Showing papers on "Continuum mechanics published in 2006"
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01 Jan 2006
TL;DR: In this paper, the authors present an overview of the history of the field of polymers in terms of elementary steps and shaping methods, and present and future perspectives of this field.
Abstract: 1 History, Structural Formulation of the Field Through Elementary Steps, and Future Perspectives 11 Historical Notes 12 Current Polymer Processing Practice 13 Analysis of Polymer Processing in Terms of Elementary Steps and Shaping Methods 14 Future Perspectives: From Polymer Processing to Macromolecular Engineering 2 The Balance Equations and Newtonian Fluid Dynamics 21 Introduction 22 The Balance Equations 23 Reynolds Transport Theorem 24 The Macroscopic Mass Balance and the Equation of Continuity 25 The Macroscopic Linear Momentum Balance and the Equation of Motion 26 The Stress Tensor 27 The Rate of Strain Tensor 28 Newtonian Fluids 29 The Macroscopic Energy Balance and the Bernoulli and Thermal Energy Equations 210 Mass Transport in Binary Mixtures and the Diffusion Equation 211 Mathematical Modeling, Common Boundary Conditions, Common Simplifying Assumptions, and the Lubrication Approximation 3 Polymer Rheology and Non-Newtonian Fluid Mechanics 31 Rheological Behavior, Rheometry, and Rheological Material Functions of Polymer Melts 32 Experimental Determination of the Viscosity and Normal Stress Difference Coefficients 33 Polymer Melt Constitutive Equations Based on Continuum Mechanics 34 Polymer Melt Constitutive Equations Based on Molecular Theories 4 The Handling and Transporting of Polymer Particulate Solids 41 Some Unique Properties of Particulate Solids 42 Agglomeration 43 Pressure Distribution in Bins and Hoppers 44 Flow and Flow Instabilities in Hoppers 45 Compaction 46 Flow in Closed Conduits 47 Mechanical Displacement Flow 48 Steady Mechanical Displacement Flow Aided by Drag 49 Steady Drag-induced Flow in Straight Channels 410 The Discrete Element Method 5 Melting 51 Classification and Discussion of Melting Mechanisms 52 Geometry, Boundary Conditions, and Physical Properties in Melting 53 Conduction Melting without Melt Removal 54 Moving Heat Sources 55 Sintering 56 Conduction Melting with Forced Melt Removal 57 Drag-induced Melt Removal 58 Pressure-induced Melt Removal 59 Deformation Melting 6 Pressurization and Pumping 61 Classification of Pressurization Methods 62 Synthesis of Pumping Machines from Basic Principles 63 The Single Screw Extruder Pump 64 Knife and Roll Coating, Calenders, and Roll Mills 65 The Normal Stress Pump 66 The Co-rotating Disk Pump 67 Positive Displacement Pumps 68 Twin Screw Extruder Pumps 7 Mixing 71 Basic Concepts and Mixing Mechanisms 72 Mixing Equipment and Operations of Multicomponent and Multiphase Systems 73 Distribution Functions 74 Characterization of Mixtures 75 Computational Analysis 8 Devolatilization 81 Introduction 82 Devolatilization Equipment 83 Devolatilization Mechanisms 84 Thermodynamic Considerations of Devolatilization 85 Diffusivity of Low Molecular Weight Components in Molten Polymers 86 Boiling Phenomena: Nucleation 87 Boiling-Foaming Mechanisms of Polymeric Melts 88 Ultrasound-enhanced Devolatilization 89 Bubble Growth 810 Bubble Dynamics and Mass Transfer in Shear Flow 811 Scanning Electron Microscopy Studies of Polymer Melt Devolatilization 9 Single Rotor Machines 91 Modeling of Processing Machines Using Elementary Steps 92 The Single Screw Melt Extrusion Process 93 The Single Screw Plasticating Extrusion Process 94 The Co-rotating Disk Plasticating Processor 10 Twin Screw and Twin Rotor Processing Equipment 101 Types of Twin Screw and Twin Rotor-based Machines 102 Counterrotating Twin Screw and Twin Rotor Machines 103 Co-rotating, Fully Intermeshing Twin Screw Extruders 11 Reactive Polymer Processing and Compounding 111 Classes of Polymer Chain Modification Reactions, Carried out in Reactive Polymer Processing Equipment 112 Reactor Classification 113 Mixing Considerations in Multicomponent Miscible Reactive Polymer Processing Systems 114 Reactive Processing of Multicomponent Immiscible and Compatibilized Immiscible Polymer Systems 115 Polymer Compounding 12 Die Forming 121 Capillary Flow 122 Elastic Effects in Capillary Flows 123 Sheet Forming and Film Casting 124 Tube, Blown Film, and Parison Forming 125 Wire Coating 126 Profile Extrusion 13 Molding 131 Injection Molding 132 Reactive Injection Molding 133 Compression Molding 14 Stretch Shaping 141 Fiber Spinning 142 Film Blowing 143 Blow Molding 15 Calendering 151 The Calendering Process 152 Mathematical Modeling of Calendering 153 Analysis of Calendering Using FEM Appendix A: Rheological and Thermophysical Properties of Polymers Appendix B: Conversion Tables to the International System of Units (SI) Appendix C: Notation Author Index Subject Index
1,163 citations
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TL;DR: In this paper, a continuum mechanics approach to model the elastic deformation of finite graphene sheets based on Brenner's potential is presented. But the authors do not consider the nonlinearity of the deformation.
