Einstein’s field equation is a highly general equation consisting of sixteen equations. However, the equation itself provides limited information about the universe unless it is solved with different boundary conditi...Einstein’s field equation is a highly general equation consisting of sixteen equations. However, the equation itself provides limited information about the universe unless it is solved with different boundary conditions. Multiple solutions have been utilized to predict cosmic scales, and among them, the Friedmann-Lemaître-Robertson-Walker solution that is the back-bone of the development into today standard model of modern cosmology: The Λ-CDM model. However, this is naturally not the only solution to Einstein’s field equation. We will investigate the extremal solutions of the Reissner-Nordström, Kerr, and Kerr-Newman metrics. Interestingly, in their extremal cases, these solutions yield identical predictions for horizons and escape velocity. These solutions can be employed to formulate a new cosmological model that resembles the Friedmann equation. However, a significant distinction arises in the extremal universe solution, which does not necessitate the ad hoc insertion of the cosmological constant;instead, it emerges naturally from the derivation itself. To the best of our knowledge, all other solutions relying on the cosmological constant do so by initially ad hoc inserting it into Einstein’s field equation. This clarification unveils the true nature of the cosmological constant, suggesting that it serves as a correction factor for strong gravitational fields, accurately predicting real-world cosmological phenomena only within the extremal solutions of the discussed metrics, all derived strictly from Einstein’s field equation.展开更多
This paper is presenting a new method for making first-order systems of nonlinear autonomous ODEs that exhibit limit cycles with a specific geometric shape in two and three dimensions, or systems of ODEs where surface...This paper is presenting a new method for making first-order systems of nonlinear autonomous ODEs that exhibit limit cycles with a specific geometric shape in two and three dimensions, or systems of ODEs where surfaces in three dimensions have attractor behavior. The method is to make the general solutions first by using the exponential function, sine, and cosine. We are building up the general solutions bit for bit according to constant terms that contain the formula of the desired limit cycle, and differentiating them. In Part One, we used only formulas for closed curves where all parts of the formula were of the same degree. In order to use many other formulas for closed curves, the method in this paper is to introduce an additional variable, and we will get an additional ODE. We will choose the part of the formula with the highest degree and multiply the other parts with an extra variable, so that all parts of the formula have the same degree, creating a constant term containing this new formula. We will place it under the fraction line in the solutions, building up the rest of the solutions according to this constant term and differentiating. Keeping this extra variable constant, we will achieve almost the desired result. Using the methods described in this paper, it is possible to make some systems of nonlinear ODEs that are exhibiting limit cycles with a distinct geometric shape in two or three dimensions and some surfaces having attractor behavior, where not all parts of the formulas are the same degree. The pictures show the result.展开更多
By asing the nonclassical method of symmetry reductions, the exact solutions for general variable coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and non...By asing the nonclassical method of symmetry reductions, the exact solutions for general variable coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and nonuniformity terms don't exist, the multisoliton solutions are found and the corresponding Painleve II type equation for the variable coefficient KdV equation is given.展开更多
First, based on the basic equations of two-dimensional piezoelectroelasticity, a displacement function is introduced and the general solution is then derived. Utilizing the generalized Almansi's theorem, the gener...First, based on the basic equations of two-dimensional piezoelectroelasticity, a displacement function is introduced and the general solution is then derived. Utilizing the generalized Almansi's theorem, the general solution is so simplified that all physical quantities can be expressed by three 'harmonic functions'. Second, solutions of problems of a wedge loaded by point forces and point charge at the apex are also obtained in the paper. These solutions can be degenerated to those of problems of point forces and point charge acting on the line boundary of a piezoelectric half-plane.展开更多
