Showing papers on "Linear elasticity published in 1971"

The Elastic Analysis of Compressible Piles and Pile Groups

Abstract: Synopsis The response of rigid and compressible single piles embedded in a homogeneous isotropic linear elastic medium has been obtained by a rigorous analysis based on Mindlin's solutions for a po...

Uniqueness Theorems in Linear Elasticity

Abstract: 1 Introduction.- 2 Basic Equations.- 2.1 Formulation of Initial-Boundary Value Problems.- 2.2 The Classical and Weak Solutions.- 2.3 The Homogeneous Isotropic Body. Plane Elasticity.- 2.4 Definiteness Properties of the Elasticities.- 3 Early Work.- 4 Modern Uniqueness Theorems in Three-Dimensional Elastostatics.- 4.1 The Displacement Boundary Value Problem for Bounded Regions.- 4.1.1 General Anisotropy.- 4.1.2 A Homogeneous Anisotropic Material.- 4.1.3 A Homogeneous Isotropic Material.- 4.1.4 The Implication of Strong Ellipticity for Uniqueness.- 4.1.5 The Non-Homogeneous Isotropic Material with no Definiteness Assumptions on the Elasticities.- 4.1.6 The Displacement Boundary Value Problem for a Homogeneous Isotropic Sphere.- 4.1.7 Fichera's Maximum Principle.- 4.2 Exterior Domains.- 4.3 The Traction Boundary Value Problem.- 4.3.1 General Anisotropy.- 4.3.2 A Homogeneous Isotropic Material.- 4.3.3 The Traction Boundary Value Problem for a Homogeneous Isotropic Elastic Sphere.- 4.3.4 Necessary Conditions for Uniqueness in the Traction Boundary Value Problem for Three-Dimensional Homogeneous Isotropic Elastic Bodies.- 4.4 Mixed Boundary Value Problems.- 4.4.1 General Anisotropy.- 4.4.2 A Homogeneous Isotropic Material.- 5 Uniqueness Theorems in Homogeneous Isotropic Two-Dimensional Elastostatics.- 5.1 Kirchhoff's Theorem in Two-Dimensions. The Displacement and Traction Boundary Value Problems.- 5.2 Uniqueness in Plane Problems with Special Geometries.- Appendix: Uniqueness of Three-Dimensional Axisymmetric Solutions.- 6 Problems in the Whole- and Half-Space.- 6.1 Specification of the Various Boundary Value Problems. Continuity onto the Boundary and in the Neighbourhood of Infinity.- 6.2 Uniqueness of Problems (a)-(d). Corollaries for the Space EN.- 6.3 Uniqueness for the Mixed-Mixed Problem of Type (e).- 6.3.1 A Complete Representation of the Biharmonic Displacement in a Homogeneous Isotropic Body Occupying the Half-Space.- 6.3.2 Uniqueness in the Mixed-Mixed Problem (e).- 7 Miscellaneous Boundary Value Problems.- 7.1 Problems for a Sphere.- 7.2 The Cauchy Problem for Isotropic Elastostatics.- 7.3 The Signorini Problem. Other Problems with Ambiguous Conditions.- 8 Uniqueness Theorems in Elastodynamics. Relations with Existence, Stability, and Boundedness of Solutions.- 8.1 The Initial Displacement and Mixed-Boundary Value Problems. Energy Arguments.- 8.2 The Initial-Displacement Boundary Value Problem. Analyticity Arguments.- 8.3 The Initial-Mixed Boundary Value Problem for Bounded Regions. Further Arguments.- 8.4 Summary of Existing Results in the Uniqueness of Elastodynamic Solutions.- 8.5 Non-Standard Problems, including those with Ambiguous Conditions.- 8.6 Stability, Boundedness, Existence and Uniqueness.- References.