Abstract: This paper presents a continuum mechanics approach to modelling the elastic deformation of finite graphene sheets based on Brenner's potential. The potential energy of the graphene sheet is minimized for determining the equilibrium configuration. The four edges of the initially rectangular graphene sheet become curved at the equilibrium configuration. The curving of the sides is attributed to smaller coordination number for the atoms at the edges compared to that of the interior atoms. Considering two graphene models, with only two or all four edges constrained to be straight, the continuum Young's moduli of graphene are computed applying the Cauchy–Born rule. The computed elastic constants of the graphene sheet are found to conform to orthotropic material behaviour. The computed constants differ considerably depending on whether a minimized or unminimized configuration is used for computation.
340 citations
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TL;DR: In this article, a nonlocal continuum mechanics model is developed and applied to study the vibration of both single-walled nanotubes (SWNTs) and double-weled nanotsubes (DWNTs), via elastic beam theories.
Abstract: A nonlocal continuum mechanics model is developed and applied to study the vibration of both single-walled nanotubes (SWNTs) and double-walled nanotubes (DWNTs) via elastic beam theories. The small-scale effects on vibration characteristics of carbon nanotubes are explicitly derived through a complete mechanics analysis. A qualitative validation study shows that the results based on nonlocal continuum mechanics are in agreement with the published experimental reports in this field. Numerical simulations are conducted to quantitatively show the small-scale effect on vibrations of both SWNTs and DWNTs with different lengths and diameters.
318 citations
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TL;DR: In this paper, a lower-order cable element is introduced for thin structures where bending stiffness can be important in some applications, and the performance of this cable element was evaluated by comparing it with existing formulations using several examples.
Abstract: The purpose of this paper is to present formulations for beam elements based on the absolute nodal co-ordinate formulation that can be effectively and efficiently used in the case of thin structural applications. The numerically stiff behaviour resulting from shear terms in existing absolute nodal co-ordinate formulation beam elements that employ the continuum mechanics approach to formulate the elastic forces and the resulting locking phenomenon make these elements less attractive for slender stiff structures. In this investigation, additional shape functions are introduced for an existing spatial absolute nodal co-ordinate formulation beam element in order to obtain higher accuracy when the continuum mechanics approach is used to formulate the elastic forces. For thin structures where bending stiffness can be important in some applications, a lower order cable element is introduced and the performance of this cable element is evaluated by comparing it with existing formulations using several examples. Cables that experience low tension or catenary systems where bending stiffness has an effect on the wave propagation are examples in which the low order cable element can be used. The cable element, which does not have torsional stiffness, can be effectively used in many problems such as in the formulation of the sliding joints in applications such as the spatial pantograph/catenary systems. The numerical study presented in this paper shows that the use of existing implicit time integration methods enables the simulation of multibody systems with a moderate number of thin and stiff finite elements in reasonable CPU time.
298 citations
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TL;DR: The results show that classical continuum mechanics and simple mechanistic models fail to describe the complex mechanics of the GFP protein structure and offer insights into the mechanical design of protein materials.
Abstract: Single-molecule methods have given experimental access to the mechanical properties of single protein molecules. So far, access has been limited to mostly one spatial direction of force application. Here, we report single-molecule experiments that explore the mechanical properties of a folded protein structure in precisely controlled directions by applying force to selected amino acid pairs. We investigated the deformation response of GFP in five selected directions. We found fracture forces widely varying from 100 pN up to 600 pN. We show that straining the GFP structure in one of the five directions induces partial fracture of the protein into a half-folded intermediate structure. From potential widths we estimated directional spring constants of the GFP structure and found values ranging from 1 N/m up to 17 N/m. Our results show that classical continuum mechanics and simple mechanistic models fail to describe the complex mechanics of the GFP protein structure and offer insights into the mechanical design of protein materials.
