Showing papers on "Linear elasticity published in 1978"
TL;DR: In this paper, the authors describe the modes that have a steady state sin(ωt−βz) dependence on (t,z), with emphasis on the modes and ranges of parameters that are of interest for delay lines.
Abstract: Clad rods have been investigated for use as long delay lines because they offer isolation of the signal from the surface and low dispersion. In addition, single‐mode propagation is achieved with a larger (and hence more conveniently transduced) cross section than is possible with a homogeneous rod at the same frequency. This paper describes the modes that have a steady‐state sin(ωt−βz) dependence on (t,z), with emphasis on the modes and ranges of parameters that are of interest for delay lines. Only rods of circular cross section, and isotropic, linear elastic materials are considered. Attention is drawn to correspondences with homogeneous rods and with the hypothetical case of infinite thickness cladding, which is most useful as a model for understanding the behavior of corresponding modes in an actual clad rod. Written for the nonspecialist, the paper includes a tutorial review of the concepts and results needed to understand wave propagation in rods and clad rods. In addition, the following new results are reported: (1) for homogeneous rods, a representation of the lowest flexural mode dispersion curve that is for practical purposes independent of Poisson’s ratio; (2) for homogeneous rods, displacement distributions of the first three high‐frequency shear modes, i.e., the ’’flexural’’ modes that are asymptotic to the shear velocity (it was found that the distribution previously attributed to the lowest such mode actually belongs to the next); (3) for clad rods, the first demonstration of interface modes of the Stoneley type for nonaxially symmetric waves; (4) classification of clad rods according to the nature and ordering of their asymptotic velocities showing that there are eight types of clad rod, rather than four as stated in the previous literature; (5) for infinitely clad rods having the same shear modulus, proof that the dispersion of torsional waves and their penetration into cladding are universal functions of f/fc∞. The ratio of the shear velocities of the two materials affects the cutoff frequency fc∞, but not the universal penetration and dispersion functions. Subjects on which significant tutorial or descriptive material is given include typical waveguide dispersion, characteristic velocities of an isotropic elastic material, the effect of coupling of dilatational and shear waves at a boundary, waves in homogeneous rods, the connection of isolation to total internal reflection (with Love waves and SH waves in a clad plate as an example), Stoneley waves at a plane interface, and previous results on the clad rod.
284 citations
TL;DR: In this article, a technique for applying flexible boundaries to an atomic region in computer simulation of dislocations or other line defects is described, which results in continuity of equilibrium, under the chosen interatomic potential, across the interface between the atomic region and the outer region described in terms of anisotropic elastic continuum solutions.
Abstract: A technique is described for applying flexible boundaries to an atomic region in computer simulation of dislocations or other line defects. The method results in continuity of equilibrium, under the chosen interatomic potential, across the interface between the atomic region and the outer region described in terms of anisotropic elastic continuum solutions. The technique has high numerical efficiency. It is shown that when the crystal is initially dislocated according to the Volterra solution for displacements, the finite strains give rise to geometrical nonlinear effects, usually disregarded in linear elasticity, which contribute to a volume change of the crystal. Allowance for this effect, and for elastic nonlinearity in the crystal beyond the boundary region, allows the overall dilatation of a finite body due to the dislocation to be rigorously computed. For illustration of the geometric nonlinear effect, and for comparison with earlier modeling methods, examples of computations are given for the [100]...
143 citations
TL;DR: Stable bend initiation and propagation, without internal viscous resistance, is demonstrated for a flagellum containing a linear elastic bending resistance and an elastic shear resistance that depends on sliding.
117 citations
TL;DR: In this paper, a brief introduction to the nonlinear theory of elasticity is given, followed by a detailed discussion of the underlying boundary value problems and a discussion demonstrates why this theory is far more difficult than most nonlinear theories of mathematical physics.
Abstract: Publisher Summary This chapter gives a brief introduction to the nonlinear theory of elasticity It presents a concise derivation of the basic equations followed by a detailed discussion of the underlying boundary value problems The discussion demonstrates why this theory is far more difficult than most nonlinear theories of mathematical physics The notation used in the chapter is summarized in a tabulated form The material time derivative is defined in the chapter A discussion is presented in the chapter on mass distribution, reference density, system of forces, surface forces, body forces, and the balance of momentum The salient features of the equation of elasticity are compared with those of the Navier–Stokes equations The elasticity tensor is defined, the boundary value problems of elastostatics are discussed, and the displacement problem is also elaborated in the chapter
93 citations
TL;DR: In this paper, exact expressions for the stresses and the displacements in an infinite elastic solid containing two spherical inclusions are presented when an arbitrary linear strain field is applied at infinity, and the analysis is based on the Boussinesq-Papkovich stress function approach and makes use of the multipole expansion technique in which the solutions are expanded into series of spherical harmonics with respect to the centers of the two spheres.