Compressive stress pulse times in flexible pavements for use in dynamic testing

Abstract: TRAFFIC MOVING OVER A PAVEMENT STRUCTURE RESULTS IN A LARGE NUMBER OF RAPIDLY APPLIED STRESS PULSES BEING APPLIED TO THE MATERIAL COMPRISING EACH LAYER. THIS CONDITION IS BEST CHARACTERIZED BY DYNAMIC TESTS SUCH AS THE REPEATED LOAD TRIAXIAL TEST OR FATIGUE TESTS. THE SHAPE AND DURATION OF THE COMPRESSIVE STRESS PULSE RESULTING AT DIFFERENT DEPTHS BENEATH THE SURFACE ARE STUDIED FOR SEVERAL FLEXIBLE PAVEMENT SYSTEMS AND VEHICLE SPEEDS VARYING BETWEEN 1 AND 45 MPH. A COMPARISON IS MADE OF THE COMPRESSIVE STRESS PULSES CAUSED BY A VEHICLE MOVING OVER A LAYERED PAVEMENT SYSTEM AT A SPEED OF 1 MPH USING BOTH LINEAR AND NONLINEAR ELASTIC FINITE ELEMENT THEORY. VISCOUS EFFECTS AND INERTIA FORCES ARE NEGLECTED. BECAUSE THE NORMALIZED STRESS PULSES CALCULATED USING THE LINEAR ELASTIC THEORY ARE FOUND TO AGREE CLOSELY WITH THOSE CALCULATED USING THE NONLINEAR THEORY, LINEAR ELASTIC FINITE ELEMENT THEORY IS USED THROUGHOUT THE STUDY. THE RESULTS SHOW THAT THE SHAPE OF THE COMPRESSIVE STRESS PULSE VARIES FROM APPROXIMATELY A SINUSOIDAL ONE AT THE SURFACE TO MORE NEARLY A TRIANGULAR PULSE AT DEPTHS BELOW APPROXIMATELY THE MIDDLE OF THE BASE. TYPICALLY, THE COMPRESSIVE PULSE TIME ALSO VARIES ALMOST INVERSELY WITH VEHICLE SPEED UP TO AT LEAST A SPEED OF 45 MPH, THE CALCULATED STRESS PULSE TIMES WERE EMPIRICALLY CORRECTED FOR VISCOUS EFFECTS AND INERTIA FORCES USING THE RESULTS OF FIELD MEASUREMENTS MADE AT THE AASHO ROAD TEST. CURVES ARE PRESENTED FROM WHICH APPROXIMATE COMPRESSIVE STRESS PULSE TIMES CAN BE SELECTED FOR USE IN DYNAMIC TESTING. /AUTHOR/

Photoelastic determination of stress-intensity factors

Abstract: The increasing number of analytical and numerical solutions for the crack-tip stress-intensity factor has greatly widened the scope of application of linear elastic fracture-mechanics technology. Experimental verification of a particular solution by elastic stress analysis is often a necessary supplement to provide the criteria for proper application to actual design problems. In this paper, it is shown that the photoelastic technique can be used to obtain rather good estimates of the stress-intensity factor for various specimen geometries and loading conditions. Treated are the following cases: wedge-opening load specimen, several notched rotating-disk configurations, and a notched pressure vessel. A sharp crack is simulated by a relatively narrow notch terminating in a root radius of 0.010 in or less. Stress distributions along the section of symmetry ahead of the notch tip are obtained using three-dimensional frozen-stress photoelasticity. The results are used to determine the stress-intensity factor, cK I , by three methods. Two of these are based on Irwin's expressions for the elastic stress field at the tip cf a crack, and the other is a result of Neuber's hyperbolic-notch analysis. Agreement, with available analytical solutions is good.

Further study of composite laminates under cylindrical bending.

Abstract: In a series of three papers [1,2,3], the range of applicability of classical laminated plate theory (CPT) in describing the response of composite laminates under static bending has been examined. Briefly, exact solutions within the framework of linear elasticity theory were developed and compared to the respective solutions governed by CPT [4,5,6]. Numerical data calculated based on simple harmonic load distributions have indicated rather wide discrepancy between the two solutions for laminates having low span-to-depth ratios. At high aspect ratios however, the CPT solution is in good agreement with the elasticity solution.

