Showing papers on "Linear elasticity published in 1980"

Elasticity: Theory and Applications

Abstract: Mathematical Preliminaries Stress Deformation and Strain Elasticity and its Limits Formulation and ''Exact'' Solutions of some Linear Elasticity Problems Structural Mechanics Energy Principles Numerical Methods The Initially Stressed Solid-Elastic Instability References Index

A theory of viscoplasticity based on infinitesimal total strain

Abstract: A viscoplasticity theory based upon a nonlinear viscoelastic solid, linear in the rates of the strain and stress tensors but nonlinear in the stress tensor and the infinitesimal strain tensor, is being investigated for isothermal, homogeneous motions. A general anisotropic form and a specific isotropic formulation are proposed. A yield condition is not part of the theory and the transition from linear (elastic) to nonlinear (inelastic) behavior is continuous. Only total strains are used and the constant volume hypothesis is not employed. In this paper Poisson's ratio is assumed to be constant. The proposed equation can represent: initial linear elastic behavior; initial elastic response in torsion (tension) after arbitrary prestrain (prestress) in tension (torsion); linear elastic behavior for pure hydrostatic loading; initial elastic slope upon large instantaneous changes in strain rate; stress (strain)-rate sensitivity; creep and relaxation; defined behavior in the limit of very slow and very fast loading. Stress-strain curves obtained at different loading rates will ultimately have the same “slope” and their spacing is nonlinearly related to the loading rate. The above properties of the equation are obtained by qualitative arguments based on the characteristics of the solutions of the resulting nonlinear first-order differential equations. In some instances numerical examples are given. For metals and isotropy we propose a simple equation whose coefficient functions can be determined from a tensile test [Eqs. (31), (35), (37), (38)]. Specializations suitable for materials other than metals are possible. The paper shows that this nonlinear viscoelastic model can represent essential features of metal deformation behavior and reaffirms our previous assertion that metal deformation is basically rate-dependent and can be represented by piecewise nonlinear viscoelasticity. For cyclic loading the proposed model must be modified to account for history dependence in the sense of plasticity.

Direct analysis of elastic–plastic structures with ‘overlay’ materials during cyclic loading

Abstract: Several computer codes have incorporated the ‘overlay’ material models: the volume element, which is characteristic of the material, is composed of sub-elements with different kinematic hardening, perfectly plastic or even viscoplastic flow rules and different elastic properties, these sub-elements exhibiting all, however, the same total strain.1,2 In this paper it is demonstrated how the simple mathematical framework we first proposed for elastic-plastic structures with kinematic hardening material,3 and we extended to some elastic viscoplastic ones,4 can easily be applied to these particular ‘overlay’ materials. One of the interesting advantages of this approach is a straightforward analysis of structural response under cyclic loadings by applying the linear elastic analysis.

Flexural vibrations of strongly anisotropie beams

Abstract: If in a transversally vibrating beam of fiber-reinforced material the fibers are much stiffer than the matrix material, the influence of shear might be dominant even for wave lengths which are large as compared with the thickness of the beam. It is shown here by asymptotic solutions of the threedimensional equations of linear elasticity for transversal isotropy that a dimensionless parameterp: = π 2H/Λ √E/G characterizes the dynamical behaviour (2H: thickness, Λ: wave length,E: longitudinal elasticity modulus,G: transversal shear modulus). The influence of shear becomes apparent forp ∼ 1 even though 2H/Λ ≪ 1. Ifp is much larger than 1 (strong anisotropy), the wave velocity is almost equal to the shear velocity even for large wave lengths, the main part of the cross section is subject to uniformly distributed shear stresses, boundary layers of large bending normal stresses vanishing rapidly towards the core of the cross section travel along with the shear wave, the beam acts as a “sandwich” structure. For all degrees of anisotropy, explicite analytical expressions for stress and displacement distributions as well as for the wave velocity can easily be derived for various shapes of cross sections. This has been illustrated for a circle and a rectangle.

On Elementary Theories of Linear Elastic Beams, Plates and Shells (Review Paper)

Abstract: This paper presents a review of the elementary theories on the bending of straight and curved beams, on plates and shells, using asymptotic approximations of the basic linearized equations of elasticity in three dimensions. The maximun norm has been chosen to specify the orders of magnitude of the quantities involved. The expansions are given as usual in terms of the small geometrical parameter characterizing the thinness of the structure. Most of the ideas and results are well known. Nevertheless, in the cases where more than one small parameter may be involved, such as small curvatures (shallow structures) or the small loading parameter used to linearize the equations of elasticity, the discussion on the limits of validity of the different theories lead to some interesting newer aspects. Moreover, the main ideas presented in this paper concerning multiple parameter expansions may be applied to discuss the behaviour of the structures and to obtain valuable analytical results in more complicated situations such as moderate and strong anisotropy, dynamic problems, stability etc.

