Showing papers on "Continuum mechanics published in 2000"

Nonlinear Finite Elements for Continua and Structures

Abstract: Preface. List of Boxes. Introduction. Lagrangian and Eulerian Finite Elements in One Dimension. Continuum Mechanics. Lagrangian Meshes. Constitutive Models Solution Methods and Stability. Arbitrary Lagrangian Eulerian Formulations. Element Technology. Beams and Shells. Contact--Impact. Appendix 1: Voigt Notation. Appendix 2: Norms. Appendix 3: Element Shape Functions. Glossary. References. Index.

Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces

Abstract: Some materials may naturally form discontinuities such as cracks as a result of deformation. As an aid to the modeling of such materials, a new framework for the basic equations of continuum mechanics, called the "peridynamic" formulation, is proposed. The propagation of linear stress waves in the new theory is discussed, and wave dispersion relations are derived. Material stability and its connection with wave propagation is investigated. It is demonstrated by an example that the reformulated approach permits the solution of fracture problems using the same equations either on or off the crack surface or crack tip. This is an advantage for modeling problems in which the location of a crack is not known in advance.

Size-dependent elastic properties of nanosized structural elements

Abstract: Effective stiffness properties (D) of nanosized structural elements such as plates and beams differ from those predicted by standard continuum mechanics (Dc). These differences (D-Dc)/Dc depend on the size of the structural element. A simple model is constructed to predict this size dependence of the effective properties. The important length scale in the problem is identified to be the ratio of the surface elastic modulus to the elastic modulus of the bulk. In general, the non-dimensional difference in the elastic properties from continuum predictions (D-Dc)/Dc is found to scale as αS/Eh, where α is a constant which depends on the geometry of the structural element considered, S is a surface elastic constant, E is a bulk elastic modulus and h a length defining the size of the structural element. Thus, the quantity S/E is identified as a material length scale for elasticity of nanosized structures. The model is compared with direct atomistic simulations of nanoscale structures using the embedded atom method for FCC Al and the Stillinger-Weber model of Si. Excellent agreement between the simulations and the model is found.

Constitutive models of rubber elasticity: A review

Abstract: A review of constitutive models for the finite deformation response of rubbery materials is given. Several recent and classic statistical mechanics and continuum mechanics models of incompressible rubber elasticity are discussed and compared to experimental data. A hybrid of the Flory—Erman model for low stretch deformation and the Arruda—Boyce model for large stretch deformation is shown to give an accurate, predictive description of Treloar's classical data over the entire stretch range for all deformation states. The modeling of compressibility is also addressed.

Continuum Mechanics and Theory of Materials

Abstract: 1 Kinematics.- 2 Balance Relations of Mechanics.- 3 Balance Relations of Thermodynamics.- 4 Objectivity.- 5 Classical Theories of Continuum Mechanics.- 6 Experimental Observation and Mathematical Modelling.- 7 General Theory of Mechanical Material Behaviour.- 8 Dual Variables.- 9 Elasticity.- 10 Viscoelasticity.- 11 Plasticity.- 12 Viscoplasticity.- 13 Constitutive Models in Thermomechanics.- References.

A finite-element model of vocal-fold vibration.

Abstract: A finite-element model of the vocal fold is developed from basic laws of continuum mechanics to obtain the oscillatory characteristics of the vocal folds. The model is capable of accommodating inhomogeneous, anisotropic material properties and irregular geometry of the boundaries. It has provisions for asymmetry across the midplane, both from the geometric and tension point of view, which enables one to simulate certain kinds of voice disorders due to vocal-fold paralysis. It employs the measured viscoelastic properties of the vocal-fold tissues. The detailed construction of the matrix differential equations of motion is presented followed by the solution scheme. Finally, typical results are presented and validated using an eigenvalue method and a commercial finite-element package (ABAQUS).

Mechanics of Deformable Solids: Linear, Nonlinear, Analytical and Computational Aspects

Abstract: Basic mechanics Variational formulations, work and energy theorems Theory of beams (strength of materials) Torsion of beams Theory of thin plates Bending of thin plates in polar coordinates Two-dimensional problems in Cartesian coordinates Two-dimensional problems in polar coordinates Thermo-elasticity Elastic stability Theory of thin shells Elasto-plasticity Elasto-viscoplasticity Nonlinear continuum mechanics Nonlinear elasticity Finite-strain elasto-plasticity Cyclic plasticity Damage mechanics Strain locolization Micro-mechanics of materials Cylindrical coordinates Cardan's formulae Matrices for the representation of second- and fourth-order tensors Zero-stress constraints.

