A THEORETICAL SOLUTION FOR AXIALLY SYMMETRIC PROBLEMS IN ELASTODYNAMICS

This paper presents a theoretical solution for the basic equation of axisymmetric problems in elastodynamics.The solution is composed of a quasi-static solution which satisfies inhomogeneous boundary conditions and a dynamic solution which satisfies homogeneous boundary conditions.After the quasi-static so- lution has been obtained an inhomogeneous equation for dynamic solution is found from the basic equation. By making use of eigenvalue problem of a corresponding homogeneous equation,a finite Hankel transform is defined.A dynamic solution satisfying homogeneous boundary conditions is obtained by means of the finite Hankel transform and Laplace transform.Thus,an exact solution is obtained.Through an example of hollow cylinders under dynamic load,it is seen that the method,and the process of computing are simple,effective and accurate.

INITIAL BOUNDARY VALUE PROBLEM FOR A NONCONSERVATIVE SYSTEM IN ELASTODYNAMICS

This article is concerned with the initial boundary value problem for a nonconservative system of hyperbolic equation appearing in elastodynamics in the space time domain x 〉 0, t 〉 0. The number of boundary conditions, to be prescribed at the boundary x = 0,depends on the number of characteristics entering the domain. Because our system is nonlinear, the characteristic speeds depends on the unknown and the direction of the characteristics curves are known apriori. As it is well known, the boundary condition has to be understood in a generalised way. One of the standard way is using vanishing viscosity method. We use this method to construct solution for a particular class of initial and boundary data, namely the initial and boundary datas that lie on the level sets of one of the Riemann invariants.

Unconventional Hamilton-type variational principles for nonlinear elastodynamics of orthogonal cable-net structures

According to the basic idea of classical yin-yang complementarity and modem dual-complementarity, in a simple and unified new way proposed by Luo, the unconventional Hamilton-type variational principles for geometrically nonlinear elastodynamics of orthogonal cable-net structures are established systematically, which can fully characterize the initial-boundary-value problem of this kind of dynamics. An ifnportant integral relation is made, which can be considered as the generalized principle of virtual work for geometrically nonlinear dynamics of orthogonal cable-net structures in mechanics. Based on such relationship, it is possible not only to obtain the principle of virtual work for geometrically nonlinear dynamics of orthogonal cable-net structures, but also to derive systematically the complementary functionals for five-field, four-field, three-field and two-field unconventional Hamilton-type variational principles, and the functional for the unconventional Hamilton-type variational principle in phase space and the potential energy functional for one-field unconventional Hamilton-type variational principle for geometrically nonlinear elastodynamics of orthogonal cable-net structures by the generalized Legendre transformation given in this paper, Furthermore, the intrinsic relationship among various principles can be explained clearly with this approach.

THE EXISTENCE OF SOLUTIONS IN NONLINEAR ELASTODYNAMICS

Under the small deformation assumption this paper shows the existence of solution for the system of elastic dynamics with the general nonlinear constitutive laws, and the existence of classical solution can be found under weaker conditions.

GURTIN-TYPE REGION-WISE VARIATIONAL PRINCIPLES FOR THERMOPIEZOELECTRIC ELASTODYNAMICS

The variation of new Gurtin-type region-wise variational principles results in continuous conditions, boundary conditions, all equations and relations in linear thermopiezoelectric elastodynamics. Gurtin-type region-wise variational principles comprise very important parts of linear thermopiezoelectric elastodynamics, and can fully characterize the initial-boundary-value problem in linear thermopiezoelectric elastodynamics.

AN APPLICATION OF UNGAR’S DIFFERENTIAL TRANSFORM TO ELASTODYNAMICS

In recent years, a lot of writers have used Cagniard-de Hoop's method[1][] to solve some problems of elastic wave. But it is a difficult and complicated task to change the path of integration when we use this method. A differential transform by A.Ungar[3,6] can obviate this difficulty. In this paper, weuse Ungar 's differential transform to solve a case of Lamb's problem [1][2]

AN INVESTIGATION ON A NEW INTEGRAL TRANSFORM-BOUNDARY ELEMENT METHOD FOR ELASTODYNAMICS

The numerical methods of Fourier eigen transform FET and its inversion are discussed and applied to the boundary element method for elastodynamics. The program for solving elastodynamic problems with the boundary element method is developed and some examples are given. From the numerical results of the examples, we know the method can increase the computing speed 5 similar to 10 times and the accuracy is guaranteed.

