Showing papers on "Linear elasticity published in 1979"

Solutions for effective shear properties in three phase sphere and cylinder models

Abstract: S olutions are presented for the effective shear modulus of two types of composite material models. The first type is that of a macroscopically isotropic composite medium containing spherical inclusions. The corresponding model employed is that involving three phases: the spherical inclusion, a spherical annulus of matrix material and an outer region of equivalent homogeneous material of unlimited extent. The corresponding two-dimensional, polar model is used to represent a transversely isotropic, fiber reinforced medium. In the latter case only the transverse effective shear modulus is obtained. The relative volumes of the inclusion phase to the matrix annulus phase in the three phase models are taken to be the given volume fractions of the inclusion phases in the composite materials at large. The results are found to differ from those of the well-known Kerner and Hermans formulae for the same models. The latter works are now understood to violate a continuity condition at the matrix to equivalent homogeneous medium interface. The present results are compared extensively with results from other related models. Conditions of linear elasticity are assumed.

On the stress singularities in the plane elasticity of the composite wedge

Abstract: This paper examines the composite, two-dimensional, linear elastic wedge for singular stresses at its vertex. A full range of wedge boundary and matching conditions is considered. Using separation of variables on the Airy stress function, the usual determinant conditions for singularities of the formO(r -λ) asr → 0 are established and further conditions are derived for singularities of the formO(r -λ lnr) asr → 0. The order of the determinant involved in these conditions depends upon the number of materials comprising the wedge. Two systematic methods of expanding the determinant for theN-material wedge are presented.

Derivation of a new stiffness matrix for helically armoured cables considering tension and torsion

Abstract: A new element stiffness matrix is derived for straight cable elements subjected to tension and torsion The cross-section of a cable, which may consist of many different structural components, is treated in the following as a single composite element The derivation is quite general; consequently, the results can be used for a broad category of cable configurations Individual helical armourning wires, for instance, may have unique geometric and material properties In addition, no limit is placed on the number of wire layers Furthermore, compressibility of the central core element can also be considered The equations of equilibrium are first derived to include ‘internal’ geometric non-linearties produced by large deformations (axial elongation and rotatioin) of a straight cable element These equations are then linearized in a consistent manner to give a liner stiffness matrix Linear elasticity is assumed throughout Excellent agreement with experimental results for two different cables validates the correctness of the analysis

Equilibrium finite elements for the linear elastic problem

Abstract: We consider a class of equilibrium finite element methods for elasticity problems. The approximate stresses satisfy the equilibrium equations but the symmetry of the stress tensor is relaxed. Optimal error bounds for the stresses and numerical examples are given.

A non-linear uniaxial integral constitutive equation incorporating rate effects, creep and relaxation

Abstract: A previously proposed first order non-linear differential equation for uniaxial viscoplasticity, which is non-linear in stress and strain but linear in stress and strain rates, is transformed into an equivalent integral equation. The proposed equation employs total strain only and is symmetric with respect to the origin and applies for tension and compression. The limiting behavior for large strains and large times for monotonic, creep and relaxation loading is investigated and appropriate limits are obtained. When the equation is specialized to an overstress model it is qualitatively shown to reproduce key features of viscoplastic behavior. These include: initial linear elastic or linear viscoelastic response: immediate elastic slope for a large instantaneous change in strain rate normal strain rate sensitivity and non-linear spacing of the stress-strain curves obtained at various strain rates; and primary and secondary creep and relaxation such that the creep (relaxation) curves do not cross. Isochronous creep curves are also considered. Other specializations yield wavy stress-strain curves and inverse strain rate sensitivity. For cyclic loading the model must be modified to account for history dependence in the sense of plasticity.

