Showing papers on "Linear elasticity published in 1970"

Inelastic Analysis of a Unidirectional Composite Subjected to Transverse Normal Loading

Abstract: Composite material behavior is divisible into three distinct re gimes, linear elastic response up to the elastic limit, inelastic be havior beyond the elastic limit and up to that point at which first failure occurs locally, and subsequent crack propagation and total composite failure. The first regime has been thoroughly explored in previous investigations, while the second is the main topic of the present paper. The third is briefly discussed and is a subject under investigation.Utilizing a finite element analysis, numerical results are obtained for composite behavior beyond the elastic limit and up to first failure. Examples representative of several actual material systems are presented and discussed in detail.Results indicate that extensive local yielding and redistribution of stresses can occur in a composite with very little indication of such behavior being apparent by observation of the total stress- strain response of the composite alone. Thus, the establishment and interpretation of design crit...

Screw dislocation in a model sodium lattice

Abstract: The deformation stacking-fault energies and screw dislocation core structures in a model sodium lattice have been calculated for body-centred-cubic and hexagonal close-packed phases, using a fundamental effective ion-ion potential, the reliability of which has been established elsewhere. Particular emphasis is placed upon deviations from continuum elasticity, and it is shown that the use of linear elasticity theory over distances smaller than about ten lattice spacings can lead to serious error. It is shown that the screw dislocation core structures in b.c.c. and h.c.p. lattices have markedly different characteristics, and relative dislocation mobilities in the two phases are discussed in the light of these differences.

Schaum's outline of theory and problems of continuum mechanics

Abstract: Mathematical Foundations Analysis of Stress Deformation and Strain Motion and Flow Fundamental Laws of Continuum Mechanics Linear Elasticity Fluids Plasticity Viscoelasticity

The flow of fluid through an elastic solid

Abstract: The discussions of constitutive equations and their use in related studies by a number of writers for interacting continua, which is based on the form of theory given byGreen andNaghdi [1], is shown to be satisfactory for many purposes even though the expressions given for the partial stresses are incomplete. Complete constitutive equations are given in this paper, and a uniqueness theorem is stated, for the slow flow of a linear viscous fluid through an anisotropic linear elastic solid.

Transient response of a sphere to torsional loading— A head injury model

Abstract: A mathematical model representation of the head subjected to torsional impact accelerations is investigated by seeking the transient response of a linear elastic sphere to a step acceleration about a diametrical axis. The displacement is prescribed to be zero at the surface of the sphere. Expressions for dynamic torsional displacement and shear stress in series form are obtained and computed. In addition, the linear viscoelastic response of the sphere is studied by using a superposition principle, and numerical results for a fluid Maxwell model are presented.

An atomistic study of fracture

Abstract: The atomistic relaxation method developed by the author to study the core structures of edge and screw dislocations in crystalline solids was used to investigate the configurations of atoms and crack-tip atom cohesion and stress fields in BCC (body-centered cubic) and FCC (face-centered cubic) iron. Single crystallites containing initially an atomically sharp and through crack (a vacancy sheet) of various orientations were subjected to tensile strain applied normal to the crack. It is shown that the crack tip stress fields depend importantly on the orientation of the crack with respect to the parent lattice, a result unexpected of existing linear elasticity and continuum theories.

Finite-element stress analysis of a non-linear problem: A connecting-road eye loaded by means of a pin

Abstract: The stress analysis of a connecting rod under a tensile load represents a non-linear problem, since the distribution of the forces, acting between the eye of the rod and the pin, changes as the loading proceeds. This is due to the tendency of the eye to deform, thus wrapping itself around the pin as the clearance between the parts is reduced. In consequence, the stress pattern in the eye also changes.Such a problem has been solved with a linear elastic finite-element computer programme. A connecting-rod eye was idealized as a plane-stress problem and special elements were introduced to connect the pin and rod together. These elements were assigned stiffnesses which were changed in a prescribed manner after each iteration until compatibility of forces and displacements, with due allowance for initial clearances, was obtained at each connecting point.Although eight iterations were required to achieve the correct displacements, it was found that the circumferential stresses did not change appreciably...

Longitudinal wave propagation on a traveling threadline—II

Abstract: The non-linear equations for wave propagation in both linear and non-linear materials are considered. An elastic shock condition is derived and the general solution for the linear elastic material is obtained by employing a von Mises type of transformation. The general non-linear elastic material is shown to have solutions of a non-linear ‘traveling wave’ character. A time to occurrence for the formation of discontinuities is obtained and compared with that of the Lax-Jeffrey theory. Solutions for both the Eulerian and Lagrangian equations are obtained.

