Dynamic Elasticity Problems Research Articles

Boundary Element Methods in Dynamic Analysis

A review of boundary element methods for the numerical solution of dynamic problems of linear elasticity is presented. The integral formulation and the corresponding numerical solution of three- and two-dimensional elastodynamics from the direct boundary element method viewpoint and in both the frequency and time domains are described. The special case of the anti-plane motion governed by the scalar wave equation is also considered. In all the cases both harmonic and transient dynamic disturbances are taken into account. Special features of material behavior such as viscoelasticity, inhomogeneity, anisotropy, and poroelasticity are briefly discussed. Some other nonconventional boundary element methods as well as the hybrid scheme that results from the combination of boundary and finite elements are also reviewed. All these boundary element methodologies are applied to: soil-structure interaction problems that include the dynamic analysis of underground and above-ground structures, foundations, piles, and vibration isolation devices; problems of crack propagation and wave diffraction by cracks; and problems dealing with the dynamics of beams, plates, and shells. Finally, a brief assessment of the progress achieved so far in dynamic analysis is made and areas where further research is needed are identified.

Stress Functions for Electrodynamics in Consideration of Viscous Damping

This paper deals with an extension of a method of solution in terms of stress function for dynamic problems of elasticity to the case of an elastic solid with viscous damping. It is Assumed that the damping forces considered in this paper are proportional to the velocity of motion of particles in a solid, and Hooke's law for stresses an strains holds. Stress functions of the so-called Galerkin type and Neuber-Papkovich type in the three dimensional theory of elasticity and the Love type in axisymmetric problems of elasticity are investigated. Stress functions are also shown for axisymmetric dynamic body force problems of elasticity, taking into account the viscous damping, and an example of their application is presented.

A numerical method for the solution of static and dynamic three-dimensional elasticity problems

Kupradze's functional equation, reduced to a regular Fredholm integral equation of the first kind, was solved by applying a new numerical method, based on numerical integration, whose collocation points are chosen in self-similar surfaces. An application of the method to a particular problem of elasticity demonstrated a sufficient accuracy and stability of the method. It was shown that the proposed method is faster, simpler and more easily programmable than the existing classical methods. Finally, suggestions were made for a better use of the method and for possible improvements.

On the solution of elastic-plastic static and dynamic postbuckling collapse of general structure

This paper develops a solution algorithm which enables the analysis of both static and dynamic large deformation elastic and elastic-plastic postbuckling problems of general structure. This is made possible through the use of a hyperellipsoidal constraint surface which self-adaptively controls the iteration process associated with the BFGS updated Newton Raphson scheme. Coupling this algorithm with implicit and explicit time integration operators, several numerical examples are presented which illustrate the efficiency and inherent stability of the scheme developed.

The equations of caustics for crack and other dynamic plane elasticity problems

The equations of caustics for crack and other dynamic plane elasticity problems

A Neumann series representation for solutions to boundary-value problems in dynamic elasticity

A regularized integral equation formulation for two exterior fundamental boundary-value problems in elastodynamics is presented. In either case, the displacement vector is assumed to be harmonic in time with a small frequency. It is shown that the solution can be expressed as a Neumann series in terms of the prescribed function; moreover, a sufficient condition for the convergence of the series is established.

Plane near-field response of an elastic plate

The plane near-field response of an elastic plate is investigated both experimentally and theoretically. Experimental observations of the displacement field within the plate are obtained by means of high-speed photography of grid lines imbedded in a clear specimen molded from a low-modulus material. The analytical solution is obtained by using a recently proposed numerical procedure developed for plane problems in dynamic elasticity.

Dynamic problems of asymmetrical elasticity

Dynamic problems of asymmetrical elasticity

A Numerical Solution of Three-Dimensional Problems in Dynamic Elasticity

The equations governing the dynamic deformation of an elastic solid are considered as a symmetric hyperbolic system of linear first-order partial-differential equations. The characteristic properties of the system are determined and a numerical method for obtaining the solution of mixed initial and boundary-value problems in elastodynamics is presented. The method, based on approximate integral relations along bicharacteristics, is an extension of the method proposed by Clifton for plane problems in dynamic elasticity and provides a system of difference equations, with second-order accuracy, for the explicit determination of the solution. Application of the method to a problem which has a known solution provides numerical evidence of the convergence and stability of the method.