A general solution for 3D Stokes flow is given which is different from, and more compact than the exist ing ones and more compact than them in that it involves only two scalar harmonic functions. The general solution ...A general solution for 3D Stokes flow is given which is different from, and more compact than the exist ing ones and more compact than them in that it involves only two scalar harmonic functions. The general solution deduced is combined with the potential theory method to study the Stokes flow induced by a rigid plate of arbitrary shape trans lating along the direction normal to it in an unbounded fluid. The boundary integral equation governing this problem is derived. When the plate is elliptic, exact analytical results are obtained not only for the drag force but also for the ve locity distributions. These results include and complete the ones available for a circular plate. Numerical examples are provided to illustrate the main results for circular and ellip tic plates. In particular, the elliptic eccentricity of a plate is shown to exhibit significant influences.展开更多
We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ||po||L∞ is appropriate...We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ||po||L∞ is appropriate small and 1 〈 γ 〈 6/5. Here the initial density could have vacuum and we do not require that the initial energy is small.展开更多
The effect of kinematic hardening behavior on the shakedown behaviors of structure has been investigated by performing shakedown analysis for some specific problems. The results obtained only show that the shakedown l...The effect of kinematic hardening behavior on the shakedown behaviors of structure has been investigated by performing shakedown analysis for some specific problems. The results obtained only show that the shakedown limit loads of structures with kinematic hardening model are larger than or equal to those with perfectly plastic model of the same initial yield stress. To further investigate the rules governing the different shakedown behaviors of kinematic hardening structures, the extended shakedown theorem for limited kinematic hardening is applied, the shakedown condition is then proposed, and a general analytical solution for the structural shakedown limit load is thus derived. The analytical shakedown limit loads for fully reversed cyclic loading and non-fully reversed cyclic loading are then given based on the general solution. The resulting analytical solution is applied to some specific problems: a hollow specimen subjected to tension and torsion, a flanged pipe subjected to pressure and axial force and a square plate with small central hole subjected to biaxial tension. The results obtained are compared with those in literatures, they are consistent with each other. Based on the resulting general analytical solution, rules governing the general effects of kinematic hardening behavior on the shakedown behavior of structure are clearly.展开更多
The thermoelastic plane problems of two-dimensional decagonal quasicrystals(QCs)are systematically investigated.By introducing a displacement function,the problem of thermoelastic plane problems can be simplified to a...The thermoelastic plane problems of two-dimensional decagonal quasicrystals(QCs)are systematically investigated.By introducing a displacement function,the problem of thermoelastic plane problems can be simplified to an eighth-order partial differential governing equation,and then general solutions are presented through an operator method.By virtue of the Almansi′s theorem,the general solutions are further established,and all expressions for the phonon,phason and thermal fields are described in terms of the potential functions.As an application of the general solution,for a steady point heat source in a semi-infinite quasicrystal plane,the closed form solutions are presented by four newly induced harmonic functions.展开更多
In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution ...In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution to the special case for z=x.Furthermore,a more general form of lump solution of the equation is found which possesses seven arbitrary parameters and four constraint conditions.By cutting the lump by the induced soliton(s),lumpoff and instanton/rogue wave solutions are also constructed by the more general form of lump solution.展开更多
In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are est...In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × (n + pr). The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair (A, B) and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.展开更多
In this paper we consider one dimensional mean-field backward stochastic differential equations(BSDEs)under weak assumptions on the coefficient.Unlike[3],the generator of our mean-field BSDEs depends not only on the s...In this paper we consider one dimensional mean-field backward stochastic differential equations(BSDEs)under weak assumptions on the coefficient.Unlike[3],the generator of our mean-field BSDEs depends not only on the solution(Y,Z)but also on the law PY of Y.The first part of the paper is devoted to the existence and uniqueness of solutions in Lp,1<p≤2,where the monotonicity conditions are satisfied.Next,we show that if the generator/is uniformly continuous in(μ,y,z),uniformly with respect to(t,ω) and if the terminal valueξbelongs to Lp(Ω,F,P)with 1<p≤2,the mean-field BSDE has a unique Lp solution.展开更多
