Showing papers on "Continuum mechanics published in 2007"

Breakage mechanics—Part I: Theory

Abstract: Different measures have been suggested for quantifying the amount of fragmentation in randomly compacted crushable aggregates. A most effective and popular measure is to adopt variants of Hardin's [1985. Crushing of soil particles. J. Geotech. Eng. ASCE 111(10), 1177–1192] definition of relative breakage ‘ B r ’. In this paper we further develop the concept of breakage to formulate a new continuum mechanics theory for crushable granular materials based on statistical and thermomechanical principles. Analogous to the damage internal variable ‘ D ’ which is used in continuum damage mechanics (CDM), here the breakage internal variable ‘ B ’ is adopted. This internal variable represents a particular form of the relative breakage ‘ B r ’ and measures the relative distance of the current grain size distribution from the initial and ultimate distributions. Similar to ‘ D ’, ‘ B ’ varies from zero to one and describes processes of micro-fractures and the growth of surface area. However, unlike damage that is most suitable to tensioned solid-like materials, the breakage is aimed towards compressed granular matter. While damage effectively represents the opening of micro-cavities and cracks, breakage represents comminution of particles. We term the new theory continuum breakage mechanics (CBM), reflecting the analogy with CDM. A focus is given to developing fundamental concepts and postulates, and identifying the physical meaning of the various variables. In this part of the paper we limit the study to describe an ideal dissipative process that includes breakage without plasticity. Plastic strains are essential, however, in representing aspects that relate to frictional dissipation, and this is covered in Part II of this paper together with model examples.

The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes.

Abstract: In this paper, the constitutive relations of nonlocal elasticity theory are presented for application in the analysis of carbon nanotubes (CNTs) when modelled as Euler-Bernoulli beams, Timoshenko beams or as cylindrical shells. In particular, the shear stress and strain relation for the nonlocal Timoshenko beam theory is discussed in great detail due to a misconception by some researchers that the nonlocal effect should appear in this constitutive relation. Different theories for proposing the value of the small scale parameter are also introduced and a recommendation for the value from the standpoint of wave propagation of CNTs is given.

Microstructural Randomness and Scaling in Mechanics of Materials

Abstract: PREFACE BASIC RANDOM MEDIA MODELS Probability Measure of Geometric Objects Basic Point Fields Directional Data Random Fibers, Random Line Fields, Tessellations Basic Concepts and Definitions of Random Microstructures RANDOM PROCESSES AND FIELDS Elements of One-Dimensional Random Fields Mechanics Problems on One-Dimensional Random Fields Elements of Two- and Three-Dimensional Random Fields Mechanics Problems on Two- and Three-Dimensional Random Fields Ergodicity The Maximum Entropy Method PLANAR LATTICE MODELS: PERIODIC TOPOLOGIES AND ELASTOSTATICS One-Dimensional Lattices Planar Lattices: Classical Continua Applications in Mechanics of Composites Planar Lattices: Nonclassical Continua Extension-Twist Coupling in a Helix LATTICE MODELS: RIGIDITY, RANDOMNESS, DYNAMICS, AND OPTIMALITY Rigidity of Networks Spring Network Models for Disordered Topologies Particle Models Michell Trusses: Optimal Use of Material TWO- VERSUS THREE-DIMENSIONAL CLASSICAL ELASTICITY Basic Relations The CLM Result and Stress Invariance Poroelasticity TWO- VERSUS THREE-DIMENSIONAL MICROPOLAR ELASTICITY Micropolar Elastic Continua Classical vis-a-vis Nonclassical (Elasticity) Models Planar Cosserat Elasticity The CLM Result and Stress Invariance Effective Micropolar Moduli and Characteristic Lengths of Composites MESOSCALE BOUNDS FOR LINEAR ELASTIC MICROSTRUCTURES Micro-, Meso-, and Macroscales Volume Averaging Spatial Randomness Hierarchies of Mesoscale Bounds Examples of Hierarchies of Mesoscale Bounds Moduli of Trabecular Bone RANDOM FIELD MODELS AND STOCHASTIC FINITE ELEMENTS Mesoscale Random Fields Second-Order Properties of Mesoscale Random Fields Does There Exist a Locally Isotropic, Smooth Elastic Material? Stochastic Finite Elements for Elastic Media Method of Slip-Lines for Inhomogeneous Plastic Media Michell Trusses in the Presence of Random Microstructure HIERARCHIES OF MESOSCALE BOUNDS FOR NONLINEAR AND INELASTIC MICROSTRUCTURES Physically Nonlinear Elastic Microstructures Finite Elasticity of Random Composites Elastic-Plastic Microstructures Rigid-Perfectly Plastic Microstructures Viscoelastic Microstructures Stokes Flow in Porous Media Thermoelastic Microstructures Scaling and Stochastic Evolution in Damage Phenomena Comparison of Scaling Trends MESOSCALE RESPONSE IN THERMOMECHANICS OF RANDOM MEDIA From Statistical Mechanics to Continuum Thermodynamics Extensions of the Hill Condition Legendre Transformations in (Thermo)Elasticity Thermodynamic Orthogonality on the Mesoscale Complex versus Compound Processes: The Scaling Viewpoint Toward Continuum Mechanics of Fractal Media WAVES AND WAVEFRONTS IN RANDOM MEDIA Basic Methods in Stochastic Wave Propagation Toward Spectral Finite Elements for Random Media Waves in Random 1D Composites Transient Waves in Heterogeneous Nonlinear Media Acceleration Wavefronts in Nonlinear Media BIBLIOGRAPHY INDEX