On the Unification of Normal-Mode Techniques for Solving Dynamic Elasticity Problems

Normal-mode techniques for solving dynamic elasticity problems with inhomogeneous boundary conditions are divided into two categories—namely, displacement (separation of variables, integral transforms) and acceleration (mode acceleration, Mindlin-Goodman, Williams) methods. In general, acceleration methods exhibit superior convergence properties relative to displacement methods. In this study, the solution of the complete linear dynamic elasticity equations of motion with inhomogeneous boundary conditions in an arbitrarily shaped regions is obtained by two methods: (1) a general integral transform technique (based upon Green's extended identity), and (2) a general Mindlin-Goodman process. It is then shown that both of these solutions are actually just different ways of writing the convolution integral in the complex (Laplace transform) domain thereby establishing the unity of all normal mode techniques. Finally, a means for obtaining the quasistatic solution from the displacement (integral transform) solution without the necessity for solving the quasistatic problem, as required in acceleration methods, is then given. [Work supported by the U. S. Atomic Energy Commission.]

On the solution of problems of dynamic plane elasticity for sandy soil: A. Mukhopadhay.J. Sci. Engng Res., India9, 2, 213–221, July 1965

On the solution of problems of dynamic plane elasticity for sandy soil: A. Mukhopadhay.J. Sci. Engng Res., India9, 2, 213–221, July 1965

The growth of a disk-shaped crack

Recent work has made it possible to treat axially symmetric problems in dynamic elasticity by a method closely analogous to that used for plane strain. This paper takes advantage of this technique to compare the growth of symmetrical plane cracks in axial symmetry and plain strain.

Advances in the theory of anisotropic elastic wave propagation

The qualitative effects of anisotropy on elastic waves propagating in a solid medium were well known to Lord Kelvin. Having been recognized, however, these effects were neglected as being of secondary importance in the dynamics of elastic mediums. This relegation of anisotropy to a secondary role in the dynamics of elastic mediums was undoubtedly justified, particularly in view of the relatively primitive state of experimental elasticity and seismology during Kelvin's time. It was not until after World War 2 that the effects of anisotropy again received serious attention. This was primarily because of the development of ultrasonic techniques for the measurement of dynamic elastic constants of pure crystals. In such experimental problems anisotropy no longer plays a secondary role. The study of how a disturbance, generated by a transducer on the surface of a crystal, spreads through the crystal led to the discovery, by Musgrave, that the wave surface, which forms the boundary of the spreading disturbance, could have cuspidal singularities. This had not been previously predicted, although it could have been predicted by Kelvin had he been more familiar with algebraic geometry. In another area of research (seismology), the post World War 2 years also saw a rise of interest in anisotropy, particularly in the effect of possible continental anisotropy on the propagation of Rayleigh waves. The increased experimental activity in crystal dynamics and the improvement of experimental seismology to the point where secondary effects became important resulted in a number of theoretical investigations into the propagation of plane, time harmonic waves in anisotropic mediums. By 1959 the state of the theoretical and experimental understanding of anisotropic elastic wave propagation had advanced to the point where rigorous wave theoretical calculations were in order. All the simple, solvable problems of isotropic dynamic elasticity, i.e. the initial value problem for an unbounded homogeneous medium, the mixed initial and boundary value problem for a surface line source on a half‐space, the normal mode problem for an elastic wave guide, etc., can be formulated and solved in detail in the anisotropic case. Furthermore, the solutions can be obtained by extensions of the usual transform methods used in isotropic problems. The results can be physically interpreted by means of classical differential geometry. This review summarizes those anisotropic problems treated since 1959 and the techniques developed to solve them.

On the solution of problems of dynamic plane elasticity

J. R. M. Radok [1] has applied complex variable methods to problems of dynamic plane elasticity. The object of this paper is to show that his results may be obtained in a somewhat simpler way by a more systematic use of complex variable analysis.In fact it is shown that the problems may be reduced to a form similar to that of the static aelotropic plane strain problems considered by Green and Zerna [2].

On the solution of problems of dynamic plane elasticity

On the solution of problems of dynamic plane elasticity