Showing papers on "Continuum mechanics published in 2004"

Point based animation of elastic, plastic and melting objects

Abstract: We present a method for modeling and animating a wide spectrum of volumetric objects, with material properties anywhere in the range from stiff elastic to highly plastic. Both the volume and the surface representation are point based, which allows arbitrarily large deviations form the original shape. In contrast to previous point based elasticity in computer graphics, our physical model is derived from continuum mechanics, which allows the specification of common material properties such as Young's Modulus and Poisson's Ratio.In each step, we compute the spatial derivatives of the discrete displacement field using a Moving Least Squares (MLS) procedure. From these derivatives we obtain strains, stresses and elastic forces at each simulated point. We demonstrate how to solve the equations of motion based on these forces, with both explicit and implicit integration schemes. In addition, we propose techniques for modeling and animating a point-sampled surface that dynamically adapts to deformations of the underlying volumetric model.

Peridynamic modeling of membranes and fibers.

Abstract: The peridynamic theory of continuum mechanics allows damage, fracture, and long-range forces to be treated as natural components of the deformation of a material. In this paper, the peridynamic approach is applied to small thickness two- and one-dimensional structures. For membranes, a constitutive model is described appropriate for rubbery sheets that can form cracks. This model is used to perform numerical simulations of the stretching and dynamic tearing of membranes. A similar approach is applied to one-dimensional string like structures that undergrow stretching, bending, and failure. Long-range forces similar to van der Waals interactions at the nanoscale influence the equilibrium configurations of these structures, how they deform, and possibly self-assembly.

A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics

Abstract: Growth (and resorption) of biological tissue is formulated in the continuum setting. The treatment is macroscopic, rather than cellular or sub-cellular. Certain assumptions that are central to classical continuum mechanics are revisited, the theory is reformulated, and consequences for balance laws and constitutive relations are deduced. The treatment incorporates multiple species. Sources and fluxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (i) a fluid phase, and (ii) the precursors and byproducts of the reactions that create and break down tissue. A notable feature is that the full extent of coupling between mass transport and mechanics emerges from the thermodynamics. Contributions to fluxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a unified fashion from previous formulations of the problem. The present work demonstrates these effects via a physically consistent treatment. The presence of multiple, interacting species requires that the formulation be consistent with mixture theory. This requirement has far-reaching consequences. A preliminary numerical example is included to demonstrate some aspects of the coupled formulation.

