Showing papers on "Linear elasticity published in 1986"

Studies of the influence of non-linear stress-strain characteristics in soil-structure interaction

Abstract: Recent field and laboratory studies have shown that, even at very small strains, many soils exhibit non-linear stress–strain behaviour. Nevertheless, because of its convenience, linear elasticity will continue to play an important role in the analysis of such problems as settlement, deformation and soil–structure interaction. In this Paper the measured non-liqear stress–strain properties of a low plasticity clay are used in the finite element analysis of footings, piles, excavations and pressuremeter tests to assess the influence of small strain non-linearity in comparison with linear elastic behaviour. In all cases non-linear behaviour results in the concentration of strain and deformation towards the loading boundaries. This is shown to have important consequences for soil–structure interaction problems such as settlement profiles, pile group interaction and contact stress distributions. Small strain non-linearity also has a significant influence on the interpretation in terms of equivalent elastic modu...

Homogenization and optimal bounds in linear elasticity

Abstract: In their celebrated paper of 1963, HASHIN & SHTRIKMAN [1] addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry. They derived a set of bounds for these moduli in the physically meaningful case of three-dimensional elasticity. In the case of a two phase composite, let K 1,/.1 and K 2,/*2 respectively denote the bulk and shear moduli for the first and the second phase, let K,/* denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite. Under the ordering restriction that both

Vibration isolation using open or filled trenches

Abstract: The problem of structural isolation from ground transmitted vibrations by open or infilled trenches under conditions of plane strain is numerically studied. The soil medium is assumed to be linear elastic or viscoelastic, homogeneous and isotropic. Horizontally propagating Rayleigh waves or waves generated by the motion of a rigid foundation or by surface blasting are considered in this work. The formulation and solution of the problem is accomplished by the boundary element method in the frequency domain for harmonic disturbances or in conjunction with Laplace transform for transient disturbances. The proposed method, which requires a discretisation of only the trench perimeter, the soil-foundation interface and some portion of the free soil surface on either side of the trench appears to be better than either finite element or finite difference techniques. Some parametric studies are also conducted to assess the importance of the various geometrical, material and dynamic input parameters and provide useful guidelines to the design engineer.

Nonlinear theory for plates and shells including the effects of transverse shearing

Abstract: Nonlinear strain displacement relations for three-dimensional elasticity are determined in orthogonal curvilinear coordinates. To develop a two-dimensiona l theory, the displacements are expressed by trigonometric series representation through the thickness. The nonlinear strain-displacement relations are expanded into a series that contains all first- and second-degree terms. In the series for the displacements only the first few terms are retained. Insertion of the expansions into the three-dimensional virtual work expression leads to nonlinear equations of equilibrium for laminated and thick plates and shells that include the effects of transverse shearing. Equations of equilibrium and buckling equations are derived for flat plates and cylindrical shells. The shell equations reduce to conventional transverse shearing shell equations when the effects of the trigonometric terms are omitted and to classical shell equations when the trigonometric terms are omitted and the shell is assumed to be thin. Numerical results are presented for the buckling of a thick simply supported flat rectangular plate in longitudinal compression.

On the construction of optimal mixed finite element methods for the linear elasticity problem

Abstract: The mixed finite element method for the linear elasticity problem is considered. We propose a systematic way of designing methods with optimal convergence rates for both the stress tensor and the displacement. The ideas are applied in some examples.

Dynamics of Articulated Structures. Part I. Theory

Abstract: A finite element based method is developed for geometrically nonlinear dynamic analysis of spatial articulated structures; i.e., structures in which kinematic connections permit large relative displacement between components that undergo small elastic deformation. Vibration and static correction modes are used to account for linear elastic deformation of components. Kinematic constraints between components are used to define boundary conditions for vibration analysis and loads for static correction mode analysis. Constraint equations between flexible bodies are derived in a systematic way and a Lagrange multiplier formulation is used to generate the coupled large displacement-small deformation equations of motion. A lumped mass finite element structural analysis formulation is used to generate deformation modes. An intermediate-processor is used to calculate time-independent terms in the equations of motion and to generate input data for a large-scale dynamic analysis code that includes coupled e...

