Showing papers on "Linear elasticity published in 1981"

An introduction to continuum mechanics

Abstract: Preface. Acknowledgments. Tensor Algebra. Tensor Analysis. Kinematics. Mass. Momentum. Force. Constitutive Assumptions. Inviscid Fluids. Change in Observer. Invariance of Material Response. Newtonian Fluids. The NavierStokes Equations. Finite Elasticity. Linear Elasticity. Appendix. References. Hints for Selected Exercises. Index.

Poroelasticity equations derived from microstructure

Abstract: Equations are derived which govern the linear macroscopic mechanical behavior of a porous elastic solid saturated with a compressible viscous fluid. The derivation is based on the equations of linear elasticity in the solid, the linearized Navier–Stokes equations in the fluid, and appropriate conditions at the solid–fluid boundary. The scale of the pores is assumed to be small compared to the macroscopic scale, so that the two‐space method of homogenization can be used to deduce the macroscopic equations. When the dimensionless viscosity of the fluid is small, the resulting equations are those of Biot, who obtained them by hypothesizing the form of the macroscopic constitutive relations. The present derivation verifies those relations, and shows how the coefficients in them can be calculated, in principle, from the microstructure. When the dimensionless viscosity is of order one, a different equation is obtained, which is that of a viscoelastic solid.

Dynamic stress concentration studies by boundary integrals and Laplace transform

Abstract: The dynamic stress field and its concentrations around holes of arbitrary shape in infinitely extended bodies under plane stress or plane strain conditions are numerically determined. The material may be linear elastic or viscoelastic, while the dynamic load consists of plane compressional waves of harmonic or general transient nature. The method consists of applying the Laplace transform with respect to time to the governing equations of motion and formulating and solving the problem numerically in the transfomed domain by the boundary integral equation method. The stress field can then be obtaind by a numerical inversion of the trasformed solution. The correspondence principle is invoked for the case of viscoelastic material behavious. The method is simplified for the case of harmonic waves where no numerical inversion is involved.

Topics in finite elasticity

Abstract: Kinematics: Stress Elastic materials, Change of observer Material symmetry Simple shear The Piola-Kirchhoff Stress Hyperelasticity The elasticity tensor The boundary-value problem Variational formulational stability and uniqueness Incompressible materials Deformations of a cube Anti-Plane Shear.

Exterior Penalty Methods for Contact Problems in Elasticity

Abstract: This paper reviews recent results of Oden, Kikuchi, and Song [4] on the use of exterior penalty methods as a basis for finite element approximations of contact problems in linear elasticity.

Consolidation of Anelastic Porous Media

Abstract: The transient response of saturated anelastic porous media is analyzed. The porous skeleton is modeled as a piecewise linear time-independent solid, and the fluid is assumed to behave macroscopically, like a perfect fluid. The coupled field equations are presented and solved numerically by the use of the finite element method. The numerical stability of the proposed time integration scheme is investigated. The use of the proposed formulation for solving boundary value problems is illustrated by applying it to analyze the one- and two-dimensional consolidation of a linear elastic porous medium. Good agreement between numerical and analytical results is obtained. Further, the formulation allows solutions to be obtained for cases in which large strains/deformations occur in the porous medium. Finally, the use of the proposed formulation is illustrated by applying it to analyze the time-dependent response of a rigid footing.

On static helical instabilities of screw dislocations in a SmA phase and on their consequence on plasticity

Abstract: The stability of a screw dislocation in a SmA phase submitted to a dilative or compressive strain normal to the layers is investigated. It is shown that the screw line transforms to a helical line, whose handedness depends on the signs of the Burgers’ vector and of the strain. For small strains the process necessitates some activation energy; there is a critical strain above which it is spontaneous. Some consequences are proposed, like : a) the nucleation of edge dislocation lines in the so-called undulation instability, and b) the easy crossing without pinning of edge and screw dislocation lines. The analysis stays entirely in the limit of linear elasticity, and it is