In this paper, we will present a new method for making first-order systems of nonlinear autonomous ODEs that exhibit limit cycles with a specific geometric shape in two and three dimensions, or systems of ODEs where s...In this paper, we will present a new method for making first-order systems of nonlinear autonomous ODEs that exhibit limit cycles with a specific geometric shape in two and three dimensions, or systems of ODEs where surfaces in three dimensions have attractor behavior. The method is to make the general solutions first by using the exponential function, sine and cosine. We are building up the general solutions bit for bit according to the constant terms that contain the formula of the desired limit cycle, and differentiating them. We will obtain a system of ODEs with the desired behavior. We design the general solutions for a distinct purpose. Using the methods described in this paper, it is possible to make some systems of nonlinear ODEs that are exhibiting limit cycles with a distinct geometric shape in two or three dimensions, and some surfaces having attractor behavior. The pictures show the result.展开更多
Li et al. (2015) claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations. In this paper, we demonstrate that this is not true in a general case and that we in fact...Li et al. (2015) claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations. In this paper, we demonstrate that this is not true in a general case and that we in fact need three scalar harmonic functions to represent the general solution of Stokes equations (Venkatalaxmi et al., 2004).展开更多
In this paper, a general solution for three-dimensional staticpiezothermoleastic prob- lems of crystal class 6mm solids ispresented. The general solution involves four piezoelastic potentialfunctions and a piezothermo...In this paper, a general solution for three-dimensional staticpiezothermoleastic prob- lems of crystal class 6mm solids ispresented. The general solution involves four piezoelastic potentialfunctions and a piezothermoelastic potential function, of which fourpiezoelastic potential functions are governed by weighted harmonicdifferential equations. Compared with the general solution given byAshida et al., in which seven potential functions are introduced, thegeneral solution proposed in the Present paper is more rigorouslyderived.展开更多
The coupling feature of transversely isotropic magnetoelectroelastic solids are governed by a system of five partial differential equations with respect to the elastic displacements, the electric potential and the mag...The coupling feature of transversely isotropic magnetoelectroelastic solids are governed by a system of five partial differential equations with respect to the elastic displacements, the electric potential and the magnetic potential. Based on the potential theory, the coupled equations are reduced to the five uncoupled generalized Laplace equations with respect to five potential functions. Further, the elastic fields and electromagnetic fields are expressed in terms of the potential functions. These expressions construct the general solution of transversely isotropic magnetoelectroelastic media.展开更多
Solving partial differential equations Has not only theoretical significance, but also practical value. In this paper, by the property of conjugate operator, we give a method to construct the general solutions of a sy...Solving partial differential equations Has not only theoretical significance, but also practical value. In this paper, by the property of conjugate operator, we give a method to construct the general solutions of a system of partial differential equations.展开更多
It is proved mathematically in this paper that the strain-stress function F onthe cylindrical shell theory suggested by Vlasov [5] will give out the general solution of the simultaneous partial differential equations...It is proved mathematically in this paper that the strain-stress function F onthe cylindrical shell theory suggested by Vlasov [5] will give out the general solution of the simultaneous partial differential equations of cylindrical shell problem. That is to say. there is no any solution Of the simultaneous partial differential equations can be omitted due to Vlasov's suggestion. The conclusion suggested in this paper is helpful to the well-known Vlasov's method.展开更多
A new type of general solution of thermoelasticity is derived from the linearized basic equations for coupled thermoelastic problem. In the case of quasi-static problem, the present general solution is simpler since i...A new type of general solution of thermoelasticity is derived from the linearized basic equations for coupled thermoelastic problem. In the case of quasi-static problem, the present general solution is simpler since it involves one less potential function than Blot's solution.展开更多
In this paper, a new method, the exact analytic method, is presented on the basis of step reduction method. By this method, the general solution for the bending of nonhomogenous circular plates and circular plates wit...In this paper, a new method, the exact analytic method, is presented on the basis of step reduction method. By this method, the general solution for the bending of nonhomogenous circular plates and circular plates with a circular hole at the center resting, on an elastfc foundation is obtained under arbitrary axial symmetrical loads' and boundary conditions. The uniform convergence of the solution is proved. This general solution can also he applied directly to the bending of circular plates without elastic foundation. Finally, it is only necessary to solve a set of binary linear algebraic equation. Numerical examples are given at the end of this paper which indicate satisfactory results of stress resultants and displacements can be obtained by the present method.展开更多