Weight functions from virtual crack extension

Abstract: The stiffness-derivative method of Parks1 for calculating the linear elastic crack tip stress intensity factor for any symmetric crack configuration and a particular loading is extended to calculate the weight function vector field2,3 which serves as a Green's function for the stress intensity factor. The method, which combines the observations of Rice3 on the weight function and of Zienkiewicz4 on the differential stiffness method, permits very efficient determination of the weight function, requiring only one additional back-substitution on the already-factored stiffness matrix. Thus, the stress intensity factor for arbitrary loading of this configuration can subsequently be determined by quadrature alone. The promising extension of the method to three-dimensional configurations is outlined. While this manuscript was under review, the authors became aware of the recent work of Vanderglas,21 in which the same approach as ours is used to extend the stiffness derivative method. The present work was then voluntarily revised in order to address further certain aspects of the topic of shape function perturbation, which Vanderglas noted.

A Course in Elasticity

Abstract: 1. Kinematics of Continuous Media.- 1.1. Material and Spatial Coordinates.- 1.2. Neighborhood Transformations.- 1.3 Composition of Changes of Configuration.- 1.4 Measure of the State of Local Deformation. Green's and Jaumann's Strain.- 1.5 Rigid-Body Rotations of a Neighborhood.- 1.6 The Kinematical Decomposition of the Jacobian Matrix.- 1.7 Geometric Interpretation of Infinitesimal Strains.- 1.8 The Eulerian Viewpoint in Kinematics. Almansi's Strain.- 1.9 Eulerian Measures of Rates of Deformation and Rotation.- 1.10 Temporal, Variation of the Polar Decomposition of the Jacobian Matrix.- 2. Statics and Virtual Work.- 2.1. The Concept of Stress. True Stress.- 2.2. The Piola Stresses.- 2.3. Translational Equilibrium Equations.- 2.4. Rotational Equilibrium Equations.- 2.5. Statics and Virtual Work.- 2.6. Commutativity of the Operators ? and Di.- 2.7 Virtual Work in a Continuous Medium.- 2.8. Statics and Virtual Power for True Stresses.- 2.9. Statics and Virtual Work in Infinitesimal Changes of Configuration.- 3. Conservation of Energy.- 3.1. Constitutive Equations for Piola's Stresses.- 3.2. The Kirchhoff-Trefftz Stresses.- 3.3 The Constitutive Equations of Geometrically Linear Elasticity.- 4. Cartesian Tensors.- 4.1. Bases and Change of Basis.- 4.2 Tensors.- 4.3 Some Special Tensors.- 4.4 The Vector Product.- 4.5. Structure of Symmetric Cartesian Tensors of Order Two. Principal Axes.- 4.6. Fundamental Invariants and the Deviator.- 4.7. Structure of Skew-Symmetric Cartesian Tensors of the Second Order.- 4.8. Matrix Representation of Tensor Operations.- 5. The Equations of Linear Elasticity.- 5.1. Compatibility of Strains in a Simply Connected Region.- 5.2. Compatibility of Strains in a Multiply Connected Region.- 5.3. Principal Elongations and Fundamental Invariants of Strain.- 5.4. Principal Stresses and Fundamental Invariants of the Stress State.- 5.5. Octahedral Stresses and Strains.- 5.6. Mohr's Circles.- 5.7. Statics and Virtual Work.- 5.8. Taylor's Development of the Strain Energy.