Continuum Mechanics

In this article the principles of the field operation and manipulation ͑FOAM͒ Cϩϩ class library for continuum mechanics are outlined. Our intention is to make it as easy as possible to develop reliable and efficient computational continuum-mechanics codes: this is achieved by making the top-level syntax of the code as close as possible to conventional mathematical notation for tensors and partial differential equations. Object-orientation techniques enable the creation of data types that closely mimic those of continuum mechanics, and the operator overloading possible in Cϩϩ allows normal mathematical symbols to be used for the basic operations. As an example, the implementation of various types of turbulence modeling in a FOAM computational-fluid-dynamics code is discussed, and calculations performed on a standard test case, that of flow around a square prism, are presented. To demonstrate the flexibility of the FOAM library, codes for solving structures and magnetohydrodynamics are also presented with appropriate test case results given.

A continuum mechanics approach to two-phase flow is reviewed. An averaging procedure is discussed and applied to the exact equations of motion. Constitutive equations are supplied and discussed for the stresses, pressure differences and the interfacial force. The nature of the resulting equations is studied.

The behaviour of granular material in motion is studied from a continuum point of view. Insofar as possible, individual grains are treated as the 'molecules' of a granular 'fluid '. Besides the obvious contrast in shape, size and mass, a key difference between true molecules and grains is that collisions of the latter are inevitably inelastic. This, together with the fact that the fluctuation velocity may be comparable to the flow velocity, necessitates explicit incorporation of the energy equation, in addition to the continuity and momentum equations, into the theoretical description. Simple ' microscopic ' kinetic models are invoked for deriving expressions for the ' coefficients ' of viscosity, thermal diffusivity and energy absorption due to collisions. The 'coefficients' are not constants, but are functions of the local state of the medium, and therefore depend on the local 'temperature ' and density. I n general the resulting equations are nonlinear and coupled. However, in the limit s -4 d , where s is the mean separation between neighbouring grain surfaces and d is a grain diameter, the above equations become linear and can be solved analytically. An important dependent variable, in this formulation, in addition to the flow velocity u, is the mean random fluctuation ('thermal') velocity ii of an individual grain. With a sufficient flux of energy supplied to the system through the boundaries of the container, ii can remain non-zero even in the absence of flow. The existence of a non-uniform is the means by which energy can be 'conducted' from one part of the system to another. Because grain collisions are inelastic, there is a natural (damping) lengthscale, governed by the value of d , which strongly influences the functional dependence of ~rs on position.

In this study, a technique is presented for developing constitutive models for polymer composite systems reinforced with single-walled carbon nanotubes (SWNT). Because the polymer molecules are on the same size scale as the nanotubes, the interaction at the polymer/nanotube interface is highly dependent on the local molecular structure and bonding. At these small length scales, the lattice structures of the nanotube and polymer chains cannot be considered continuous, and the bulk mechanical properties can no longer be determined through traditional micromechanical approaches that are formulated by using continuum mechanics. It is proposed herein that the nanotube, the local polymer near the nanotube, and the nanotube/polymer interface can be modeled as an effective continuum fiber using an equivalentcontinuum modeling method. The effective fiber serves as a means for incorporating micromechanical analyses for the prediction of bulk mechanical properties of SWNT/polymer composites with various nanotube shapes, sizes, concentrations, and orientations. As an example, the proposed approach is used for the constitutive modeling of two SWNT/LaRC-SI (with a PmPV interface) composite systems, one with aligned SWNTs and the other with three-dimensionally randomly oriented SWNTs. The Young's modulus and shear modulus have been calculated for the two systems for various nanotube lengths and volume fractions.

This paper examines the use of a continuum damage model to predict strength and size effects in notched carbon-epoxy laminates. The effects of size and the development of a fracture process zone before final failure are identified in an experimental program. The continuum damage model is described and the resulting predictions of size effects are compared with alternative approaches: the point stress and the inherent flaw models, the Linear-Elastic Fracture Mechanics approach, and the strength of materials approach. The results indicate that the continuum damage model is the most accurate technique to predict size effects in composites. Furthermore, the continuum damage model does not require any calibration and it is applicable to general geometries and boundary conditions.