THE EXISTENCE OF SOLUTION TO THE FINITE ELASTODYNAMICS WITH MIXEDBOUNDARY CONDITIONS

In this paper the existence of solution to finite elastodynamics constrainted by mixed boundary conditions is derived when the hyperpotential and its gradient (for Green's strain) satisfy adequate conditions.

An insight into the pass effect of the planet gear from an elastodynamics perspective

Distinguished from parallel-axis gear systems,the revolution of the carrier brings out the pass effect of the planet gear.Compared with the phenomenological descriptions in conventional studies,this paper aims to provide more abundant physical information and the elastodynamics mechanism of the pass effect of the planet gear.To this end,a continuous-discrete model of helical planetary gears is proposed.Considering both the in-plane and out-of-plane vibration,a semi-analytical model is established to simulate the elastic ring gear.The elastic and the lumped parameter submodels are synthesized using the moving elastic coupling boundary.The simulated modal and dynamic characteristics are verified using a planetary gear test rig.Based on the proposed dynamic model,the pass effect of the planet gear,the effects of bolt constraint,and the out-of-plane vibration of helical planetary gears are investigated.It is revealed that the“realistic rotation”strategy adopted in the proposed dynamic model can better reflect the physical essence of the pass effect of the planet gear compared with the“pseudo rotation”strategy utilized in the traditional model.

Boundary element-free method for elastodynamics

The moving least-square approximation is discussed first. Sometimes the method can form an ill-conditioned equation system, and thus the solution cannot be obtained correctly. A Hilbert space is presented on which an orthogonal function system mixed a weight function is defined. Next the improved moving least-square approximation is discussed in detail. The improved method has higher computational efficiency and precision than the old method, and cannot form an ill-conditioned equation system. A boundary element-free method (BEFM) for elastodynamics problems is presented by combining the boundary integral equation method for elastodynamics and the improved moving least-square approximation. The boundary element-free method is a meshless method of boundary integral equation and is a direct numerical method compared with others, in which the basic unknowns are the real solutions of the nodal variables and the boundary conditions can be applied easily. The boundary element-free method has a higher computational efficiency and precision. In addition, the numerical procedure of the boundary element-free method for elastodynamics problems is presented in this paper. Finally, some numerical examples are given.

Unconventional Hamilton-type variational principles for electromagnetic elastodynamics

According to the basic idea of classical yin-yang complementarity and modern dual-complementarity, in a simple and unified new way proposed by Luo, the unconventional Hamilton-type variational principles for electromagnetic elastodynamics can be established systematically. This new variational principles can fully characterize the initial-boundary-value problem of this dynamics. In this paper, the expression of the generalized principle of virtual work for electromagnetic dynamics is given. Based on this equation, it is possible not only to obtain the principle of virtual work in electromagnetic dynamics, but also to derive systematically the complementary functionals for eleven-field, nine-field and six-field unconventional Hamilton-type variational principles for electromagnetic elastodynamics, and the potential energy functionals for four-field and three-field ones by the generalized Legendre transformation given in this paper. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly.

Some basic principles for the linear theory of piezoelectric micropolar elastodynamics

According to the basic idea of classical yin-yang complementarity and modern dual-complementarity,in a simple and unified way proposed by Luo,some basic principles in the linear theory of piezoelectric micropolar elastodynamics can be established systematically. In this paper,an important integral relation in terms of convolutions is given,which can be considered as the generalized principle of virtual work in mechanics. Based on this relation,it is not only possible to obtain the principle of virtual work and the reciprocal theorem,but also to systematically derive the complementary functionals for the eleven-field,nine-field and six-field simplified Gurtin-type variational principles and the potential energy-functional for the three-field one in the linear theory of piezoelectric micropolar elastodynamics by the generalized Legendre transformations given in this paper. Furthermore,with this approach,the intrinsic relationships among various principles can be explained clearly.

New Insights on Convergence Properties of Peridynamic Models for Transient Diffusion and Elastodynamics

Recent analytical solutions for peridynamic(PD)models of transient diffusion and elastodynamics allow us to revisit convergence of 1D PD models to their classical counterparts.We find and explain the reasons for some interesting differences between the convergence behavior for transient diffusion and elastodynamics models.Except for very early times,PD models for transient diffusion converge monotonically to the classical one.In contrast,for elastodynamic problems this convergence is more complex,with some periodic/almost-periodic characteristics present.These special features are investigated for sine waves used as initial conditions.The analysis indicates that the convergence behavior of PD solutions to the classical one can be understood in terms of convergence properties for models using the Fourier series expansion terms of a particular initial condition.The results obtained show new connections between PD models and their corresponding classical versions.