The Finite Element Method in Thermomechanics

Abstract: 1 Fundamentals of the Finite Element Method.- 1.1 Introduction.- 1.2 The concept of discretization.- 1.3 Steps in the finite element method.- References.- 2 Finite Element Analysis in Heat Conduction.- 2.1 Introduction.- 2.2 Review of basic formulations.- 2.3 Finite element formulation of transient heat conduction in solids.- 2.4 Transient heat conduction in axisymmetric solids.- 2.5 Computation of the thermal conductivity matrix.- 2.6 Computation of the heat capacitance matrix.- 2.7 Computation of thermal force matrix.- 2.8 Transient heat conduction in the time domain.- 2.9 Boundary conditions 45 2.10 Solution procedures for axisymmetric structures.- References.- 3 Thermoelastic-Plastic Stress Analysis.- 3.1 Introduction.- 3.2 Mechanical behavior of materials.- 3.3 Review of basic formulations in linear elasticity theory.- 3.4 Basic formulations in nonlinear elasticity.- 3.5 Elements of plasticity theory.- 3.6 Strain hardening.- 3.7 Plastic potential (yield) function.- 3.8 Prandtl-Reuss relation.- 3.9 Derivation of plastic stress-strain relations.- 3.10 Constitutive equations for thermoelastic-plastic stress analysis.- 3.11 Derivation of the [Cep] matrix.- 3.12 Determination of material stiffness (H').- 3.13 Thermoelastic-plastic stress analysis with kinematic hardening rule.- 3.14 Finite element formulation of thermoelastic-plastic stress analysis.- 3.15 Finite element formulation for the base TEPSAC code.- 3.16 Solution procedure for the base TEPSA code.- References.- 4 Creep Deformation of Solids by Finite Element Analysis.- 4.1 Introduction.- 4.2 Theoretical background.- 4.3 Constitutive equations for thermoelastic-plastic creep stress analysis.- 4.4 Finite element formulation of thermoelastic-plastic creep stress analysis.- 4.5 Integration schemes.- 4.6 Solution algorithm.- 4.7 Code verification.- 4.8 Closing remarks.- References.- 5 Elastic-Plastic stress analysis with Fourier Series.- 5.1 Introduction.- 5.2 Element equation for elastic axisymmetric solids subject to nonaxisymmetric loadings.- 5.3 Stiffness matrix for elastic solids subject to nonaxisymmetric loadings.- 5.4 Elastic-plastic stress analysis of axisymmetric solids subject to nonaxisymmetric loadings.- 5.5 Derivation of element equation.- 5.6 Mode mixing stiffness equations.- 5.7 Circumferential integration scheme.- 5.8 Numerical example.- 5.9 Discussion of the numerical example.- 5.10 Summary.- References.- 6 Elastodynamic stress analysis with Thermal Effects.- 6.1 Introduction.- 6.2 Theoretical background.- 6.3 Hamilton's variational principle.- 6.4 Finite element formulation.- 6.5 Direct time integration scheme.- 6.6 Solution algorithm.- 6.7 Numerical illustration.- References.- 7 Thermofracture Mechanics.- 1: Review of fracture mechanics concept.- 7.1 Introduction.- 7.2 Linear elastic fracture mechanics.- 7.3 Elastic-plastic fracture mechanics.- 7.4 Application of the finite element method to fracture mechanics.- 2: Thermoelastic-plastic fracture analysis page.- 7.5 Introduction.- 7.6 Fracture criteria.- 7.7 J integral with thermal effect.- 7.8 Numerical illustrations of J integrals with thermal effect.- 7.9 The "breakable element".- 7.10 Numerical illustrations of stable crack growth.- 3: Thermoelastic-plastic creep fracture analysis.- 7.11 Literature review.- 7.12 Generalized creep fracture model.- 7.13 Path dependence of the Cg* integral.- 7.14 Creep crack growth simulated by "breakable element"1 Fundamentals of the Finite Element Method.- 1.1 Introduction.- 1.2 The concept of discretization.- 1.3 Steps in the finite element method.- References.- 2 Finite Element Analysis in Heat Conduction.- 2.1 Introduction.- 2.2 Review of basic formulations.- 2.3 Finite element formulation of transient heat conduction in solids.- 2.4 Transient heat conduction in axisymmetric solids.- 2.5 Computation of the thermal conductivity matrix.- 2.6 Computation of the heat capacitance matrix.- 2.7 Computation of thermal force matrix.- 2.8 Transient heat conduction in the time domain.- 2.9 Boundary conditions 45 2.10 Solution procedures for axisymmetric structures.- References.- 3 Thermoelastic-Plastic Stress Analysis.- 3.1 Introduction.- 3.2 Mechanical behavior of materials.- 3.3 Review of basic formulations in linear elasticity theory.