Showing papers on "Linear elasticity published in 1993"

Micromechanics: Overall Properties of Heterogeneous Materials

Abstract: Part 1 Overall properties of heterogeneous solids: aggregate properties and averaging methods aggregate properties, averaging methods elastic solids with microcavities and microcracks linearly elastic solids, elastic solids with traction-free defects, elastic solids with micrcavities, elastic solids with microcracks elastic solids with micro-inclusions overall elastic modulus and compliance tensors, examples o elastic solids with elastic micro-inclusions, upper and lower bounds for overall elastic moduli, self-consistent differential and related averaging methods, Eshelby's tensor and related topics solids with periodic microstructure general properties and field equations, overall properties of solids with periodic microstructure, mirror-image decomposition of periodic fields. Part 2 Introduction to basic elements of elasticity theory: foundations geometric foundations, kinematic foundations, dynamic foundations, constitutive relations elastostatic problems of linear elasticity boundary-value problems and extremum principles three-dimensional problems solution of singular problems. Appendix: references.

On the numerical integration of interface elements

Abstract: SUMMARY Eigenmode analyses of the element stiffness matrices have been used to assess the impact of the applied integration scheme on the stress predictions of two- and three-dimensional plane interface elements. It is demonstrated that large stress gradients over the element and coupling of the individual node-sets of the interface element may result in an oscillatory type of response. For line elements and linear plane interface elements the performance can be improved by using either a nodal lumping scheme or Newton-Cotes or Lobatto integration schemes instead of the more traditional Gauss scheme. For quadratic interface elements the same holds true except for a nodal lumping scheme. Interface elements are a powerful tool in the modelling of geometrical discontinuities in different kinds of structures. In finite element analysis of civil engineering structures a large variety of applications for interface elements is present. Interface elements can be used to model soil reinforcement interaction,' to model the intermediate layer between rock and concrete, e.g. in arch dam or in the analysis of rock joints.536 Applications in concrete mechanics cover the modelling of discrete cracking,'~~ aggregate interlock9 and bond between concrete and reinforcement."-I4 In rubber parts, interface elements can be of importance when disintegration of rubber and texture is concerned, e.g. in conveyor belts. Furthermore, interface elements are suited to model delamination in layered composite structures' ', or frictional contact in forming processes. Interface elements can bc divided into two elementary classes. The first class contains the continuous interface elements (line, plane and shell interface^),^-^^"-'*, 2"+23 whereas the second class of elements contains the nodal or point interface elernent~,'~'~,~ ~ which, to a certain extent, are identical to spring elements. In this contribution we shall only consider the numerically and lumped integrated continuous interface elements, since nodal interfaces are integrated explicitly. A basic requirement of interface elements is that during the elastic mge of the loading process no significant additional deformations occur due to the presence of these elements in the finite element model. Therefore, a sufficiently high initial dummy stiffness has to be supplied for the interface elemenls. Depending upon the applied numerical integration scheme, this high dummy stiffness may result in undesired spurious oscillations of the stress field. In this paper th'e impact of Gauss, Newton-Cotes, Lobatto and lumped integration schemes on the stress prediction in interfaces is investigated for three-dimensional linear and non-linear analyses. Eigenvalue analyses of the linear elastic and non-linear element stiffness matrices have been carried out to explain the observed oscillatory performance of interface elements. Since we shall focus on 0029-598 1/93/010043+ 24$17.00 0 1993 by John Wiley & Sons, Ltd.

Acoustic emissions from rapidly moving cracks.

Abstract: Linear elasticity is unable to predict completely the dynamics of a rapidly moving crack without the addition of a phenomenological fracture energy. Our measurements of acoustic emission, crack velocity, and surface structure demonstrate quantitatively similar dynamical fracture behavior in two very different materials, polymethylmethacrylate and soda-lime glass. This unexpected agreement suggests that there exist universal features of the fracture energy that result from dissipation of energy in a dynamical instability.