The tensile creep behaviour of cold-drawn low-density polyethylene

Abstract: Sheets of highly anisotropic low-density polyethylene possessing transverse symmetry were prepared by the simple tensile drawing at room temperature of initially isotropic sheets. Classical elasticity theory shows that five constants are necessary to characterize the deformation behaviour of a linear elastic material possessing such symmetry. The time-dependent equivalents of these constants were determined from the simultaneous measurement of longitudinal and lateral strain during tensile creep of specimens cut at 0°, 45° and 90° to the draw direction. Creep measurements were also made on specimens cut at intermediate angles from the drawn sheets and on specimens cut at various angles from the isotropic sheets. A wide range of tensile creep strains, from 0·1% upwards, was used for all specimens to enable the extent of linear behaviour to be examined. For the oriented material, the tensile strain at the onset of non-linear behaviour varied systematically with angle, being lower than 0·2% for the 90° specimen. The isotropic material exhibited non-linear behaviour down to at least 0·1% tensile strain. At low strains it is found that the formalism of classical elasticity theory holds if isochronous creep data are used, and the extension and contraction results are combined to calculate volume changes occurring during tensile creep. It is shown that at angles away from 0° and 90° the deformation behaviour of the oriented material is dominated by an easy-shear mechanism, parallel to the oriented chains, for a wide range of tensile strains.

Reliability Of Finite Element Methods For TheNumerical Computation Of Waves

Abstract: where k is a real parameter (scalar wavenumber). Unlike the case of linear elasticity, stable dependence of both analytical (if existing) and numerical solutions on the data is not straightforward. Stability estimates of the form \\u\\t < Cij\\f\\j do hold for various norms i,j but the constants dj depend in general on the parameter k. Hence also the quality of the discrete solution depends on fc, as well as on the parameters of the numerical model (stepwidth ft, degree of approximation p). For practical application it is essential to have reliable "rules of the thumb" for the choice of the numerical parameters as a function of physical parameters. It is well known from computations that, for the "classical" Galerkin FEM, the linear rule for mesh-design kh = const, leads to reliable results only in the low frequency range. This leads to two questions:

Homogeneous constitutive equations for materials with permanent memory

Abstract: : Nonlinear homogeneous constitutive equations are developed in this thesis for highly filled polymeric materials such as solid propellants. In the range of strains below vacuole dilatation these materials obey the homogeneity rule of linearity but do not obey the superposition rule. Such materials typically exhibit an irreversible 'stress softening' called the 'Mullins' Effect.' The development in this dissertation stems from attempting to mathematically describe the failing microstructure of these composite materials in terms of a linear cumulative damage model. It is demonstrated that pth order Lebesgue norms of the strain history can be used to describe the state of damage in these materials and can also be used in the constitutive equation to characterize their time dependent mechanical response to strain disturbances. Stress analysis procedures for materials having nonlinear homogeneous constitutive equations are developed for two and three dimensional proportional boundary value problems. A series of correspondence principles are derived wherein half of the solution, either the stresses or the strains, can be obtained by solving an equivalent linear elastic problem. The remaining half of the solution can be obtained by substituting the linear elastic solution into the nonlinear homogeneous constitutive equation.

Forced Plane Strain Motion of Cylindrical Shells-A Comparison of Shell Theory with Elasticity Theory

Abstract: A radially directed load is suddenly applied to a portion of the outer surface of a circular cylindrical shell which responds in a state of plane strain. An analytical solution for the resulting dynamic response is obtained within the context of linear elasticity theory, Flugge shell theory, and an improved shell theory. A comparison of results for specific loading conditions indicates that the improved theory is far superior to the Flugge theory in terms of predicting both the magnitude and characteristics of the response. However, as expected, neither shell theory satisfactorily predicts the wave character of the initial response. (Author)