The physics of granular media

Abstract: PrefaceList of ContributorsI Static Properties1 Stress in Dense Granular Materials (I Goldhirsch and C Goldenberg)11 Introduction12 Continuum Mechanics: A Brief Review13 Constitutive Relations for Dense Granular Materials14 A Microscopic Approach15 Forces, Stress and Response Functions16 Concluding RemarksReferences2 Response Functions in Isostatic Packings (C F Moukarzel)21 Introduction22 Rigidity Considerations for Contact Networks23 Consequences of Isostaticity24 Specific Examples25 DiscussionReferences3 Statistical Mechanics of Jammed Matter (H A Makse, J Bruji-c, and S F Edwards)31 Introduction to the Concept of Jamming32 New Statistical Mechanics for Granular Matter33 Jamming with the Confocal34 Jamming in a Periodic BoxReferencesII Granular Gas4 The Inelastic Maxwell Model (E Ben-Naim and P Krapivsky)41 Introduction42 Uniform Gases: One Dimension43 Uniform Gases: Arbitrary Dimension44 Impurities45 Mixtures46 Lattice Gases47 ConclusionsReferences5 Cluster Formation in Compartmentalized Granular Gases (K van der Weele, R Mikkelsen, D van der Meer, and D Lohse)51 Introduction52 The Vertically Vibrated Experiment53 Eggers' Flux Model54 Extension to More than two Compartments55 Urn Model56 Horizontally Vibrated System57 Double Well Model58 Further DirectionsReferencesIII Dense Granular Flow6 Continuum Modeling of Granular Flow and Structure Formation (I S Aranson and L S Tsimring)61 Introduction62 Order Parameter Description of Partially Fluidized Granular Flows63 Avalanchesonan Inclined Plane64 Fitting the Theory with Molecular Dynamics Simulations65 Surface-driven Shear Granular Flow Under Gravity66 Stick-Slips and Granular Friction67 ConclusionsReferences7 Contact Dynamics Study of 2D Granular Media: Critical States and Relevant Internal Variables (F Radjai and S Roux)71 A Geometry-Mechanics Dialogue72 Agranular Model73 Macroscopic Continuum Description74 Numerical Results75 ConclusionReferences8 Collision of Adhesive Viscoelastic Particles (N V Brilliantov and T Poschel)81 Introduction82 Forces Between Granular Particles83 Collision of Granular Particles84 ConclusionReferencesIV Hydrodynamic Interactions9 Fluidized Beds: From Waves to Bubbles (E Guazzelli)91 Introduction92 Flow Regimes and Instabilities93 Instability Mechanism94 Governing Equations95 Primary Instability96 Rheology of the Particle Phase97 Secondary Instability and the Formation of Bubbles98 ConclusionsReferences10 Wind-blown Sand (H J Herrmann)101 Introduction102 The Wind Field103 Aeolian Sand Transport104 Dunes105 ConclusionReferencesV Charged and Magnetic Granular Matter11 Electrostatically Charged Granular Matter (S M Dammer, J Werth, and H Hinrichsen)111 Introduction112 Charged Granular Matter in Vacuum113 Charged Granular Matter in Suspension114 Agglomeration of Monopolar Charged Suspensions115 Coating Particles in Bipolarly Charged Suspensions116 SummaryReferences12 Magnetized Granular Materials (D L Blair and A Kudrolli)121 Introduction122 Background: Dipolar Hard Spheres123 Experimental Technique124 The Phase Diagram125 The Non-equipartition of Energy126 Cluster Growth Rates127 Compactness of the Cluster128 Migration of Clusters129 SummaryReferencesVI Computational Aspects13 Molecular Dynamics Simulations of Granular Materials (S Luding)131 Introduction132 The Soft-particle Molecular Dynamics Method133 Hard-sphere Molecular Dynamics134 The Link between ED and MD via the TC Model135 The Stress in Particle Simulations136 2D Simulation Results137 Large-scale Computational Examples138 ConclusionReferences14 Contact Dynamics for Beginners (L Brendel, T Unger, and D E Wolf)141 Introduction142 Discrete Dynamical Equations143 Volume Exclusion in a One-dimensional Example144 The Three-dimensional Single Contact Case Without Cohesion145 Iterative Determination of Constraint Forces in a Multi-contact System146 Computational Effort: Comparison Between CD and MD147 Rolling and Torsion Friction148 Attractive Contact Forces149 ConclusionReferencesIndexCD-ROMThe enclosed CD-ROM contains the figures of the articles, many of them colored, as well as related movies

Finite element methods for the non‐linear mechanics of crystalline sheets and nanotubes

Abstract: The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy-Born rule. The constitutive model for a two-dimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyper-elastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thin-shell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multi-million systems illustrate the computational savings which can be achieved.

Micro-macro methods for the multiscale simulation of viscoelastic flow using molecular models of kinetic theory

Abstract: We survey the field of micro-macro numerical techniques for predicting complex flows of viscoelastic fluids. The micro-macro approach couples the mesoscopic scale of kinetic theory to the macroscopic scale of continuum mechanics. A numerical solution is sought to the coupled non-linear problem involving the conservation laws and a microstructural model of kinetic theory. Although micro-macro techniques are much more demanding in terms of computer resources than conventional continuum computations, they allow the direct use of kinetic theory models in flow simulations, thus avoiding potentially inaccurate closure approximations. The focus of our survey is mainly put on mathematical formulations and numerical approaches. Applications to polymer solutions and melts, liquid crystalline polymers, and fibre suspensions, are briefly reviewed.

The role of stress in the growth of a multicell spheroid

Abstract: Rather recent experimental results demonstrate the non-negligible role of mechanical stress in the growth of a multicell spheroid. In this paper we discuss a theoretical framework for volumetric growth suitable for modeling the growth of soft tissues exhibiting the properties of a solid. After a proper kinematic decomposition, balance equations for mass, momentum and energy are discussed together with constitutive relationships. The mathematical model is then applied to avascular tumor growth. We show by numerical simulation that, under assumption of spherical symmetry, the mathematical model is able to reproduce the experimental data with a satisfying qualitative agreement.

