Showing papers on "Continuum mechanics published in 2001"

Random Heterogeneous Materials: Microstructure and Macroscopic Properties

Abstract: Motivation and overview * PART I Microstructural characterization * Microstructural descriptors * Statistical mechanics of particle systems * Unified approach * Monodisperse spheres * Polydisperse spheres * Anisotropic media * Cell and random-field models * Percolation and clustering * Some continuum percolation results * Local volume fraction fluctuation * computer simulation and image analysis * PART II Microstructure property connections * Local and homogenized equations * Variational Principles * Phase-interchange relations * Exact results * Single-inclusion solutions * Effective medium approximations * Cluster expansions * Exact contrast expansions * Rigorous bounds * Evaluation of bounds * Cross-property relations * Appendix A Equilibrium Hard disk program * Appendix B Interrelations among 2-3D moduli* References * Index

Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics

Abstract: We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical modelling of viscoelastic bodies (Sect. 1), and unsteady motion of a particle in a viscous fluid, i.e. the Basset problem (Sect. 2). In the former analysis fractional calculus leads us to introduce intermediate models of viscoelasticity which generalize the classical spring-dashpot models. The latter analysis induces us to introduce a hydrodynamic model suitable to revisit in Sect. 3 the classical theory of the Brownian motion, which is a relevant topic in statistical mechanics. By the tools of fractional calculus we explain the long tails in the velocity correlation and in the displacement variance. In Sect. 4 we consider the fractional diffusion-wave equation, which is obtained from the classical diffusion equation by replacing the first-order time derivative by a fractional derivative of order $0< \beta <2$. Led by our analysis we express the fundamental solutions (the Green functions) in terms of two interrelated auxiliary functions in the similarity variable, which turn out to be of Wright type (see Appendix), and to distinguish slow-diffusion processes ($0 < \beta < 1$) from intermediate processes ($1 < \beta < 2$).

A computational model of viscoplasticity and ductile damage for impact and penetration

Abstract: A coupled constitutive model of viscoplasticity and ductile damage for penetration and impact related problems has been formulated and implemented in the explicit finite element code LS-DYNA. The model, which is based on the constitutive model and fracture strain model of Johnson and Cook, and on continuum damage mechanics as proposed by Lemaitre, includes linear thermoelasticity, the von Mises yield criterion, the associated flow rule, non-linear isotropic strain hardening, strain-rate hardening, temperature softening due to adiabatic heating, isotropic ductile damage and failure. For each of the physical phenomena included in the model, one or several material constants are required. However, all material constants can be identified from relatively simple uniaxial tensile tests without the use of numerical simulations. In this paper the constitutive model is described in detail. Then material tests for Weldox 460 E steel and the calibration procedure are presented and discussed. The calibrated model is finally verified and validated through numerical simulations of material and plate perforation tests investigated experimentally.

Advanced Mechanics of Materials

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A Critical Evaluation of Cohesive Zone Models of Dynamic Fracture

Abstract: Finite element calculations of dynamic fracture based on embedding cohesive surfaces in a continuum indicate that the predictions are sensitive to the cohesive law used. Simulations were performed on a square block in plane strain with an initial edge crack loaded at a constant rate of strain, Cohesive laws that have an initial elastic response were observed to produce spontaneous branching at high velocity, but to modify the linear elastic properties of the body As a consequence the cohesive surface spacing cannot be refined arbitrarily and becomes an important length scale in the simulations. Cohesive laws that are initially rigid do not alter the linear elastic response of the body. However, crack branching behavior was not observed when such a cohesive relation was implemented using a regular finite element mesh.

A micromechanical model for a viscoelastic cohesive zone

Abstract: A micromechanical model for a viscoelastic cohesive zone is formulated herein. Care has been taken in the construction of a physically-based continuum mechanics model of the damaged region ahead of the crack tip. The homogenization of the cohesive forces encountered in this region results in a damage dependent traction-displacement law which is both single integral and internal variable-type. An incrementalized form of this traction-displacement law has been integrated numerically and placed within an implicit finite element program designed to predict crack propagation in viscoelastic media. This research concludes with several example problems on the response of this model for various displacement boundary conditions.