Effect of nonlinear material response on the behavior of pavements under traffic

Abstract: A REVIEW OF LABORATORY TEST DATA REVEALS THAT MOST COMMON HIGHWAY MATERIALS, UNDER CONDITIONS REPRESENTATIVE OF MOVING TRAFFIC ON AN INSERVICE PAVEMENT, EXHIBIT A NONLINEAR RESPONSE TO STRESS. THE REPORTED STRESS-STRAIN RESPONSE OF PAVEMENTS CONSTRUCTED WITH SUCH MATERIALS VARIES FROM THE STRESS-SOFTENING TO THE STRESS-STIFFENING TYPE, IN ACCORDANCE WITH THE RESPONSE OF THE CONSTITUTENT MATERIALS. A NONLINEAR ELASTIC INCREMENTAL FINITE ELEMENT ANALYSIS OF A UNIFORM SAND MASS SUBJECTED TO A UNIFORM CIRCULAR SURFACE LOAD , USING A CONSTITUTIVE EQUATION BASED ON PUBLISHED LABORATORY DATA, REVEALED A PRONOUNCED STIFFENING RELATIONSHIP BETWEEN THE APPLIED PRESSURE AND SURFACE DEFLECTION AND SLIGHTLY NONLINEAR RELATIONSHIPS BETWEEN THE APPLIED PRESSURE AND THE VERTICAL STRESSES INDUCED IN THE MASS. AN APPROXIMATE NONLINEAR ELASTIC ANALYSIS OF A FULL-DEPTH ASLPHAT CONCRETE PAVEMENT OVER A SANDY CLAY SUBGRADE, USING STRESS-STRAIN COEFFICIENT MATRICES MEASURED IN LABORATORY TRIAXIAL TESTS ON THE MATERIALS, GAVE ALMOST LINEAR RELATIONSHIPS BETWEEN THE APPLIED PRESSURE AND THE RESULTING DEFLECTION, AND DISTRIBUTIONS OF STRESSES AND STRAINS WITH THE STRUCTURE VERY SIMILAR TO THOSE YIELDED BY A LINEAR ELASTIC ANALYSIS USING STRESS-STRAIN COEFFICIENTS AT REALISTIC STRESS LEVELS. TO AN ENGINEERING APPROXIMATION, A LINEAR ANALYSIS WAS SUFFICIENTLY ACCURATE IN THE CASE OF THIS PARTICULAR FULL-DEPTH ASPHALT CONCRETE PAVEMENT BUT APPEARED UNACCEPTABLE IN THE CASE OF A PAVEMENT WITH UNBOUND GRANULAR MATERIALS CLOSE TO THE SURFACE. /AUTHOR/

Deformation and Stability of Viscoelastic Soil Media

Abstract: Linear viscoelasticity is a more resalistic representation than linear elasticity in studying the deformation and stability problems of saturated cohesive soil media The linear viscoelastic model is incorporated in terms of volumetric (a single Kelvin unit) and deviatoric (a single Maxwell Unit) components Elastic-viscoelastic correspondence is utilized to obtain the time-dependent displacement and stress fields Limitations of the correspondence principle are investigated quantitatively in parallel to stress level and overconsolidation ratio Factor of safety for stability is considered as the ratio of the time-dependent octahedral shear strength to the time dependent octahedral shear stress This in turn leads to the observation that a viscoelastic medium may experience a stable state (ie, factor of safety > 1) upon the application of a particular boundary loading and yet, it may bring itself into an unstable state with time (ie, factor of safety = 1) even though the shear strength is (or is not) exhibiting an increase during the same interval This is demonstrated for a strip surface loading and a plane strain problem

Application of Fracture Mechanics to Predicting Failures in Solid Propellants.

Abstract: : Analytical and experimental methods were developed to measure the structural and fracture properties of a linear elastic epoxy and a linear viscoelastic polyurethane material (Solithane). These properties were used in stress and fracture analysis to predict the initiation, growth, and trajectory of cracks in circular port and star grain designs. Experimental confirmation of these predictions was obtained by testing thin two-dimensional (plane stress) and three-dimensional grains of both materials under thermal and pressurization loading conditions. Crack initiation conditions with the epoxy material were in excellent agreement with theoretical predictions using elastic numerical analysis methods. Crack initiation and velocity measurements with the Solithane two-dimensional grains were in good agreement with numerical analysis techniques combined with a time-dependent crack analysis which was programmed for efficient use. An approximate three-dimensional fracture analysis for circular port grains with elliptical flaws was developed and programmed. Results of the crack prediction computer program were in good agreement with three-dimensional grain test data. Experimental work with the epoxy and Solithane grains showed that cracks in two- and three-dimensional grains will propagate under thermal and pressurization loading conditions and these fracture conditions can be predicted using analytic methods presented in this report. Photoelastic analysis was used to confirm the stress intensity factors at the crack tip for thermal and pressurization loaded grains. Photoelastic results were in good agreement with both the numerical predicitons and the crack velocity measurements with Solithane grains. (Author)