Continuum damage mechanics-based fatigue model of asphalt concrete

Abstract: A fatigue performance prediction model of asphalt concrete is developed from a uniaxial constitutive model based on the elastic-viscoelastic correspondence principle and continuum damage mechanics through mathematical simplifications. This fatigue model has a form similar to the phenomenological tensile strain-based fatigue model. Therefore, a comparison between the new model and the phenomenological model yields that the regression coefficients in the phenomenological model are functions of viscoelastic properties of the materials, loading conditions, and damage characteristics. The experimental study on two mixtures with compound loading histories demonstrates that the fatigue model maintains all of the strengths of the constitutive model such as its accuracy and abilities to account for the effects of rate of loading, stress/strain level dependency, rest between loading cycles, and mode-of-loading on fatigue life of asphalt concrete.

A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods

Abstract: We propose a scheme for the direct passage from atomic level to continuum level. The scheme is applicable to geometries, like films, rods and tubes, in which one or more dimensions are large relative to atomic scale but other dimensions may be of atomic scale. The atomic theory is assumed to be governed by a variational principle resting on the Born–Oppenheimer approximation. The atomic level energy is further assumed to satisfy certain decay properties when evaluated for disjoint sets of atoms. The scheme is based on two hypotheses: (1) distortions are limited, and (2) there are many atoms in certain directions. The scheme produces in a natural way the variables of the continuum theory. In the case of a film, the continuum theory that emerges is a Cosserat membrane theory with (ν−1) Cosserat vectors, ν being the number of atomic layers in the film. The arguments presented are not mathematically rigorous. One difficulty is that it is not clear under which circumstances our decay hypothesis on interatomic interactions is consistent with quantum mechanics or density functional theory.

Nonlinear continuum mechanics

Abstract: In all previous chapters, we have worked within the small perturbation hypothesis (SPH) As a consequence, we wrote (and solved for) equilibrium and boundary condition equations on the initial, undeformed (thus known) configuration of a body The only exception was the study of possible buckling modes in Chap 10 In this chapter, we present basic continuum mechanics in the presence of geometric non-linearities, namely finite strains, displacements and rotations Practical examples are metal forming problems or large displacements of slender beams and thin shells

General Equations Describing Elastic Indentation Depth and Normal Contact Stiffness versus Load.

Abstract: Continuum mechanics models describing the contact between two adhesive elastic spheres, such as the JKR and DMT models, provide a relationship between the elastic indentation depth and the normal load, but the general intermediate case between these two limiting cases requires a more complex analysis. The Maugis–Dugdale theory gives analytical solutions, but they are difficult to use when comparing to experimental data such as those obtained by scanning force microscopy. In this paper we propose a generalized equation between elastic indentation depth and load that approximates Maugis' solution very closely. If the normal contact stiffness can be described as the force gradient, that is the case of the force modulation microcopy, then a generalized equation between normal contact stiffness and load can be deduced. Both general equations can be easily fit to experimental data, and then interfacial energy and elastic modulus of the contact can be determined if the radius of the indenting sphere is known.

A new material model for concrete in high-dynamic hydrocode simulations

Abstract: The development of material laws for concrete subjected to highly dynamic loadings is a topic of current research. Explosive charges or high-velocity impacts produce high pressures in the kilobar region within microseconds. Hydrocode simulations by coupling of Lagrangian with Eulerian grids have been carried out, considering the interaction between explosive loading and the structure. Concrete is a composite material with a variety of inhomogenities. By homogenization of the microstructure, a macroscopic approach in the framework of continuum mechanics has been adopted. Appropriate constitutive laws that enable the nonlinear rate-dependent as well as the local damage behaviour to be modelled had to be introduced. A new damage law that describes void compaction as well as the classical theory of plasticity had been taken into account. An equation of state had to be provided to ensure the compliance with conservation laws on which hydrocodes are based. To obtain the necessary material data, experimental investigations were indispensable. Therefore, a series of field tests with specimens which were concrete slabs exposed to explosive contact charges has been conducted.

Dynamic Fracture of Silicon: Concurrent Simulation of Quantum Electrons, Classical Atoms, and the Continuum Solid

Abstract: Our understanding of materials phenomena is based on a hierarchy of physical descriptions spanning the space-time regimes of electrons, atoms, and matter and given by the theories of quantum mechanics, statistical mechanics, and continuum mechanics. The pioneering work of Clementi and co-workers provides a lucid example of the traditional approach to incorporating multiscale phenomena associated with these three mechanics. Using quantum mechanics, they evaluated the interactions of several water molecules. From this data base, they created an empirical potential for use in atomistic mechanics and evaluated the viscosity of water. From this computed viscosity, they performed a fluid-dynamics simulation to predict the tidal circulation in Buzzard's Bay. This is a powerful example of the sequential coupling of length and time scales: a series of calculations is used as input to the next rung up the length/time-scale ladder.

Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients

Abstract: Conservative linear equations arise in many areas of application, including continuum mechanics or high-frequency geometrical optics approximations. This kind of equations admits most of the time solutions which are only bounded measures in the space variable known as duality solutions. In this paper, we study the convergence of a class of finite-differences numerical schemes and introduce an appropriate concept of consistency with the continuous problem. Some basic examples including computational results are also supplied.

Numerical Models of Translational Landslide Rupture Surface Growth

Abstract: We analyze the initiation and enlargement of the rupture surface of translational landslides as a fracture phenomenon using a two-dimensional boundary-element method. Both processes are governed largely by the stress field and the pre-existing planes of weakness in a slope. Near the ground surface, the most compressive stress becomes either parallel or perpendicular to the slope, depending on the topography and regional stresses. The shear stress available to drive slope-parallel sliding in a uniform slope thus is small, and therefore pre-existing weaknesses are required in many cases for sliding. Stresses in a uniform slope favor the initiation of sliding near the slope base. Sliding can progress upslope from there in retrogressive fashion. Most slopes are not uniform and notches in a slope will concentrate stresses and generally promote sliding there. As the region of sliding at depth enlarges, the stress concentration near the edge of the area of slip will tend to rise. Stress concentrations can become sufficient to open fractures above and below a basal slide plane, in keeping with observations. If one tip of a slide plane intersects the ground surface, then stresses near the other tip can increase markedly, as can slip. Our analyses show that slope-parallel sliding along a plane at depth will cause downslope extension in the upslope half of a slide mass and shortening in the downslope half, consistent with observations. Displacement profiles that could be interpreted as rotational can result from sliding along such a plane, however careful analysis of surface deformation can be used to understand sliding at depth.

Multi-scale computational method for elastic bodies with global and local heterogeneity

Abstract: A multi-scale computational method using the homogenization theory and the finite element mesh superposition technique is presented for the stress analysis of composite materials and structures from both micro- and macroscopic standpoints. The proposed method is based on the continuum mechanics, and the micro–macro coupling effects are considered for a variety of composites with very complex microstructures. To bridge the gap of the length scale between the microscale and the macroscale, the homogenized material model is basically used. The classical homogenized model can be applied to the case that the microstructures are periodically arrayed in the structure and that the macroscopic strain field is uniform within the microscopic unit cell domain. When these two conditions are satisfied, the homogenization theory provides the most reliable homogenized properties rigorously to the continuum mechanics. This theory can also calculate the microscopic stresses as well as the macroscopic stresses, which is the most attractive advantage of this theory over other homogenizing techniques such as the rule of mixture. The most notable feature of this paper is to utilize the finite element mesh superposition technique along with the homogenization theory in order to analyze cases where non-periodic local heterogeneity exists and the macroscopic field is non-uniform. The accuracy of the analysis using the finite element mesh superposition technique is verified through a simple example. Then, two numerical examples of knitted fabric composite materials and particulate reinforced composite material are shown. In the latter example, a shell-solid connection is also adopted for the cost-effective multi-scale modeling and analysis.

On modelling thermal oxidation of Silicon I: Theory

Abstract: This work focuses on a new mathematical framework to model the process of thermal oxidation in silicon. The mathematical model is derived from the fundamental conservation equations of mechanics. The mass balance law provides the description of the oxidant transport and the Si–SiO2 interface motion, and momentum balance provides the framework to model the displacements and stresses in the bulk and the oxide. The displacements define the geometry of the final oxide structure. The large expansion is treated within a mathematically exact formulation following a split of the deformation gradient. A thermodynamically consistent constitutive equation for silicon dioxide is suggested to represent recent experimental data. Copyright © 2000 John Wiley & Sons, Ltd.

A further study on the postbuckling of extensible elastic rods

Abstract: The postbuckling of extensible elastic rods is studied using non-linear geometric models. Accordingly the kinematics and equilibrium are stated. Nine different strain–stress relationships are analyzed. The classical Strength of Materials approach is compared and discussed with other eight constitutive laws stated with Lagrangian and Eulerian descriptions. The well-known Cauchy and Green methods in Continuum Mechanics are alternatively employed. Four of the approaches are worked out until an explicit solution of the secondary equilibrium path is obtained. The analysis is applicable to small strain problems. The linearized problem is presented for all the laws together with numerical results for rods with various values of the extensibility parameter. The secondary equilibrium paths are numerically evaluated to illustrate the degree of discrepancy. A specific example that displays unexpected unstable behavior is shown. Both critical loads and postbuckling curves are coincident when the theoretical problem of an inextensible rod is solved. It is shown that even when small strains are addressed, the extensibility influence gives rise to disagreement of the postbuckling response when using the different alternatives.