- 5.9. Infinitesimal Stability.- 5.10. Hadamard's Condition for Infinitesimal Stability.- 5.11. Isotropy and Anisotropy.- 5.12. Criteria for Elastic Limits.- 5.13. Navier's Equations.- 5.14. The Beltrami-Michell Equations.- 6. Extension, Bending, and Torsion of Prismatic Beams.- 6.1. Green's and Stokes' Formulas.- 6.2. The Centroid.- 6.3. Moments of Inertia.- 6.4. The Semi-Inverse Method of Saint-Venant.- 6.5. Resultants of Stresses on a Cross Section.- 6.6. Calculation of the Transverse Displacements.- 6.7. Equations Governing the Shear Stresses.- 6.8. Calculation of the Longitudinal Displacement.- 6.9. Separation of Solutions.- 6.10. Pure Torsion.- 6.11. The Center of Torsion for a Fully Constrained Section.- 6.12. Bending without Torsion.- 6.13. The Stiffness Relation for the Twist.- 6.14. Total Energy as a Function of the Deformations of the Fibers.- 6.15. Total Energy as a Function of Generalized Forces.- 6.16. The Generalized Constitutive Equations for Bending and Torsion of Beams.- 6.17. One-Dimensional Formulation of Bending and Torsion of Beams.- 6.18. Applications.- A. Stress function for torsion of the elliptic bar.- B. Stress functions for torsion of the circular bar.- C. Stress functions with poles.- D. Torsion of a triangular bar.- E. Torsion of a rectangular bar.- F. Bending of a circular bar.- G. Bending of a circular tube.- H. Bending of a rectangular bar.- 7. Plane Stress and Plane Strain.- 7.1. Lemmas for the Integration of Partial Differential Equations in Complex Form.- 7.2. The Structure of a Biharmonic Function.- 7.3. Structure of the Solution of the Problems of Plane Strain.- 7.4.Structure of the Solution of the Problem of Plane Stress.- 7.5. Generalized Plane Stress.- 7.6. Airy's Stress Function.- 7.7. Complex Representation of Airy's Function.- 7.8. Polar Coordinates.- 7.9. Applications in Cartesian Coordinates.- A. The state of hydrostatic stress.- B. Uniform gradient of areal dilation.- C. Pure uniform shear.- D. Linear variation of a normal stress.- E. Simple extension.- F. Pure bending.- G. Shear lag.- H. Bending by shear forces.- I. Saint-Venant's bending of a rectangular beam with flanges.- J. Transverse loading of a beam with flanges.- 7.10. Applications in Polar Coordinates.- A. Circular aperture with traction-free circumference in a plate in plane stress.- B. Volterra's dislocation of the circular ring.- C. Bending of beams with constant curvature.- D. The annular ring loaded by shear tractions.- E. The thick tube under pressure.- F. Concentric cylindrical tubes and rings.- G. Force concentrated at the origin in an infinite plate.- 8. Bending of Plates.- 8.1. Basic Hypotheses.- 8.2. Application of the Canonical Variational Principle.- 8.3. The Two-Dimensional Canonical Principle.- 8.4. Further Connections Between the Two- and Three-Dimensional Theories.- 8.5. Other Types of Approximations.- 8.6. Kirchhoff's Hypothesis.- 8.7. Boundary Conditions in Kirchhoff's Theory.- 8.8. Kirchhoff's Variational Principle.- 8.9. Structure of the Solution of the Equations of Plates of Moderate Thickness.- 8.10. The Edge Effect.- 8.11. Torsion of a Plate.- 8.12. Saint-Venant's Bending of a Plate.- 8.13. Particular Solutions for Transverse Load.- 8.14. Solutions in Polar Coordinates.- 8.15. Axisymmetric Bending.