This is one of the applications of Part (Ⅰ), in which the angular stiffness, the lateral stiffness and the corresponding stress distributions of C_shaped bellows were calculated. The bellows was divided into protrudi...This is one of the applications of Part (Ⅰ), in which the angular stiffness, the lateral stiffness and the corresponding stress distributions of C_shaped bellows were calculated. The bellows was divided into protruding sections and concave sections for the use of the general solution (Ⅰ), but the continuity of the stress resultants and the deformations at each joint of the sections were entirely satisfied. The present results were compared with those of the other theories and experiments, and are also tested by the numerically integral method. It is shown that the governing equation and the general solution (Ⅰ) are very effective.展开更多
文摘Einstein’s field equation is a highly general equation consisting of sixteen equations. However, the equation itself provides limited information about the universe unless it is solved with different boundary conditions. Multiple solutions have been utilized to predict cosmic scales, and among them, the Friedmann-Lemaître-Robertson-Walker solution that is the back-bone of the development into today standard model of modern cosmology: The Λ-CDM model. However, this is naturally not the only solution to Einstein’s field equation. We will investigate the extremal solutions of the Reissner-Nordström, Kerr, and Kerr-Newman metrics. Interestingly, in their extremal cases, these solutions yield identical predictions for horizons and escape velocity. These solutions can be employed to formulate a new cosmological model that resembles the Friedmann equation. However, a significant distinction arises in the extremal universe solution, which does not necessitate the ad hoc insertion of the cosmological constant;instead, it emerges naturally from the derivation itself. To the best of our knowledge, all other solutions relying on the cosmological constant do so by initially ad hoc inserting it into Einstein’s field equation. This clarification unveils the true nature of the cosmological constant, suggesting that it serves as a correction factor for strong gravitational fields, accurately predicting real-world cosmological phenomena only within the extremal solutions of the discussed metrics, all derived strictly from Einstein’s field equation.
文摘This paper is presenting a new method for making first-order systems of nonlinear autonomous ODEs that exhibit limit cycles with a specific geometric shape in two and three dimensions, or systems of ODEs where surfaces in three dimensions have attractor behavior. The method is to make the general solutions first by using the exponential function, sine, and cosine. We are building up the general solutions bit for bit according to constant terms that contain the formula of the desired limit cycle, and differentiating them. In Part One, we used only formulas for closed curves where all parts of the formula were of the same degree. In order to use many other formulas for closed curves, the method in this paper is to introduce an additional variable, and we will get an additional ODE. We will choose the part of the formula with the highest degree and multiply the other parts with an extra variable, so that all parts of the formula have the same degree, creating a constant term containing this new formula. We will place it under the fraction line in the solutions, building up the rest of the solutions according to this constant term and differentiating. Keeping this extra variable constant, we will achieve almost the desired result. Using the methods described in this paper, it is possible to make some systems of nonlinear ODEs that are exhibiting limit cycles with a distinct geometric shape in two or three dimensions and some surfaces having attractor behavior, where not all parts of the formulas are the same degree. The pictures show the result.
基金Supported by the Develop Programme Foundation of the National Basic research(G1 9990 3 2 80 1 )
文摘By asing the nonclassical method of symmetry reductions, the exact solutions for general variable coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and nonuniformity terms don't exist, the multisoliton solutions are found and the corresponding Painleve II type equation for the variable coefficient KdV equation is given.
文摘First, based on the basic equations of two-dimensional piezoelectroelasticity, a displacement function is introduced and the general solution is then derived. Utilizing the generalized Almansi's theorem, the general solution is so simplified that all physical quantities can be expressed by three 'harmonic functions'. Second, solutions of problems of a wedge loaded by point forces and point charge at the apex are also obtained in the paper. These solutions can be degenerated to those of problems of point forces and point charge acting on the line boundary of a piezoelectric half-plane.