Plasticity and Creep Theory Examples and Problems

Abstract: BASIC DEFINITIONS Stress and Strain State Stress tensor Strain tensor Finite Deformations Finite strain tensors in material or spatial coordinates Strain rates tensors Stress tensors in material or spatial descriptions FOUNDATIONS OF PLASTICITY Basic Equations of Perfect Plasticity Uniaxial stress-strain behavior Criteria for yielding in perfect plasticity Stress-strain relations for perfect plasticity Methods of reduction of equations of perfect plasticity Problems Basic Equations of Plastic Hardening Drucker's postulate and the associated flow rule Subsequent yield surfaces for hardening material Theories of plastic hardening Problems Methods of the Theory of Plasticity Analysis of the level of a cross-section Interaction curves on levels of a cross-section or a body Extremum theorems of limit analysis: statically or kinematically admissible solutions Shakedown analysis Integration along characteristics in plane strain problems Problems SOL UTIONS OF ELASTIC-PLASTIC PROBLEMS Elastic-Plastic Torsion and Bending Elastic-plastic torsion of prismatic bars Problems Elastic-plastic bending of prismatic beams and plane frames Problems Elastic-Plastic Analysis of Cylinders, Disks, and Plates Thick-walled tubes, spherical shells and disks Problems Limit analysis of Plates Problems FOUNDATIONS OF CREEP Basic Equations of Uniaxial Creep Models Creep phenomenon Schematizations of creep at constant uniaxial stress Modelling of creep at varying uniaxial stress Linear uniaxial viscoelastic models Modelling of viscoplastic materials Problems Creep Constitutive Equations Under Multiaxial Loading Classical multiaxial creep theories Developed multiaxial creep theories Linear multiaxial viscoelastic equations SOLUTION OF CREEP PROBLEMS Bending, Buckling, and Torsion of Bars Under Creep Conditions Bending and buckling of a prismatic bar made of the linear viscoelastic material Bending of a prismatic be am made of the piece-wise linear elastic/viscoplastic material Bending of a prismatic beam made of the time hardening material Torsion of a circular bar made of the elastic-Bingham material Problems Rotationally Symmetric Creep Problems Creep of a thick-walled tube General formulae for the rotationally-symmetric transient creep problems CREEP RUPTURE Constitutive Equations of Creep Rupture Creep rupture phenomenon Classical creep rupture theories Problems Rotationally Symmetric Creep Rupture Problems Mechanisms of brittle rupture of tubes and disks Design of disks with respect to creep rupture References Author Index Subject Index

A nonconforming mixed multigrid method for the pure displacement problem in planar linear elasticity

Abstract: An optimal order multigrid method is developed for the pure displacement problem in two-dimensional linear elasticity. It is based on a nonconforming mixed formulation, where the displacement is approximated by weakly continuous piecewise linear vector functions, and the pressure is approximated by piecewise constants. The full multigrid convergence is proved. The performance of the multigrid method does not deteriorate as the material becomes nearly incompressible.

On the mathematical modelling of material behavior in continuum mechanics

Abstract: The classical theories of continuum mechanics — linear elasticity, viscoelasticity, plasticity and hydrodynamics — are defined by special constitutive equations. These can be understood to be asymptotic approximations of a quite general constitutive model, valid under restrictive assumptions for the stress functional or the input processes. The general theory of material behavior develops systematic methods to represent material properties in a context of physical evidence and mathematical consistency. According to experimental observations material behavior may be rate independent or rate dependent with or without equilibrium hysteresis. This motivates four different constitutive theories, namely elasticity, plasticity, viscoelasticity and viscoplasticity. Constitutive equations can be formulated explicitly as functionals. Then, the particular constitutive models correspond to continuity properties of these functionals, related to convenient function spaces. On the other hand, a system of differential equations may lead to an implicit definition of a stress functional. In this case additional variables are introduced, which are called internal variables. For these variables additional evolution equations must be formulated, specifying the rate of change of the internal variables in dependence on their present values and the strain (or stress) input. In the context of different models of inelastic material behavior the evolution equations have different mathematical characteristics. These concern the existence of equilibrium solutions and their stability properties. Rate independent material behavior is modelled by means of evolution equations, which are related to an arclength instead of the time as independent variable. It can be shown that the rate independent constitutive equations of elastoplasticity are the asymptotic limit of rate dependent viscoplasticity for slow deformation processes.