291 citations
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TL;DR: In this paper, a non-local elastic beam and shell model was developed and applied to investigate the small scale effect on buckling analysis of carbon nanotubes (CNTs) under compression.
192 citations
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TL;DR: In this paper, a new energy functional for multi-phase hyperelastic media with interface energy effect is proposed, which takes into account the interface energy and the interface stress-induced residual elastic field, which reflects the intrinsic physical properties of the material.
Abstract: In addition to the classical governing equations in continuum mechanics, two kinds of governing equations are necessary in the solution of boundary-value problems for the stress fields in multi-phase hyperelastic media with the surface/interface energy effect. The first is the interface constitutive relation, and the second are the discontinuity conditions of the traction across the interface, namely, the Young-Laplace equations. In this paper, the interface consitutive relations are presented in terms of the interface energy in both Lagrangian and Eulerian descriptions within the framework of finite deformation, and the expressions of the interface stress for an isotropic interface are given as a special case. Then, by introducing a fictitious stress-free configuration, a new energy functional for multi-phase hyperelastic media with interface energy effect is proposed. The functional takes into account the interface energy and the interface stress-induced ``residual'' elastic field, which reflects the intrinsic physical properties of the material. All field equations, including the generalized Young-Laplace equation, can be derived from the stationary condition of this functional. The present theory is illustrated by simple examples. The results in this paper provide a theoretical framework for studying the elastostatic problems of multi-phase hyperelastic bodies that involve surface/interface energy effects at finite deformation.
177 citations
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01 Jan 2006TL;DR: In this paper, the impact-induced transition in two-phase elastic materials is modeled as a two-well potential, and the effect of the transition layers on the potential is investigated.
Abstract: Part I. Introduction: 1. What this monograph is about 2. Some experiments 3. Continuum mechanics 4. Quasilinear systems 5. Outline of monograph Part II. Two-Well Potentials, Governing Equations and Energetics: 1. Introduction 2. Two-phase nonlinearly elastic materials 3. Field equations and jump conditions 4. Energetics of motion, driving force and dissipation inequality Part III. Equilibrium Phase Mixtures and Quasistatic Processes: 1. Introduction 2. Equilibrium states 3. Variational theory of equilibrium mixtures of phases 4. Quasistatic processes 5. Nucleation and kinetics 6. Constant elongation rate processes 7. Hysteresis Part IV. Impact-Induced Transitions in Two-Phase Elastic Materials: 1. Introduction 2. The impact problem for trilinear two-phase materials 3. Scale-invariant solutions of the impact problem 4. Nucleation and kinetics 5. Comparison with experiment 6. Other types of kinetic relations 7. Related work Part V. Multiple-Well Free Energy Potentials: 1. Introduction 2. Helmholtz free energy potential 3. Potential energy function and the effect of stress 4. Example 1: the van der Waals fluid 5. Example 2: two-phase martensitic material with cubic and tetragonal phases Part VI. The Continuum Theory of Driving Force: 1. Introduction 2. Balance laws, field equations and jump conditions 3. The second law of thermodynamics and the driving force Part VII. Thermoelastic Materials: 1. Introduction 2. The thermoelastic constitutive law 3. Stability of a thermoelastic material 4. A one-dimensional special case: uniaxial strain Part VIII. Kinetics and Nucleation: 1. Introduction 2. Nonequilibrium processes, thermodynamic fluxes and forces, kinetic relation 3. Phenomenological examples of kinetic relations 4. Micromechanically-based examples of kinetic relations 5. Nucleation Part IX. Models for Two-Phase Thermoelastic Materials in One Dimension: 1. Preliminaries 2. Materials of Mie-Gruneisen type 3. Two-phase Mie-Gruneisen materials Part X. Quasistatic Hysteresis in Two-Phase Thermoelastic Tensile Bars: 1. Preliminaries 2. Thermomechanical equilibrium states for a two-phase material 3. Quasistatic processes 4. Trilinear thermoelastic material 5. Stress cycles at constant temperature 6. Temperature cycles at constant stress 7. The shape-memory cycle 8. The experiments of Shaw and Kyriakides 9. Slow thermomechanical processes Part XI. Dynamics of Phase Transitions in Uniaxially Strained Thermoelastic Solids: 1. Introduction 2. Uniaxial strain in adiabatic thermoelasticity 3. The impact problem Part XII. Statics: Geometric Compatibility: 1. Preliminaries 2. Examples Part XIII. Dynamics: Impact-Induced Transition in a CuA1Nl Single Crystal: 1. Introduction 2. Preliminaries 3. Impact without phase transformation 4. Impact with phase transformation 5. Application to austenite-B1 martensite transformation in CuA1Nl Part XIV. Quasistatics: Kinetics of Martensitic Twinning: 1. Introduction 2. The material and loading device 3. Observations 4. The model 5. The energy of the system 6. The effect of the transition layers: further observations 7. The effect of the transition layers: further modeling 8. Kinetics.