Rayleigh Waves in Linear Elasticity as a Propagation of Singularities Phenomenon

Abstract: We examine the propagation of surface waves known as Rayleigh waves from the perspective of microlocal analysis. This paper is a TeXed version of Taylor (1979).

Failure analysis of composite laminates with free edges

Abstract: Linear elastic stress distributions obtained from a refined finite element mesh are used in conjunction with the tensor polynomial failure criterion to predict the initiation of failure in symmetric, finite-width graphite-epoxy laminates under tensile loading Results are presented for a wide variety of laminates including: (+ and - theta)s angle-ply; cross-ply (0/90)s and (90/0)s; and quasi-isotropic (90/0/+ and - 45)s and (+ and -45/0/90)s It is shown that the elastic stress distributions generally compare favorably with other published results, but also indicate improved satisfaction of the stress-free boundary conditions and indicate some differences in the singular behavior of selected stress components at the free edge The tensor polynomial failure criterion is used to predict the location and mode of first failure Examination of the individual terms of the polynomial indicates different modes of failure depending upon the laminate configuration

On the dynamic behavior of poroelastic materials

Abstract: Poroelastic materials are two phase material systems consisting of a porous linear elastic solid phase filled with a Newtonian viscous fluid. Analytical investigations have demonstrated that poroelastic structures can, depending on loading and geometry, exhibit elastic response or the creep and relaxation response associated with various models of linear viscoelastic materials. This paper examines the dynamic response to harmonic loading of a disk or slab of poroelastic material. Biot’s dynamic theory for deformable poroelastic media is applied to derive expressions for a complex modulus of the material in terms of poroelastic material coefficients. Both a quasistatic analysis, accounting for dissipation but neglecting inertia, and a dynamic analysis, which neglects dissipation, are presented. For a choice of poroelastic coefficients roughly appropriate to water filled sandstone or compact bone, the poroelastic layer is shown to exhibit rubber‐to‐glass transition in a low‐frequency range. Resonance effect...

On the impossibility of linear Cauchy and Piola-Kirchhoff constitutive theories for stress in solids

Abstract: We investigate the possibility of linear elasticity as an infinitesimal theory based on a genuinely linear response function which retains its validity even for finite deformations. Careful consideration of the domain of definition of the stress response function, the definition of linearity and the notion of material frame-indifference leads to our main result that an exact linear constitutive theory for elastic solids is impossible. We then generalize our result to viscoelasticity theory where the stress response is dependent on deformation gradient histories.

Some Upper Bound Principles to Plastic Strains in Dynamic Shakedown of Elastoplastic Structures

Abstract: Suitable measures of plastic strains after adaptation (shakedown) of elastoplastic continuous structures subjected to dynamic loading are shown to be bounded from above by quantities which can be evaluated on the basis of a linear elastic response of the body to prescribed initial conditions The results achieved are extended to elastic work-hardening bodies under the assumption of a piecewise linear yield surface The practical use of the results is checked through application to a simple example

Meridionally Imperfect Cooling Towers

Abstract: With the recent collapse of a cooling tower in Scotland having been attributed to the presence of imperfections in the positioning of the shell surface, the influences on the linear elastic behavior of axi-symmetric meridional imperfections in cooling tower shells are reassessed. It is shown that even moderate imperfections can result in horizontal membrane tension stresses of similar orders of magnitude as the vertical membrane compression stresses that would exist in the perfect tower. The geometrically imperfect shell is modelled as an assemblage of piecewise continuous second order shells of rotation. The differential system describing the behavior of this shell assemblage is briefly outlined, and a selection of numerical experiments using a finite difference approximation are used to isolate the dominant parameters controlling the response of the imperfect shell.

Dynamic stiffness for rectangular rigid foundations on a semi‐infinite elastic medium

Abstract: The present paper deals with the dynamic steady-state force-displacement relationships (complex stiffness) for rectangular rigid foundations resting on a semi-infinite medium, consisting of homogenous, isotropic, linear elastic materials. The foundations are considered to be excited under harmonic vertical and rocking vibration. This gives mixed boundary value problems which cannot be easily solved by analytical approaches. Therefore, a numerical method is proposed here. The method is based on quite, simple equations, and is straightforward in computation, compared with other methods. Although the proposed method gives just approximate solutions, it is satisfactory for engineering practices, and the soluations become highly accurate for a small value of ωB/Vs. The results obtained by the proposed method are compared with those of other methods to evaluate the accuracy of the results. The effects of length/width ratio and the area of the contact plane of the foundations are also discussed.

On the representation of elastic displacement fields in terms of three harmonic functions

Abstract: In this paper the representation of displacement fields in linear elasticity in terms of harmonic functions is considered. In the original work of Papkovich and Neuber four harmonic functions were presented with a subsequent reduction to three on the grounds that only three are sufficient for the representation of displacements fields. This reduction is unsubstantiated and several authors have investigated the generality of the Papkovich-Neuber solutions. The paper derives by simple means the conditions under which it is possible to omit one of the four harmonic functions and considers the significance of the subsequent three function form.

