Dynamic point sources in laminated media via the thin-layer method

This paper presents closed-form expressions for the Greens functions associated with harmonic point sources acting within horizontally layered media. These expressions are intended for use with the highly efficient Thin-Layer Method (TLM) described elsewhere, which is now being used widely for diverse engineering purposes. Among the dynamic sources considered are point forces, force dipoles (cracks and moments) , blast loads, seismic double couples with no net resultant, and bimoments (moment dipoles) . Comparisons with known analytical solutionsfor homogenous media demonstrate the accuracy of the formulation. However, the main field ofapplication is laminated media, for which no analytical solutions can be obtained. On the otherhand, it should be noted that the computational effort in this method does not depend on whetherthe system is layered. The resulting Greens functions could be used to efficiently model elasticwaves in complex media by means of the Boundary Integral Method.     

Doubly curved shallow shells with rectangular base elastically supported by edge arched beams and tie-rods (I)

The equations of equilibrium of shallow shells with rectangular base elastically supported with edge arched beams are obtained through the variational principle together with corresponding boundary conditions and corner conditions. It is assumed that edge arched beams are of narrow plate form, so that only the rigidities in their own planes are taken into consideration, torsional rigidities and bending rigidities out of their own planes are neglected. In this paper, two kinds of corner conditions are discussed. First of these is pinned corner conditions. Second of these is simply supported corner conditions, such that the corner can be moved freely in horizontal directions. The former corresponds to the conditions of those with heavy tension beams, in which the tension rigidities of the rods can be assumed infinite. The latter corresponds to the conditions of elastically supported edge arched beams without tension rods. In the former case, the edge tangential displacement of shallow shells is assumed to be zero everywhere, so that the vertical displacement of the edge arched beams gives the only elastic supported forces. This kind of supporting conditions is a good approximation for practical roof design.In this paper, the solutions of the problem of shallow spherical shell of square base supported elastically by edge arched beams and tie-rods under the conditions, such that the corners are restricted, are solved by the method of double trigonometric series. The edge conditions are integrated along their respective edge, and the conditions at corner are satisfied by proper choices of integral constants. The integrated edge conditions are then used to determine the unknown constants in the double trigonometric series. The result of this paper gives the tension in the tie-rods directly, which is an important quantity in the shell roof design practice.The method of integrated form of boundary conditions used in this paper in general is useful for the treatment of problem of plates and shells elastically supported by edge frames and tie-rods or by other means.This paper also gives the results of numerical calculation based upon the method of double trigonometric series on the problem of shallow spherical shell with square bases elastically supported by arched beams. The corner are pinned supported or simply supported. The calculated results for =11.5936 show that the trigonometric series converges rapidly. The effect of elastic deformation in the arched beams to the components of membrane tensions, moments, and deflections of the shell are given.     

In-plane/out-of-plane concentrated forces and moments on composite laminates with elliptical elastic inclusions

The problems of composite laminates containing elliptical elastic inclusions subjected to concentrated forces and moments are considered in this paper. By employing Stroh-like formalism for the coupled stretching–bending analysis, analytical closed form solutions are obtained explicitly. The generality of the solutions provided in this paper can be shown as follows: (1) The laminates include any kinds of laminate lay-ups, symmetric or unsymmetric, which allow the stretching and bending deformations couple each other. (2) The concentrated forces and moments can be applied in in-plane and/or out-of-plane directions, located inside and/or outside the inclusions. (3) The elliptical elastic inclusions can be any kinds of elastic materials including the limiting cases such as holes, rigid inclusions, cracks, line inclusions, etc. Since no such general solution has been found in the literature, the solutions are checked and verified by the special cases that no inclusions are embedded in the laminates, and that the inclusions are replaced by holes. Moreover, with various hardness ratios of inclusion and matrix some numerical examples showing the stress resultants along the interface are presented. Like the Green’s functions for the infinite laminates and those containing holes/cracks, the present solutions associated with the in-plane concentrated forces and out-of-plane concentrated moments have exactly the same mathematical form as those of the corresponding two-dimensional problems, in which the only difference is the contents of the symbols. While for the other loading cases, new types of solutions are obtained explicitly.     