Numerical analyses of deformability tests in jointed rock — ``Joint Perturbation'' and ``No Tension'' Finite Element solutions

Abstract: Numerical Analyses of Deformability Tests in Jointed Rock — “Joint Perturbation” and “No Tension” Finite Element Solutions Sound design of structures in or upon rock requires a thorough knowledge of the rock mass deformability. Field test data interpretation has generally relied upon analytical or Finite Element linear elastic solutions. However, when testing in jointed rock, these can no longer be readily used since the medium cannot resist the tension induced upon loading. Accordingly, other solutions are required. If the discontinuities of the rock mass can all be mapped and their surface properties determined, the Finite Element model will use a “Joint Perturbation” solution where the “joint” elements have variable stiffnesses to account for movements along the fractures. If the discontinuities cannot be satisfactorily mapped, a global approach is provided by a “No Tension” analysis of the “stress transfer” type. The two techniques are compared in the case of bore hole jack deformability tests and found to agree remarkably well, indicating an appreciable difference from results of linear elasticity.

Mohr circles for large and small strains in two-dimensional deformations:

Abstract: The mathematical apparatus is provided for the analysis of large deformations in two dimensions. Deformations are considered as co-ordinate transformations. For homogeneous deformations, a method of finding the directions and magnitudes of the principal strains is presented. The formulae used may be expressed either in terms of the elements of a transformation matrix or in those of displacement gradients. It is shown that, when the displacements are small, the results are reduced to the familiar formulae for small strains in linear elasticity. The Mohr circles for Green's and Cauchy's deformation tensors are also discussed, and their relations with the Mohr circle for the matrix of pure deformation shown. Examples of Mohr circles are provided for some commonly known types of deformation.

Dependence of linear elasticity solutions on the elastic constants: I. Dependence on Poisson's ratio in elastostatics

Abstract: Necessary and sufficient conditions on the dilatation for the displacement and/or the stress to be independent of Poisson's ratio in the standard boundary-value problems of three-dimensional classical elastostatics are determined. Formulas for the volume averages of the changes induced in the stress and strain by a variation of Poisson's ratio are also given.RésuméOn détermine des conditions nécessaires et suffisantes sur le changement de volume pour que les déplacements et (ou) les contraintes soient indépendants du coefficient de Poisson dans les problèmes des conditions des limites classiques de l'élasticité tri-dimensionnelle. On donne également des formules pour les moyennes voluminiques des changements subis par les contraintes et les déformations lors d'une variation du coefficient de Poisson.

Dynamic response of an elastic cylindrical shell-solid core composite subject to time- dependent loading

Abstract: A long cylindrical shell bonded to a solid elastic core is subject to external time-dependent loading. The loading is uniform along the generator of the cylinder but may have arbitrary variation around the circumference. The solution is obtained by means of the Laplace transform. Inversion is accomplished by the Residue Theorem, giving the solution in the form of a modal series. Numerical results for impulsive loads demonstrate the wave nature of the motion in the core. The dynamic stress concentration due to the focusing effect is evident. The effect of wave propagation in the core on the motion of the shell varies widely depending upon the material properties of the core. These properties also affect the rate of convergence of the modal solution.

Body Force Equivalents as Sources of Anelastic Processes

Abstract: Once a failure criterion and a rule for conversion of mechanical into thermal energy are established, linear elastic materials can be made to behave, theoretically at least, in a highly anelastic f...

Optimization of a Standard Linear Viscoelastic Material for Use in a Seat Belt

Abstract: : It is shown that a properly designed viscoelastic seat-belt material could have a low stiffness at low strain rates while still affording considerable improvement in the maximum safe cruising speed in comparison with a linear elastic belt. (Author)

Approximate creep analysis for thermoplastic beams and struts

Abstract: Creep deformation up to 106 s has been measured for beams subjected to pure and three-point bending and for buckling struts made from polymethyl methacrylate, polyvinyl chloride and polypropylene. Of the analytical methods considered for predicting creep, the most accurate was that in which the isochronous stress-strain curves in tension and compression were used in conjunction with equilibrium and strain displacement equations. However, this involved more computation than the other two methods which simply substituted a secant-modulus value from the isochronous curves (1) at the expected maximum stress and (2) at a limit of 1 per cent strain in the established linear elastic formulae for the components. In (1) there was only slight over-estimation of experimental deformations but in (2) there were quite serious differences, particularly at longer times, and small strains.