The continuum mechanical treatment of biological growth and remodeling has attracted considerable attention over the past fifteen years. Many aspects of these problems are now well-understood, yet there remain areas in need of significant development from the standpoint of experiments, theory, and computation. In this perspective paper we review the state of the field and highlight open questions, challenges, and avenues for further development.

Methods for examining structure at the microscopic scale have seen a number of recent developments. Many scanning probe microscopies have emerged, aberration correctors are revolutionizing traditional electron microscopy, and optical microscopy has seen the arrival of fluorescent, confocal, and two-photon variants. The contributions in this book each review a form of microscopy, describing the instruments and their areas of application.

This work presents a rigorous continuum mechanics model of solvent diffusion describing the growth of solid-electrolyte interfaces ͑SEIs͒ in Li-ion cells incorporating carbon anodes. The model assumes that a reactive solvent component diffuses through the SEI and undergoes two-electron reduction at the carbon-SEI interface. Solvent reduction produces an insoluble product, resulting in increasing SEI thickness. The model predicts that the SEI thickness increases linearly with the square root of time. Experimental data from the literature for capacity loss in two types of prototype Li-ion cells validates the solvent diffusion model. We use the model to estimate SEI thickness and extract solvent diffusivity values from the capacity loss data. Solvent diffusivity values have an Arrhenius temperature dependence consistent with solvent diffusion through a solid SEI. The magnitudes of the diffusivities and activation energies are comparable to literature values for hydrocarbon diffusion in carbon molecular sieves and zeolites. These findings, viewed in the context of recent SEI morphology studies, suggest that the SEI may be viewed as a single layer with both micro-and macroporosity that controls the ingress of electrolyte, anode passivation by the SEI, and cell performance during initial cycling as well as long-term operation.

This paper is concerned with micromechanics of Schneebeli material specimens composed of wooden roller stacks. Several laboratory tests are carried out to analyse the material behaviour under complex loading conditions, involving loading-unloading cycles and principal axes rotations. In order to characterize micromechanical deformation features and structure evolution, a series of pictures is taken during loading. Pictures are then digitized using a stereo device, obtaining the position of each roller. Starting from these data a number of computer programs, conceived for the purpose, allow us to measure micromechanical variables and to analyse their evolution. In the following, after the description of the devices employed in this research, macromechanical results are analysed to evaluate the reliability of the laboratory model. Then, local variables are introduced and the use of continuum mechanics to describe granular materials behaviour is discussed. Finally, the evolution of local kinematic variables is described, focusing interest on the evolution of specimen anisotropy.

This paper presents the theoretical analysis and the experimental validation of the force sensing capabilities of continuum robots. These robots employ super-elastic NiTi backbones and actuation redundancy. The paper uses screw theory to analyze the limitations and to provide geometric interpretation to the sensible wrenches. The analysis is based on the singular value decomposition of the Jacobian mapping between the configuration space and the twist space of the end effector. The results show that the sensible wrenches belong to a two-dimensional screw system and the insensible wrenches belong to a four-dimensional screw system. The theory presented in this paper is validated through simulations and experiments. It is shown that the force sensing errors have an average of 0.34 grams with a standard deviation of 0.83 grams. Another experiment of generating the stiffness map of a silicone strip suggests possible medical application of palpation for tumor detection. The presented study allows force sensing in challenging environments where placing force sensors at the distal end of a robot is not possible due to limitations such as size and MRI compatibility. x x x x J J J J T T J J I J E J W APPENDIX B Since x J is formulated as p J J , p J can be found by taking time derivative of p b , as in , while J is found by formulating the angular velocity of GCS, (43). p

The paper deals with the strong discontinuity approach and shows the links with the decohesive fracture mechanics provided by that approach. On the basis of 1D continuum damage models it is shown that, by introducing some few ingredients like the strong discontinuity kinematics, discrete constitutive models (traction vs. displacement jumps) are automatically induced. For the general 2D±3D cases it is shown that the weak discontinuity concept is an additional ingredient, necessary in order to ful®ll the strong discontinuity conditions, which allows to establish additional links with the fracture process zone concept. Also classical fracture mechanics properties as the fracture energy are related to the continuum model properties in a straightforward manner. Ó