Numerical Modeling of Yield Zones in Weak Rock

Abstract: Publisher Summary This chapter discusses numerical modeling of yield zones in weak rock. Weakened or partly fractured materials can be modeled by a linear elastic material with a low modulus of elasticity. The major difficulty in the numerical modeling of geological materials is in the determination of the input parameters. Results of laboratory tests for such parameters as the modulus of elasticity and Mohr–Coulomb or Hoek–Brown failure properties are not always relevant to the rock in situ. Equivalent elastic properties of a weakened or yielded rock can be related, via the deformation theory of plasticity, to failure and postfailure properties such as pre-and postfailure cohesion and angle of friction of the rock. A major advantage of the deformation theory is ease of computation. The formulation is done in such a way that an iterative procedure can be set up with a linear elastic finite element program. In the incremental theories, the boundary tractions are reduced in increments until the material adjacent to the opening yields.

R-curve modeling of rate and size effects in quasibrittle fracture

Abstract: The equivalent linear elastic fracture model based on an R-curve (a curve characterizing the variation of the critical energy release rate with the crack propagation length) is generalized to describe both the rate effect and size effect observed in concrete, rock or other quasibrittle materials. It is assumed that the crack propagation velocity depends on the ratio of the stress intensity factor to its critical value based on the R-curve and that this dependence has the form of a power function with an exponent much larger than 1. The shape of the R-curve is determined as the envelope of the fracture equilibrium curves corresponding to the maximum load values for geometrically similar specimens of different sizes. The creep in the bulk of a concrete specimen must be taken into account, which is done by replacing the elastic constants in the linear elastic fracture mechanics (LEFM) formulas with a linear viscoelastic operator in time (for rocks, which do not creep, this is omitted). The experimental observation that the brittleness of concrete increases as the loading rate decreases (i.e. the response shifts in the size effect plot closer to LEFM) can be approximately described by assuming that stress relaxation causes the effective process zone lenght in the R-curve expression to decrease with a decreasing loading rate. Another power function is used to describe this. Good fits of test data for which the times to peak range from 1 sec to 250000 sec are demonstrated. Furthermore, the theory also describes the recently conducted relaxation tests, as well as the recently observed response to a sudden change of loading rate (both increase and decrease), and particularly the fact that a sufficient rate increase in the post-peak range can produce a load-displacement response of positive slope leading to a second peak.

Localized Elasticae for the Strut on the Linear Foundation

Abstract: Localized solutions, for the. classical problem of the nonlinear strut (elastica) on the linear elastic foundation, are predicted from double-scale analysis, and confirmed from nonlinear volume-preserving Runge-Kutta runs. The dynamical phase-space analogy introduces a spatial Lagrangian function, valid over the initial post-buckling range, with kinetic and potential energy components. The indefinite quadratic form of the spatial kinetic energy admits unbounded solutions, corresponding to escape from a potential well. Numerical experimentation demonstrates that there is a fractal edge to the escape boundary, resulting in spatial chaos.

Linear elasticity theory of cubic quasicrystals.

Abstract: With group-representation theory all quadratic invariants and the expressions of elastic energy have been derived for quasicrystals with cubic point-group symmetry Using the generalized elasticity theory of quasicrystals, we have also obtained the expressions of the generalized Hooke's law and equilibrium for cubic quasicrystals

Mixed finite element methods with continuous stresses

Abstract: We present a modified mixed formulation for second order elliptic equations and linear elasticity problems which automatically satisfies the “ellipticity on the kernel” condition, i.e., one of the two compatibility conditions necessary to ensure stability and optimal error bounds (the other being the Inf-Sup condition). This modification differs from similar ones introduced by other authors in that it is independent of the mesh size. Moreover, it allows the use of continuous stresses.

