Showing papers on "Linear elasticity published in 1983"

Linear elastic materials with voids

Abstract: A linear theory of elastic materials with voids is presented. This theory differs significantly from classical linear elasticity in that the volume fraction corresponding to the void volume is taken as an independent kinematical variable. Following a discussion of the basic equations, boundary-value problems are formulated, and uniqueness and weak stability are established for the mixed problem. Then, several applications of the theory are considered, including the response to homogeneous deformations, pure bending of a beam, and small-amplitude acoustic waves. In each of these applications, the change in void volume induced by the deformation is determined. In the final section of the paper, the relationship between the theory presented and the effective moduli approach for porous materials is discussed. In the two year period between the submission of this manuscript and the receipt of the page proof, there have been some extensions of the results reported here. In the context of the theory described, the classical pressure vessel problems and the problem of the stress distribution around a circular hole in a field have uniaxial tension have been solved [19,22]. The solution given in the present paper for the pure bending of a beam when the rate effect of the theory is absent is extended to case when the rate effect is present in [21]. The various implications of the rate effect in the void volume deformation are pursued all the subsequent works [19,20,21,22].

An introduction to thermomechanics

Abstract: Preface. Chapters: 1. Mathematical Preliminaries. 2. Kinematics. 3. Kinetics. 4. Thermodynamics. 5. Material Properties. 6. Ideal Liquids. 7. Linear Elasticity. 8. Inviscid Gases. 9. Viscous Fluids. 10. Plastic Bodies. 11. Viscoelasticity. 12. General Tensors. 13. Large Displacements. 14. Thermodynamic Orthogonality. 17. Plasticity. 18. Viscoelastic Bodies. Bibliography. Subject Index.

Introduction to Linear Elasticity

Abstract: Preface 1 Introduction and Mathematical Preliminaries 2 Traction, Stress and Equilibrium 3 Deformations 4 Material Behavior 5 Formulation, Uniqueness and Solution Strategies 6 Extension, Bending and Torsion 7 Two-Dimensional Elaticity 8 Bending of Thin Plates 9 Time-Dependent Effects 10 Energy Principles 11 Strength and Failure Criteria 12 Something New Index

An analysis of thep-version of the finite element method for nearly incompressible materials

Abstract: In this paper we analyze the behavior of the so-calledp-version of the finite element method when applied to the equations of plane strain linear elasticity. We establish optimal rate error estimates that are uniformly valid, independent of the value of the Poisson ratio,v, in the interval ]0, 1/2[. This shows that thep-versiondoes not exhibit the degeneracy phenomenon which has led to the use of various, only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material.

Boundary element methods in solid mechanics: With applications in rock mechanics and geological engineering

Abstract: Review of linear elasticity introduction to boundary element methods the fictitious stress method the displacement discontinuity methods the direct boundary integral methods improvements and extensions applications in rock mechanics and geological engineering.

Predictions of Nonlinear Fracture Process Zone in Concrete

Abstract: Crack propagation in concrete is associated with a nonlinear zone around the crack-tip. the size of this fracture process zone length may be large depending upon the size of the aggregates and the geometry of the specimen. A theoretical model to predict the extent of this nonlinear zone and a method to include the effects of this nonlinearity in predicting the fracture resistance of concrete are described. The model is based on some simple and approximate extensions of the concepts of linear elastic fracture mechanics. The model is successfully used to analyze the results of the experiments on double cantilever, double torsion and the notched-beam specimens.