Non-local elastic plate theories

Abstract: A non-local plate model is proposed based on Eringen's theory of non-local continuum mechanics. The basic equations for the non-local Kirchhoff and the Mindlin plate theories are derived. These non-local plate theories allow for the small-scale effect which becomes significant when dealing with micro-/nanoscale plate-like structures. As illustrative examples, the bending and free vibration problems of a rectangular plate with simply supported edges are solved and the exact non-local solutions are discussed in relation to their corresponding local solutions.

Boundary Conditions for the Moving Contact Line Problem

Abstract: The physical processes near a moving contact line are investigated systematically using molecular dynamics and continuum mechanics. Constitutive relations for the friction force in the contact line region, the fluid-fluid interfacial force, and the stresses in the fluid-solid interfacial region are studied. Verification of force balance demonstrates the importance of the normal stress jump across the contact line region. Effective boundary conditions are derived using force balance. It is found that in the flow regime studied, the deviation of the wall contact angle from the equilibrium contact angle is proportional to the velocity of the contact line. The effective continuum model is solved numerically and the behavior of the apparent contact angle and the wall contact angle is studied. It is found that the fluid-fluid interface near the wall exhibits a universal behavior. The onset of the nonlinear response for the contact line motion is studied within the framework of Blake’s molecular kinetic theory.

A molecular mechanics approach for analyzing tensile nonlinear deformation behavior of single-walled carbon nanotubes

Abstract: In this paper, by capturing the atomic informa- tion and reflecting the behaviour governed by the nonlin- ear potential function, an analytical molecular mechanics approach is proposed. A constitutive relation for single- walled carbon nanotubes (SWCNT's) is established to describe the nonlinear stress-strain curve of SWCNT's and to predict both the elastic properties and breaking strain of SWCNT's during tensile deformation. An analysis based on the virtual internal bond (VIB) model proposed by P. Zhang et al. is also presented for comparison. The results indicate that the proposed molecular mechanics approach is indeed an acceptable analytical method for analyzing the mechanical behavior of SWCNT's. of CNT's. The Young's modulus of CNT's was found to be about 1 TPa (2-5). Many theories of mechanics have also been proposed to study the mechanical properties of CNT's. Zhang et al. (6) developed a continuum mechanics approach to model elastic properties of single-walled carbon nanotubes (SWCNT's), and the Young's modulus of SWCNT's was pre- dicted to be 0.705 TPa. Li and Chou (7) presented a structural mechanics approach to model the deformation of CNT's, and calculated the Young's moduli for CNT's with different radii. A similar approach was presented by Chang and Gao (8), and the chirality- and size-dependent elastic properties such as Young's modulus, Poisson's ratio and shear modulus were predicted (9,10). Moreover, the nonlinear effect of SWCNT's was taken into account (11) recently. In view of the unrealistic demand of computational power to study materials of practical size, atomistic simulations are deemed unsuitable for the study of large scaled nanometer materials in large time spans. Therefore, various attempts have been made by researchers to introduce atomic character- istics into the mechanical theory. For example, the molecular mechanics originally developed by chemical scientists (12) can be considered one of the successful attempts. According to the definition of Burkert and Allinger (12), the total potential energy, U , is constitutive of several individual energy terms corresponding to bond stretching, angle bend- ing, torsion, and van der Waals interactions, respectively: U = � Ustretch + � Ubend

Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes

Abstract: Wave propagation in carbon nanotubes (CNTs) is studied based on the proposed nonlocal elastic shell theory. Both theoretical analyses and numerical simulations have explicitly revealed the small-scale effect on wave dispersion relations for different CNT wavenumbers in the longitudinal and circumferential directions and for different wavelengths in the circumferential direction. The applicability of the proposed nonlocal elastic shell theory is especially explored and analyzed based on the differences between the wave solutions from local and nonlocal theories in numerical simulations. It is found that the newly proposed nonlocal shell theory is indispensable in predicting CNT phonon dispersion relations at larger longitudinal and circumferential wavenumbers and smaller wavelength in the circumferential direction when the small-scale effect becomes dominant and hence noteworthy. In addition, the asymptotic frequency, phase velocities and cut-off frequencies are also derived from the nonlocal shell theory. Moreover, an estimation of the scale coefficient is provided based on the derived asymptotic frequency. The research findings not only demonstrate great potential of the proposed nonlocal shell theory in studying vibration and phonon dispersion relations of CNTs but also signify limitations of local continuum mechanics in analysis of small-scale effects, and thus are of significance in promoting the development of nonlocal continuum mechanics in the design of nanostructures.

The Mechanics and Physics of Defect Nucleation

Abstract: The following article is based on the Outstanding Young Investigator Award presentation given by Ju Li on April 19, 2006, at the Materials Research Society Spring Meeting in San Francisco. Li received the award “for his innovative work on the atomistic and first-principles modeling of nanoindentation and ideal strength in revealing the genesis of materials deformation and fracture.”Defect nucleation plays a critical role in the mechanical behavior of materials, especially if the system size is reduced to the submicron scale. At the most fundamental level, defect nucleation is controlled by bond breaking and reformation events, driven typically by mechanical strain and electronegativity differences. For these processes, atomistic and first-principles calculations are uniquely suited to provide an unprecedented level of mechanistic detail. Several connecting threads incorporating notions in continuum mechanics and explicit knowledge of the interatomic energy landscape can be identified, such as homogeneous versus heterogeneous nucleation, cleavage versus shear-faulting tendencies, chemomechanical coupling, and the fact that defects are singularities at the continuum level but regularized at the atomic scale. Examples are chosen from nano-indentation, crack-tip processes, and grain-boundary processes. In addition to the capacity of simulations to identify candidate mechanisms, the computed athermal strength, activation energy, and activation volume can be compared quantitatively with experiments to define the fundamental properties of defects in solids.

Using the material-point method to model sea ice dynamics

Abstract: [1] The material-point method (MPM) is a numerical method for continuum mechanics that combines the best aspects of Lagrangian and Eulerian discretizations. The material points provide a Lagrangian description of the ice that models convection naturally. Thus properties such as ice thickness and compactness are computed in a Lagrangian frame and do not suffer from errors associated with Eulerian advection schemes, such as artificial diffusion, dispersion, or oscillations near discontinuities. This desirable property is illustrated by solving transport of ice in uniform, rotational and convergent velocity fields. Moreover, the ice geometry is represented by unconnected material points rather than a grid. This representation facilitates modeling the large deformations observed in the Arctic, as well as localized deformation along leads, and admits a sharp representation of the ice edge. MPM also easily allows the use of any ice constitutive model. The versatility of MPM is demonstrated by using two constitutive models for simulations of wind-driven ice. The first model is a standard viscous-plastic model with two thickness categories. The MPM solution to the viscous-plastic model agrees with previously published results using finite elements. The second model is a new elastic-decohesive model that explicitly represents leads. The model includes a mechanism to initiate leads, and to predict their orientation and width. The elastic-decohesion model can provide similar overall deformation as the viscous-plastic model; however, explicit regions of opening and shear are predicted. Furthermore, the efficiency of MPM with the elastic-decohesive model is competitive with the current best methods for sea ice dynamics.

Drift and breakup of spiral waves in reaction-diffusion-mechanics systems.

Abstract: Rotating spiral waves organize excitation in various biological, physical, and chemical systems. They underpin a variety of important phenomena, such as cardiac arrhythmias, morphogenesis processes, and spatial patterns in chemical reactions. Important insights into spiral wave dynamics have been obtained from theoretical studies of the reaction-diffusion (RD) partial differential equations. However, most of these studies have ignored the fact that spiral wave rotation is often accompanied by substantial deformations of the medium. Here, we show that joint consideration of the RD equations with the equations of continuum mechanics for tissue deformations (RD-mechanics systems), yield important effects on spiral wave dynamics. We show that deformation can induce the breakup of spiral waves into complex spatiotemporal patterns. We also show that mechanics leads to spiral wave drift throughout the medium approaching dynamical attractors, which are determined by the parameters of the model and the size of the medium. We study mechanisms of these effects and discuss their applicability to the theory of cardiac arrhythmias. Overall, we demonstrate the importance of RD-mechanics systems for mathematics applied to life sciences.