- 3.4 Basic formulations in nonlinear elasticity.- 3.5 Elements of plasticity theory.- 3.6 Strain hardening.- 3.7 Plastic potential (yield) function.- 3.8 Prandtl-Reuss relation.- 3.9 Derivation of plastic stress-strain relations.- 3.10 Constitutive equations for thermoelastic-plastic stress analysis.- 3.11 Derivation of the [Cep] matrix.- 3.12 Determination of material stiffness (H').- 3.13 Thermoelastic-plastic stress analysis with kinematic hardening rule.- 3.14 Finite element formulation of thermoelastic-plastic stress analysis.- 3.15 Finite element formulation for the base TEPSAC code.- 3.16 Solution procedure for the base TEPSA code.- References.- 4 Creep Deformation of Solids by Finite Element Analysis.- 4.1 Introduction.- 4.2 Theoretical background.- 4.3 Constitutive equations for thermoelastic-plastic creep stress analysis.- 4.4 Finite element formulation of thermoelastic-plastic creep stress analysis.- 4.5 Integration schemes.- 4.6 Solution algorithm.- 4.7 Code verification.- 4.8 Closing remarks.- References.- 5 Elastic-Plastic stress analysis with Fourier Series.- 5.1 Introduction.- 5.2 Element equation for elastic axisymmetric solids subject to nonaxisymmetric loadings.- 5.3 Stiffness matrix for elastic solids subject to nonaxisymmetric loadings.- 5.4 Elastic-plastic stress analysis of axisymmetric solids subject to nonaxisymmetric loadings.- 5.5 Derivation of element equation.- 5.6 Mode mixing stiffness equations.- 5.7 Circumferential integration scheme.- 5.8 Numerical example.- 5.9 Discussion of the numerical example.- 5.10 Summary.- References.- 6 Elastodynamic stress analysis with Thermal Effects.- 6.1 Introduction.- 6.2 Theoretical background.- 6.3 Hamilton's variational principle.- 6.4 Finite element formulation.- 6.5 Direct time integration scheme.- 6.6 Solution algorithm.- 6.7 Numerical illustration.- References.- 7 Thermofracture Mechanics.- 1: Review of fracture mechanics concept.- 7.1 Introduction.- 7.2 Linear elastic fracture mechanics.- 7.3 Elastic-plastic fracture mechanics.- 7.4 Application of the finite element method to fracture mechanics.- 2: Thermoelastic-plastic fracture analysis page.- 7.5 Introduction.- 7.6 Fracture criteria.- 7.7 J integral with thermal effect.- 7.8 Numerical illustrations of J integrals with thermal effect.- 7.9 The "breakable element".- 7.10 Numerical illustrations of stable crack growth.- 3: Thermoelastic-plastic creep fracture analysis.- 7.11 Literature review.- 7.12 Generalized creep fracture model.- 7.13 Path dependence of the Cg* integral.- 7.14 Creep crack growth simulated by "breakable element" algorithm.- References.- 8 Thermoelastic-Plastic Stress Analysis By Finite Strain Theory.- 8.1 Introduction.- 8.2 Lagrangian and Eulerian coordinate systems.- 8.3 Green and Almansi strain tensors.- 8.4 Lagrangian and Kirchhoff stress tensors.- 8.5 Equilibrium in the large.- 8.6 Equilibrium in the small.- 8.7 The boundary conditions.- 8.8 The constitutive equation.- 8.9 Equations of equilibrium by the principle of virtual work.- 8.10 Finite element formulation.- 8.11 Stiffness matrix [K2].- 8.12 Stiffness matrix [K3].- 8.13 Constitutive equations for thermoelastic-plastic stress analysis.- 8.14 The finite element formulation.- 8.15 The computer program.- 8.16 Numerical examples.- References.- 9 Coupled Thermoelastic-Plastic Stress Analysis.- 9.1 Introduction.- 9.2 The energy balance concept.- 9.3 Derivation of the coupled heat conduction equation.- 9.4 Coupled thermoelastic-plastic stress analysis.- 9.5 Finite element formulation.- 9.6 The y matrix.- 9.7 The thermal moduli matrix ?.- 9.8 The internal dissipation factor.- 9.9 Computation algorithm.- 9.10 Numerical illustration.- 9.11 Concluding remarks.- References.- 10 Application of Thermomechanical Analyses in Industry.- 10.1 Introduction.- 10.2 Thermal analysis involving phase change.- 10.3 Thermoelastic-plastic stress analysis.- 10.4 Thermoelastic-plastic stress analysis by TEPSAC code.- 10.5 Simulation of thermomechanical behavior of nuclear reactor fuel elements.- References.- Appendix 1 Area coordinate system for triangular simplex elements.- Appendix 2 Numerical illustration on the implementation of thermal boundary conditions.- Appendix 3 Integrands of the mode-mixing stiffness matrix.- Appendix 4 User's guide for TEPSAC.- Appendix 5 Listing of TEPSAC code.- Author Index.