基金supported by the National Natural Science Foundation of China(11102171)the Program for New Century Excellent Talents in University of Ministry of Education of China(NCET-13-0973)
文摘A general solution for 3D Stokes flow is given which is different from, and more compact than the exist ing ones and more compact than them in that it involves only two scalar harmonic functions. The general solution deduced is combined with the potential theory method to study the Stokes flow induced by a rigid plate of arbitrary shape trans lating along the direction normal to it in an unbounded fluid. The boundary integral equation governing this problem is derived. When the plate is elliptic, exact analytical results are obtained not only for the drag force but also for the ve locity distributions. These results include and complete the ones available for a circular plate. Numerical examples are provided to illustrate the main results for circular and ellip tic plates. In particular, the elliptic eccentricity of a plate is shown to exhibit significant influences.
基金supported by National Natural Science Foundation of China (11001090)the Fundamental Research Funds for the Central Universities(11QZR16)
文摘We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ||po||L∞ is appropriate small and 1 〈 γ 〈 6/5. Here the initial density could have vacuum and we do not require that the initial energy is small.
基金Supported by National Science and Technology Major Project of China(Grant No.2013ZX04003031)National Natural Science Foundation of China(Grant No.51575474)+1 种基金Hebei Provincial College Innovation Team Leader Training Program of China(Grant No.LJRC012)Hebei Provincial Natural Science Foundation of China(Grant No.E2015203223)
文摘The effect of kinematic hardening behavior on the shakedown behaviors of structure has been investigated by performing shakedown analysis for some specific problems. The results obtained only show that the shakedown limit loads of structures with kinematic hardening model are larger than or equal to those with perfectly plastic model of the same initial yield stress. To further investigate the rules governing the different shakedown behaviors of kinematic hardening structures, the extended shakedown theorem for limited kinematic hardening is applied, the shakedown condition is then proposed, and a general analytical solution for the structural shakedown limit load is thus derived. The analytical shakedown limit loads for fully reversed cyclic loading and non-fully reversed cyclic loading are then given based on the general solution. The resulting analytical solution is applied to some specific problems: a hollow specimen subjected to tension and torsion, a flanged pipe subjected to pressure and axial force and a square plate with small central hole subjected to biaxial tension. The results obtained are compared with those in literatures, they are consistent with each other. Based on the resulting general analytical solution, rules governing the general effects of kinematic hardening behavior on the shakedown behavior of structure are clearly.
基金supported by the National Natural Sci-ence Foundation of China(11172319)the Chinese Univer-sities Scientific Fund(2011JS046,2013BH008)+2 种基金the Opening Fund of State Key Laboratory of Nonlinear Mechanicsthe Program for New Century Excellent Talents in Univer-sity(NCET-13-0552)the National Science Foundation for Post-doctoral Scientists of China(2013M541086)
文摘The thermoelastic plane problems of two-dimensional decagonal quasicrystals(QCs)are systematically investigated.By introducing a displacement function,the problem of thermoelastic plane problems can be simplified to an eighth-order partial differential governing equation,and then general solutions are presented through an operator method.By virtue of the Almansi′s theorem,the general solutions are further established,and all expressions for the phonon,phason and thermal fields are described in terms of the potential functions.As an application of the general solution,for a steady point heat source in a semi-infinite quasicrystal plane,the closed form solutions are presented by four newly induced harmonic functions.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11675084 and 11435005)the Fund from the Educational Commission of Zhejiang Province,China(Grant No.Y201737177)+1 种基金Ningbo Natural Science Foundation(Grant No.2015A610159)the K C Wong Magna Fund in Ningbo University
文摘In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution to the special case for z=x.Furthermore,a more general form of lump solution of the equation is found which possesses seven arbitrary parameters and four constraint conditions.By cutting the lump by the induced soliton(s),lumpoff and instanton/rogue wave solutions are also constructed by the more general form of lump solution.
基金This work was supported by the Chinese Outstanding Youth Foundation(No.69925308)Program for Changjiang Scholars and Innovative ResearchTeam in University.
文摘In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × (n + pr). The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair (A, B) and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.
