Penalization(SIMP),and continuation methods are studied on this platform.Interestingly,in most cases,the grayscale solutions obtained by SEMDOT demonstrate better performance in dealing with the ZR problem than SIMP.The reasons are investigated and attributed to the usage of two different regularization techniques,namely,the Heaviside smooth function in SEMDOT and the power-law penalty in SIMP.More importantly,a simple-to-use benchmarking graph is proposed for evaluating newly developed topology optimization methods.展开更多
Aim The general arbitrary cracked problem in an elastic plane was discussed. Methods For the purpose of acquiring the solution of the problem, a new formulation on the problem was proposed. Compared with the classic...Aim The general arbitrary cracked problem in an elastic plane was discussed. Methods For the purpose of acquiring the solution of the problem, a new formulation on the problem was proposed. Compared with the classical plane elastic crack model, only the known conditions were revised in the new formulation, which are greatly convenient to solve the problem, and no other new condition was given. Results and Conclusion The general exact analytic solution is given here based on the formulation though the problem is very complicated. Furthermore, the stress intensity factors K Ⅰ, K Ⅱ of the problem are also given.展开更多
文摘In this study,a trifunctional strategy was developed to prepare a confined Ni-based catalyst(Ni-CeO_(2)@SiO_(2))for dry reforming of methane(DRM)of two main greenhouse gases-CO_(2)and CH_(4).The Ni-CeO_(2)@SiO_(2)catalyst was fabricated by utilizing the confinement effect of the SiO_(2)shell and the synergistic interaction between Ni-Ce and the decoking effect of CeO_(2).The catalysts were systematically characterized via X-ray diffraction,N_(2 )adsorption/desorption,transmission electron microscopy,energy dispersive X-ray spectroscopy,hydrogen temperature reduction and desorption set by program,oxygen temperature program desorption,Raman spectroscopy,thermogravimetric analysis,and in situ diffuse reflectance infrared Fourier transform spectroscopy measurements to reveal their physicochemical properties and reaction mechanism.The Ni-CeO_(2)@SiO_(2)catalyst exhibited higher activity and stability than the catalyst synthesized via the traditional impregnation method.In addition,no carbon deposition was detected over Ni-CeO_(2)@SiO_(2)after a 100 h durability test at 800℃,and the average particle size of Ni nanoparticles(NPs)in the catalyst increased from 5.01 to 5.77 nm.Remarkably,Ni-CeO_(2)@SiO_(2)also exhibited superior low-temperature stability;no coke deposition was observed when the catalyst was reacted at 600℃ for 20 h.The high coking and sintering resistance of this confined Ni-based DRM catalyst can be attributed to its trifunctional effect.The trifunctional strategy developed in this study could be used as a guideline to design other high-performance catalysts for CO_(2)and CH4 dry forming and accelerate their industrialization.
基金Projects(50175035 50375024) supported by the National Natural Science Foundation of China+1 种基金 Project(E0210024) sup ported by the Natural Science Foundation of Fujian Province of China Project(2002H044) supported by the Program of High Science andTec
文摘A new technology of functionally graded materials(FGM) mould fabricated by plasma spraying and arc spraying was developed. According to applied characteristic of plastic mould, the reasonable coatings of FGM were designed and their microstructures were analyzed. At the same time, some key problems were solved including spray mould fabricating, FGM forming and demoulding, etc. The results show that the service performance of the FGM mould is much more excellent than the one composed of the traditional materials, and the life span can also be greatly increased. The technology will have a significant influence on materials development in mould industry.
基金supported by National Foundation of Natural Science under the Grant 11071216
文摘In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems （with same stiffness matrix but different right-hand side） on the fine mesh space. The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the Mgorithm produces a numerical solution with the optimal asymptotic H^2-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.