On Reissner's variational theorem for boundary values in linear elasticity

Abstract: E. Reissner suggested a variational theorem for the theory of elasticity, related closely to the well-known Trefftz method. In the present paper, the Reissner's theorem is discussed within the range of linear anisotropic and non-homogeneous elasticity. For the traction boundary-value problem, the minimal property of the functional and the convergence of any minimizing sequence are proved. For the displacement boundary-value problem and sime mixed problems, it is shown that a modification is necessary. Then, in case of the displacement problem, the maximal property of the functional on the modified class of admissible functions and the convergence of maximizing sequence are proved.

A static and dynamic finite element shell analysis with experimental verification

Abstract: A system of finite element shell analysis codes, called SABOR/DRASTIC, is used to analyse a complex two-layered shell of revolution under static and dynamic asymmetric loads. The dynamic analysis is compared with experimentally measured response. In this linear elastic analysis, emphasis is placed on the inherent flexibility of the finite element method in modelling the complex structural geometry of a given test specimen. Static studies, which involve variations in important shell parameters, and dynamic studies, which provide a successful correlation with experiment, are used to illustrate both the detail and the generality with which shell analyses may now be performed with confidence.

Evaluation of the second order elastic moduli

Abstract: The expressions of the apparent linear elastic moduli and their first and second derivatives, with respect to hydrostatic pressure, are obtained according to the second order elasticity theory. As a particular case when the material is hyperelastic, formulae of the first derivatives of the linear elastic moduli reduce to those obtained by Seeger and Buck. The general formulae for determining the four second order elastic moduli are derived and expressed in terms of four measurable quantities. The numerical values of these moduli for steel and copper are calculated according to Poynting's experimental data. They are compared with Seeger and Buck's results with good agreement when the material is assumed to be hyperelastic.

On some problems in a theory of thermally and mechanically interacting continuous media

Abstract: Using a linearized theory of thermally and mechanically interacting mixture of linear elastic solid and viscous fluid, we derive a fundamental relation in an integral form called a reciprocity relation. This reciprocity relation relates the solution of one initial-boundary value problem with a given set of initial and boundary data to the solution of a second initial-boundary value problem corresponding to a different initial and boundary data for a given interacting mixture. From this general integral relation, reciprocity relations are derived for a heat-conducting linear elastic solid, and for a heat-conducting viscous fluid. An initial-boundary value problem is posed and solved for the mixture of linear elastic solid and viscous fluid. With the aid of the Laplace transform and the contour integration, a real integral representation for the displacement of the solid constituent is obtained as one of the principal results of the analysis.

A potential representation for two-dimensional waves in elastic materials of harmonic type

Abstract: In the present note we consider two-dimensional finite dynamical deformations for the class of homogeneous, isotropic elastic materials introduced by F. John in [1] and referred to by him as materials of harmonic type. The theory of such materials, developed in [1] and [2], appears to be simpler in many respects than that of more general elastic materials, and it may offer the possibility of investigating some features of nonlinear elastic behavior more explicitly than is possible in general. For plane motions of such materials, we derive here a representation for the displacements in terms of two potentials which is analogous to the theorem of Lame in classical linear elasticity (see [3]) for the case of plane strain. The two nonlinear differential equations satisfied by the potentials reduce upon linearization to the wave equations associated with irrotational and equivoluminai waves in the linear theory. In the following section we state without derivation the equations governing two-dimensional waves in an elastic material of harmonic type. The reader is referred to [1] for details. In Sec. 3 we derive the representation in terms of potentials described briefly above.

The Influence of Construction Step Sequence and Nonlinear Material Behavior on Cracking of Earth and Rock-Fill Dams.

Abstract: : The WES-Palmerton finite element computer program was used to make a preliminary study of the effects of construction sequence and nonlinear modulus on cracking in earth dams. Results of the nonlinear analysis are compared with linear elastic one-step analyses and linear elastic construction step analyses including low tension modulus. Linear elastic, one-step analyses gave the largest tension zone, and no tension zone occurred in the nonlinear analyses. No provisions were included for tensile strength of the soil.