An invariant infinitesimal theory of motions superposed on a given motion

Abstract: : The development is applicable to any material but special attention is given to elastic solids. Included as a special case is an infinitesimal theory of elasticity with the following properties: (1) It is properly invariant under arbitrary (not necessarily infinitesimal) superposed rigid body motions, (2) it reduces by specialization to the theory of rigid bodies undergoing finite motion, and (3) it can be brought into correspondence with the classical linear elasticity through a suitable reinterpretation of the symbols in the constitutive equation of the latter. (Author)

Existence of Solutions in Finite Elasticity

Abstract: Very few exact solutions are known to static and dynamic problems of finite elasticity, particularly in the case when the material is compressible. General theorems on existence of solutions provide reassurance that the theory is mathematically sound; for example it is important to understand whether or not solutions of the basic equations have singularities consistent with assumptions used in deriving the equations. But there are several other, equally important, reasons for studying questions of existence of solutions. One such is the establishment of convergence properties for numerical methods in elasticity (in this connection it should be noted that finite-difference schemes for certain partial differential equations may converge to solutions of different equations). Experience with other partial differential equations has also taught us that existence theorems are an essential prerequisite for the study of various qualitative properties of solutions (for example, bifurcation, stability and asymptotic behaviour). In a broader context, we today face problems in elasticity similar to unsolved questions in other branches of mechanics and physics, and the unifying nature of the theory of partial differential equations can thus lead us to hope, as has been the case in the past, that advances in elasticity will lead to corresponding progress in other fields. Here, however, we concentrate on a more specific reason for proving, or attempting to prove, existence theorems in elasticity, namely that it leads to information concerning the relationship between constitutive hypotheses (i.e. assumptions on the stored-energy function, or stress-strain law of the material) and smoothness properties of solutions.

Calculating Dislocation Spacings in Pile-Ups at Grain Boundaries

Abstract: Dislocation pile-ups near planar grain boundaries have been analyzed employing full anisotropic elastic solutions, single-crystal anisotropic approximations, and isotropic approximations. The calculations were performed for a series of iron-silicon bicrystal configurations and results from the various methods compared. Three of the bicrystal/slip system combinations exhibited repulsive image forces, allowing elastically self-consistent position calculations. Based on these results, either linear elastic or nonlinear effects can dominate the behavior of these pile-ups, depending on specific boundary conditions.

Non linear pattern selection in a problem of elasticity

Abstract: 2014 The buckling of long rectangular elastic plates offers the possibility of testing a recent proposal. The wavenumber of cellular structures in slightly supercritical conditions is determined by the boundary conditions. In the present case a supercritical decrease of the wavelength is predicted. Tome 42 ? 1 ler JANVIER 1981 LE JOURNAL DE PHYSIQUE LETTRES J. Physique LETTRES 42 (1981) LI L-4 1 er JANVIER 1981, Classification Physics Abstracts 03.40D Having in mind the wavelength selection in cellular flows, as in Rayleigh-Benard convection in large horizontal layers, S. Zaleski and the author [1] solved the question of non linear pattern selection in slightly supercritical conditions for one dimensional models. This sort of problem is formulated as follows : space dependent fluctuations with a (horizontal) fixed wavenumber, say qo, become linearly unstable around a homogeneous rest state whenever a control parameter, say 8, exceeds some critical value, which can be taken at s = 0. The growth of these fluctuations is limited by non linear effects and a new steady state is reached via a supercritical (or normal) bifurcation. In a large class of problems, for slightly positive values of 8 linearly unstable fluctuations grow from the homogeneous state whenever their wavenumber belongs to a band of width of order 81/2 near the threshold value qo. However, owing to the boundary conditions limiting the lateral extent of the structure, the supercritical steady pattern has its wavenumber in a much narrower band of width of order 8 near qo. The applicability of this sort of consideration to the Rayleigh-Benard problem is not obvious, since long rolls, when parallel to a lateral boundary, are unstable against a cross roll instability localized near this boundary [1]. One should account for the structure of these boundary rolls parallel to their axis. This structure is due either to the cross roll modulation or to lateral boundaries inhibiting their growth. A realistic treatment of this problem is not an easy task. Therefore it is of interest to look at a physical situation involving a non linear selection of the wavelength, but nevertheless permitting quantitative predictions from simple ab initio calculations. That is why I have considered the following version of the von Karman problem in elasticity of thin plates [2]. This is the buckling of long rectangular elastic plates [3] submitted to a load along their long axis (see Fig. 1). According to the general considerations of reference [1], whenever the length of the long axis, say L, is much larger than a quantity of order E-1, one may limit oneself to the consideration of a half infinite problem. The possible buckling patterns for a large (but finite) L are obtained by gluing together two half infinite solutions in a convenient way. We shall not consider this specific problem here ; it is treated in reference [1]. Fig. 1. At the centre of the figure the plate is represented from above; its cuts along the short (A) and long (B) are represented on the left and right. A similar figure is in reference [4]. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019810042010100 L-2 JOURNAL DE PHYSIQUE LETTRES The von Karman [2] equations for the static buckling are [4, 5] :