Echinocyte Shapes: Bending, Stretching and Shear Determine Spicule Shape and Spacing

Abstract: We study the shapes of human red blood cells using continuum mechanics. In particular, we model the crenated, echinocytic shapes and show how they may arise from a competition between the bending energy of the plasma membrane and the stretching/shear elastic energies of the membrane skeleton. In contrast to earlier work, we calculate spicule shapes exactly by solving the equations of continuum mechanics subject to appropriate boundary conditions. A simple scaling analysis of this competition reveals an elastic length which sets the length scale for the spicules and is, thus, related to the number of spicules experimentally observed on the fully developed echinocyte.

Regularization of the Burnett Equations via Relaxation

Abstract: The classical Chapman–Enskog expansions for the pressure deviator P and heat flux q provide a natural bridge between the kinetic description of gas dynamics as given by the Boltzmann equation and continuum mechanics as given by the balance laws of mass, momentum, energy supplemented by the expansions for P and q. Truncation of these expansions beyond first (Navier–Stokes) order yields instability of the rest state and is inconsistent with thermodynamics. In this paper we propose a visco-elastic relaxation approximation that eliminates the instability paradox. This system is weakly parabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman–Enskog expansion.

Definition of the Elastic Forces in the Finite-Element Absolute Nodal Coordinate Formulation and the Floating Frame of Reference Formulation

Abstract: The equivalence of the finite-element formulations used inflexible multibody dynamics is the focus of this investigation. Thisequivalence will be used to address several fundamental issues related tothe deformations, flexible body coordinate systems, and the geometriccentrifugal stiffening effect. Two conceptually different finite-elementformulations that lead to exact modeling of the rigid body dynamics will beused. The first one is the absolute nodal coordinateformulation in which beams and plates can be treated as isoparametricelements. This formulation leads to a constant and symmetric mass matrix andhighly nonlinear elastic forces. In this study, it is demonstrated thatdifferent element coordinate systems which are used for the convenience ofdescribing the element deformations lead to similar results as the elementsize is reduced. In particular, two element frames are used;the pinned and the tangent frames. The pinned frame has one ofits axes passing through two nodes of the element, while the tangent frame isrigidly attached to one of the ends of the element. Numerical resultsobtained using these two different frames are found tobe in good agreement as the element size decreases. The relationshipbetween the coordinates used in the absolute nodal coordinate formulationand the floating frame of reference formulation is presented. Thisrelationship can be used to obtain the highly nonlinear expression of thestrain energy used in the absolute nodal coordinate formulation from thesimple energy expression used in the floating frame of referenceformulation. It is also shown that the source of the nonlinearityis due to the finite rotation of the element. The result of the analysispresented clearly demonstrates that the instability observedin high-speed rotor analytical models due to the neglect of the geometriccentrifugal stiffening is not a problem inherent to a particular finite-element formulation. Such a problem can only be avoided by considering the known linear effect of the geometric centrifugal stiffening or by using a nonlinear elastic model as recently demonstrated. Fourier analysis of the solutions obtained in this investigation also sheds new light on the fundamental problem of the choice of the deformable body coordinate system in the floating frame of reference formulation. Another method forformulating the elastic forces in the absolute nodal coordinate formulationbased on a continuum mechanics approach is also presented.

Statistical continuum theory for large plastic deformation of polycrystalline materials

Abstract: This paper focuses on the application of statistical continuum mechanics to the prediction of mechanical response of polycrystalline materials and microstructure evolution under large plastic deformations. A statistical continuum mechanics formulation is developed by applying a Green's function solution to the equations of stress equilibrium in an infinite domain. The distribution and morphology of grains (crystals) in polycrystalline materials is represented by a set of correlation functions that are described by the corresponding probability functions. The elastic deformation is neglected and a viscoplastic power law is employed for crystallographic slip in single crystals. In this formulation, two- and three-point probability functions are used. A secant modulus-based formulation is used. The statistical analysis is applied to simulate homogeneous deformation processes under uniaxial tension, uniaxial compression and plane strain compression of an FCC polycrystal. The results are compared to the well-known Taylor upper bound model and discussed in comparison to experimental observations.