A predictive model for the mechanical behavior of particulate composites. Part I: Model derivation

Abstract: A method for predicting the highly nonlinear stress-strain behavior and dilatation induced by cavitation of highly filled particulate composites from constituent properties has been developed. The approach presented uses a variation of linear elasticity throughout and has no adjustable parameters, unlike the methods currently used, which require large numbers of fitting factors and complicated nonlinear analyses. An energy balance derived from the first law of thermodynamics calculates critical strain values at which filler particles will debond when subjected to deformation. Repeated calculations of critical strain values using re-evaluated material properties accounting for the damage caused by debonding give very nonlinear stress-strain and dilatation curves. Experimentally observed dependencies on particle size, filler concentration, adhesion, and matrix and filler properties are correctly predicted. The method can be generalized for any state of stress or particle shape. Comparisons of experimental data with the model results give good agreement.

Numerical study of the hydromechanical behavior of two rough fracture surfaces in contact

Abstract: Flow in fractures is traditionally modeled by characterizing the aperture distribution with some deterministic function or set of stochastic parameters. Other models generate the aperture distribution by the closure of two stochastic surfaces. The objective of this research is to develop a model where the aperture distribution is determined during the closure of two random elastic surfaces with complete hydromechanical interaction. Because stress and strain conditions required to generate a given aperture distribution are calculated during closure, the model is used to couple the mechanical and hydraulic characteristics of the fracture. Stochastic realizations of clay fracture surfaces are generated by measuring one-dimensional profiles of a fracture surface. Next, the spectral representation of the profile is related to the fractal dimension of the fracture. Using the fractal dimension determined from one-dimensional clay profiles, an equivalent two-dimensional fractal surface is generated. Conceptually, each surface consists of linear elastic rectangular asperities resting upon a linear elastic half-space. During closure, asperities that come into contact deform and punch into the half space creating mechanical interaction between all the asperities on the grid. Once we determine the aperture distribution at an applied stress level, a hydraulic gradient is applied across the fracture and fluid flow is determined. Nodal pressures created by flow deform the aperture distribution coupling hydraulic to mechanical behavior. Stress versus relative closure results indicate that stress increases nonlinearly with relative closure. Fluid pressures in the aperture distribution exert a significant influence on the mechanical characteristics of a fracture. Fluid discharge through the fracture decreases exponentially with an increase in relative closure. Flow calculated in the rough walled aperture distribution deviates increasingly from the parallel plate model with the geometric mean aperture as the percent contact area increases. The deviation results from an increase in tortuosity and channelling of the flow field in the aperture distribution. We can use this model to develop stress and flow versus relative closure constitutive relationships for a single fracture as a function of fracture surface geometry.

Small-contrast perturbation expansions for the effective properties of nonlinear composites

Abstract: This Note deals with nonlinear composite materials with local constitutive behaviour controlled by a convex potential ω, which varies slightly from point to point in the composite, as determined by a small parameter t. The effective behaviour of the composite is in turn controlled by a macroscopic potential W, which is assumed to depend smoothly on the contrast t. Exact expressions are obtained for the first three terms in a perturbation expansion of W about t=0; thee derivation being reduced to the solution of a standard linear elasticity problem for a homogeneous isotropic body with body forces determined by the relevant polarization tensors. An explicit expansion, exact to second order in the contrast, and depending only on the volume fractions for the N phases, is obtained for the special class of power-law incompressible composites with a statistically homogeneous and isotropic distribution of the phases. One interesting feature of the result is the explicit dependence of the second-order term on the third invariant of the applied strain

