Complete Closed-form Solution Research Articles

Thermal effects on the development of plastic zones in thin axisymmetric plates

A simple thermoelastic-plastic problem is solved analytically to show a thermal effect on the development of plastic zones in thin plates. The temperature field is assumed to be uniform but monotonically varying with the time. The material of the plate obeys the von Mises yield criterion with a constant yield stress. It is shown that a negligible rise of temperature may lead to a significant growth of the plastic zone. Even though stress analysis only is needed in the plastic zone to illustrate this effect, a complete closed-form solution to the problem is given.

Lagrangian multipliers for coast-arcs of optimum space trajectories

Some properties of generalized canonical systems - special dynamical systems described by a Hamiltonian function linear in the adjoint variables - are applied in determining the solution of the two-dimensional coast-arc problem in an inverse-square gravity field. A complete closed-form solution for Lagrangian multipliers - adjoint variables - is obtained by means of such properties for elliptic, circular, parabolic and hyperbolic motions. Classic orbital elements are taken as constants of integration of this solution in the case of elliptic, parabolic and hyperbolic motions. For circular motion, a set of nonsingular orbital elements is introduced as constants of integration in order to eliminate the singularity of the solution.

Elastic solutions for a transversely isotropic half-space subjected to a point load

We rederive and present the complete closed-form solutions of the displacements and stresses subjected to a point load in a transversely isotropic elastic half-space. The half-space is bounded by a horizontal surface, and the plane of transverse isotropy of the medium is parallel to the horizontal surface. The solutions are obtained by superposing the solutions of two infinite spaces, one acting a point load in its interior and the other being free loading. The Fourier and Hankel transforms in a cylindrical co-ordinate system are employed for deriving the analytical solutions. These solutions are identical with the Mindlin and Boussinesq solutions if the half-space is homogeneous, linear elastic, and isotropic. Also, the Lekhnitskii solution for a transversely isotropic half-space subjected to a vertical point load on its horizontal surface is one of these solutions. Furthermore, an illustrative example is given to show the effect of degree of rock anisotropy on the vertical surface displacement and vertical stress that are induced by a single vertical concentrated force acting on the surface. The results indicate that the displacement and stress accounted for rock anisotropy are quite different for the displacement and stress calculated from isotropic solutions. © 1998 John Wiley & Sons, Ltd.

Solutions of generalized proportional navigation with maneuvering and nonmaneuvering targets

In this generalized proportional navigation (GPN), the commanded acceleration is applied in a direction with a bias angle to the normal direction of line-of-sight (LOS) and its magnitude is proportional to the product of closing speed and LOS rate between interceptor and its target. Some solutions of GPN were obtained before under the assumption that the magnitude of commanded acceleration is proportional to the LOS rate only. Now in this article, the exact and complete closed-form solutions are derived under this modified guidance scheme for both maneuvering and nonmaneuvering targets. Some related important characteristics, such as capture capability and energy cost, are investigated and discussed. The variation of bias angle will induce the change of capture area and energy cost required. Also, a typical example of target maneuver is introduced to describe the effect of target maneuver easily. It shows that the target maneuver will decrease the capture area and increase the energy cost for effective intercept of target.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The elastic field of an internal circular crack subjected to a linear normal loading

A complete closed form solution in terms of elementary functions is given to the problem of the elastic field of an internal circular crack subjected to a linear normal loading. Explicit expressions are derived for the stresses and displacements in both transversely isotropic and isotropic space weakened by a penny-shaped crack and subjected to a linear normal loading. Some numerical results are presented.

Influence of non-classical elastic-plastic constitutive features on shock wave existence and spectral solutions

T he influence of non-classical elastic-plastic constitutive features on dynamically moving discontinuities in stress, strain and material velocity is investigated. Non-classical behavior here includes non-normality of the plastic strain increment to the yield surface, plastic compressibility, pressure sensitivity of yield and dependence of the elastic moduli on plastic strain. D rugan and S hen's (1987) analysis of dynamically moving discontinuities with strain as well as stress jumps in classical materials is shown to be valid for a broad class of non-associative material models until deviation from normality exceeds a critical (non-infinitesimal) level. For these non-classical materials, an inequality that bounds the magnitude of the stress jump is derived, which is information not obtainable from a standard spectral analysis of a shock. For the special case of stress discontinuities with continuous strain or for quasi-static deformations, this inequality is shown to rule out jumps in specific projections of the stress tensor unless the non-normality is sufficiently large. These results invalidate a recent claim in the literature that an infinitesimal amount of non-normality permits moving surfaces of discontinuity in stress (with no strain jump) near the tip of a dynamically advancing crack tip. Using a very general plastic constitutive law that subsumes most non-classical (and classical) descriptions currently in use, a complete closed form solution is obtained for the plastic wave speeds and eigenvectors. A novel feature of the analysis is the clarity and completeness of the solutions. If the elastic part of the response is isotropic, one plastic wave speed equals the elastic shear wave speed, while the other two possible wave speeds depend in general on the stress and plastic strain within the shock transition layer. Concise necessary and sufficient conditions for real eigenvalues and for vanishing eigenvalues are derived. The real eigenvalues are classified by numerical sign and ordering relative to the elastic eigenvalues. The geometric multiplicity of plastic eigenvectors associated with elastic eigenvalues is shown to depend on the stress state within the shock transition layer. These solutions, several of which hold for arbitrary elastic anisotropy, are also applicable to acceleration waves and localization problems and to materials with dependence of the elastic moduli on plastic strain. Such elastic-plastic coupling is shown to imply a non-self-adjoint fourth order tangent stiffness tensor even if the plastic constitutive law is associative.

The input admittance of a TEM excited annular slot antenna

The admittance problem solution of a TEM excited annular slot antenna embedded in an infinite perfectly conducting ground plane is accomplished in the rigorous Maxwell equation-boundary value sense. The net result is a complete closed form solution for the admittance which includes the effects of all beyond-cutoff higher order TM modes. Infinite integrals related to the TEM admittance and to perturbations produced by the presence of higher order TM modes are solved in an approximate manner. The TEM admittance solution obtained is shown to compare favorably with admittance values accurately determined by numerical integration. The presence of beyond-cutoff higher order TM modes is shown to produce a minor effect to the annular slot admittance. The form of the admittance solution reported upon herein is such that the results may be readily applied in estimating the input admittance of physical annular slot antennas without recourse to the usual infinite integral reduction problem.

On the analysis of discontinuous systems

An effective procedure, employing an operational approach, is demonstrated for the analysis of constant parameter continuous systems with multiple lumped parameter attachments and concentrated forces. Convenience of the method and of the form in which the results are obtained are evaluated in comparison with other applicable methods. Systems with piecewise constant parameters are also investigated. The desirability of obtaining a complete closed-form steady-state solution and various methods of obtaining it are examined. Finally, an example is treated which illustrates the application of the operational method to system with jump discontinuities in deflection and slope.

Rupture Instability of Plane Membranes and Solids

A fundamentally new approach to the rupture-fracture problem is presented. Because of the particular type of dynamic phenomena being investigated, the formulation is given in terms of the conservation equations of continuum mechanics instead of in the usual elasticity-plasticity relations. The introduction of a similarity co-ordinate permits a complete closed-form solution to a particular problem of practical interest subject to certain compatibility conditions which depend upon the specific properties of the material under consideration.