Determining stress parameters of formations from multi-mode velocity data

Abstract: A method for determining unknown stress parameters in earth formation measures velocities in four sonic transmissions modes (compression, fast shear, slow shear and Stoneley) at a series of depths. Relationships between measured velocities and other measured values, two independent linear constants, and three nonlinear constant associated with equations of motion for pre-stressed isotropic materials are expressed in a set of four velocity difference equations derived from non-linear continuum mechanics. The velocity difference equations are solved using inversion for useful stress parameters, including maximum horizontal stress, minimum horizontal stress, pore pressure, and change in pore pressure over time.

Assessment of Mechanical Properties of Cohesive Particulate Solids. Part 1: Particle Contact Constitutive Model

Abstract: The fundamentals of cohesive particulate solids' consolidation and flow properties using a reasonable combination of particle and continuum mechanics are explained. By means of the model "stiff particles with soft contacts," the combined influence of elastic, elastoplastic, and viscoplastic repulsion in particle contacts is derived. As a result, contact normal force displacement F N (h K ) and adhesion force models F H (F N ) are presented that describe the instantaneous and time consolidation behavior at characteristic (averaged) particle contacts. The approach has been shown to be effective for the data evaluation of a very cohesive titania nanopowder (surface diameter d S = 200 nm , solid density „ s = 3870 kg/m 3 ).

Decomposition of Damage Tensor in Continuum Damage Mechanics

Abstract: The principles of continuum damage mechanics are reviewed first for the case of uniaxial tension. The damage variable is then decomposed into two variables called crack damage variables and void damage variables. A consistent mathematical formulation is presented to decompose the damage tensor into two parts: one caused by voids and the other caused by cracks. In the first part of this work, isotropic damage in the uniaxial tension case is assumed. However, the generalization to three-dimensional states of damage is presented in the second part of this work, using tensorial damage variables. It is shown that the components of tensorial crack damage variables and void damage variables are not independent of each other, implying a coupling between the two damage mechanisms. This coupling may be obvious based on the physics of the problem, but a rigorous mathematical proof is given for it. Also, explicit relations governing the components of the crack and void damage variables are derived.

Some issues about anisotropic elastic-plastic models at finite strain

Abstract: The description of anisotropic elastic-plastic behaviour at large strain is difficult. In the rigid plastic case, anisotropic plasticity is now reasonably well understood and is based on formulating anisotropic constitutive equations in a moving frame which, correctly chosen, will ensure objectivity and which can be postulated: (i) a priori to follow in some sense the material rotations (kinematical rotating frame); (ii) a posteriori by an appropriate constitutive evolution equation (plastic spin equation). With respect to the classical continuum models this is in fact an alternative for the formulation of frame indifferent constitutive equations. This rotating frame formulation of continuum mechanics is first formally presented and applied to anisotropic elasticity to show that in this case the proper rotating frame must be used to get a true elasticity. The remaining part of the paper is then devoted to the elastic-plastic case starting from a direct extension of the formalism with derivation of the elastic constitutive equation and plastic flow rule which is best written in the so-called isoclinic configuration. Finally the kinematic plastic rotating frames will be introduced and exemplified as the appropriate choice for a correct description of an anisotropic elastic- plastic behaviour.

Mechanics of random and multiscale microstructures

Abstract: Preface Statistical Continuum Mechanics: an Introduction by M. J. Beran.- Random Structure Models for Homogenization and Fracture Statistics by D. Jeulin.- Mechanics of Random Materials: Stochastics, Scale Effects, and Computation by M. Ostoja-Starzewski.- The Randomness of Fatigue and Fracture Behaviour in Metallic Materials and Mechanical Structures by A. Pineau.- Lectures on Mechanics of Random Media by J. R. Willis.