Double plate system with a discontinuity in the elastic bonding layer

The double plate system with a discontinuity in the elastic bonding layer of Winker type is studied in this paper. When the discontinuity is small, it can be taken as an interface crack between the bi-materials or two bodies (plates or beams). By comparison between the number of multifrequencies of analytical solutions of the double plate system free transversal vibrations for the case when the system is with and without discontinuity in elastic layer we obtain a theory for experimental vibration method for identification of the presence of an interface crack in the double plate system. The analytical analysis of free transversal vibrations of an elastically connected double plate systems with discontinuity in the elastic layer of Winkler type is presented. The analytical solutions of the coupled partial differential equations for dynamical free and forced vibration processes are obtained by using method of Bernoulli’s particular integral and Lagrange’s method of variation constants. It is shown that one mode vibration corresponds an infinite or finite multi-frequency regime for free and forced vibrations induced by initial conditions and one-frequency or corresponding number of multi-frequency regime depending on external excitations. It is shown for every shape of vibrations. The analytical solutions show that the discontinuity affects the appearance of multi-frequency regime of time function corresponding to one eigen amplitude function of one mode, and also that time functions of different vibration basic modes are coupled. From final expression we can separate the new generalized eigen amplitude functions with corresponding time eigen functions of one frequency and multi-frequency regime of vibrations. The English text was polished by Keren Wang.     

Interaction of one-periodic disk-shaped cracks under an incident elastic harmonic wave

We study a symmetric problem of harmonic wave propagation in an elastic space with a one-periodic array of interacting disk-shaped cracks. Using the Green function obtained by the Fourier transform, we reduce the problem to a boundary integral equation (BIE) for the function characterizing the displacement discontinuity on one of the cracks and numerically determine the desired function by solving the BIE. We present graphs of the dynamic stress intensity factors near a circular crack versus the wave number for various distances between the defects.     

Horizontal Redistribution of Two Fluid Phases in a Porous Medium: Experimental Investigations

Classical models for flow and transport processes in porous media employ the so-called extended Darcy’s Law. Originally, it was proposed empirically for one-dimensional isothermal flow of an incompressible fluid in a rigid, homogeneous, and isotropic porous medium. Nowadays, the extended Darcy’s Law is used for highly complex situations like non-isothermal, multi-phase and multi-component flow and transport, without introducing any additional driving forces. In this work, an alternative approach by Hassanizadeh and Gray identifying additional driving forces were tested in an experimental setup for horizontal redistribution of two fluid phases with an initial saturation discontinuity. Analytical and numerical solutions based on traditional models predict that the saturation discontinuity will persist, but a uniform saturation distribution will be established in each subdomain after an infinite amount of time. The pressure field, however, is predicted to be continuous throughout the domain at all times and is expected to become uniform when there is no flow. In our experiments, we also find that the saturation discontinuity persists. But, gradients in both saturation and pressure remain in both subdomains even when the flow of fluids stops. This indicates that the identified additional driving forces present in the truly extended Darcy’s Law are potentially significant.     

SINGULAR SOLUTIONS OF ANISOTROPIC PLATE WITH AN ELLIPTICAL HOLE OR A CRACK

In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments.They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.     

Nonstationary hydrodynamic characteristics of a double grid of profiles

The flow about a double grid of solid profiles of arbitrary shape which vibrate in a stream of an ideal incompressible fluid is considered. Behind the grid profiles, the nonstationary vortex traces simulated by the lines of contact velocity discontinuity are taken into account. The problem is reduced to the solution of a system of two integral equations relative to the fluid velocity on the initial profiles of the double grid under the assumption that the vibration amplitudes are small. Formulas for calculating the nonstationary forces and moments are derived. The dependences of these forces on the shape, mutual positions, and laws of vibration of the grid profiles are studied. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 150–155, July–August, 1999.     