72 citations
TL;DR: In this paper, a correlation between two notations currently used in the formulation of geometrically nonlinear analysis in a Lagrangian co-ordinate system is defined and an explicit example of the correlation is presented for a pin-jointed bar element.
Abstract: A correlation is geven between two notations currently used in the formulation of geometrically non-linear analysis in a Lagrangian co-ordinate system. The first referred to as the B-notation results in the linear elastic, initial displacement and initial stress matrices, whilst the second, here called the N-notation features matrices that repeat in the total potential energy, equilibrium and incremental equilibrium equations. An explicit example of the correlation is presented for a pin-jointed bar element.
58 citations
TL;DR: In the classical theory of elasticity, one of the celebrated methods for solving boundary value problems regarding an isotropic homogeneous linear elastic continuum is the method of Galerkin's vector as discussed by the authors.
Abstract: In the classical theory of elasticity, one of the celebrated methods for solving boundary value problems regarding an isotropic homogeneous linear elastic continuum is the method of Galerkin's vector. However, the set of governing equations to be solved is a system of N partial differential equations, N being the number of spatial dimension.
51 citations
TL;DR: In this article, the problem of a spherical cavity which is embedded in a linear, fluid-infiltrated, elastic porous medium and which is subjected to the sudden quasi-static application of a stress at the cavity boundary is solved.
35 citations
26 citations
TL;DR: In this paper, a numerical simulation of ballistic impact and penetration on woven textile panels is described which can easily incorporate a wide variety of realistic constitutive and fracture models, and is further extended to include non-linear Viscoelastic effects.
Abstract: A numerical simulation of ballistic impact and penetration on woven textile panels is described which can easily incorporate a wide variety of realistic constitutive and fracture models. The use of this model in assessing Viscoelastic relaxation effects is illustrated, and is further extended to include non-linear Viscoelastic effects. Since a variety of non-linear models is presently available and there is insufficient evidence to indicate the superiority of any single one in this instance, the Eyring non-linear model was chosen arbitrarily to indicate the ease with which these models may be implemented into the numerical treatment. The results obtained using the non-linear model are compared with comparable computer experiments using linear elastic and linear Viscoelastic models.
20 citations
TL;DR: In this paper, a finite element program for computer automated synthesis of linear elastic structures under static loading is presented, where the objective of the synthesis is to obtain minimum structural weight with stress and local instability constraints of large structures.
TL;DR: In this paper, three types of analysis are considered: (1) limit analysis, (2) slip line theory, and (3) finite element analysis of work-hardening material.
Abstract: Stress distribution and relevant formula for computing the tensile strength of a split-concrete or rock cylinder were derived on the basis of the linear elasticity theory. Three types of analysis are considered: (1)Limit analysis; (2)slip-line theory; and (3)finite element analysis of work-hardening material. In the finite element analysis, three types of plasticity models are used: (1)von Mises isotropic work-hardening model; (2)Drucker-Prager isotropic hardening model; and (3)concrete plasticity model. Each type of analysis corresponds to a somewhat different stress-strain idealization. It is demonstrated here that analyses 2 and 3 give tensile stress distribution similar to that of linear elasticity theory. The relevant formula for computing the tensile strength of various split tests obtained by different plasticity analyses are found to be similar to that of the elasticity solution. The equation derived on the basis of elasticity theory is sufficiently accurate for estimating the maximum tensile strength of nonlinear fracture materials.
TL;DR: In this paper, the Griffith model for crack propagation is superposed on the periodic energy field of the crystal lattice and a complete kinetics analysis is carried out for the description of the rigid and kinked crack-front propagation rate over the consecutive energy-barrier system.
Abstract: Simple rigid crack‐front propagation and the mechanism of double‐kink nucleation and spreading are analyzed in a periodic crystallographic potential‐energy field. For the analysis the classical Griffith model for crack propagation is superposed on the periodic energy field of the crystal lattice. The energy‐condition theory of the time‐dependent crack propagation is developed in terms of the absolute rate theory. A complete kinetics analysis is carried out for the description of the rigid and kinked crack‐front propagation rate over the consecutive energy‐barrier system of the crystal lattice. The rate is expressed as a function of the applied stress, temperature, the free energy of fracture, and the lattice parameters.
01 Jul 1978
TL;DR: In this article, a conformal mapping technique was used to obtain an exact linear elastic solution for the problem of a crack in a semi-infinite plate subjected, at infinity, to antiplane stress (Mode III) loading.
TL;DR: In this paper, some partial differntial equations are developed to describe the diffusion of a dilute solute in a linear elastic solid supporting a static deformation, based exclusively on the use of the conservation laws of mass and momentum, as well as on the introduction of an internal diffusive force to evaluate the diffusion effects.
Abstract: Some new partial differntial equations are developed to describe the diffusion of a dilute solute in a linear elastic solid supporting a static deformation. The analysis is based exclusively on the use of the conservation laws of mass and momentum, as well as, on the introduction of an internal diffusive force to evaluate the diffusion effects.