Geometrically Exact Analysis of Spatial Frames

Abstract: Department of Structural and Foundation Engineering, Escola Politecnica Universidade de Sao Paulo, CP 61548, 05424-970 Sao Paulo, SP, Brazil A fully nonlinear, geometrically exact, finite strain rod model is derived from basic kinematical assumptions. The model incorporates shear distortion in bending and can take account of torsion warping. Rotation in 3D space is handled with the aid of the Euler-Rodrigues formula. The accomplished parametrization is simple and does not require update algorithms based on quaternions parameters. Weak and strong forms of the equilibrium equations are derived in terms of cross section strains and stresses, which are objective and suitable for constitutive description. As an example, an invariant linear elastic constitutive equation based on the small strain theory is presented. The attained formulation is very convenient for numerical procedures employing Galerkin projection like the finite element method and can be readily implemented in a finite element code. A mixed formulation of Hu-Washizu type is also derived, allowing for independent interpolation of the displacement, strain and stress fields within a finite element. An exact expression for the Frechet derivative of the weak form of equilibrium is obtained in closed form, which is always symmetric for conservative loading, even far from an equilibrium state and is very helpful for numerical procedures like the Newton method as well as for stability and bifurcation analysis. Several numerical examples illustrate the usefulness of the formulation in the lateral stability analysis of spatial frames. These examples were computed with the code FENOMENA, which is under development at the Computational Mechanics Laboratory of the Escola Politecnica. INTRODUCTION The interest on geometrically nonlinear analysis of struc­tures has increased in the recent few years. Besides the practical importance of nonlinear static and dynamic analysis of flexible rod and shell assemblages, the de­velopment of convenient geometrically exact models has contributed to this fact. These models show many ben­efits, which have been emphasized by many authors, as one can verify in a non-exhaustive list reproduced in the references. This work derives a geometrically exact rod model from the kinematic assumption that cross sections, which are initially orthogonal to the axis, remain plane and undistorted during the deformation. The theory accom­modates finite strains, large displacements and rotations, and accounts for shear distortion in bending. Torsion warping can be effortlessly acquainted for, provided elas­tic behavior is assumed. On the other hand, the intro­duction of elastic-plastic, visco-plastic and visco-elastic constitutive equations in terms of cross section general­ized strains and stresses is straightforward. The consid­eration of cross section inertia is direct as well. The accomplished formulation can be readily applied to the nonlinear analysis of spatial frames through the finite element method and presents the following advan­tages: (a) rotations in 3D space are treated in a consistent but convenient way through the Euler-Rodrigues for­mula: update algorithms based on quaternion pa­rameters are not required; (b) there is no need of approximate strain-displacement relationships or additional assumptions like moder­ate rotations, small curvatures and small cross sec­tion dimensions; (c) generalized cross section strains and stresses, which are energetically conjugate, can be consistently de­fined; (d) generalized cross section displacements and external loadings, which are energetically conjugate, can be consistently defined; (e) equilibrium and motion equations are consistently derived in weak form as well as in strong form; (f) boundary conditions are obtained by variational ar­guments;

Structural mechanics in reactor technology

Abstract: The paper deals with the effect of crack gro1lJth rate and creep in static fracture of concretc. The available experimental data and various modeling approaches, including a rate-dependent generalization of the cohesive crack model based on the activation energy concept are reviewed. Attention is then focused on the description of these effects by means of a generalization of the equivalent linear elastic fracture model based on the R-curve concept, and a new model of this type is presented in detail. The crack propagation velocity is assumed to depend on the ratio of the stress intensity factor to its critical value from the R-curve. This dependence can be assumed as a power function 1IJith an exponent much larger than 1. The shape of the R-curve is determined as the envelope of the fracture equilibrium c:., u;,s con-esponding to the maximum load values for geometrically similar specimens of different sizes. The creep in the bulk of a concrete specimen must be taken into account in the case of static loading, which is done by replacing the elastic constants with a linear viscoelastic operator in time. The model fits the existing data on concrete (as well as rock) reasonably well. It exhibits not only the effects of size and rate, but for concrete it also exhibits an increase of brittleness with a decrease of loading rate, manifested a.' a shifl of the maximum load points in the size effect plot toward linear elastic fracture mechanics (LEFM).