基金support from the Portuguese Foundation for Science and Technology (FCT) via the projects PTDC/EMS-TEC/2404/2012, and PTDC/EMS-TEC/1805/2012FEDER funds through the program COMPETE-‘‘Programa Operacional Factores de Competitividade’’ is greatly acknowledged
文摘The forming limit diagram（FLD） is an important tool to be used when characterizing the formability of metallic sheets used in metal forming processes. Experimental measurement and determination of the FLD is timeconsuming and therefore the analytical prediction based on theory of plasticity and instability criteria allows a direct and efficient methodology to obtain critical values at different loading paths, thus carrying significant practical importance.However, the accuracy of the plastic instability prediction is strongly dependent on the choice of the material constitutive model [1–3]. Particularly for materials with hexagonal close packed（HCP） crystallographic structure, they have a very limited number of active slip systems at room temperature and demonstrate a strong asymmetry between yielding in tension and compression [4, 5]. Not only the magnitude of the yield locus changes, but also the shape of the yield surface is evolving during the plastic deformation [4]. Conventional phenomenological constitutive models of plasticity fail to capture this unconventional mechanical behavior [4, 6]. Cazacu and Plunkett [6] have proposed generic yield criteria, by using the transformed principal stress, to account for the initial plastic anisotropy and strength differential（SD） effect simultaneously. In this contribution, a generic FLD MATLAB script was developed based on Marciniak–Kuczynski analytical theory and applied to predict the localized necking. The influence of asymmetrical effect on the FLD was evaluated. Several yield functions such as von Mises, Hill, Barlat89, and Cazacu06 were incorporated into analysis. The paper also presents and discusses the influence of different hardening laws on the formability of materials with HCP crystal structures. The findings indicate that the plastic instability theory coupled with Cazacu model can adequately predict the onset of localized necking for HCP materials under different strain paths.
文摘Long before,the research of form has already entered its heyday,but the study of function is a somewhat new branch of language study.This essay is an attempt to take a brief look at the most significant distinction between"form and function",and then aims at making clear the definition of"form and function"in discourse analysis.
基金Key project(02103) supported by National Education Department of ChinaKey project(02A008) supported by the Education Department of Hunan Province,China+3 种基金Project(2005090) supported by Central South University of Forestry and TechnologyProject(03JJY3007) supported by the Natural Science Foundation of Hunan Province,ChinaProject supported by the Rewarding Project for Excellent PhD Thesis of Hunan Province,ChinaProject(07031B) supported by Scientific Research Fund of Central South University of Forestry and Technology
文摘The element-free method is a new numerical technique presented in recent years.It uses the moving least square(MLS) approximation as its shape function,and it is determined by the basic function and weight function.The weight function is the mainly determining factor,so it greatly affects the accuracy of the computational results.The element-free Galerkin method(EFGM) was applied for the solution to plastic large deformation.The simulation of metal rheological forming was successfully done by programming and its results were visualized by using the plotting and data analyses software Tecplot.Then plastic strain under different stages during rheological forming and the three principal stresses at the last deformation were obtained.The example shows the feasibility of EFGM used for metal rheological forming and provides a new method for numerical simulation of rheological forming of complex parts.
文摘A new form function involving parameters Pi is presented. On the basis of the form function, an initial form of tension structure was Sound by interpolating through the control points on boundary of the structure. The form function can be controlled by changing beta (1) according to the pre-tension and the boundary of the structure. The final form of a tension structure should be an equilibrium system under the pretension. To examine the nature of the initial form, the FEM was used. Many examples show that the initial form gives a very ideal result for equal or unequal pre-tension in two directions of the structure. In general cases, there is little difference between the initial form and the final one.
基金Partially supported by the National Natural Science Foundation of China via research project 10605016
文摘A multi-group pin power reconstruction method that fully exploits nodal information obtained from global coarse mesh solution has been developed.It expands the intra-nodal flux distributions into nonseparable semi-analytic basis functions,and a colorset based form function generating method is proposed,which can accurately model the spectral interaction occurring at assembly interface.To demonstrate its accuracy and applicability to realistic problems, the new method is tested against two benchmark problems,including a mixed-oxide fuel problem.The results show that the new method is comparable in accuracy to fine-mesh methods.
基金in partby the National Natural Science Founda-tion of China(1 950 1 0 36)
文摘In the present paper the transformation of symmetric Markov processes by symmetric martingale multiplicative functionals is studied and the corresponding Dirichlet form is formulated.
文摘A non-incremental time-space algorithm is proposed for numerical. analysis of forming process with the inclusion of geometrical, material, contact-frictional nonlinearities. Unlike the widely used Newton-Raphson counterpart, the present scheme features an iterative solution procedure on entire time and space domain. Validity and feasibility of foe present scheme are further justiced by the numerical investigation herewith presented.