Dynamic response of 3-D embedded foundations by the boundary element method

Abstract: The dynamic response of three-dimensional rigid embedded foundations of arbitrary shape, resting on a linear elastic, homogeneous, and isotropic half-space is numerically obtained. The foundations are subjected either to externally applied forces or to obliquely incident seismic body or surface waves of arbitrary time variation. The time domain boundary element method (BEM) is utilized to simulate the soil medium with the aid of Stokes' fundamental solutions. The dynamic response of the foundation-soil system is obtained in a step-by-step time-marching solution. Use of this time domain BEM requires a minimum amount of surface discretization only and provides the basis for an extension to nonlinear soil-structure interaction (SSI) problems.

Elasto-Plastic Failure Analysis of Composite Bolted Joints:

Abstract: An elasto-plastic finite element analysis of pin loaded joints in laminated composites has been investigated and comparisons made with both existing 2-D linear elastic plane stress analytical solutions and experimental results for a graphite/epoxy laminate. The finite element analysis included nonlinear material behavior after initial failure by assuming an elastic-perfectly plastic bimodular material model. Laminated plate theory was used to obtain lamina stresses and the Hill yield criterion applied in each layer to create a ply-by-ply failure analysis. The effect of including friction forces along the fastener hole interface on the stress distribution around the hole was also studied. Based upon the results, failure criteria are proposed for each of the basic failure modes, bearing, shearout and net-tension. Failure maps for each ply were developed to characterize the damage progression and identify critical failure strength and mode. For the [0i/±45j/90k] family of laminates the elasto-plastic finite ...

Thermodynamic Consequences of Strain Softening in Tension

Abstract: The strain softening behavior of a tension bar loaded by an increasing elongation is analyzed. The constitutive model consists of linear elasticity in combination with associated plasticity theory using a maximum tensile stress criterion as yield surface. The resulting mechanical stability criterion is augmented by considerations of the use of the second law of thermodynamics. These thermodynamical considerations imply a significant reduction in the possible strain softening responses. Moreover, for very brittle material behavior, it is shown that the softening region cannot be considered to have a specific strain state, but rather is described by a strrss-elongation relation. This result provides strong physical support for a fictitious crack model. This crack model is then reevaluated in the spirit of a smeared crack approach and the resulting expressions turn out to be identical with those of a composite fracture model.

