By the analysis for the vectors of a wave field in the cylindrical coordinate and Sommerfeld＇s identity as well as Green＇s functions of Stokes＇ solution pertaining the conventional elastic dynamic equation, the res...By the analysis for the vectors of a wave field in the cylindrical coordinate and Sommerfeld＇s identity as well as Green＇s functions of Stokes＇ solution pertaining the conventional elastic dynamic equation, the results of Green＇s function in an infinite space of an axisymmetric coordinate are shown in this paper. After employing a supplementary influence field and the boundary conditions in the free surface of a senti-space, the authors obtain the solutions of Green＇s function for Lamb＇s dynamic problem. Besides, the vertical displacement uzz and the radial displacement urz can match Lamb＇s previous results, and the solutions of the linear expansion source u^r and the linear torsional source uee are also given in the paper. The authors reveal that Green＇s function of Stokes＇ solution in the semi-space is a comprehensive form of solution expressing the dynamic Lamb＇s problem for various situations. It may benefit the investigation of deepening and development of Lamb＇s problems and solution for pertinent dynamic problems conveniently.展开更多
When dynamic force is applied to a saturated porous soil, drainage is common. In this paper, the saturated porous soil with a two-phase saturated medium is simulated, and Lamb＇s integral formulas with drainage and st...When dynamic force is applied to a saturated porous soil, drainage is common. In this paper, the saturated porous soil with a two-phase saturated medium is simulated, and Lamb＇s integral formulas with drainage and stress formulas for a two-phase saturated medium are given based on Biot＇s equation and Betti＇s theorem （the reciprocal theorem）. According to the basic solution to Biot＇s equation, Green＇s function Gij and three terms of Green＇s function G4i, Gi4, and G44 of a two-phase saturated medium subject to a concentrated force on a spherical coordinate are presented. The displacement field with drainage, the magnitude of drainage, and the pore pressure of the center explosion source are obtained in computation. The results of the classical Sharpe＇s solutions and the solutions of the two-phase saturated medium that decays to a single-phase medium are compared. Good agreement is observed.展开更多
Novel exact solutions of one-dimensional transient dynamic piezoelectric problems for thickness polarized layers and disks, or length polarized rods, are obtained. The solutions are derived using a time-domain Green’...Novel exact solutions of one-dimensional transient dynamic piezoelectric problems for thickness polarized layers and disks, or length polarized rods, are obtained. The solutions are derived using a time-domain Green’s function method that leads to an exact analytical recursive procedure which is applicable for a wide variety of boundary conditions including nonlinear cases. A nonlinear damper boundary condition is considered in more detail. The corresponding nonlinear relationship between stresses and velocities at a current time moment is used in the recursive procedure. In addition to the exact recursive procedure that is effective for calculations, some new practically important explicit exact solutions are presented. Several examples of the time behavior of the output electric potential difference are given to illustrate the effectiveness of the proposed exact approach.展开更多
The paper is devoted to dynamic design of thick orthotropic cantilever plates by applying the bimoment theory of plates, which takes into account the forces, moments and bimoments;and the theory takes into account non...The paper is devoted to dynamic design of thick orthotropic cantilever plates by applying the bimoment theory of plates, which takes into account the forces, moments and bimoments;and the theory takes into account nonlinear law of displacements distribution in cross section of the plate. The methods for constructing bimoment theory are based on Hooke’s Law, three-dimensional equations of the theory of dynamic elasticity and the method of displacements expansion into Maclaurin series. The article gives the expressions to determine the forces, moments and bimoments. Bimoment theory of plates is described by two unrelated two-dimensional systems with nine equations in each. On each edge of the plate, depending on the type of fastening, nine boundary conditions are given. As an example, the solution of the problem of dynamic bending of thick isotropic and orthotropic plate under the influence of transverse dynamic loads in the form of the Heaviside function is given. The equations of motion of the plate are solved by numerical method of finite differences. The numerical results are obtained for isotropic and orthotropic plate. The graphs of changes of displacements and stresses of faces surfaces of the plate are presented. Maximum values of these displacements are found and analyzed. It is shown that by Timoshenko theory numerical values of stresses are much smaller compared to the ones obtained by bimoment theory of plates. Maximum numerical values of generalized displacements, forces, moments, and bimoments are obtained and presented in tabular form. The analysis of numerical results is done and the conclusions are drawn.展开更多