Asymptotic fields for dynamic crack growth in non-associative pressure sensitive materials

Asymptotic near-tip fields are analyzed for a plane strain Mode I crack propagating dynamically in non-associative elastic–plastic solids of the Drucker–Prager type with an isotropic linear strain hardening response. Eigen solutions are obtained over a range of material parameters and crack speeds, based on the assumption that asymptotic solutions are variable-separable and fully continuous. A limiting speed, beyond which a tendency to slope discontinuity in angular distributions of stresses and velocities is detected, is found to deviate from the associative models. At low strain-hardening rates, the onset of the plastic potential corner zone ahead of the crack-tip imposes another limit to the crack speed. Correspondingly, those limits imply the limits to the degree of non-associativity at a given crack speed. In addition, a tendency to slope discontinuity in the angular radial stress distribution sets another limit on the non-associativity at vanishing hardening rates.     

Fracture Analysis for a Frictional Fastener Hole with Edge Cracks in an Anisotropic Plate∗

Abstract

Stress intensity factors are evaluated for a singly or doubly cracked fastener hole with frictional traction in an anisotropic plate, using a special kernel boundary integral equation (BIE) approach. The integration kernel (Green's function) used in this BIE approach has already taken the presence of the crack (or cracks) into account, thus.avoiding the need for element discretization to model the stress singularity at the crack tip. The Green's function employed is that of an infinite anisotropic plate containing an elliptical hole or crack, and subjected to an arbitrarily positioned point force. Several types of normal and shear traction conditions at the pinhole interface are considered. Numerical results are obtained for various geometrical and loading conditions and are compared with known solutions, where available, for their isotropic counterparts.     

Exact solutions of the kinetic-moment equations of a mixture of monatomic gases

An extension is given of the class of exact solutions of the kinetic-moment equations for a monatomic gas in the absence of external forces [1] to the case of a mixture of monatomic Maxwellian gases with account for external forces. Very simple solutions of this class are obtained which are examples of the normal solutions of the Boltzmann equations in the Chapman-Enskog sense [2]. Conclusions are summarized concerning the applicability of the various methods of solving the Boltzmann equations and their properties, obtained on the basis of an analysis of the indicated exact solutions.The author wishes to thank M. N. Kogan and A. A. Nikol'skii for their interest in the study.     

Displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space

This paper describes a displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space or full space. The formulation is based on hypersingular integral equations that relate displacement jumps and tractions along the crack. The integral kernels, which represent stress influence functions for ring dislocation dipoles, are derived from available axisymmetric dislocation solutions. The crack is discretized into constant-strength displacement discontinuity elements, where each element represents a slice of a cone. The influence integrals are evaluated using a combination of numerical integration and a recursive procedure that allows for explicit integration of hyper- and Cauchy singularities. The accuracy of the solution at the crack tip is ensured by adding corrective stresses across the tip element. The method is validated by a comparison with analytical and numerical reference solutions.     

THREE-DIMENSION INTERACTIONS OF CIRCULAR CRACK IN TRANSVERSELY ISOTROPIC PIEZOELECTRIC SPACE WITH RESULTANT SOURCES

Exact solutions in form of elementary functions were derived for the stress and electric displacement intensity factors of a circular crack in a transversely isotropic piezoelectric space interacting with various stress and charge sources: force dipoles, electric dipoles, moments, force dilatation and rotation. The circular crack includes penny-shaped crack and external circular crack and the locations and orientations of these resultant sources with respect to the crack are arbitrary. Such stress and charge sources may model defects like vacancies, foreign particles, and dislocations. Numerical results are presented at last.     

Dynamic modeling of the front structure of an excavator

This paper presents a new mathematical model of 4 degrees of freedom of links to qualitatively describe the dynamic behavior of the front structure of an excavator. In the model, the effects of couple of forces as new additional effects are involved. The exact forms and solutions for position-varying moments of inertia used in this model are presented. A topologic structure is used for the kinematic analysis of the body frame. The numerical results show that the new additional effects can change the angular kinetic energy of all links to a significant degree when the upper structure swings. The results suggest that the new additional effects should be taken into account for analysis of excavator dynamics.     