Some Invariant Solutions of Two-Dimensional Elastodynamics in Linear Homogeneous Isotropic Materials

Invariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoreticalmethod.The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators.Three invariant solutions are constructed.Their graphical figures are presented and physical meanings are elucidated in some cases.

Multi-axial unsplit frequency-shifted perfectly matched layer for displacement-based anisotropic wave simulation in infinite domain

Multi-axial perfectly matched layer(M-PML),known to have lost the perfect-matching property owing to multi-axial coordinate stretching,has been numerically validated to be long-time stable and it is thus used extensively in linear anisotropic wave simulation and in isotropic cases where the PML becomes unstable.We are concerned with the construction of the M-PML for anisotropic wave simulation based on a second order wave equation implemented with the displacement-based numerical method.We address the benefit of the incorrect chain rule,which is implicitly adopted in the previous derivation of the M-PML.We show that using the frequency-shifted stretching function improves the absorbing efficiency of the M-PML for near-grazing incident waves.Then,through multi-axial complex-coordinate stretching the second order anisotropic wave equation in a weak form,we derive a time-domain multi-axial unsplit frequency-shifted PML(M-UFSPML)using the frequency-shifted stretching function and the incorrect chain rule.A new approach is provided to reduce the number of memory variables needed for computing convolution terms in the M-UFSPML.The obtained M-UFSPML is well suited for implementation with a finite element or the spectral element method.By providing several typical examples,we numerically verify the accuracy and long-time stability of the implementation of our M-UFSPML by utilizing the Legendre spectral element method.

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 .

Critical velocity of sandwich cylindrical shell under moving internal pressure

Critical velocity of an infinite long sandwich shell under moving internal pressure is studied using the sandwich shell theory and elastodynamics theory. Propagation of axisymmetric free harmonic waves in the sandwich shell is studied using the sandwich shell theory by considering compressibility and transverse shear deformation of the core, and transverse shear deformation of face sheets. Based on the elastodynamics theory, displacement components expanded by Legendre polynomials, and position-dependent elastic constants and densities are introduced into the equations of motion. Critical velocity is the minimum phase velocity on the desperation relation curve obtained by using the two methods. Numerical examples and the finite element （FE） simulations are presented. The results show that the two critical velocities agree well with each other, and two desperation relation curves agree well with each other when the wave number k is relatively small. However, two limit phase velocities approach to the shear wave velocities of the face sheet and the core respectively when k limits to infinite. The two methods are efficient in the investigation of wave propagation in a sandwich cylindrical shell when k is relatively small. The critical velocity predicted in the FE simulations agrees with theoretical prediction.

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅲ)—NOETHER'S THEOREM

The existing various couple stress theories have been carefully restudied.The purpose is to propose a coupled Noether's theorem and to reestablish rather complete conservation laws and balance equations for couple stress elastodynamics. The new concrete forms of various conservation laws of couple stress elasticity are derived. The precise nature of these conservation laws which result from the given invariance requirements are established. Various special cases are reduced and the results of micropolar continua may be naturally transited from the results presented in this paper.

DYNAMIC THERMAL SHOCK IN A LAYERED CYLINDER WITH INITIALINTERFACE PRESSURE

An analytical method is developed to determine the transient response of dynamic thermostress in a two-layered cylinder with initial interface pressure. At first, the initial interface pressure in a two-layered cylinder caused by a heat-assembling method is considered as the initial condition of a thermal elastodynamic equilibrium equation. Thus, a thermal elastodynamic solution for a separate hollow cylinder with the initial stress field is found out by means of a series of simply mathematical transform. By making use of the boundary conditions and continuity conditions of a layered cylinders, the solution for the thermal shock exerting an influence on the initial interface pressure in a two-layered cylinder is also discussed.

STUDY ON DYNAMICS CHARACTER-ISTICS OF AN ELASTODYNAMIC SYSTEM

The system dynamics model has been done. The contracting process of thecurved-surface of the rubber disc is analyzed carefully, which is the main research aspect of theelastodynamics in the system. The coupling equation of the 'elastic surface and fluid' is resolvedby inverse resolution method. The main characteristics of the system, for instance, thedeparting-tube velocity, with which a carrier is to depart a launch tube, are estimated throughresolving the simulation model of the system dynamics. The simulation is based on some knownparameters and the experiment outcome of the elasticity modulus of a special rubber. On the otherhand, the influences on the system characteristics accompanying with the changing of some systemparameters are discussed. Finally, the conclusions are given that the elastodynamic system has asuperior performance and that the system deserves to be developed further as a kind of launchdevice.