A unilateral contact problem in linear elasticity

Abstract: In this paper we study the equilibrium of an elastic body which is simply supported, without friction, by a soft elastic plane. We prove that, if the exterior loads satisfy certain conditions of compatibility, the solution exists and is unique. Moreover, we find which regularity properties of the solution hold, provided the data are sufficiently smooth.

On Singular Problems in Linearized and Finite Elastostatics.

Abstract: : The predictions of linear elasticity theory for various basic types of singular equilibrium problems are illustrated and issues associated with such solutions are discussed. Attention is then turned to recent studies concerning the implications of finite elastostatics for certain singular problems, including some that have no counterpart in the linearized theory. (Author)

Dynamic linear elastic crack propagation in anti-plane shear by finite differences

Abstract: The dynamic propagation of a crack in an anti-plane shear deformation field is analyzed by second-order-accurate finite differences. The finite difference equations are obtained by integrating the dynamic linear elastic equations of motion along the bicharacteristic strips in four perpendicular directions and the time axis to 0(Δt 3). The singularity in stresses around the crack is calculated by performing a global energy balance on small region containing the crack tip and approximating the stresses and velocity in this region by a one term asymptotic expansion about the crack tip. Results for stresses and stress intensity factor are presented for a semi-infinite crack propagating steadily in an infinite strip, from which errors in the numerical calculations are identified. Four cases of typical non-steady crack propagation in an infinite strip following steady propagation are also considered.

On the well-posedness of the equilibrium problem for linear elasticity in unbounded regions

Abstract: In this paper we establish some continuous dependence and uniqueness theorems for equilibrium solutions of the equations of general anisotropic linear elasticity in exterior domains. The method we employ is that of the weight function which we introduced in previous papers. However, this is the first example where the method is applied to a static problem. The above theorems are obtained by allowing the strain to be unbounded at large spatial distances. In some cases, no growth condition is assumed. Moreover, the displacement and the elasticities are also possibly allowed to grow.

Diametral modulus testing on non linear pavement materials

Abstract: This investigation takes the extreme case of diametral loading of a typical, uncured, open graded emulsified asphalt specimen subjected to a range of confining pressures. Theoretical stresses and deformations of this non linear elastic specimen are calculated using the finite element method and then compared with corresponding values for linear elastic behaviour to highlight possible discrepancies in interpretation of test results in terms of linear elastic theory. Analysis of test results for uncured open graded emulsified asphalt mixtures provides some confirmation of the theoretical results. The significance of anisotropy and inhomogeneity is also considered in a brief comparison between triaxial and diametral testing.

On the non-existence of semi-groups for some equations of continuum mechanics

Abstract: Linear semi-group theory can be used to prove the existence of solutions to the equations of linear elasticity when the elasticity tensor is positive definite. Here, it is shown that this condition is also necessary for the existence of a semi-group. The method is also applied to linear dissipative equations.

Rectangular Plates on Linear Viscoelastic Foundations

Abstract: This paper deals with the quasistatic bending problems of the rectangular plates and the infinite strips on the linear viscoelastic foundations of the Kelvin, the Maxwell and the standard linear solid types. The general solutions for them are developed by using the eigenfunctions derived from a free lateral-vibration problem of the plates with the same geometries and the same boundary conditions and by utilizing the correspondence principle between linear elastic boundary value problem and linear viscoelastic one. Numerical results for the variations of the deflection in space and time are illustrated for a rectangular plate and an infinite strip on the viscoelastic foundation of the standard linear solid type.

On failure modes in finite width angle ply laminates

Abstract: The results of a failure analysis of finite width (+ or - theta)s T300/5208 graphite-epoxy laminates subjected to thermal curing loads and applied tensile strain are presented. Stress distributions obtained from linear elastic and linear thermoelastic finite element stress analyses are used in conjunction with the tensor polynomial failure criterion to predict the mode and location of first failure in laminates with fiber angles ranging from five to seventy-five degrees. It is shown that the mode of failure is predominantly interlaminar shear for small fiber angles, shifts to combined interlaminar normal, in-plane shear and in-plane normal for intermediate angles, and is primarily transverse tension for large fiber angles. The location of initial failure shifts from the (+ or - theta)s interface for small fiber angles to the midplane for large fiber angles.