The cyclicJ-integral as a criterion for fatigue crack growth

Abstract: The definition of the cyclic J-integral is offered and its physical significance for fatigue crack growth is discussed using the Dugdale model on the assumption that the crack closure, cycle dependent creep deformation, and crack extension under cycling can be neglected. It is shown that the cyclic J-integral for small scale yielding is equivalent to theJ-integral for linear elastic crack independent of loading processes, while the value for large scale yielding varies with the loading processes. However, in both cases, the cyclicJ-integral remains constant during the reversal of loading under a constant stress range, if the first monotonic loading stage is excluded. In this situation, the cyclicJ-integral can be applied as a criterion for fatigue crack growth, since it is evaluated as a generalized force on dislocations to be moved or the energy flow rate to be dissipated to heat by the dislocation movements in an element just attached to the fatigued crack tip during one cycle of loading. It is suggested that the available experimental data of different materials for fatigue crack growth can be generalized to a unified formulation on the basis of the energy criterion. It is also deduced that the threshold ΔJ corresponding to ΔKth should be larger than 4γ where γ is the surface energy of the material. Finally the operational definition of the cyclicJ-integral on single loadversus displacement curves is given for center cracked plate with wide uncracked ligaments in tension.

The role of Terzaghi effective stress in linearly elastic deformation

Abstract: It has been shown that the overall strain of a fluid-filled porous elastic solid is not governed by the Terzaghi effective stress law. The authors show, in the context of anisotropic linear elasticity, that the overall strain may be resolved into a component which is the average strain of the solid matrix and a component due to change in relative pore geometry, and that the latter is determined by the Terzaghi effective stress. This leads to a simple form of the response laws and, in particular, to effective stress laws for overall strain (obtained previously) and for strain of the pore space.

A viscoplastic analysis of rapid crack propagation experiments in steel

Abstract: I n order to investigate rapid Mode I crack propagation in steel under large scale yielding conditions a finite element program was developed in which the material is treated as elastic-viscoplastic, based on a theory by P. P erzyna . In this model, rate effects in the plastic deformation process are accounted for. The parameters of the constitutive model were determined from ordinary tensile tests on small specimens by varying the loading rate. It was found that the strain rate can satisfactorily be related to the increase of the inelastic stress-strain relation raised to some power n . The FEM-program was applied to a number of crack propagation experiments on highly loaded SEN specimens of a high strength steel. For comparison some experiments were also analysed assuming linear elastic and simulated elastic-plastic material behaviour with no rate sensitivity. It was found that combined plastic and viscous effects are of great importance for an adequate description of rapid crack growth when large scale yielding is at hand. The energy flow to the crack tip region in the viscoplastic case, which is used as a fracture parameter, seems to converge to finite and non-trivial values for sufficiently small values of the element size along the crack tip. This may be due to the asymptotic dominance of the elastic field at the crack-tip for the particular rate-dependent model used. Geometry effects, which are present in a linear elastic analysis, are shown to be negligible in the present elastic-viscoplastic analysis with respect to different specimen heights.

Boundary Element Method in Geomechanics

Abstract: 1 Introduction.- 2 Material Behaviour and Numerical Techniques.- 2.1 Introduction.- 2.2 Linear Elastic Material Problems.- 2.3 Nonlinear Elastic Material Problems.- 2.4 Inelastic Material Problems.- 2.5 Time-Dependent Problems.- 3 Boundary Integral Equations.- 3.1 Introduction.- 3.2 Governing Equations and Fundamental Solutions.- 3.3 Integral Equations.- 3.4 Body Force Problem.- 3.5 Prestress Force Problem.- 3.6 Temperature Shrinkage and Swelling.- 4 Boundary Integral Equations for Complete Plane Strain Problems.- 4.1 Introduction.- 4.2 Governing Equations and Fundamental Solutions.- 4.3 Integral Equations for Interior Points.- 4.4 Boundary Integral Equation.- 5 Boundary Element Method.- 5.1 Introduction.- 5.2 Discretization of the Integral Equations.- 5.3 Subregions.- 5.4 Traction Discontinuities.- 5.5 Thin Subregions.- 5.6 Solution Technique.- 5.7 Practical Application of Boundary Element on Linear Problems.- 6 Notension Boundary Elements.- 6.1 Introduction.- 6.2 Rock Material Behaviour.- 6.3 Method of Solution.- 6.4 Application of No-Tension in Rock Mechanics.- 7 Discontinuity Problems.- 7.1 Introduction.- 7.2 Plane of Weakness.- 7.3 Analysis of Discontinuity Problems.- 7.4 Numerical Applications.- 8 Boundary Element Technique for Plasticity Problems.- 8.1 Introduction.- 8.2 Elastoplastic Problems in One Dimension.- 8.3 Theory of Plasticity for Continuum Problems.- 8.4 Numerical Approach for the Plastic Solution.- 8.5 Practical Applications in Geomechanics.- 9 Elasto/Viscoplastic Boundary Element Approach.- 9.1 Introduction.- 9.2 Time-Dependent Behaviour in One Dimension.- 9.3 Elasto/Viscoplastic Constitutive Relations for Continuum Problems.- 9.4 Outline of the Solution Technique.- 9.5 Time Interval Selection and Convergence.- 9.6 Elasto/Viscoplastic Applications.- 10 Applications of the Nonlinear Boundary Element Formulation.- 10.1 Introduction.- 10.2 Strip Footing Problem.- 10.3 Slope Stability Analysis.- 10.4 Tunnelling Stress Analysis.- 11 Conclusions.- References.- Appendices.