Non‐parabolic band hydrodynamical model of silicon semiconductors and simulation of electron devices

Abstract: A consistent hydrodynamical model for electron transport in silicon semiconductors, free of any fitting parameter, has been formulated in Anile and Romano (Continuum Mechanics Thermodynamics 1999; 11:307–325) and Romano (Continuum Mechanics Thermodynamics 1999; 12:31–51) on the basis of the maximum entropy principle, by considering the energy band described by the Kane dispersion relation. Explicit constitutive functions for fluxes and production terms in the macroscopic balance equations of density, crystal momentum, energy and energy flux have been obtained. Scatterings of electrons with non-polar optical phonons (both for intervalley and intravalley interactions), acoustic phonons and impurities have been taken into account. In this article we show the link with other macroscopic models describing the motion of charge carriers. In particular, under suitable scaling assumptions, an energy transport model is recovered. An analysis of the formal properties is given by showing that the evolution equations form a hyperbolic system in the physically relevant region of the space of the dependent variables. At last, by using the numerical method developed in Liotta et al. (International Series of Numerical Mathematics 1999; 130:651–660) and Liotta et al. (SIAM Journal on Numerical Analysis 1999, to appear) simulations for bulk silicon and n+–n–n+ silicon diode are performed. The obtained results are in good agreement with the Monte Carlo data. Copyright © 2001 John Wiley & Sons, Ltd.

Continuum simulations of directional dependence of crack growth along a copper/sapphire bicrystal interface. Part I: experiments and crystal plasticity background

Abstract: Cracks that exhibit relative amounts of ductility along a copper/sapphire bicrystal interface are simulated within the context of continuum mechanics. The specimen in question exhibits a directional dependence of fracture; that is a crack oriented in one direction along the copper/sapphire interface propagates much more during a given load increment than does the crack oriented to propagate in the opposite direction along the interface. This phenomenon had previously been explained on the basis of an energetic competition between dislocation nucleation and cleavage failure at the two crack tips using both the Rice and Thomson (Philos. Mag. 29 (1974) 73) model as well as the more recent type of dislocation nucleation analysis by Rice (J. Mech. Phys. Solids 40 (1992) 239) based on a Peierls-like stress vs. displacement relationship on the slip plane. However, recent experiments by Kysar (Acta Mater. 48 (2000) 3509) have shown that the orientation of the directional dependence of fracture in the copper/sapphire bicrystal is opposite to that predicted on the basis of dislocation nucleation arguments. The goal of the present work is to attempt to explain the directional dependence of fracture solely on the basis of continuum mechanics. In Part I of this pair of papers we review the main results of the experiments and then set the stage for a series of finite element analyses of the bicrystal specimen by reviewing the fundamentals of single crystal plasticity and the general features of crack tip fields in single crystals. We then discuss two different constitutive hardening models for single crystals and predict that, depending on the hardening model, regions of single-slip around the crack tip may degenerate into regions of triple slip. This leads to a discussion of how the near-tip displacement field can change dramatically with constitutive models. Next the constitutive properties used in the simulations are fit to experimental data. Finally we describe the finite element meshes and procedures for simulating the stationary and quasistatically growing cracks. The simulation results are reported in Part II.

Discrete dislocation and continuum descriptions of plastic flow

Abstract: Conventional continuum mechanics models of inelastic deformation processes are size scale independent. In contrast, there is considerable experimental evidence that plastic flow in crystalline materials is size-dependent over length scales of the order of tens of microns and smaller. Geometrically necessary dislocations play a key role in this regard. At present there is no generally accepted framework for analyzing the size-dependent response of a plastically deforming crystal. Dislocation-based plasticity can provide information on the form of the governing equations and the boundary conditions, as well as material properties. Two model problems that highlight the limitations of conventional continuum plasticity are considered. For each problem, solutions using discrete dislocation plasticity and a recently proposed nonlocal continuum formulation are compared. The capabilities and limitations of the currently available nonlocal continuum theories are discussed.

Design sensitivity analysis and optimization of nonlinear transient dynamics

Abstract: A shape design sensitivity analysis (DSA) and optimization of structural transient dynamics are proposed for the finite deformation elastoplastic materials under impact with a rigid surface A shape variation of the structure is considered using the material derivative approach in continuum mechanics Hyperelasticitybased multiplicatively decomposed elastoplasticity is used for the constitutive model The implicit Newmark time integration scheme is used for the structural dynamics The design sensitivity equation is solved at each converged time step with the same tangent stiffness matrix as response analysis without iteration The cost of sensitivity computation is more efficient than the cost of response analysis for the implicit time integration method The efficiency and the accuracy of the proposed method are shown through the design optimization of a vehicle bumper