Buckling eigenvalues for a clamped plate embedded in an elastic medium and related questions

Abstract: This paper considers the dependence of the sum of the first m eigenvalues of three classical problems from linear elasticity on a physical parameter in the equation. The paper also considers eigenvalues $\gamma _i (a)$ of a clamped plate under compression, depending on a lateral loading parameter $a;\Lambda i(a)$, the Dirichlet eigenvalues of the elliptic system describing linear elasticity depending on a combination a of the Lame constants, and eigenvalues $\Gamma _i (a)$ of a clamped vibrating plate under tension, depending on the ratio a of tension and flexural rigidity. In all three cases $a \in [0,\infty )$. The analysis of these eigenvalues and their dependence on a gives rise to some general considerations on singularly perturbed variational problems.

On almost constant contact stress distributions by shape optimization

Abstract: This paper addresses the problem of finding shapes of contacting bodies avoiding undesirable stress concentrations It has previously been shown that designing the shape of a rigid body in contact with a fixed linear elastic body by minimizing the equilibrium potential energy under an isoparametric constraint results in a uniform contact pressure distribution As an extension of this result, it is shown here that when the shape of an elastic body in contact with a flat rigid foundation is chosen on the same premises, the uniform pressure distribution is found only if displacement gradients can be considered small From the point of view of applications, an important conclusion is that this smallness holds in a case when linear elasticity is physically valid

A hollow cylinder torsional simple shear apparatus capable of a wide range of shear strain measurement

Abstract: The lack of high-quality data at small strains under monotonic loads has led to the assumption of linear elasticity in small strain analysis and the use of low values of stiffness for analysis of geotechnical structures under working loads. In addition, since most simple shear apparatuses cannot measure lateral stresses, various assumptions have to be made in order to define the failure strength in terms of principal stresses. In this paper a hollow cylinder torsional simple shear system that is capable of measuring shear strains accurately from a strain level as small as 10−6 to failure in addition to measuring all stress components is described. This system is also capable of consolidating specimens along both normal and overconsolidated stress paths and during shearing is able to apply extremely small load reversals with little or no backlash effect. Using the special features of this system, the behavior of kaolin specimens under various stress histories from very small strains to failure was studied. The results show that irrespective of the consolidation stress history, the major part of the principal stress rotation occurs at the early stages of undrained simple shear, and the final direction near the peak strength is always a few degrees below 45°. The small strain behavior indicates that there exists a very small elastic zone of the order of 2 × 10−5 shear strain, and that the behavior beyond this region is highly nonlinear. The trend in the variation of the peak strength with OCR obtained in this system is similar to that obtained in the conventional simple shear apparatus as reported in the literature.

Stabilization of beam parametric vibrations

Abstract: A theoretical investigation of dynamic stability for linear elastic beams due to time dependent harmonic anal forces is presented. The concept of intelligent structure is used to insure the active damping. In the present paper the applicability of active vibration control is extended to linear continuous systems with parametric harmonic excitations. The study is based on the application of distributed sensors, actuators, and an appropriate feedback and is adopted for stability problems of system consisting of beam with control part governed by uniform partial differential equations with time, dependent coefficients. To estimate deviations of solutions from the equilibrium state (the distance between a solution with nontrivial initial conditions and the trivial solution) a scalar measure of distance equal to the square root of the functional is introduced. The tyapunoy method is used to derive a velocity feedback implying nonincreasing of the functional along an arbitrary beam motion and in conseqiaeftce to balance the supplied energy by the parametric excitation and the dissipated ejiergy by the inner and control damping. In order to calculate the energetic norm of disturbed solution as a function of time the partial differential equation is solved numencally. The numerical tests performed for the simply supported beam with surface bonded actuators and sensors show the influence of the fedback constant on the vibration decrease.