The classical pressure vessel problems for linear elastic materials with voids

Abstract: The traditional problems of the thick walled spherical and circular cylindrical shells under internal and external pressure are solved in the context of the theory of linear elastic materials with voids. The solutions are quasi-static. The stress distributions are those predicted by isotropic linear elasticity. The displacement and solid volume fraction charge fields exhibit a volumetric viscoelasticity induced by a rate dependence of the volume fraction change.

Stress and Seepage Analysis of Earth Dams

Abstract: A finite element procedure is developed for stress, seepage and stability analysis of dams and earthbanks. The seepage analysis is based on a residual flow scheme involving saturated and unsaturated zones in which the original mesh remains invariant during transient flow and iterations. By using the same mesh for stress analysis including sequential embankment construction, it is possible to superimpose directly the effects of external and seepage forces. The soil behavior is modelled by using linear elastic, piecewise linear elastic and Drucker-Prager models. The procedure is applied to a number of problems involving free surface seepage, stress analysis with stability, and combination of both. The predictions are compared with closed form solutions and field observations.

A detailed comparison of experimental and theoretical stress-analysis of a human femur

Abstract: Experimental strain-gauge and theoretical stress analysis methods are used to evaluate the mechanical behavior of the femur as a structural element under loading. It is shown that when the cortical bone material is assumed to behave linear elastic, homogeneous and transversely isotropic, excellent agreement between experimental results and theoretical predictions is obtained. Also that the bone shaft can with reasonable approximation be represented by an axisymmetric model, even when intramedullary hip joint prostheses are present. The implications of these results for the analysis of intramedullary bone-prosthesis structures are discussed.

Elastic Surface Waves with Transverse Horizontal Polarization

Abstract: Publisher Summary This chapter discusses elastic surface waves with transverse horizontal polarization. Generally speaking, surface waves are time-varying, spatially nonuniform perturbations that exhibit spatial variation in the field amplitude, markedly confined to the vicinity of the limiting surface of a body and practically nil outside this relatively narrow zone. The treatment of surface waves is the simplest dynamic problem concerning a body of finite extent in one of its dimensions. It requires considering boundary conditions so that, in principle at least, any field theory and resulting continuous field equations that are accompanied by well-set boundary conditions can be made to exhibit a surface wave phenomenon to a greater or lesser degree. The stability in time of such a phenomenon is not guaranteed in all cases. This chapter discusses a few mathematical features of surface wave propagation in Section II, and the notion of resonance coupling between modes in Section III. The exemplary case of Bleustein-Gulyaev waves in linear piezoelectrics is then briefly examined in Section IV. SH surface waves in inhomogeneous isotropic linear elasticity are the subject of Section V. Surface modes with SH elastic polarization in elastic ferroelectrics and elastic ferromagnets are examined in Sections VI and VII, respectively.