Photoelastic and finite element analysis of embankments constructed over soft soils

Abstract: STRESSES AND DISPLACEMENTS WERE DETERMINED EXPERIMENTALLY AND NUMERICALLY FOR MODEL EMBANKMENTS THAT UNDERWENT LARGE DEFLECTIONS WHEN CONSTRUCTED OVER MODELS OF TRENCHED AND UNTRENCHED SOFT ORGANIC SOILS. IN THE EXPERIMENTAL PHASE OF THE STUDY, GRID AND PHOTOELASTIC METHODS WERE USED TO DETERMINE DEFORMATIONS AND STRESSES IN THE MODEL SOIL, WHICH CONSISTED OF A SOFT GELATIN MIX WITH AN EMBEDDED INK GRID. A GELATIN SLAB, WHICH MODELED A CROSS SECTION OF THE SOIL SUPPORTING AN EMBANKMENT, WAS FORMED IN A TANK WITH GLASS SIDES. MODEL EMBANKMENTS WERE PLACED IN THE TANK ON THE GELATIN, AND RESULTING GRID DEFORMATIONS AND PHOTOELASTIC FRINGES IN THE GELATIN WERE PHOTOGRAPHED. EXPERIMENTAL AND NUMERICAL (FINITE ELEMENT) RESULTS FOR MAXIMUM NORMAL AND SHEARING STRESS ALONG THE CENTERLINE BENEATH THE EMBANKMENTS WERE COMPARED WITH RESULTS OBTAINED BY CLASSICAL LINEAR ELASTICITY. FOR EMBANKMENTS UNDERGOING LARGE DEFORMATIONS, WHERE ANALYSIS BY THE CLASSICAL THEORY USING SMALL DEFORMATION RESTRICTIONS AND SUPERPOSITION PRINCIPLES IS SUSPECT, IT WAS FOUND THAT THE STRESS VALUES WERE BOUNDED ABOVE BY THE CLASSICAL SOLUTION IN THE CASE OF MAXIMUM SHEAR STRESS AND WERE VERY CLOSE TO THOSE PREDICTED BY THE CLASSICAL SOLUTION IN THE CASE OF NORMAL STRESS. IT IS CONCLUDED THAT THE DESIGNER IS CONSERVATIVE IN USING THE CLASSIC BOUSSINESQ APPROACH IN PREDICTING BOTH MAXIMUM SHEAR STRESS AND MAXIMUM VERTICAL NORMAL STRESS ALONG THE CENTERLINE AND BENEATH AN EMBANKMENT UNDERGOING LARGE DEFORMATIONS. WHERE TRENCHING IS USED, NEGATIVE (UPWARD) AS WELL AS POSITIVE (DOWNWARD) DISPLACEMENTS SHOULD BE TAKEN INTO ACCOUNT IN ESTIMATING THE AREA OF THE NET CROSS SECTION OF AN EMBANKMENT. TRENCHING IS NOT RECOMMENDED FOR THE PLACEMENT OF EMBANKMENTS ON SOFT FOUNDATIONS UNLESS (A) THE TRENCH IS TO BE USED AS AN ECONOMICAL WATERWAY TO TRANSPORT BULK EMBANKMENT MATERIAL AND EQUIPMENT FOR THE CONSTRUCTION, OR (B) THE TRENCHING OPERATION WILL REMOVE SOFT MATERIAL AND EXPOSE STIFFER LAYERS WITH HIGHER BEARING CAPACITY. UNTRENCHED EMBANKMENT CONFIGURATIONS PRODUCED NO STRESS CONCENTRATIONS BENEATH THE EMBANKMENT. /AUTHOR/

Stress and cracking in concrete due to shrinkage

Abstract: A method has been developed for calculating the stress in concrete which arises from linear-elastic restraint to shrinkage. Such stress depends on the rate and amount of shrinkage and creep and on the stiffnesses of the concrete and restraint. The relative importance of these factors is examined. Experiments on a particular concrete have verified that the method can predict stress and hence can indicate the likelihood of cracking. /TRRL/

A new method for measuring the complex shear modulus of a viscoelastic material coupled with an elastic material

Abstract: In the mechanical-dynamic characterization of viscoelastic materials as a function of temperature, considerable difficulties are encountered, due to the change of joint, to the strong variation of the modulus of elasticity and the increase ofQ−1.