Recent advances in analytical and computational modelling frameworks to describe the mechanics of materials on scales ranging from the atomistic, through the microstructure or transitional, and up to the continuum are reviewed. It is shown that multiscale modelling of materials approaches relies on a systematic reduction in the degrees of freedom on the natural length scales that can be identified in the material. Connections between such scales are currently achieved either by a parametrization or by a 'zoom-out' or 'coarse-graining' procedure. Issues related to the links between the atomistic scale, nanoscale, microscale and macroscale are discussed, and the parameters required for the information to be transferred between one scale and an upper scale are identified. It is also shown that seamless coupling between length scales has not yet been achieved as a result of two main challenges: firstly, the computational complexity of seamlessly coupled simulations via the coarse-graining approach and, secondly, the inherent difficulty in dealing with system evolution stemming from time scaling, which does not permit coarse graining over temporal events. Starting from the Born-Oppenheimer adiabatic approximation, the problem of solving quantum mechanics equations of motion is first reviewed, with successful applications in the mechanics of nanosystems. Atomic simulation methods (e.g. molecular dynamics, Langevin dynamics and the kinetic Monte Carlo method) and their applications at the nanoscale are then discussed. The role played by dislocation dynamics and statistical mechanics methods in understanding microstructure self-organization, heterogeneous plastic deformation, material instabilities and failure phenomena is also discussed.

Analytical formulations are presented to predict the elastic moduli of graphene sheets and carbon nanotubes using a linkage between lattice molecular structure and equivalent discrete frame structure. The obtained results for a graphene sheet show an isotropic behavior, in contrast to limited molecular dynamic simulations. Young's modulus of CNT represents a high dependency of stiffness on tube thickness, while dependency on tube diameter is more tangible for smaller tube diameters. The presented closed-form solution provides an insight to evaluate finite element models constructed by beam elements. The results are in a good agreement with published data and experimental results.

In this Part I, of a two-part paper, we present a detailed continuummechanical development of a thermo-mechanically coupled elasto-viscoplasticity theory to model the strain rate and temperature dependent large-deformation response of amorphous polymeric materials. Such a theory, when further specialized (Part II) should be useful for modeling and simulation of the thermo-mechanical response of components and structures made from such materials, as well as for modeling a variety of polymer processing operations.

To achieve quantitative interpretation of piezoresponse force microscopy (PFM), including resolution limits, tip bias-and strain-induced phenomena and spectroscopy, analytical representations for tip-induced electroelastic fields inside the material are derived for the cases of weak ...

We consider the motion of red blood cells and other non-spherical microcapsules dilutely suspended in a simple shear flow. Our analysis indicates that depending on the viscosity, membrane elasticity, geometry and shear rate, the particle exhibits either tumbling, tank-treading of the membrane about the viscous interior with periodic oscillations of the orientation angle, or intermittent behavior in which the two modes occur alternately. For red blood cells, we compute the complete phase diagram and identify a novel tank-treading-to-tumbling transition at low shear rates. Observations of such motions coupled with our theoretical framework may provide a sensitive means of assessing capsule properties.

We have used an ultrahigh vacuum atomic-force microscope to study the variation in contact radius and friction with applied force between a silicon tip and a NbSe 2 sample. The data are compared to the Maugis-Dugdale theory, which is the appropriate continuum mechanics model for the properties and size of the tip-sample contact. The lateral stiffness of the tip-sample contact is related to the radius of the tip-sample contact through the shear moduli of the materials and we have used this relationship to measure directly the variation in contact radius with applied load. The contact radius measured in this way is found to be in agreement with the Maugis-Dugdale theory using the bulk values of the shear moduli. We also measured the variation in friction force with applied load using the same silicon tip. The variation in friction force with applied normal force is found to follow the variation of the contact area as predicted by the Maugis-Dugdale theory ͓D. Maugis, J. Colloid Interface Sci. 150, 243 ͑1992͔͒, which supports the hypothesis that for a single asperity contact, the frictional shear stress is constant. The value of the shear stress is found to be Ϸ6ϫ10 8 N/m 2 , which is comparable to the estimated theoretical shear strength of NbSe 2. ͓S0163-1829͑97͒05916-X͔