On continuous dependence in finite elasticity

Abstract: We investigate the relation between stability and continuous dependence for a nonlinearly elastic body at equilibrium. We show that solutions of the governing equations that lie in a convex, stable set of deformations depend continuously on the body forces and the surface tractions. The definition of stability used is essentially due to Hadamard.

Nonlinear Dynamic Analysis of Frames and Axisymmetric Shell Structures

Abstract: New beam bending element was proposed in Ref. 1) including the effect of shearing deformation, which was composed of two rigid bars connected by two different types of springs. In case of uniform division this discrete element derived by physical consideration is equivalent to the beam element in the conventional finite element method, which was introduced in Ref. 2) by employing linear displacement and rotation functions in conjunction with one-point integration based on the penalty method. Almost the same situation holds for the conical frustum element used for the discrete analysis of axisymmetric shell structures.These discrete elements give accurate linear elastic solutions in spite of their simplicity and they can be effectively used in the limit load analysis because they can construct the plastic collapse mechanisms.In this paper the application of these discrete elements is treated to the nonlinear dynamic analysis of frames and axisymmetric shell structures. As numerical examples nonlinear behavior is analyzed of plane and space frames, circular plates, circular cylindrical shells, and spherical shells loaded impulsively. The obtained results are compared with theoretical solutions, other numerical ones, and experimental results in order to justify the validity of the present method.

Stress analysis of multi layered anisotropic elastic systems subject to rectangular loads

Abstract: This paper outlines the application of the dual fourier transform to the solution of boundary value problems in three dimensional linear elasticity. The system considered consists of an arbitrary number of horizontal layers of cross anisotropic or isotropic materials. The loads are assumed to be applied on horizontal areas on the free surface or within the layered system. Loading can be stress defined or displacement defined. By taking integral transforms the equilibrium equations are reduced to simultaneous ordinary differential equations for the transformed displacements in terms of the vertical coordinate. Basic solutions for the transformed displacements are derived. Solutions for displacements, strains and stresses are shown to be expressible in terms of double integrals involving the transform of the loading multiplied by influence functions dependent on the geometry and material properties of the layered system. Examples are given of problems for which closed form solutions can be obtained. Methods for the efficient numerical evaluation of the integrals for uniform rectangular loading are discussed. The solution techniques presented are directly applicable to a wide range of stress analysis problems in geomechanics such as foundations, pavements, dams, embankments and underground openings. In addition, particular solutions can be incorporated in integral equation and finite element methods for solution of problems involving complicated geometries and boundary conditions (a).

Failure analyses of composite bolted joints

Abstract: The complex failure behavior exhibited by bolted joints of graphite epoxy (Hercules AS/3501) was investigated for the net tension, bearing and shearout failure modes using combined analytical and experimental techniques. Plane stress, linear elastic, finite element methods were employed to determine the two dimensional state of stress resulting from a loaded hole in a finite width, semiinfinite strip. The stresses predicted by the finite element method were verified by experiment to lend credence to the analysis. The influence of joint geometric parameters on the state of stress and resultant strength of the joint was also studied. The resulting functional relationships found to exist between bolted joint strength and the geometric parameters, were applied in the formulation of semiempirical strength models for the basic failure modes. A point stress failure criterion was successfully applied as the failure criterion for the net tension and shearout failure modes.

A Coupled, Isotropic Theory of Thermoviscoplasticity and its Prediction for Stress and Strain Controlled Loading in Torsion

Abstract: A coupled, isotropic, infinitesimal theory of thermoviscoplasticity was developed in [32,33] and applied to a variety of loadings including thermal monotonic and cyclic straining [32]. In this paper we rederive the coupled equatioris using the first law of thermodynamics. The predictions of the theory in torsion are examined qualitatively and by numerical experiments. They simulate monotonic loading at loading rates differing by four orders of magnitude. Jumps in loading rate are also included. The theory exhibits initial linear elastic response followed by nonlinear, rate-dependent plastic behavior. The adiabatic temperature changes are initially isothermal followed by heating. The theory exhibits rate-dependence, a difference in strain and stress controlled loading and deformation induced temperature changes which are qualitatively in agreement with recent experiments.