Part-Elliptical Cracks Emanating From Open and Loaded Holes in Plates

Abstract: A linear elastic analysis using the finite element-alternating method is conducted for problems of single semi-elliptical and double quarter-elliptical cracks near fastener holes. Mode-one stress intensity factors are presented along the crack periphery for cases of open and loaded holes and crack opening displacements are calculated. Results are shown for a variety of crack geometries and loading conditions and for two ratios of hole diameter to plate thickness.

Flow of elastic compressible spheres in tubes

Abstract: The flow of closely fitting neutrally buoyant elastic spheres through a circular cylindrical tube is considered under the assumptions that the Reynolds’ equation is valid in the fluid and equations of linear elasticity hold in the solid. Computations are carried out for several values of Poisson's ratio. The results are compared with the results of previous models on elastic compressible particles.

Creep of 2618 Aluminum Under Step Stress Changes Predicted by a Viscous-Viscoelastic Model

Abstract: : Nonlinear constitutive equations are developed and used to predict from constant stress data the creep behavior of 2618 Aluminum at 200 C (392 F) for tension or torsion stresses under varying stress history including step-up, step-down, and reloading stress changes. The strain in the constitutive equation employed includes the following components: linear elastic, time-independent plastic, nonlinear time-dependent recoverable (viscoelastic), nonlinear time-dependent nonrecoverable (viscous) positive, and nonlinear time-dependent nonrecoverable (viscous) negative. The modified superposition principle, derived from the multiple integral representation, and strain hardening theory were used to represent the recoverable and nonrecoverable components, respectively, of the time-dependent strain in the constitutive equations. (Author)

Strength of Wood Beams with End Splits.

Abstract: : A method of analysis for determining crack propagation loads on wood beams with end splits is presented. The method is based on (1) linear elastic orthotropic fracture mechanics concepts, (2) theory of complex variables, and (3) least squares boundary value collocation (BVC). Using this method, the critical stress intensity factor is determined. Also, a simple failure equation for end split beams is proposed.

Dynamic variational principles for discontinuous elastic fields

Abstract: Various forms of variational principles are derived for the three-dimensional theory of elastodynamics. The continuity requirements on the fields of stresses or strains and/or displacements are relaxed through Friedrichs's transformation. Thus, the generalized forms of certain types of earlier variational principles are systematically constructed using a basic principle of physics. The variational principles derived herein are shown to generate, as the appropriate Euler equations, the complete set of the governing equations of linear elastodynamics, that is, the stress equations of motion, the strain displacement relations, the mixed natural boundary conditions, the constitutive equations, the natural initial conditions, and the jump conditions. Similarly, generalized variational principles are established for the nonlinear theory of elastodynamics, for the incremental motions in linear elasticity, and for an elastic Cosserat continuum, as well.

On the estimation of the deformability characteristics of an isotropic elastic soil medium by means of a vane test

Abstract: The conventional use of the shear vane test is primarily restricted to the in-situ measurement of the undrained shear strength characteristics of saturated cohesive soils. Scant attention has been devoted to the use of this test as a means of measuring further properties of geotechnical interest. This paper presents an analytical study which illustrates the possible use of a shear vane test as a technique for the measurement of in-situ deformability characteristics of a soil medium. Certain plausible assumptions have been invoked for the analytical treatment of the shear vane problem. The vane blades are represented as elliptical shapes, the soil disturbance associated with the vane penetration is neglected and the soil mass enclosedwithin the swept boundary of the vane is represented as a rigid region. These, together with assumptions of classical isotropic elastic soil behaviour, enable the development of certain exact solutions for the torque–twist relationships of vanes fully or, partially embedded in the soil. The results indicate that the elastic deformability characteristics of a soil medium can be directly recovered from an examination of the initial stages of an experimental torque–twist curve. In particular, the measured parameter would correspond to the linear elastic shear modulus of the soil medium.