The role of surface energy and surface stress has been a topic of extensive discussion since the seminal work by Gibbs [Gibbs JW. The scientific papers of J. Willard Gibbs. Vol. I: Thermodynamics. New York and Bombay:Longmans, Green, and Co; 1906; Gibbs JW. Collected works. New Haven:Yale University Press; 1957]. Both quantities have the same value for liquids, but not for solids. The distinction between these terms is of special importance for phase transforming nanoparticles (precipitates, transforming or melting/solidifying single particles), since surface quantities scale as the inverse of the particle size relative to volume quantities. Continuum mechanics and, especially, the concept of configurational forces (stresses) provide a convenient framework for distinguishing between ''surface energy'', ''surface tension'' and ''surface stress''. Therefore, this progress report gives a rather detailed introduction into the continuum mechanics and thermodynamics of a moving surface. The transformation conditions for the cases where an entire nanoparticle transforms suddenly and when the transformation is interface-driven are discussed. A global transformation condition for a sudden phase-transforming nanoparticle is explained. For the interface-driven transformation, the concept of configurational forces is applied to derive a local transformation condition in a material point at the phase interface. Four examples of nanoparticles (growing precipitate, growing solid nucleus in liquid, melting particle, solidifying particle) are studied in detail. The surface energy and surface stress are shown to

Surgical simulators present a safe and potentially effective method for surgical training, and can also be used in robot-assisted surgery for pre-and intra-operative planning. Accurate modeling of the interaction between surgical instruments and organs has been recognized as a key requirement in the development of high-fidelity surgical simulators. Researchers have attempted to model tool-tissue interactions in a wide variety of ways, which can be broadly classified as (1) linear elasticity-based, (2) nonlinear (hyperelastic) elasticity-based finite element (FE) methods, and other techniques not based on FE methods or continuum mechanics. Realistic modeling of organ deformation requires populating the model with real tissue data (which are difficult to acquire in vivo) and simulating organ response in real time (which is computationally expensive). Further, it is challenging to account for connective tissue supporting the organ, friction, and topological changes resulting from tool-tissue interactions during invasive surgical procedures. Overcoming such obstacles will not only help us to model tool-tissue interactions in real time, but also enable realistic force feedback to the user during surgical simulation. This review paper classifies the existing research on tool-tissue interactions for surgical simulators specifically based on the modeling techniques employed and the kind of surgical operation being simulated, in order to inform and motivate future research on improved tool-tissue interaction models.

Higher order shear deformation theory (HSDT) is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived. The developed equations of motion have been applied to study buckling characteristics of nanoplates such as graphene sheets. Navier's approach has been used to solve the governing equations for all edges simply supported boundary conditions. Analytical solutions for critical buckling loads of the graphene sheets are presented. Nonlocal elasticity theories are employed to bring out the small scale effect on the critical buckling load of graphene sheets. Effects of (i) nonlocal parameter, (ii) length, (iii) thickness of the graphene sheets and (iv) higher order shear deformation theory on the critical buckling load have been investigated. The theoretical development as well as numerical solutions presented herein should serve as reference for nonlocal theories as applied to the stability analysis of nanoplates and nanoshells.

In the mechanics of multiphase (or multicomponent) mixtures, one of the outstanding issues is the formulation of constitutive relations for the interaction force. In this paper, we give a brief review of the various relations proposed for this interaction force. The review is tilted toward presenting the works of those who have used the mixture theory (or the theory of interacting continua) to derive or to propose a relationship for the interaction (or di usive) force. We propose a constitutive relation which is general and frame-indi erent and thus suitable for use in many ow conditions. At the end, we provide an alternative approach for ÿnding the drag force on a particle in a particulate mixture. This approach has been used in the non-Newtonian uid mechanics to ÿnd the drag force on surfaces. Published by Elsevier Science Ltd.

Structures of nanoparticles in Fe-16Cr-4.5Al-0.3Ti-2W-0.37Y 2 O 3 (K3) and Fe-20Cr-4.5Al-0.34 Ti-0.5Y 2 O 3 (MA956) oxide dispersion strengthened (ODS) ferritic steels produced by mechanical alloying (MA) and followed by hot extrusion have been studied using high-resolution ...

The paper studies local and global in time solutions to a class of multidimensional generalized Burgers-type equations with a fractional power of the Laplacian in the principal part and with general algebraic nonlinearity. Such equations naturally appear in continuum mechanics. Our results include existence, uniqueness, regularity and asymptotic behavior of solutions to the Cauchy problem as well as a construction of self-similar solutions. The role of critical exponents is also explained.