169 citations
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TL;DR: In this article, the authors introduce fractional calculus into the continuum mechanics area describing non-local constitutive relations, and propose an elastic model with nonlocal stress-strain behavior.
163 citations
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TL;DR: Atomistic simulations are used to test the equations of continuum contact mechanics in nanometer scale contacts and the atomic scale roughness present on any tip made of discrete atoms is shown to have profound effects on the results.
Abstract: Atomistic simulations are used to test the equations of continuum contact mechanics in nanometer scale contacts. Nominally spherical tips, made by bending crystals or cutting crystalline or amorphous solids, are pressed into a flat, elastic substrate. The normal displacement, contact radius, stress distribution, friction, and lateral stiffness are examined as a function of load and adhesion. The atomic scale roughness present on any tip made of discrete atoms is shown to have profound effects on the results. Contact areas, local stresses, and the work of adhesion change by factors of 2 to 4, and the friction and lateral stiffness vary by orders of magnitude. The microscopic factors responsible for these changes are discussed. The results are also used to test methods for analyzing experimental data with continuum theory to determine information, such as contact area, that cannot be measured directly in nanometer scale contacts. Even when the data appear to be fit by continuum theory, extracted quantities can differ substantially from their true values.
147 citations
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TL;DR: Based on fiber reinforced continuum mechanics theory, an anisotropic hyperelastic constitutive model for the human annulus fibrosus is developed in this paper, where a strain energy function representing the elastic material behavior of the annulus fiber is additively decomposed into three parts nominally representing the energy contributions from the matrix, fiber and fiber-matrix shear interaction, respectively.
Abstract: Based on fiber reinforced continuum mechanics theory, an anisotropic hyperelastic constitutive model for the human annulus fibrosus is developed. A strain energy function representing the anisotropic elastic material behavior of the annulus fibrosus is additively decomposed into three parts nominally representing the energy contributions from the matrix, fiber and fiber-matrix shear interaction, respectively. Taking advantage of the laminated structure of the annulus fibrosus with one family of aligned fibers in each lamella, interlamellar fiber-fiber interaction is eliminated, which greatly simplifies the constitutive model. A simple geometric description for the shearing between the fiber and the matrix is developed and this quantity is used in the representation of the fiber-matrix shear interaction energy. Intralamellar fiber-fiber interaction is also encompassed by this interaction term. Experimental data from the literature are used to obtain the material parameters in the constitutive model and to provide model validation. Determination of the material parameters is greatly facilitated by the partition of the strain energy function into matrix, fiber and fiber-matrix shear interaction terms. A straightforward procedure for computation of the material parameters from simple experimental tests is proposed.
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TL;DR: In this paper, the authors reviewed the formulations used over the last 40 years for the solution of problems involving tensile cracking, with both the discrete and the smeared crack approaches, identifying as its main drawbacks the observed mesh-size and mesh-bias spurious dependence when applied "straightly".
Abstract: This paper briefly reviews the formulations used over the last 40 years for the solution of problems involving tensile cracking, with both the discrete and the smeared crack approaches. The paper focuses on the smeared approach, identifying as its main drawbacks the observed mesh-size and mesh-bias spurious dependence when the method is applied ‘straightly’. A simple isotropic local damage constitutive model is considered, and the (exponential) softening modulus is regularized according to the material fracture energy and the element size. The continuum and discrete mechanical problems corresponding to both the weak discontinuity (smeared cracks) and the strong discontinuity (discrete cracks) approaches are analysed and the question of propagation of the strain localization band (crack) is identified as the main difficulty to be overcome in the numerical procedure. A tracking technique is used to ensure stability of the solution, attaining the necessary convergence properties of the corresponding discrete finite element formulation. Numerical examples show that the formulation derived is stable and remarkably robust. As a consequence, the results obtained do not suffer from spurious mesh-size or mesh-bias dependence, comparing very favourably with those obtained with other fracture and continuum mechanics approaches. Copyright © 2006 John Wiley & Sons, Ltd.
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TL;DR: In this paper, the peridynamic formulation of continuum mechanics is applied to the motion of phase boundaries in one dimension and the authors show that phase boundaries remain essentially planar with little bowing.