TL;DR: In this article, it was shown that the load-deflection relationship for a deep rigid anchor can be directly recovered from the expression for the drag induced on an impermeable object with the same size and shape as the anchor, which is appropriately placed in a slow viscous flow region of uniform velocity.
Abstract: The equations governing the undrained linear elastic behaviour of a saturated soil are formally similar to the equations governing slow of an incompressible Newtonian viscous fluid. This principle of equivalence can then be effectively employed to obtain the load-deflection reiationship for a deep rigid anchor with the shape of a solid of revolution which is embedded in bonded contact with an unbounded incompressible elastic medium. It is found that the load-deflection relationship for the deep rigid anchor can be directly recovered from the expression for the drag induced on an impermeable object with the same size and shape as the anchor, which is appropriately placed in a slow viscous flow region of uniform velocity.
01 Jan 1978
TL;DR: In this paper, the authors studied harmonic wave propagation in an infinite elastic bar of elliptical cross section with stress-free surface by using Mathieu functions and modified Mathieu function which are the exact solutions of the equation of motion from linear elasticity in an elliptical cylinder coordinate system.
Abstract: This paper studies harmonic wave propagation in an infinite elastic bar of elliptical cross section with stress-free surface by using Mathieu functions and modified Mathieu functions which are the exact solutions of the equation of motion from linear elasticity in an elliptical cylinder coordinate system. We give a procedure leading to the frequency equations for longitudinal, torsional and flexural waves by making use of the orthogonal properties of Mathieu functions. Numerical calculation for each mode is carried out.
01 Jan 1978
TL;DR: In this paper, the difference between a nonlinear elastic model and a dilatancy model that exhibits hysteresis is noted, and small changes in stress produced by the earth tide act as a probe of the properties of crustal rocks, and observations of earth tide tilts and strains in such high stress zones may provide keys to the constitutive properties and the tectonic stress rate tensor of these zones.
Abstract: In the vincinity of earthquake focal regions, conditions may not be equal. Crustal rocks stressed to more than approximately 0.6 of their failure strength exhibit material properties over and above that of linear elasticity. Interactions between the earth tide and crustal rocks that are under high tectonic stress are discussed in terms of simple phenomenological models. In particular, the difference between a nonlinear elastic model of dilatancy and a dilatancy model that exhibits hysteresis is noted. It is concluded that the small changes in stress produced by the earth tide act as a probe of the properties of crustal rocks. Observations of earth tide tilts and strains in such high stress zones may, therefore, provide keys to the constitutive properties and the tectonic stress rate tensor of these zones.
TL;DR: In this article, a general derivation of expressions for lower bounds to fundamental frequencies and buckling loads is given for the class of structures governed by linear elastic theory in the prebuckling state.
TL;DR: In this article, the linear elastic behavior of a unidirectional composite and an isotropic material with an oriented microstructure is described and the physical interpretation of additional stresses and constitutive constants present in the theory of micro-elasticity is given and the shear stress between matrix and fibre is presented in terms of these stresses.
TL;DR: In this article, the solutions of the quasistatic bending problem of circular plates on linear viscoelastic foundations under arbitrary loading conditions are presented, and the method of solution is developed by using the eigenfunctions found in the free lateral vibration problem of the circular plates with the same geometry and the same boundary condition.
Abstract: The solutions of the quasistatic bending problem of circular plates on linear viscoelastic foundations under arbitrary loading conditions are presented. The method of solution is developed by using the eigenfunctions found in the free lateral vibration problem of the circular plates with the same geometry and the same boundary condition, and the correspondence principle between linear elastic boundary value problem and viscoelastic one is utilized. Some results of numerical calculations for the variations in deflection and reactive force in space and with time are illustrated for the viscoelastic foundations of Kelvin (Voigt), Maxwell, and Standard linear solid types.
01 Feb 1978
TL;DR: The CHILES 2 program as discussed by the authors is a finite-element computer program that calculates the strength of singularities in linear elastic bodies using a generalized quadrilateral finite element with singular point at a corner node.
Abstract: CHILES 2 is a finite-element computer program that calculates the strength of singularities in linear elastic bodies A generalized quadrilateral finite element that includes a singular point at a corner node is incorporated in the code The displacement formulation is used and interelement compatibility is maintained so that monotone convergence is preserved Plane stress, plane strain, and axisymmetric conditions are treated Isotropic and orthotropic crack tip singularity problems are solved by this version of the code, but any type of singularity may be properly modeled by modifying selected subroutines in the program
TL;DR: In this paper, an efficient method of historical analysis using an initial strain approach coupled with the matrix flexibility method is developed for elastoplastic structures, where the spread of inelastic straining along members is taken into account using a trilinear moment-curvature relationship with linear elastic unloading.
TL;DR: In this paper, a system of linear elastic equations for fiber reinforced composite materials is applied to some simple problems concerning the transfer of load from the reinforcement to the matrix, and the same equations are applied to surface waves propagating in the direction of the fiber reinforcement.