A viscoelastic damage model with applications to stable and unstable fracturing

Abstract: SUMMARY A viscoelastic damage rheology model is presented that provides a generalization of Maxwell viscoelasticity to a non-linear continuum mechanics framework incorporating material degradation and recovery, transition from stable to unstable fracturing and gradual accumulation of non-reversible deformation. The model is a further development of the damage rheology framework of Lyakhovsky et al. for evolving effective elasticity. The framework provides a quantitative treatment for macroscopic effects of evolving distributed cracking with local density represented by an intensive state variable. The formulation, based on thermodynamic principles, leads to a system of kinetic equations for the evolution of damage. An effective viscosity inversely proportional to the rate of damage increase is introduced to account for gradual accumulation of irreversible deformation due to dissipative processes. A power-law relation between the damage variable and elastic moduli leads to a non-linear coupling between the rate of damage evolution and the damage variable itself. This allows the model to reproduce a transition from stable to unstable fracturing of brittle rocks and the Kaiser effect. 3-D numerical simulations based on the model formulation for homogeneous and heterogeneous materials account for the main features of rock behaviour under large strain. The model coefficients are constrained, using triaxial laboratory experiments with low-porosity Westerly granite and high-porosity Berea sandstone samples.

Power-law slip profile of the moving contact line in two-phase immiscible flows.

Abstract: Large-scale molecular dynamics (MD) simulations on two-phase immiscible flows show that, associated with the moving contact line, there is a very large 1=x partial-slip region where x denotes the distance from the contact line. This power-law partial-slip region is verified in large-scale adaptive continuum calculations based on a local, continuum hydrodynamic formulation, which has proved successful in reproducing MD results at the nanoscale. Both MD simulations and numerical solutions of continuum equations indicate the existence of a universal slip profile in the Stokes-flow regime.

Fundamentals of cohesive powder consolidation and flow

Abstract: The fundamentals of cohesive powder consolidation and flow behaviour are explained to combine reasonably particle and continuum mechanics. The influence of elastic-plastic repulsion and, consequently, stressing pre-history dependent adhesion is demonstrated by the new model “stiff particles with soft contacts” and the contact force equilibrium. With this as the physical basis, incipient powder consolidation, yield and cohesive steady-state flow are explained. These models are used to evaluate shear cell test results as constitutive functions for computer aided apparatus design for reliable flow.

Continuum Methods of Physical Modeling: Continuum Mechanics, Dimensional Analysis, Turbulence

Abstract: Introduction.- Continuum Mechanics: Basic Kinematics.- Balance Equations.- Jump Conditions.- Moving Reference Systems.- Material Equations.- Phase Transition in Viscous Heat Conducting Compressible Fluids.- Theory of Mixtures.- Dimensional Analysis: Theoretical Foundation of Dimensional Analysis.- Similitude and Model Experiments.- Turbulence: Fundamental Concepts of Turbulence.- k-e Model for Density Preserving and Boussinesq Fluids.- Algebraic Reynolds Stress Models.- Application of k-e Model.- References.- Indices.

Microstructure design of a two phase composite using two-point correlation functions

Abstract: Two-point distribution functions are used here as to introduce “Microstructure Sensitive Design” in two-phase composites. Statistical distribution functions are commonly used for the representation of microstructures and also for homogenization of materials properties. The use of two-point statistics allows the composite designer to include the morphology and distribution in addition to the properties of the individual phases and components. Statistical continuum mechanics is used to make a direct link between the microstructure and properties (elastic and plastic) in terms of these two-point statistical functions. An empirical form of the two-point statistical function is used which allows the construction of a composite hull. Two different composites (isotropic and anisotropic) are considered and the effect of anisotropy for the prediction of the elastic properties is discussed

Peridynamic Modeling of Impact Damage

Abstract: The peridynamic theory is an alternative formulation of continuum mechanics oriented toward modeling discontinuites such as cracks. It differs from the classical theory and most nonlocal theories in that it does not involve spatial derivatives of the displacement field. Instead, it is formulated in terms of integral equations, whose validity is not affected by the presence of discontinuities such as cracks. It may be thought of as a “continuum version of molecular dynamics” in that particles interact directly with each other across a finite distance. This paper outlines the basis of the peridynamic theory and its numerical implementation in a three-dimensional code called EMU. Examples include simulations of a Charpy V-notch test, accumulated damage in concrete due to multiple impacts, and crack fragmentation of a glass plate.Copyright © 2004 by ASME