基金supported in part by the NSFC(11222110,11871037)Shandong Province(JQ201202)+1 种基金NSFC-RS(11661130148,NA150344)111 Project(B12023)。
文摘In this paper we consider one dimensional mean-field backward stochastic differential equations(BSDEs)under weak assumptions on the coefficient.Unlike[3],the generator of our mean-field BSDEs depends not only on the solution(Y,Z)but also on the law PY of Y.The first part of the paper is devoted to the existence and uniqueness of solutions in Lp,1<p≤2,where the monotonicity conditions are satisfied.Next,we show that if the generator/is uniformly continuous in(μ,y,z),uniformly with respect to(t,ω) and if the terminal valueξbelongs to Lp(Ω,F,P)with 1<p≤2,the mean-field BSDE has a unique Lp solution.
文摘In this paper, we will present a new method for making first-order systems of nonlinear autonomous ODEs that exhibit limit cycles with a specific geometric shape in two and three dimensions, or systems of ODEs where surfaces in three dimensions have attractor behavior. The method is to make the general solutions first by using the exponential function, sine and cosine. We are building up the general solutions bit for bit according to the constant terms that contain the formula of the desired limit cycle, and differentiating them. We will obtain a system of ODEs with the desired behavior. We design the general solutions for a distinct purpose. Using the methods described in this paper, it is possible to make some systems of nonlinear ODEs that are exhibiting limit cycles with a distinct geometric shape in two or three dimensions, and some surfaces having attractor behavior. The pictures show the result.
文摘Li et al. (2015) claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations. In this paper, we demonstrate that this is not true in a general case and that we in fact need three scalar harmonic functions to represent the general solution of Stokes equations (Venkatalaxmi et al., 2004).
基金the National Natural Science Foundation of China(19872060)
文摘In this paper, a general solution for three-dimensional staticpiezothermoleastic prob- lems of crystal class 6mm solids ispresented. The general solution involves four piezoelastic potentialfunctions and a piezothermoelastic potential function, of which fourpiezoelastic potential functions are governed by weighted harmonicdifferential equations. Compared with the general solution given byAshida et al., in which seven potential functions are introduced, thegeneral solution proposed in the Present paper is more rigorouslyderived.
文摘The coupling feature of transversely isotropic magnetoelectroelastic solids are governed by a system of five partial differential equations with respect to the elastic displacements, the electric potential and the magnetic potential. Based on the potential theory, the coupled equations are reduced to the five uncoupled generalized Laplace equations with respect to five potential functions. Further, the elastic fields and electromagnetic fields are expressed in terms of the potential functions. These expressions construct the general solution of transversely isotropic magnetoelectroelastic media.
文摘Solving partial differential equations Has not only theoretical significance, but also practical value. In this paper, by the property of conjugate operator, we give a method to construct the general solutions of a system of partial differential equations.
文摘It is proved mathematically in this paper that the strain-stress function F onthe cylindrical shell theory suggested by Vlasov [5] will give out the general solution of the simultaneous partial differential equations of cylindrical shell problem. That is to say. there is no any solution Of the simultaneous partial differential equations can be omitted due to Vlasov's suggestion. The conclusion suggested in this paper is helpful to the well-known Vlasov's method.
文摘A new type of general solution of thermoelasticity is derived from the linearized basic equations for coupled thermoelastic problem. In the case of quasi-static problem, the present general solution is simpler since it involves one less potential function than Blot's solution.
文摘In this paper, a new method, the exact analytic method, is presented on the basis of step reduction method. By this method, the general solution for the bending of nonhomogenous circular plates and circular plates with a circular hole at the center resting, on an elastfc foundation is obtained under arbitrary axial symmetrical loads' and boundary conditions. The uniform convergence of the solution is proved. This general solution can also he applied directly to the bending of circular plates without elastic foundation. Finally, it is only necessary to solve a set of binary linear algebraic equation. Numerical examples are given at the end of this paper which indicate satisfactory results of stress resultants and displacements can be obtained by the present method.
文摘This is one of the applications of Part (Ⅰ), in which the angular stiffness, the lateral stiffness and the corresponding stress distributions of C_shaped bellows were calculated. The bellows was divided into protruding sections and concave sections for the use of the general solution (Ⅰ), but the continuity of the stress resultants and the deformations at each joint of the sections were entirely satisfied. The present results were compared with those of the other theories and experiments, and are also tested by the numerically integral method. It is shown that the governing equation and the general solution (Ⅰ) are very effective.