A contact mechanics model for quasi-continua

Abstract: A computational multiscale contact mechanics model is proposed to describe the interaction between deformable solids based on the interaction of individual atoms or molecules belonging to the solids. The contact model, formulated in the framework of large deformation continuum mechanics, is derived from coarsening the molecular dynamics (MD) description of a large assembly of individual atoms, and it thus bears some of the characteristics of the underlying atomic structure. The multiscale contact model distinguishes between atoms interacting within a small neighbourhood within the solids and atoms interacting over large distances between remote regions of the solids. The former furnishes a constitutive relation for the continuum, like the Cauchy-Bom Rule, while the latter is used to model the interaction between distinct bodies. The proposed contact model is formulated as a variational weak form and implemented within an updated Lagrangian finite element method. It is shown that, as the problem size increases, the description of the model can be simplified to yield more efficient computational algorithms. In this respect, the proposed multiscale formulation leads to a smooth transition from MD to continuum contact mechanics. The general behaviour of the contact model is studied, and some numerical examples are given.

Continuum Models of Carbon Nanotube-Based Composites Using the Boundary Element Method

Abstract: This paper presents some recent advances in the boundary element method (BEM) for the analysis of carbon nanotube (CNT)-based composites. Carbon nanotubes, formed conceptually by rolling thin graphite sheets, have been found to be extremely stiff, strong and resilient, and therefore may be ideal for reinforcing composite materials. However, the thin cylindrical shape of the CNTs presents great challenges to any computational method when these thin shell-like CNTs are embedded in a matrix material. The BEM, based on exactly the same boundary integral equation (BIE) formulation developed by Rizzo some forty years ago, turns out to be an ideal numerical tool for such simulations using continuum mechanics. Modeling issues regarding model selections, representative volume elements, interface conditions and others, will be discussed in this paper. Methods for dealing with nearly-singular integrals which arise in the BEM analysis of thin or layered materials and are crucial for the accuracy of such analyses will be reviewed. Numerical examples using the BEM and compared with the finite element method (FEM) will be presented to demonstrate the efficiency and accuracy of the BEM in analyzing the CNT-reinforced composites.

Convergence Analysis of a Quasi-Continuum Approximation for a Two-Dimensional Material Without Defects

Abstract: In many applications, materials are modeled by a large number of particles (or atoms), where any particle can interact with any other. The computational cost is very high since the number of atoms is huge. Recently much attention has been paid to a so-called quasi-continuum (QC) method, which is a mixed atomistic/continuum model. The QC method uses an adaptive finite element framework to effectively integrate the majority of the atomistic degrees of freedom in regions where there is no serious defect. However, numerical analysis of this method is still in its infancy. In this paper we will conduct a convergence analysis of the QC method in the case when there is no defect. We will also remark on the case when the defect region is small. The difference between our analysis and conventional analysis is that our exact atomistic solution is not a solution of a continuous partial differential equation, but a discrete lattice scale solution which is not approximately related to any conventional partial differential equation.

Lattice Boltzmann model for the simulation of multicomponent mixtures.

Abstract: A lattice Boltzmann (LB) model for the simulation of realistic multicomponent mixtures is constructed. In the hydrodynamic limit, the LB model recovers the equations of continuum mechanics within the mixture-averaged diffusion approximation. The present implementation can be used to simulate realistic mixtures with arbitrary Schmidt numbers and molecular masses of the species. The model is applied to the mixing of two opposed jets of different concentrations and the results are in excellent agreement with a continuum model. An application to the simulation of mixtures in microflows is also presented. Results compare well with existing kinetic theory predictions of the slip coefficient for mixtures in a Couette flow.

Material Inhomogeneities and their Evolution: A Geometric Approach

Abstract: Inhomogeneity in Continuum Mechanics.- An overview of inhomogeneity theory.- Uniformity of second-grade materials.- Uniformity of Cosserat media.- Functionally graded bodies.- Material Evolution.- On energy, Cauchy stress and Eshelby stress.- An overview of the theory of material evolution.- Second-grade evolution.- Mathematical Foundations.- Basic geometric concepts.- Theory of connections.- Bundles of linear frames.- Connections of higher order.