A field theory approach to stability of radial equilibria in nonlinear elasticity

Abstract: In this paper we study the stability of a class of singular radial solutions to the equilibrium equations of nonlinear elasticity, in which a hole forms at the centre of a ball of isotropic material held in a state of tension under prescribed boundary displacements. The existence of such cavitating solutions has been shown by Ball[1], Stuart [11] and Sivaloganathan[10]. Our methods involve elements of the field theory of the calculus of variations and provide a new unified interpretation of the phenomenon of cavitation.

Stresses and Deformations in Composite Tubes Due to a Circumferential Temperature Gradient

Abstract: A linear elasticity solution for determining the response of composite tubes subjected to a circumferential temperature gradient is presented. Numerical examples are used to show that, in a single layer tube, fiber orientation strongly influences response. When the fibers are aligned axially, all stress components in the tube are small. When the fibers are aligned circumferentially, the hoop stress becomes large. This difference in behavior is due to the large difference between the radial and circumferential coefficients of thermal expansion when the fibers are oriented circumferentially. In multilayer tubes, stresses are quite high and just two constants characterize the overall bending and axial deformations of the tubes.

Free Vibration of a Rectangular Parallelepiped Using the Theory of a Cosserat Point

Abstract: Free vibration of a rectangular parallelepiped composed of a homogeneous linear elastic isotropic material is studied. The parallelepiped is modeled as an isotropic Cosserat point and simple formulas are developed to predict the lowest frequencies of vibration. Within the context of the theory, extensional and shear vibrations are uncoupled. The lowest extensional frequency predicted by the Cosserat theory is compared with available exact solutions and with predictions of thin rod theory. Finally, by introducing a simple modification of the director inertia coefficient it is shown that the Cosserat predictions of the extensional frequencies are correct.

Surface waves in an elastic half-space with voids

Abstract: Surface waves of general type propagating in a plane-faced, homogeneous and isotropic linear elastic halfspace containing a distribution of vacuous pores (voids) are studied. Assuming that the face of the halfspace is free of external loads, it is found that the motion is not necessarily two-dimensional and that all physical variables associated with the waves are derivable from a single scalar function. The phase speed equation reveals that, unlike in the classical problem, the waves are dispersive, in general. If the frequency is very small, the motion is qualitatively similar to that in the classical problem, but the wave speed is modified due to the presence of voids.

Dislocation motion and vertical vorticity in Rayleigh-Benard convective structures

Abstract: The behaviour of an isolated dislocation in a Rayleigh-Benard roll structure is studied within a linear elasticity theory of topological defects on a model which includes the effect of a large-scale drift flow. The climb velocity is given as a function of the Prandtl number, Rayleigh number and wavenumber for both rigid and stress-free boundary conditions. The effect of a lateral boundary is also briefly discussed.

Generalized thermoelasticity with temperature dependent modulus of elasticity under three theories

Abstract: A new model of generalized thermoelasticity equations for isotropic media with temperature-dependent mechanical properties is established. The modulus of elasticity is taken as a linear function of reference temperature. The present model is described both generalizations, Lord-Shulman (L-S) theory with one relaxation time and Green-Lindsay (G-L) with two relaxation times, as well as the coupled theory, instantaneously. The method of the matrix exponential, which constitutes the basis of the state space approach of modern control theory, applied to two-dimensional equations. Laplace and Fourier integral transforms are used. The resulting formulation is applied to a problem of a thick plate subject to heating on parts of the upper and lower surfaces of the plate that varies exponentially with time. Numerical results are given and illustrated graphically for the problem considered. A comparison was made with the results obtained in case of temperature-independent modulus of elasticity in each theory.

Biaxial dynamic bending of elastoplastic beams

Abstract: An elastic-inelastic analogy is presented for dynamic biaxial bending of beams. Inelastic effects are treated as fictitious additional sources of self-stresses in the adjoint linear elastic structure of time-invariant stiffness. Thus, superposition applies and linear methods such as Green's functions and Duhamel's integral may be used in a convenient manner. Fictitious loadings themselves are determined from the inelastic material's law in a time-stepping procedure. As an example problem, a preloaded elastoplastic beam under harmonic excitation of support motion type is considered.