Abstract: We study the kinetics of phase transformations in solids using the peridynamic formulation of continuum mechanics. The peridynamic theory is a nonlocal formulation that does not involve spatial derivatives, and is a powerful tool to study defects such as cracks and interfaces.
We apply the peridynamic formulation to the motion of phase boundaries in one dimension. We show that unlike the classical continuum theory, the peridynamic formulation does not require any extraneous constitutive laws such as the kinetic relation (the relation between the velocity of the interface and the thermodynamic driving force acting across it) or the nucleation criterion (the criterion that determines whether a new phase arises from a single phase). Instead this information is obtained from inside the theory simply by specifying the inter-particle interaction. We derive a nucleation criterion by examining nucleation as a dynamic instability. We find the induced kinetic relation by analyzing the solutions of impact and release problems, and also directly by viewing phase boundaries as traveling waves.
We also study the interaction of a phase boundary with an elastic non-transforming inclusion in two dimensions. We find that phase boundaries remain essentially planar with little bowing. Further, we find a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase boundaries at the edges of the inclusion while the original phase boundary slows down or stops. Transformation proceeds as the freshly nucleated phase boundaries propagate leaving behind some untransformed martensite around the inclusion.
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01 Jan 2006TL;DR: The peridynamic model of solid mechanics has been developed for applications involving discontinuities as mentioned in this paper, which treats crack and fracture as just another type of deformation, rather than as a pathology that requires special mathematical treatment.
Abstract: *† ‡ Nearly all finite element codes and similar methods for the analysis of deformation in structures attempt to solve the partial differential equations of the classical theory of continuum mechanics. Yet these equations, because they require the partial derivatives of displacement to be known throughout the region modeled, are in some ways unsuitable for the modeling of cracks and other discontinuities, in which these derivatives fail to exist. As a means of avoiding this limitation, the peridynamic model of solid mechanics has been developed for applications involving discontinuities. The objective of this method is to treat crack and fracture as just another type of deformation, rather than as a pathology that requires special mathematical treatment. The peridynamic theory is based on integral equations, rather than differential equations, so there is no problem in applying the equations directly on a crack tip or crack surface. In the peridynamic model, displacements and internal forces are permitted to have discontinuities and other singularities. Particles interact with each other directly across finite distances through central forces known as “bonds”. Damage is introduced into the peridynamic model by permitting these bonds to break irreversibly. Breakage occurs when a bond is stretched in tension (or possibly compression) beyond some prescribed critical amount. After a bond breaks, it sustains no force. A distinguishing feature of this approach is its ability to treat the spontaneous formation of cracks together with their mutual interaction and dynamic growth in a consistent framework. A three-dimensional code called EMU implements the peridynamic model on parallel computers. The peridynamic method has been applied successfully to the analysis of material and structural failure in aerospace composites, particularly in graphiteepoxy laminates. For example, the method has been applied to the prediction of failure mode and crack direction in large-notch composite panels under tension loads with different layups and stacking sequences. The results have reproduced the experimentally observed dependence of crack growth direction on the relative percentage of fibers in different directions. The authors also have analyzed the damage occurring in a composite panel due to low velocity impact. The method predicts in detail the delamination and matrix damage process. Although the numerical method in EMU lends itself to parallelization, threedimensional analysis of large problems is computationally intensive. The applications reported here were run on the Columbia supercomputer at NASA Advanced Supercomputing (NAS) division. The Columbia supercomputer is proving to be invaluable in high-resolution modeling of the failure of composite materials.
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TL;DR: In this article, a multiscale molecular dynamics approach to contact mechanics was developed, which can be used also when the surfaces have roughness on many different length-scales, e.g., for self affine fractal surfaces.
Abstract: The friction and adhesion between elastic bodies are strongly influenced by the roughness of the surfaces in contact. Here we develop a multiscale molecular dynamics approach to contact mechanics, which can be used also when the surfaces have roughness on many different length-scales, e.g., for self affine fractal surfaces. As an illustration we consider the contact between randomly rough surfaces, and show that the contact area varies linearly with the load for small load. We also analyze the contact morphology and the pressure distribution at different magnification, both with and without adhesion. The calculations are compared with analytical contact mechanics models based on continuum mechanics.
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TL;DR: In this paper, a nonlocal gradient-enhanced theory coupled to visco-coasticity is presented to solve the problem of deformation and failure in ductile metal deformation.