Creep of 2618 Aluminum under Side Steps of Tension and Torsion and Stress Reversal Predicted by a Viscous-Viscoelastic Model.

Abstract: : Nonlinear constitutive equations were developed and used to predict the creep behavior of 2618-T61 Aluminum at 200 C (392 F) for combined tension and torsion stresses and under varying stress histories including side step stress changes and stress reversals. The constitutive equations consist of 5 components: linear elastic; time-independent plastic; nonlinear time-dependent plastic recoverable; nonlinear time-dependent nonrecoverable under positive stress; and nonlinear time-dependent nonrecoverable under negative stress. For time-dependent stress inputs, the modified superposition principle and strain hardening are used to describe the behavior of nonlinear time-dependent recoverable and nonlinear time-dependent nonrecoverable respectively. The theory which combines all these features, the viscous-viscoelastic theory, and other modified theories were used to predict from information from constant stress creep the creep behavior of 2618 aluminum under the above stress histories with very satisfactory agreement with the experimental results. (Author)

Internal constraints in linear elasticity

Abstract: Sufficient conditions are obtained for continuous dependence of solutions of boundary value problems of linear elasticity on internal constraints. Arbitrary hyperelastic materials with arbitrary (linear) internal constraints are included. In particular the results of Bramble and Payne, Kobelkov, Mikhlin for homogeneous, isotropic, incompressible materials are obtained as a special case. In the case of boundary value problem of place, a compatibility condition is obtained between the internal constraints and the boundary data which is necessary for the existence of solutions. With a further coercivity assumption on the compliance tensor, it is shown that the compatibility condition is also sufficient for existence. An orthogonal decomposition theorem for second order tensor fields modeled after Weyl's decomposition of solenoidal and gradient fields leads to the variational formulation of the problem and existence theorems.

Non-linear elastic limit state analysis of rigid jointed frames

Abstract: An elastic limit state analysis of rigid jointed frames is presented on the basis of the non-linear stability theory. A simple two-bar frame, eccentrically loaded at its joint, and exhibiting both snap-through buckling and post-buckling strength, depending basically on the loading eccentricity, is used as model. It is found that: (a) in case of a post-buckling strength there is an elastic limit state (ultimate) load which-contrary to a limit point load, depends directly on the yield stress of the material of the frame and (b) at the critical eccentricity a sudden jump in the load-carrying capacity of the frame occurs. Moreover, the effects of several parameters on the non-linear and linear elastic limit state load, are assessed.

Finite Element Analysis of Reinforced Concrete Slabs and Panels

Abstract: The real behaviour of reinforced concrete structures differs widely from the results of a linear elastic computation. Because of cracking of concrete, the nonlinear stress-strain relations of concrete and reinforcement, and the bond-slip between concrete and reinforcement, the deformation behaviour of reinforced concrete structures is extremely nonlinear even under working loads.