A multi‐director formulation for elastic—viscoelastic layered shells

Abstract: A linear finite element formulation for the analysis of multi-layered shells comprised of linear elastic and viscoelastic layers is presented. The elastic and viscoelastic layers may occupy arbitrary locations and the formulation is appropriate for thick and thin shells. The concept of a multi-director field defined over a reference surface is employed for the description of the initial geometry and motion of the multi-layered shell. The kinematical theory incorporated in the three-dimensional variational formulation describes, within individual layers, the effects of transverse shear and transverse normal strain to arbitrary orders in the layer thickness co-ordinate. These kinematics have ‘local support’ over a layer and prove to be convenient and accurate in application. All stresses are computed through three-dimensional constitutive equations and the usual ‘zero normal stress’ shell hypothesis is not employed. Layer material properties are assumed to be isotropic, although this is not a restriction of the formulation. Sufficiently general constitutive equations for the viscoelastic layers are presented in rate form and an accurate and efficient product algorithm is introduced for their temporal integration. Finite element formulations using both resultant and continuum approaches are developed and compared. Observations and suggestions on the use of reduced/selective integration in the presence of high-order kinematics are made and a number of numerical examples are presented to illustrate the capability of the formulation.

On the Three‐Dimensional Computation of Steel‐Belted Tires

Abstract: A geometrically‐nonlinear finite element model of a complete tire is used in an analysis for inflation and footprint loadings. Each reinforced layer of the tire is approximated as being homogeneous, orthotropic, and linear elastic. The finite element model used in the analysis allows the computation of interply shear strains due to inflation and footprint loads. Some numerical results on loaded tires are also presented and compared with those obtained experimentally.

Dynamic behaviour of sandwich plates

Abstract: This paper discusses the physical behaviour of “bending” waves in sandwich plates in which the facings are thin, stiff and heavy as compared with the core. By means of asymptotic expansions of the basic equations of linear elasticity, it is shown that different “physical mechanims” predominate in different frequency ranges. The consequence is that different, but relatively simple equations of motion may be used within limited frequency ranges. The predictions of such an equation, valid for moderate frequencies, are compared with measurements performed on glass-polyurethane sandwich plates. The agreement of theoretical and experimental dispersion curves for axisymmetric waves, and hence for plane waves as well, was found to be very satisfactory.

Rigid bottom considerations for nondestructive evaluation of pavements (discussion)

Abstract: Mechanistic analysis of dynamic deflection basins for evaluating in situ moduli of pavement-subgrade systems has become an important part of nondestructive pavement evaluation techniques. Discussed is the influence of a rock layer on the evaluation of in situ moduli by using the multilayered linear elastic theory. The value of Young's modulus of elasticity of the subgrade overlying a rock layer can be significantly overestimated if a semi-infinite subgrade is assumed in applying the linear elastic layer theory to analyze deflection basins. An algorithm has been developed to correct this type of error for two cases: (a) when the subgrade thickness is known and (b) when depth to the rock layer is unknown. For the Dynaflect and falling weight deflectometer systems, a rigid bottom can be considered for the second case by assigning a subgrade thickness as a function of the wave length of compression wave in the subgrade. The computer programs FPEDD1 (for flexible pavements) and RPEDD1 (for rigid pavements) incorporate procedures for evaluating in situ moduli with regard to rigid bottom considerations in pavement-subgrade systems.

Stress patterns in an interplate shear zone: an effective anisotropic model and implications for the transverse ranges, California

Abstract: The pervasive Quaternary strike-slip (or wrench) structure that occurs in southern California is analyzed by solving a series of planar, linear elastic boundary value problems of tractional type that assume pure shear forces at infinity. It is concluded that the cross-cutting tectonic fabric of the Transverse Ranges in southern California causes the Ranges to act as a stress concentrator. In addition, it is found that rigid rotation of the Transverse Ranges is mechanically supported by an asymmetric distribution of stress and elastic strain encircling it. It is noted that for Quaternary tectonic orientation, a clockwise sense to this rotation is expected.