By applying the integral transform method and the inverse transformation technique based upon the two types of integration, the present paper has successfully obtained an exact algebraic solution for a two-dimensional...By applying the integral transform method and the inverse transformation technique based upon the two types of integration, the present paper has successfully obtained an exact algebraic solution for a two-dimensional Lamb＇s problem due to a strip impulse loading for the first time. With the algebraic result, the excitation and propagation processes of stress waves, including the longitudinal wave, the transverse wave, and Rayleigh-wave, are discussed in detail. A few new conclusions have been drawn from currently available integral results or computational results.展开更多
Using a polarization method, the scattering problem for a two-dimensional inclusion embedded in infinite piezoelectric/piezomagnetic matrices is investigated. To achieve the purpose, the polarization method for a two-...Using a polarization method, the scattering problem for a two-dimensional inclusion embedded in infinite piezoelectric/piezomagnetic matrices is investigated. To achieve the purpose, the polarization method for a two-dimensional piezoelectric/piezomagnetic ＂comparison body＂ is formulated. For simple harmonic motion, kernel of the polarization method reduces to a 2-D time-harmonic Green＇s function, which is obtained using the Radon transform. The expression is further simplified under conditions of low frequency of the incident wave and small diameter of the inclusion. Some analytical expressions are obtained. The analytical solutions for generalized piezoelectric/piezomagnetic anisotropic composites are given followed by simplified results for piezoelectric composites. Based on the latter results, two numerical results are provided for an elliptical cylindrical inclusion in a PZT-5H-matrix, showing the effect of different factors including size, shape, material properties, and piezoelectricity on the scattering cross-section.展开更多
Biot's dynamic consolidation equations and Hankel transform were used to derive the integral solutions of stress and displacement for axisymmetric harmonic excitations in the two-phase saturated soil with subjacen...Biot's dynamic consolidation equations and Hankel transform were used to derive the integral solutions of stress and displacement for axisymmetric harmonic excitations in the two-phase saturated soil with subjacent rock-stratum. The influence of the coefficient of permeability and loading frequency on the soil displacement at the ground surface were studied. The results showed that higher loading frequency led to more dynamic characteristics; and that the effect of the soil permeability was more obvious at higher frequencies.展开更多
An analytic solution to three-dimensional problem of transient contact dynamics is developed, and a generalized Galin’s theorem is proved. Based on the theorem, contact displacements can be determined analytically wh...An analytic solution to three-dimensional problem of transient contact dynamics is developed, and a generalized Galin’s theorem is proved. Based on the theorem, contact displacements can be determined analytically when the given contact stresses are some quite general functions; in contrast to this, one can readily evaluate the corresponding contact stresses in analytic version when some quite general displacements are prescribed. The quite general form mentioned above means the prescribed quantities are any polynomials of x and y and any transient function of time. Numerical results for the contact displacements-time variation dearly demonstrate the effect of inertia induced by the dynamic stress.展开更多
Impulsive line load in a half-space (Lamb’s problem) can be solved with a closed form solution. This solution is helpful for understanding the phenomenon of Rayleigh’s waves. In this article, we use a boundary eleme...Impulsive line load in a half-space (Lamb’s problem) can be solved with a closed form solution. This solution is helpful for understanding the phenomenon of Rayleigh’s waves. In this article, we use a boundary element method to simulate the solution of an elastic solid with a curved free surface under impact loading. This problem is considered difficult for numerical methods. Lamb’s problem is calculated first to verify the method. Then the method is applied on the problems with different surface curvatures. The method simulates the phenomenon of Rayleigh’s wave propagating on a curved surface very well. The results are shown in figures.展开更多