Abstract: During dynamic loading processes, large inelastic deformation associated with high strain rates leads, for a broad class of ductile metals, to degradation and failure by strain localization. However, as soon as material failure dominates a deformation process, the material increasingly displays strain softening and the finite element computations are considerably affected by the mesh size and alignment. This gives rise to a non-physical description of the localized regions. This article presents a theoretical framework to solve this problem with the aid of nonlocal gradient-enhanced theory coupled to viscoinelasticity. Constitutive equations for anisotropic thermoviscodamage (rate-dependent damage) mechanism coupled with thermo-hypoelasto-viscoplastic deformation are developed in this work within the framework of thermodynamic laws, nonlinear continuum mechanics, and nonlocal continua. Explicit and implicit microstructural length-scale measures, which preserve the well-posedness of the differential equations, are introduced through the use of the viscosity and gradient localization limiters. The gradient- enhanced theory that incorporates macroscale interstate variables and their high- order gradients is developed here to describe the change in the internal structure and to investigate the size effect of statistical inhomogeneity of the evolution related plasticity and damage. The gradients are introduced in the hardening internal state variables and are considered dependent on their local counterparts. The derived microdamage constitutive model is destined to be applied in the context of high velocity impact and penetration damage mechanics. The theoretical framework presented in this article can be considered as a feasible thermodynamic approach that enables to derive various gradient (visco) plasticity/(visco) damage theories
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TL;DR: In this article, the authors extended the quasicontinuum approach for a multiscale analysis of silicon nanostructures at finite temperature using the classical continuum mechanics framework, but the constitutive response of the system is determined by employing an atomistic description.
Abstract: In this paper, we extend the quasicontinuum approach for a multiscale analysis of silicon nanostructures at finite temperature. The quasicontinuum method uses the classical continuum mechanics framework, but the constitutive response of the system is determined by employing an atomistic description. For finite-temperature solid systems under isothermal conditions, the constitutive response is determined by using the Helmholtz free energy density. The static part of the Helmholtz free energy density is obtained directly from the interatomic potential while the vibrational part is calculated by using the theory of quantum-mechanical lattice dynamics. Specifically, we investigate three quasiharmonic models, namely the real space quasiharmonic model, the local quasiharmonic model, and the reciprocal space quasiharmonic model, to compute the vibrational free energy. Using the finite-temperature quasicontinuum method, we compute the effect of the temperature and strain on the phonon density of states, phonon Gruneisen parameters, and the elastic properties of the Tersoff silicon. We also compute the mechanical response of silicon nanostructures for various external loads and the results are compared to molecular dynamics simulations.
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TL;DR: In this article, a methodology to model slip lines as strong displacement discontinuities within a continuum mechanics context is presented, where the loss of hyperbolicity of the IBVP is used as the criterion for switching from a classical continuum description of the constitutive behaviour to a traction-separation model acting at the discontinuity surface.
Abstract: A methodology to model slip lines as strong displacement discontinuities within a continuum mechanics context is presented. The loss of hyperbolicity of the IBVP is used as the criterion for switching from a classical continuum description of the constitutive behaviour to a traction-separation model acting at the discontinuity surface. A version of the element-free Galerkin (EFG) method is employed where the slip line is represented as a set of slipped particles. The representation of the slip line as set of cohesive segments promises to remove difficulties in the propagation of the slip line. Two-dimensional examples are studied using the Drucker–Prager material model. Copyright © 2006 John Wiley & Sons, Ltd.
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TL;DR: Ottinger's recent incorporation of fluctuations into the formulation of the friction matrix appearing in the phenomenological GENERIC theory of nonequilibrium irreversible processes is shown to furnish transport equations for single-component gases and liquids undergoing heat transfer which support the view that revisions to the Navier-Stokes-Fourier (N-S-F) momentum/energy equation set are necessary.