Mechanical properties of living cells are commonly described in terms of the laws of continuum mechanics. The purpose of this report is to consider the implications of an alternative approach that emphasizes the discrete nature of stress bearing elements in the cell and is based on the known structural properties of the cytoskeleton. We have noted previously that tensegrity architecture seems to capture essential qualitative features of cytoskeletal shape distortion in adherent cells (Ingber, 1993a; Wang et al., 1993). Here we extend those qualitative notions into a formal microstructural analysis. On the basis of that analysis we attempt to identify unifying principles that might underlie the shape stability of the cytoskeleton. For simplicity, we focus on a tensegrity structure containing six rigid struts interconnected by 24 linearly elastic cables. Cables carry initial tension (''prestress'') counterbalanced by compression of struts. Two cases of interconnectedness between cables and struts are considered: one where they are connected by pin-joints, and the other where the cables run through frictionless loops at the junctions. At the molecular level, the pinned structure may represent the case in which different cytoskeletal filaments are cross-linked whereas the looped structure represents the case where they are free to slip past one another. The system is then subjected to uniaxial stretching. Using the principal of virtual work, stretching force vs. extension and structural stiffness vs. stretching force relationships are calculated for different prestresses. The stiffness is found to increase with increasing prestress and, at a given prestress, to increase approximately linearly with increasing stretching force. This behavior is consistent with observations in living endothelial cells exposed to shear stresses (Wang & Ingber, 1994). At a given prestress, the pinned structure is found to be stiffer than the looped one, a result consistent with data on mechanical behavior of isolated, cross-linked and uncross-linked actin networks (Wachsstock et al., 1993). On the basis of our analysis we concluded that architecture and the prestress of the cytoskeleton might be key features that underlie a cell's ability to regulate its shape.

The field of dynamic fracture has been enlivened over the last 5 years or so by a series of remarkable accomplishments in different fields-earthquake science, atomistic (classical and quantum) simulations, novel laboratory experiments, materials modeling, and continuum mechanics. Important concepts either discovered for the first time or elaborated in new ways reveal wider significance. Here the separate streams of the literature of this progress are reviewed comparatively to highlight commonality and contrasts in the mechanics and physics. Much of the value of the new work resides in the new questions it has raised, which suggests profitable areas for research in the next few years and beyond. From the viewpoint of fundamental science, excitement is greatest in the struggle to probe the character of dynamic fracture at the atomic scale, using Newtonian or quantum mechanics as appropriate (a qualifier to be debated!). But lively interest is also directed towards modeling and experimentation at macroscales, including the geological, where the science of fracture is pulled at once by fundamental issues, such as the curious effects of friction, and the structural, where dynamic effects are essential to proper design or certification and even in manufacture.

ABSTRACT: Suspensions in industrial manufacturing processes are prevalent across a spectrum of diverse industries adding more emphasis to the fundamental challenge of predicting on a sound basis the rheological properties as well as the constitutive structure of non-colloidal suspensions under forcing. Thus, the raison d'être of the present work, a comprehensive up-to-date review of an important segment of the non-colloidal suspension mechanics covering the inertial diffusivity of non-colloidal particles in suspension in an unbounded Newtonian fluid medium followed by the closely related review of the numerical simulations of unbounded suspensions. The presence of the boundaries introduces extensive complications and the progress made in inertial and shear induced self-diffusivity in bounded Newtonian and viscoelastic suspending media will be covered in an upcoming review article dedicated to this topic. Efforts have not been spared to be thorough in the presentation with commentaries about the successes and failures of each theory and the reasons behind them expounded within the framework of their historical context. The link between different theories and the naturally unfolding succession of theories over time borne out of the necessity for better predictions as well as the challenges in the field at this time are given much emphasis at the expanse of a detailed in-depth account of various theories. For a detailed in-depth exposition, the reader is referred to the extensive reference list.

Various modes of failure in geomaterials have been observed in practice. Different criteria have been proposed to analyse these failures. In particular, Hill's Condition of Stability and diffuse modes of failure are considered in this paper in a dual framework: continuum mechanics and discrete mechanics. With the assumption of continuous media, experiments have shown that q constant loading paths (q characterizes the second stress invariant. For axisymmetric conditions, q is equal to: q ¼ r 1 À r 3 , where r 1 is the axial stress and r 3 the lateral stress) can exhibit non-localized failure modes and are analyzed by the second order work criterion. With the assumption of discrete media, grain avalanches are considered, and spatial and temporal correlations between bursts of kinetic energy and peaks of negative values of second order work are demonstrated from discrete computations. It is concluded that the second order work criterion (under its dual form: continuous and discrete) can be a proper tool to analyse diffuse modes of failure in geomaterials.