文摘In order to solve fuzzy mathematical programming with soft constraints,the initial models must first be converted into crisp models.Membership functions are employed to describe the fuzzy right-hand side parameters needed to achieve this conversion.In some cases,echelon form membership functions(EFMFs)are required to depict the actual fuzzy situation.However,due to their discrete properties,fuzzy programming problems with such membership functions cannot be modeled by traditional methods.Motivated by these challenges,this paper introduces a novel absolute value representation modeling approach to formulate fuzzy programming using EFMFs.This approach can translate a discrete model to a continuous one which can then be easily solved.Finally,by means of a numerical example,the effectiveness of our new approach is demonstrated.
基金Project supported by the Natural Science Foundation of Shaanxi Province (No.2002A18) and the Doctorate Foundation of Xi’an Jiao-Tong University.
文摘A new and simple method is developed to establish the pseudo orthogonal properties (POP) of the eigenfunction expansion form (EEF) of crack-tip stress complex potential functions for cracked anisotropic and piezoelectric materials, respectively. Di?erent from previous research, the complex argument separation technique is not required so that cumbersome manipulations are avoided. Moreover, it is shown, di?erent from the previous research too, that the orthogonal properties of the material characteristic matrices A and B are no longer necessary in obtaining the POP of EEF in cracked piezoelectric materials. Of the greatest signi?cance is that the method presented in this paper can be widely extended to treat many kinds of problems concerning path- independent integrals with multi-variables.
文摘Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.
基金Supported by the NNSF of China（11071186）Supported by the Science Foundation for the Excellent Youth Scholars of Shanghai（ssc08017）Supported by the Doctoral Research Fund of Shanghai Ocean University
文摘In this paper a zero-density estimate of the large sieve type is given for the automorphic L-function L f (s,χ),where f is a holomorphic cusp form and χ a Dirichlet character of mod q.
文摘Adaptive fuzzy neural inference systems are used to illustrate the primary nodal number of plant life-forms. Categorization of two candidate areas is carried out using the water-energy dynamic (for Ecuador, South America) and Macedonia, Southern Europe), within which the life-form spectra are distributed. Genetic optimization methods are used to expand the primary nodal number to the complete number of life-form categories. The distribution of the elements exhibits a stochastic, binomial distribution and the utopia line and curve are summarized which enhance accuracy of the climatic data and of the consequent numbers of plant species occurrences. Expansion of the distribution of each life-form category is approximated within the Z utopia hyperplane with use of the functional approximation algorithm. This process gives additional structure and informative value to the Z plane, enhancing our ability to make informed policy decisions concerning species and ecosystem conservation.
文摘Most material distribution-based topology optimization methods work on a relaxed form of the optimization problem and then push the solution toward the binary limits.However,when benchmarking these methods,researchers use known solutions to only a single form of benchmark problem.This paper proposes a comparison platform for systematic benchmarking of topology optimization methods using both binary and relaxed forms.A greyness measure is implemented to evaluate how far a solution is from the desired binary form.The well-known ZhouRozvany(ZR)problem is selected as the benchmarking problem here,making use of available global solutions for both its relaxed and binary forms.The recently developed non-penalization Smooth-edged Material Distribution for Optimizing Topology(SEMDOT),well-established Solid Isotropic Material with Penalization(SIMP),and continuation methods are studied on this platform.Interestingly,in most cases,the grayscale solutions obtained by SEMDOT demonstrate better performance in dealing with the ZR problem than SIMP.The reasons are investigated and attributed to the usage of two different regularization techniques,namely,the Heaviside smooth function in SEMDOT and the power-law penalty in SIMP.More importantly,a simple-to-use benchmarking graph is proposed for evaluating newly developed topology optimization methods.
文摘Aim The general arbitrary cracked problem in an elastic plane was discussed. Methods For the purpose of acquiring the solution of the problem, a new formulation on the problem was proposed. Compared with the classical plane elastic crack model, only the known conditions were revised in the new formulation, which are greatly convenient to solve the problem, and no other new condition was given. Results and Conclusion The general exact analytic solution is given here based on the formulation though the problem is very complicated. Furthermore, the stress intensity factors K Ⅰ, K Ⅱ of the problem are also given.