Stress Concentration Layers in Finite Deformation of Fibre-Reinforced Elastic Materials

Abstract: We consider finite plane strain of incompressible elastic materials reinforced by inextensible fibres. Deformations of inextensible materials often give rise to stress concentration layers, which are sheets of fibres which carry infinite direct stress but finite force, and across which the shear stress may be discontinuous. For linear elastic materials reinforced by straight parallel fibres these layers are well understood, and asymptotic methods of analysis have been developed for linear elastic materials with small but finite extensibility. When finite deformations occur, a qualitatively new feature arises because, in general, the fibres become curved and then the normal stress across them may also be discontinuous. We consider two examples, namely simple shear and shear bending of a rectangular block. In both of these examples the solution for inextensible material involves surface stress concentration layers. For ‘almost inextensible’ material we obtain approximate solutions for the displacement and stress in the neighbourhood of these surfaces, and show that the solution for an ideally inextensible material can be interpreted as the limit of the solution for material with small but finite extensibility as the ratio of shear modulus to fibre extension modulus tends to zero.

Computer simulation of dislocation cores in hcp metals

Abstract: The cores of the prism-edge, basal-edge and screw dislocations with have been simulated in two model hcp crystals with different basal stacking-fault energy γ Three aspects are considered here First, the prism edge is found not to dissociate, in contrast to several earlier predictions Second, the basal edge and screw dissociate into Shockley partials of spacing close to that given by γ and linear elasticity, suggesting that the use of elasticity to estimate γ from observed partial spacing may not be as questionable as previous simulations had indicated Finally, constriction and cross slip of the screw onto the prism plane under an applied shear stress have been simulated They occur at the site of one of the partials only, and this can be explained by the asymmetry of Shockley partial dislocations

Moment rotation characteristics of flat plate and column systems

Abstract: The response of flat plate and column structures to lateral loading is closely associated with the stiffness of the slab/column joints. Many previous theoretical and experimental investigations have studied the linear elastic response of sway systems but few experimental results are available on the non-linear behaviour of concrete slab/column joints. Results from exeriments on reinforced concrete models are presented in this paper. These tests show a non-linear response from very low load levels. The behaviour of the test specimens is compared with both linear elastic and empirical predictions. The results are also compared with an equivalent slab derived from the equivalent frame method of ACI 318-77. This is found to give a closer description of the observed behaviour than the elastic theories (A). (TRRL)

Edge dislocation trapping by a cylindrical inclusion near a traction-free surface.

Abstract: The force on an edge dislocation located near a cylindrical inclusion and a traction‐free surface is calculated within the framework of linear elasticity. It is shown that the glide equilibrium positions are very sensitive to the orientation a of the slip plane with respect to the surface; in particular, the trapping effectiveness is monotonically reduced by increasing a from 0 to 90°. It is also found that a trapping action is exterted mainly by inclusions more rigid than the matrix, whereas in the bulk it is exterted mainly by inclusions less rigid than the matrix. Possible physical implications of the obtained results are discussed.

Assessment of Defects: The c.o.d. Approach

Abstract: In welded construction particular problems arise with the application of fracture mechanics for the assessment of the effect of defects on structural performance. In many practical cases the use of plane strain linear elastic fracture mechanics methods is invalidated by the actual material thicknesses of interest, by residual stresses or by local stress concentration effects, and by local yielding. The crack opening displacement approach was originally devised as a means of extending linear elastic methods to more widespread application to welded structures. This required the development of a means of assessing fracture toughness, and a means of relating this fracture toughness to the applied loading conditions, and to sizes and types of defects which might be present. The success of this method of assessing defects over a period of some 10-12 years will be illustrated, together with a discussion of the inherent limitations of the approach and possible improvements resulting from recent research into slow tearing and design curve relationships.