Boundary integral equation stress analysis of a rotating disc with a corner crack

Abstract: The analytical and numerical formulation of the boundary integral equation (BIE) method are outlined for the general case in linear elasticity. Using this method, three-dimensional linear elastic fracture mechanics analyses of a rotating disc with a corner crack at its bore are carried out. The cases considered are for a disc with external to internal radius ratio of 8 and with thickness equal to the diameter of the central bore. Two different crack shapes, namely, a quarter-circular crack and a quarter-elliptical crack with ellipse aspect ratio of 0.75, are analysed. For each of these shapes, corner cracks penetrating 50 per cent and 75 per cent of the disc thickness are treated. Stress intensity factor solutions for these cracks are presented for the centrifugal loading condition, as well as when the disc is subjected to a radial tensile stress at its external circumferential periphery.

A Rheological Model of Nonlinear Viscoplastic Solids

Abstract: A general rheological model of a nonlinear viscoplastic solid is developed, affording better quantitative prediction of behavior of viscoplastic materials. Rheological properties are quantified in terms of three types of curves obtained by tests, commonly used for description of mechanical material properties: constant rate, force deformation, creep, and relaxation curves. The mathematical model is based on a characteristic nonlinear differential equation describing the mechanical properties of a material. This equation defines the force F acting on a test specimen as composed of a cubic elasticity force K0x+rx3, viscous damping force Cẋ, and internal friction force Ff (sgn ẋ). The elastic parameter K0 quantifies linear elasticity while the strain hardening (or softening when negative) parameter r affords prediction of nonlinear behavior of the material. The viscous and Coulomb damping forces properly account for both rate‐dependent and rate‐independent energy dissipative properties of the material. Local...

Large deflections of point loaded cantilevers with nonlinear behaviour

Abstract: Thin beams, being flexible, form a curve with large deflections when subjected to sufficiently large transverse loads. Therefore, geometrical nonlinearity occurs, and the problem must be formulated in terms of the nonlinear theory of bending. In this paper, the beam is constructed from nonlinear elastic material, and subjected to several transverse concentrated loads. Due to the large deflection of the beam, the exact expression of the curvature of the deflected shape is used in the Bernoulli-Euler relationship. Therefore, this leads to a second order nonlinear differential equation for the transverse deflection. The solution of this equation is obtained by using the fourth-order Runge-Kutta method, and the arc length is evaluated using Simpson's Rule. The results obtained from this procedure are compared with previously published results for thin beams of linear elastic materials in order to verify the theory and the method of analysis.

Thermodynamic properties of a small droplet of water around an ion in a compressible matrix

Abstract: A model is presented for the adsorption of water by ions in a deformable matrix. Only two opposing “effects” are included. The first is the mechanical resistance of the matrix to deformation which is treated within the framework of linear elasticity extended to the case of large deformations. The other effect is of electrical origin which is shown to be much more important than the purely entropic osmotic pressure. In the model only one immobile ion located at the center of a spherical droplet of water is considered. It is shown that the result does not significantly depend upon the dielectric constant of water, provided it is much larger than that of the matrix. The thermodynamic relations reveal that in fact most of the properties of the system depend mostly on the dielectric and elastic coefficients of the matrix. This simplified model should be considered as a guide for the understanding of the behavior of such systems rather than as a precise description of real materials. Some quantitative comparisons are made with a few experimental results.

Homogenization and Asymptotic Expansions for Solutions of the Elasticity System with Rapidly Oscillating Periodic Coefficients

Abstract: Homogenization for a boundary value problem in linear elasticity is discussed. Justification is given for the asymptotic expansion in powers of a small parameter for a solution to the elasticity system in a half-space with rapidly oscillating periodic coefficients.

Mechanical Properties of the Corn Cob Under Quasi-Static Radial Compression

Abstract: THEORETICAL equations were presented for calculating the apparent elastic modulus and the strength of corn cob under quasi-static radial compression. An empirical equation was derived for calculating its toughness. Experimental justification was provided for the application of Hertz linear elastic contact theory in this study, since the cob is a composite of three inelastic materials. Cob mechanical properties were found to be significantly affected by cob moisture content, while loading rate had no significant effect. Regression analysis showed that cob elastic modulus and crushing strength were each linearly related to cob moisture content, while cob modulus of toughness has a non-linear relationship with cob moisture content.