Based on the integral transformation method with which the combinatory integral transform of the displacements and the combinatory integral transform of the stresses are presented, the three-dimensional (3-D) non-axis...Based on the integral transformation method with which the combinatory integral transform of the displacements and the combinatory integral transform of the stresses are presented, the three-dimensional (3-D) non-axisymmetric governing dynamic equation in the Biot’s theory of two-phase medium is solved. Integral solutions with the soil skeleton displacements and pore pressure as the main unknown quantity are obtained. On the basis of this solution, a systematic study on Lamb’ s problems for saturated soils is performed. Considering the case of drained surface and the case of undrained surface, the integral solutions for surface radial, vertical and circumferential direction displacements under the vertical surface force and horizontal surface force are obtained, which would be reduced to the solutions of the classical Lamb’s problem. So, the correctness of the solutions would be verified. The numerical example indicates that the two-dimensional (2-D) model cannot be applied to 3-D problem accurately.展开更多
基金supported by the National Natural Science Foundation of China(No.11172268)
文摘By the analysis for the vectors of a wave field in the cylindrical coordinate and Sommerfeld＇s identity as well as Green＇s functions of Stokes＇ solution pertaining the conventional elastic dynamic equation, the results of Green＇s function in an infinite space of an axisymmetric coordinate are shown in this paper. After employing a supplementary influence field and the boundary conditions in the free surface of a senti-space, the authors obtain the solutions of Green＇s function for Lamb＇s dynamic problem. Besides, the vertical displacement uzz and the radial displacement urz can match Lamb＇s previous results, and the solutions of the linear expansion source u^r and the linear torsional source uee are also given in the paper. The authors reveal that Green＇s function of Stokes＇ solution in the semi-space is a comprehensive form of solution expressing the dynamic Lamb＇s problem for various situations. It may benefit the investigation of deepening and development of Lamb＇s problems and solution for pertinent dynamic problems conveniently.
基金Project supported by the National Natural Science Foundation of China(No.10572129)
文摘When dynamic force is applied to a saturated porous soil, drainage is common. In this paper, the saturated porous soil with a two-phase saturated medium is simulated, and Lamb＇s integral formulas with drainage and stress formulas for a two-phase saturated medium are given based on Biot＇s equation and Betti＇s theorem （the reciprocal theorem）. According to the basic solution to Biot＇s equation, Green＇s function Gij and three terms of Green＇s function G4i, Gi4, and G44 of a two-phase saturated medium subject to a concentrated force on a spherical coordinate are presented. The displacement field with drainage, the magnitude of drainage, and the pore pressure of the center explosion source are obtained in computation. The results of the classical Sharpe＇s solutions and the solutions of the two-phase saturated medium that decays to a single-phase medium are compared. Good agreement is observed.
文摘Novel exact solutions of one-dimensional transient dynamic piezoelectric problems for thickness polarized layers and disks, or length polarized rods, are obtained. The solutions are derived using a time-domain Green’s function method that leads to an exact analytical recursive procedure which is applicable for a wide variety of boundary conditions including nonlinear cases. A nonlinear damper boundary condition is considered in more detail. The corresponding nonlinear relationship between stresses and velocities at a current time moment is used in the recursive procedure. In addition to the exact recursive procedure that is effective for calculations, some new practically important explicit exact solutions are presented. Several examples of the time behavior of the output electric potential difference are given to illustrate the effectiveness of the proposed exact approach.
文摘The paper is devoted to dynamic design of thick orthotropic cantilever plates by applying the bimoment theory of plates, which takes into account the forces, moments and bimoments;and the theory takes into account nonlinear law of displacements distribution in cross section of the plate. The methods for constructing bimoment theory are based on Hooke’s Law, three-dimensional equations of the theory of dynamic elasticity and the method of displacements expansion into Maclaurin series. The article gives the expressions to determine the forces, moments and bimoments. Bimoment theory of plates is described by two unrelated two-dimensional systems with nine equations in each. On each edge of the plate, depending on the type of fastening, nine boundary conditions are given. As an example, the solution of the problem of dynamic bending of thick isotropic and orthotropic plate under the influence of transverse dynamic loads in the form of the Heaviside function is given. The equations of motion of the plate are solved by numerical method of finite differences. The numerical results are obtained for isotropic and orthotropic plate. The graphs of changes of displacements and stresses of faces surfaces of the plate are presented. Maximum values of these displacements are found and analyzed. It is shown that by Timoshenko theory numerical values of stresses are much smaller compared to the ones obtained by bimoment theory of plates. Maximum numerical values of generalized displacements, forces, moments, and bimoments are obtained and presented in tabular form. The analysis of numerical results is done and the conclusions are drawn.