Abstract: Ottinger's recent nontraditional incorporation of fluctuations into the formulation of the friction matrix appearing in the phenomenological GENERIC theory of nonequilibrium irreversible processes is shown to furnish transport equations for single-component gases and liquids undergoing heat transfer which support the view that revisions to the Navier–Stokes–Fourier (N–S–F) momentum/energy equation set are necessary, as empirically proposed by the author on the basis of an experimentally supported theory of diffuse volume transport. The hypothesis that the conventional N–S–F equations prevail without modification only in the case of “incompressible” fluids, where the density ρ of the fluid is uniform throughout, serves to determine the new phenomenological parameter α ′ appearing in the GENERIC friction matrix. In the case of ideal gases the consequences of this constitutive hypothesis are shown to yield results identical to those derived theoretically by Ottinger on the basis of a “proper” coarse-graining of Boltzmann's kinetic equation. A major consequence of the present work is that the fluid's specific momentum density v is equal to its volume velocity v v , rather than to its mass velocity v m , contrary to current views dating back 250 years to Euler. In the case of rarefied gases the proposed modifications are also observed to agree with those resulting from Klimontovich's molecularly based, albeit ad hoc, self-diffusion addendum to Boltzmann's collision integral. Despite the differences in their respective physical models—molecular vs. phenomenological—the role played by Klimontovich's collisional addition to Boltzmann's equation in modifying the N–S–F equations is noted to constitute a molecular counterpart of Ottinger's phenomenological fluctuation addition to the GENERIC friction matrix. Together, these two theories collectively recognize the need to address multiple - rather than single - encounter collisions between a test molecule and its neighbors when formulating physically satisfactory statistical–mechanical theories of irreversible transport processes in gases. Overall, the results of the present work implicitly support the unorthodox view, implicit in the GENERIC scheme, that the translation of Newton's discrete mass-point molecular mechanics into continuum mechanics, the latter as embodied in the Cauchy linear momentum equation of fluid mechanics, cannot be correctly effected independently of the laws of thermodynamics. While Ottinger's modification of GENERIC necessitates fundamental changes in the foundations of fluid mechanics in regard to momentum transport, no basic changes are required in the foundations of linear irreversible thermodynamics (LIT) beyond recognizing the need to add volume to the usual list of extensive physical properties undergoing transport in single-species fluid continua, namely mass, momentum and energy. An alternative, nonGENERICally based approach to LIT, derived from our findings, is outlined at the conclusion of the paper. Finally, our proposed modifications of both Cauchy's linear momentum equation and Newton's rheological constitutive law for fluid-phase continua are noted to be mirrored by counterparts in the literature for solid-phase continua dating back to the classical interdiffusion experiments of Kirkendall and their subsequent interpretation by Darken in terms of diffuse volume transport.
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TL;DR: In this paper, the authors developed constitutive equations to model the mechanical behavior of crystallizable shape memory polymers and used them to simulate a typical uni-axial cycle of deformation, the results of this simulation compare very well with experimental data.
Abstract: Shape memory polymers are novel materials that can be easily formed into complex shapes, retaining memory of their original shape even after undergoing large deformations. The temporary shape is stable and return to the original shape is triggered by a suitable mechanism such as heating. In this paper, we develop constitutive equations to model the mechanical behavior of crystallizable shape memory polymers. Crystallizable shape memory polymers are called crystallizable because the temporary shape is fixed by a crystalline phase, while return to the original shape is due to the melting of this crystalline phase. The modeling is done using a framework that was developed recently for studying crystallization in polymers ([28], [25], [27], [31]) and is based on the theory of multiple natural configurations. In this paper we formulate constitutive equations for the original amorphous phase and the semi-crystalline phase that is formed after the onset of crystallization. In addition we model the melting of the crystalline phase to capture the return of the polymer to its original shape. The model has been used to simulate a typical uni-axial cycle of deformation, the results of this simulation compare very well with experimental data. In addition to this we also simulate circular shear of a hollow cylinder and present results for different cases in this geometry.
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TL;DR: In this article, the elastic properties of a chiral single-walled carbon nanotube were investigated and the effects of tube chirality and tube diameter on the elasticity of the tube were investigated.
Abstract: Molecular mechanics has been widely used to analytically study mechanical behaviour of carbon nanotubes. However, explicit expressions for elastic properties of carbon nanotubes are so far confined to some special cases due to the lack of fully constructed governing equations for the molecular mechanics model. In this paper, governing equations for an analytical molecular mechanics model are fully established. The explicit expressions for five in-plane elastic properties of a chiral single-walled carbon nanotube are derived, which make properties at different length-scales directly connected. The effects of tube chirality and tube diameter are investigated. In particular, the present results show that the classic relationship from the isotropic elastic theory of continuum mechanics between Young’s modulus and shear modulus of a single-walled carbon nanotube is not retained. The present analytical results are helpful to the understanding of elastic properties of carbon nanotubes, and also useful to the topic of linking molecular mechanics with continuum mechanics.
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TL;DR: In this paper, the applicability of the absolute nodal coordinate formulation for the modeling of belt-drive systems is studied, where the interaction between the belt and the pulleys is modeled using an elastic approach in which the contact is accounted for by the inclusion of a set of external forces that depend on the penetration between the Belt and pulley.