THREE-DIMENSIONAL INTERACTIONS OF CIRCULAR CRACK IN TRANSVERSELY ISOTROPIC PIEZOELECTRIC SPACE WITH RESULTANT SOURCES

Exact solutions in form of elementary functions were derived for the stress and electric displacement intensity factors of a circular crack in a transversely isotropic piezoelectric space interacting with various stress and charge sources: force dipoles, electric dipoles, moments, force dilatation and rotation. The circular crack includes penny-shaped crack and external circular crack and the locations and orientations of these resultant sources with respect to the crack are arbitrary. Such stress and charge sources may model defects like vacancies, foreign particles, and dislocations. Numerical results are presented at last.     

Instability of self-similar ideal gas flows with shock waves

The results of the numerical simulation of three problems of ideal gas flow with shock waves, which admit self-similar solutions, are presented. These problems are the double Mach-type reflection of a shock from a wedge, the breakdown of a combined discontinuity on a 90° sharp corner, and the outflow of a supersonic jet from an expanding slot. It is shown that for certain input data the self-similar solution may become unstable and is replaced by a fluctuating flow. The reasons for the generation of these fluctuations and their mechanism are discussed. Volgograd. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 166–175, July–August, 1998.     

Validation of the Rayleigh–Ritz method for the postbuckling analysis of rectangular plates with application to delamination growth

Postbuckling solutions of laminated rectangular plates are obtained by the Rayleigh–Ritz method using von Karman’s nonlinear strain displacement relations and high-order polynomial expansions of the displacements. The potential energy function and the nonlinear algebraic equations governing the undetermined coefficients are obtained by Mathematica. Reasonably accurate solutions for the membrane forces, the bending and twisting moments and the pointwise energy-release rates generally require more undeterminated coefficients. Such refined postbuckling solutions show significant non-uniformity of the in-plane forces and strains and certain boundary effects characterized by concentration of the curvatures and the bending moment.     

Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates

The governing equation of motion of gradient elastic flexural Kirchhoff plates, including the effect of in-plane constant forces on bending, is explicitly derived. This is accomplished by appropriately combining the equations of flexural motion in terms of moments, shear and in-plane forces, the moment–stress relations and the stress–strain equations of a simple strain gradient elastic theory with just one constant (the internal length squared), in addition to the two classical elastic moduli. The resulting partial differential equation in terms of the lateral deflection of the plate is of the sixth order instead of the fourth, which is the case for the classical elastic case. Three boundary value problems dealing with static, stability and dynamic analysis of a rectangular simply supported all-around gradient elastic flexural plate are solved analytically. Non-classical boundary conditions, in additional to the classical ones, have to be utilized. An assessment of the effect of the gradient coefficient on the static or dynamic response of the plate, its buckling load and natural frequencies is also made by comparing the gradient type of solutions against the classical ones.     

Contact with a Corner for Nonlinearly Elastic Rods

In elastic contact problems it is usually required that the contact force has to be directed normally to the contact surface in the absence of friction. For an obstacle with nonsmooth surface this gives infinitely many normal directions at an edge or at a corner. For the case where a nonlinearly elastic rod under terminal loads is hanging over a needle, it is shown that the balance equations supplemented with such a normality condition have a continuum of solutions. Moreover, an additional contact condition is derived from a corresponding variational problem by means of special inner variations that preserve the shape of the rod. This way one is finally lead to a unique solution at least locally.     

Phenomenological Meniscus Model for Two-Phase Flows in Porous Media

A new macroscale model of a two-phase flow in porous media is suggested. It takes into consideration a typical configuration of phase distribution within pores in the form of a repetitive field of mobile menisci. These phase interfaces give rise to the appearance of a new term in the momentum balance equation, which describes a vectorial field of capillary forces. To derive the model, a phenomenological approach is developed, based on introducing a special continuum called the Meniscus-continuum. Its properties, such as a unique flow velocity, an averaged viscosity, a compensation mechanism and a duplication mechanism, are derived from a microscale analysis. The closure relations to the phenomenological model are obtained from a theoretical model of stochastic meniscus stream and from numerical simulations based on network models of porous media. The obtained transport equation remains hyperbolic even if the capillary forces are dominated, in contrast to the classic model which is parabolic. For the case of one space dimension, the analytical solutions are obtained, which manifest non-classical effects as double displacement fronts or counter-current fronts.