基金Project supported by the National Natural Science Foundation of China(No.10572002).
文摘By applying the integral transform method and the inverse transformation technique based upon the two types of integration, the present paper has successfully obtained an exact algebraic solution for a two-dimensional Lamb＇s problem due to a strip impulse loading for the first time. With the algebraic result, the excitation and propagation processes of stress waves, including the longitudinal wave, the transverse wave, and Rayleigh-wave, are discussed in detail. A few new conclusions have been drawn from currently available integral results or computational results.
基金supported by the National Natural Science Foundation of China (Nos. 10732100, 10572155)the Science and Technology Planning Project of Guangdong Province of China (No. 2006A11001002)the Ph. D. Programs Foundation of Ministry of Education of China (No. 2006300004111179)
文摘Using a polarization method, the scattering problem for a two-dimensional inclusion embedded in infinite piezoelectric/piezomagnetic matrices is investigated. To achieve the purpose, the polarization method for a two-dimensional piezoelectric/piezomagnetic ＂comparison body＂ is formulated. For simple harmonic motion, kernel of the polarization method reduces to a 2-D time-harmonic Green＇s function, which is obtained using the Radon transform. The expression is further simplified under conditions of low frequency of the incident wave and small diameter of the inclusion. Some analytical expressions are obtained. The analytical solutions for generalized piezoelectric/piezomagnetic anisotropic composites are given followed by simplified results for piezoelectric composites. Based on the latter results, two numerical results are provided for an elliptical cylindrical inclusion in a PZT-5H-matrix, showing the effect of different factors including size, shape, material properties, and piezoelectricity on the scattering cross-section.
文摘Biot's dynamic consolidation equations and Hankel transform were used to derive the integral solutions of stress and displacement for axisymmetric harmonic excitations in the two-phase saturated soil with subjacent rock-stratum. The influence of the coefficient of permeability and loading frequency on the soil displacement at the ground surface were studied. The results showed that higher loading frequency led to more dynamic characteristics; and that the effect of the soil permeability was more obvious at higher frequencies.
文摘An analytic solution to three-dimensional problem of transient contact dynamics is developed, and a generalized Galin’s theorem is proved. Based on the theorem, contact displacements can be determined analytically when the given contact stresses are some quite general functions; in contrast to this, one can readily evaluate the corresponding contact stresses in analytic version when some quite general displacements are prescribed. The quite general form mentioned above means the prescribed quantities are any polynomials of x and y and any transient function of time. Numerical results for the contact displacements-time variation dearly demonstrate the effect of inertia induced by the dynamic stress.
文摘Impulsive line load in a half-space (Lamb’s problem) can be solved with a closed form solution. This solution is helpful for understanding the phenomenon of Rayleigh’s waves. In this article, we use a boundary element method to simulate the solution of an elastic solid with a curved free surface under impact loading. This problem is considered difficult for numerical methods. Lamb’s problem is calculated first to verify the method. Then the method is applied on the problems with different surface curvatures. The method simulates the phenomenon of Rayleigh’s wave propagating on a curved surface very well. The results are shown in figures.
文摘Based on the integral transformation method with which the combinatory integral transform of the displacements and the combinatory integral transform of the stresses are presented, the three-dimensional (3-D) non-axisymmetric governing dynamic equation in the Biot’s theory of two-phase medium is solved. Integral solutions with the soil skeleton displacements and pore pressure as the main unknown quantity are obtained. On the basis of this solution, a systematic study on Lamb’ s problems for saturated soils is performed. Considering the case of drained surface and the case of undrained surface, the integral solutions for surface radial, vertical and circumferential direction displacements under the vertical surface force and horizontal surface force are obtained, which would be reduced to the solutions of the classical Lamb’s problem. So, the correctness of the solutions would be verified. The numerical example indicates that the two-dimensional (2-D) model cannot be applied to 3-D problem accurately.