Abstract: In this paper, the applicability of the absolute nodal coordinate formulation for the modeling of belt-drive systems is studied. A successful and effective analyzing method for belt-drive systems requires the exact modeling of the rigid body inertia during an arbitrary rigid body motion, accounting of shear deformation, description of highly nonlinear deformations, and a simple as well as realistic description of the contact. The absolute nodal coordinate formulation meets the challenge and is a promising approach for the modeling of belt-drive systems. In this study, a recently proposed two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation has been modified to obtain a belt-like element. In the original element, a continuum mechanics approach is applied to the exact displacement field of the shear deformable beam. The belt-like element allows the user to control the axial and bending stiffness through the use of two parameters. In this study, the interaction between the belt and the pulleys is modeled using an elastic approach in which the contact is accounted for by the inclusion of a set of external forces that depend on the penetration between the belt and pulley. When using the absolute nodal coordinate formulation, the contact forces can be distributed over the length of the element due to the use of high-order polynomials. This is different from other approaches that are used in the modeling of belt-drives. Static and dynamic analysis are used in this study to show the performance of the distributed contact force model and the proposed belt-like element, which is able to model highly nonlinear deformations. Applying these two contributions to the modeling of belt-drive systems, instead of contact forces applied at nodes and low-order elements, leads to a considerable reduction in the degrees of freedom.
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02 Nov 2006
TL;DR: This book takes readers through all the steps necessary for solving hard problems in continuum mechanics with smooth particle methods, with particular attention to the parallel computing necessary for large problems and to the graphic displays required for the efficient completion of computational projects.
Abstract: This book takes readers through all the steps necessary for solving hard problems in continuum mechanics with smooth particle methods. Pedagogical problems clarify the generation of initial conditions, the treatment of boundary conditions, the integration of the equations of motion, and the analysis of the results. Particular attention is paid to the parallel computing necessary for large problems and to the graphic displays, including debugging software, required for the efficient completion of computational projects. The book is self-contained, with summaries of classical particle mechanics and continuum mechanics for both fluids and solids, computer languages, the stability of numerical methods, Lyapunov spectra, and message-passing parallel computing. The main difficulties faced by meshless particle methods are discussed and the means of overcoming them are illustrated with worked examples.
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TL;DR: In this article, a derivation of the balance laws of nonlocal continuum mechanics, based on the concept of microelements, is presented, and the nonlocal balance laws are then established.
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TL;DR: In this paper, the authors present techniques, advances, problems, and new developments in modelling the progressive mechanical breakdown of, and associated fluid flow in, intact heterogeneous rock, which can be classified into three categories of discrete models based on fracture mechanics, the continuum damage mechanics approach, and statistical approaches.
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TL;DR: This work presents a formulation for coupling atomistic and continuum simulation methods for application to both quasistatic and dynamic analyses, which uses interpolation and projection operators to link the kinematics of each region, which are then used to formulate a system potential energy.
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TL;DR: In this paper, a finite element model based on the molecular mechanics theory is proposed to investigate the fracture progress in Zig-Zag and Armchair carbon nanotubes with defects under uniaxial tensile stress.
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TL;DR: In this article, a cohesive interface element for ductile tearing of thin structures modelled by plane-stress continuum or shell elements is presented which accounts for thickness reduction, which prevents localisation of plastic deformation in the adjacent continuum elements often inhibiting crack extension and leading to divergence of the numerical simulations.
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TL;DR: In this paper, a combined continuous-discontinuous approach has been used to account for the interaction between macroscopic cracks and the surrounding softening material, which is modelled using continuum damage mechanics concepts.
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TL;DR: In this article, a single hardening model is enhanced to account for the couple stress and Cosserat rotation and the internal length scales are incorporated accordingly; then, the enhanced Lade's model and its material properties are discussed in detail; then the finite element formulations in the Updated Lagrangian Frame (UL) are used.
Abstract: It has been known that classical continuum mechanics laws fail to describe strain localization in granular materials due to the mathematical ill-posedness and mesh dependency. Therefore, a non-local theory with internal length scales is needed to overcome such problems. The micropolar and high-order gradient theories can be considered as good examples to characterize the strain localization in granular materials. The fact that internal length scales are needed requires micromechanical models or laws; however, the classical constitutive models can be enhanced through the stress invariants to incorporate the Micropolar effects. In this paper, Lade's single hardening model is enhanced to account for the couple stress and Cosserat rotation and the internal length scales are incorporated accordingly. The enhanced Lade's model and its material properties are discussed in detail; then the finite element formulations in the Updated Lagrangian Frame (UL) are used. The finite element formulations were implemented into a user element subroutine for ABAQUS (UEL) and the solution method is discussed in the companion paper. The model was found to predict the strain localization in granular materials with low dependency on the finite element mesh size. The shear band was found to reflect on a certain angle when it hit a rigid boundary. Applications for the model on plane strain specimens tested in the laboratory are discussed in the companion paper. Copyright © 2006 John Wiley & Sons, Ltd.