Continuum limit of amorphous elastic bodies II: Linear response to a point source force

Abstract: The linear response of two-dimensional amorphous elastic bodies to an external delta force is determined in analogy with recent experiments on granular aggregates. For the generated forces, stress, and displacement fields, we find strong relative fluctuations of order 1 close to the source, which, however, average out readily to the classical predictions of isotropic continuum elasticity. The stress fluctuations decay (essentially) exponentially with distance from the source. Only beyond a surprisingly large distance, $b\ensuremath{\approx}30$ interatomic distances, self-averaging dominates, and the quenched disorder becomes irrelevant for the response of an individual configuration. We argue that this self-averaging length $b$ also sets the lower wavelength bound for the applicability of classical eigenfrequency calculations. Particular attention is paid to the displacements of the source, allowing a direct measurement of the local rigidity. The algebraic correlations of these displacements demonstrate the existence of domains of slightly different rigidity without, however, revealing a characteristic length scale, at least not for the system sizes we are able to probe.

Onset of failure in finitely strained layered composites subjected to combined normal and shear loading

Abstract: A limiting factor in the design of fiber-reinforced composites is their failure under axial compression along the fiber direction. These critical axial stresses are significantly reduced in the presence of shear stresses. This investigation is motivated by the desire to study the onset of failure in fiber-reinforced composites under arbitrary multi-axial loading and in the absence of the experimentally inevitable imperfections and finite boundaries. By using a finite strain continuum mechanics formulation for the bifurcation (buckling) problem of a rate-independent, perfectly periodic (layered) solid of infinite extent, we are able to study the influence of load orientation, material properties and fiber volume fraction on the onset of instability in fiber-reinforced composites. Two applications of the general theory are presented in detail, one for a finitely strained elastic rubber composite and another for a graphite–epoxy composite, whose constitutive properties have been determined experimentally. For the latter case, extensive comparisons are made between the predictions of our general theory and the available experimental results as well as to the existing approximate structural theories. It is found that the load orientation, material properties and fiber volume fraction have substantial effects on the onset of failure stresses as well as on the type of the corresponding mode (local or global).

Nonlocal Theories in Continuum Mechanics

Abstract: The purpose of this paper is to explain why the standard continuum theory fails to properly describe certain mechanical phenomena and how the description can be improved by enrichments that incorporate the influence of gradients or weighted spatial averages of strain or of an internal variable. Three typical mechanical problems that require such enrichments are presented: (i) dispersion of short elastic waves in heterogeneous or discrete media, (ii) size effects in microscale elastoplasticity, in particular with the size dependence of the apparent hardening modulus, and (iii) localization of strain and damage in quasibrittle structures and with the resulting transitional size effect. Problems covered in the examples encompass static and dynamic phenomena, linear and nonlinear behavior, and three constitutive frameworks, namely elasticity, plasticity and continuum damage mechanics. This shows that enrichments of the standard continuum theory can be useful in a wide range of mechanical problems.

Product Design of Cohesive Powders – Mechanical Properties, Compression and Flow Behavior

Abstract: The fundamentals of cohesive powder consolidation and flow behavior are explained using a reasonable combination of particle and continuum mechanics. By the model stiff particles with soft contacts, universal models are presented which include the elastic-plastic and viscoplastic particle contact behavior with adhesion, load-unload hysteresis and thus energy dissipation, a history-dependent and a nonlinear adhesion force function. With this as the physical basis, incipient powder consolidation, yield and cohesive steady-state flow, consolidation and compression functions, compression and preshear works are explained. As an example, the flow properties of an ultrafine limestone powder are shown. These constitutive models are used to evaluate shear cell test results for apparatus design to ensure reliable powder flow. Finally, conclusions are drawn concerning particle stressing, powder handling behavior and product quality assessment in processing industries.

An experimental investigation into the dynamics of a string

Abstract: We describe a detailed experimental investigation into the dynamics of a sinusoidally forced string. We find qualitative agreement with the predictions of the averaged equations of motion for a string in the high damping regime. At low damping we observe more complex phenomena not present in the averaged equations.