On the phenomenological representation of curing phenomena in continuum mechanics

Abstract: To simulate curing phenomena, for example for the purpose of optimising the manufacturing processes or to calculate the stress distribution in adhesive seams, constitutive models representing the thermomechanically-coupled behaviour of adhesives are required. During the curing reaction, the adhesive changes its thermomechanical material behaviour from a viscous fluid to a viscoelastic solid. This phase transition is an exothermal chemical reaction which is accompanied by thermal expansion, chemical shrinkage and changes in temperature. In this essay we develop a physically-based theory of finite strain thermoviscoelasticity to represent these phenomena. To this end, we introduce a multiplicative split of the deformation gradient into a thermal, a chemical and a mechanical part. We define the coordinate of chemical reaction determined by an evolution equation to describe the temporal behaviour of the curing reaction. The free energy of the model contains an additional term, the chemically-stored free energy, which depends on this internal variable. The mechanical behaviour of the adhesive is modelled using a constitutive approach of finite thermoviscoelasticity and the viscosities are functions of the coordinate of chemical reaction. We show that the model is compatible with the Clausius–Duhem inequality, derive the equation of heat conduction and illustrate the physical properties of the theory by a numerical example.

A bridging domain and strain computation method for coupled atomistic–continuum modelling of solids

Abstract: We present a multiscale method that couples atomistic models with continuum mechanics. The method is based on an overlapping domain-decomposition scheme. Constraints are imposed by a Lagrange multiplier method to enforce displacement compatibility in the overlapping subdomain in which atomistic and continuum representations overlap. An efficient version of the method is developed for cases where the continuum can be modelled as a linear elastic material. An iterative scheme is utilized to optimize the coupled configuration. Conditions for the regularity of the constrained matrices are determined. A method for computing strain in atomistic models and handshake domains is formulated based on a moving least-square approximation which includes both extensional and angle-bending terms. It is shown that this method exactly computes the linear strain field. Applications to the fracture of defected single-layer atomic sheets and nanotubes are given. Copyright © 2006 John Wiley & Sons, Ltd.

Wave propagation in single- and double-walled carbon nanotubes filled with fluids

Abstract: Wave propagation approach of single- and double-walled carbon nanotubes conveying fluid is presented through the use of the continuum mechanics. A simplified Flugge shell equations are proposed as the governing equations of motion for carbon nanotubes studied here. For the double-walled nanotubes, the deflection of nested tubes is considered to be coupled through the van der Waals interaction between two adjacent nanotubes. Effects of filled fluid property and nanotube diameter on the wave propagation are investigated and analyzed based on the proposed elastic continuum model. The theoretical investigation may give a useful reference for potential design and application of nanoelectronics and nanodevices.

On the geometric character of stress in continuum mechanics

Abstract: This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other funda- mental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented.

Studies of validity of the Cauchy Born rule by direct comparison of continuum and atomistic modelling

Abstract: The Cauchy–Born rule is widely used as a standard method in continuum mechanics in order to construct descriptions of material behaviour using atomistic information. The main objective of the present work is to investigate the validity of this kinematic assumption, i.e. to determine the state when a transition to non-affine deformations is possible due to instabilities of the underlying atomic system. To this end, the results of the Cauchy–Born rule are compared with the results of direct molecular dynamics calculations on the one hand and with the results obtained by using the acoustic tensor of continuum mechanics on the other hand.

Continuum damage mechanics for hysteresis and fatigue of quasi-brittle materials and structures

Abstract: For a material exhibiting hysteresis such as quasi-brittle materials, it is natural to consider that hysteresis and fatigue are related to each other. One shows in the present work that damage, from the continuum damage mechanics point of view, may be seen as the link between both phenomena. One attempts, hence, to set up a unified modelling of hysteresis and damage. Numerical examples are given for concrete and validate the proposed model of internal sliding and friction coupled with damage. The problem of a proper phenomenological modelling of the micro-defects closure effect leading to a dissymmetric tension/compression response and to stiffness recovery in compression is also addressed. Cyclic and fatigue applications are in mind, and also random fatigue and seismic responses. Copyright © 2006 John Wiley & Sons, Ltd.

Theory of dynamic crack branching in brittle materials

Abstract: The problem of dynamic symmetric branching of a tensile crack propagating in a brittle material is studied within Linear Elastic Fracture Mechanics theory. The Griffith energy criterion and the principle of local symmetry provide necessary conditions for the onset of dynamic branching instability and for the subsequent paths of the branches. The theory predicts a critical velocity for branching and a well defined shape described by a branching angle and a curvature of the side branches. The model rests on a scenario of crack branching based on reasonable assumptions and on exact dynamic results for the anti-plane branching problem. Our results reproduce within a simplified 2D continuum mechanics approach the main experimental features of the branching instability of fast cracks in brittle materials.