A systematic experimental and theoretical investigation of the elastic and failure properties of ZnO nanowires (NWs) under different loading modes has been carried out. In situ scanning electron microscopy (SEM) tension and buckling tests on single ZnO NWs along the polar direction [0001] were conducted. Both tensile modulus (from tension) and bending modulus (from buckling) were found to increase as the NW diameter decreased from 80 to 20 nm. The bending modulus increased more rapidly than the tensile modulus, which demonstrates that the elasticity size effects in ZnO NWs are mainly due to surface stiffening. Two models based on continuum mechanics were able to fit the experimental data very well. The tension experiments showed that fracture strain and strength of ZnO NWs increased as the NW diameter decreased. The excellent resilience of ZnO NWs is advantageous for their applications in nanoscale actuation, sensing, and energy conversion.

The importance of nonlinearities in material constitutive relations has long been appreciated in the continuum mechanics of macroscopic rods. Although the moment ͑torque͒ response to bending is almost universally linear for small deflection angles, many rod systems exhibit a high-curvature softening. The signature behavior of these rod systems is a kinking transition in which the bending is localized. Recent DNA cyclization experiments by Cloutier and Widom have offered evidence that the linear-elastic bending theory fails to describe the high-curvature mechanics of DNA. Motivated by this recent experimental work, we develop a simple and exact theory of the statistical mechanics of linear-elastic polymer chains that can undergo a kinking transition. We characterize the kinking behavior with a single parameter and show that the resulting theory reproduces both the low-curvature linear-elastic behavior which is already well described by the wormlike chain model, as well as the high-curvature softening observed in recent cyclization experiments.

Physical Mesomechanics of materials is a new branch of mechanics that focuses attention on a mesovolume of loaded material. It is a macro particle in classical continuum mechanics and its behavior under load is equivalent to the bulk. The structural elements for a particular application requires speci®c models while the computational techniques have to be developed. These research groups have been studying heterogeneous materials behavior at the mesolevel under dierent types of loading. Hierarchical models are developed to study deformation and fracture of solids at the micro-meso-and macrolevels. Taken into account are the in¯uence of micro-and mesostructure of loaded material in relation to its macro behavior. This work focuses on unifying the method of approach to be supported by tests and calculations. In particular, deformation and fracture mechanisms at the micro-, meso-and macrolevels are examined for metals and composites.

We consider continuous media in which contact edge forces are present. Introducing the notion of quasi-balanced contact force distribution, we are able to prove the conjectures by Noll and Virga [1] concerning the representation of contact edge forces. We generalize the Hamel-Noll theorem on the Cauchy postulate. Then we adapt the celebrated tetrahedron construction of Cauchy in order to obtain a representation theorem for stress states. In fact, we show that two stress tensors of order two and three are necessary for such a representation. Moreover we f nd the relationship between the notion of interstitial working introduced by Dunn and Serrin [2] and the notion of contact edge force.

We study the shapes of human red blood cells using continuum mechanics. In particular, we model the crenated, echinocytic shapes and show how they may arise from a competition between the bending energy of the plasma membrane and the stretching/shear elastic energies of ...

We present an empirical model for the effective thermal conductivity (ETC) of a polymer composite that includes dependency on the filler size distribution—chosen as the Rosin-Rammler distribution. The ETC is determined based on certain hypotheses that connect the behavior of a real composite material A, to that of a model composite material B, filled with mono-dimensional filler. The application of these hypotheses to the Maxwell model for ETC is presented. The validation of the new model and its characteristic equation was carried out using experimental data from the reference. The comparison showed that by using the size distribution law a very good fit between the equation of the new model (the size distribution model for the ETC) and the reference experimental results is obtained, even for high volume fractions, up to about 50%.

Observations of relative motion in a geodetic network in Ladakh, India, and across southern Tibet indicate slow shear on the Karakorum fault, rapid east-west extension across the whole of southern Tibet, and constant arc-normal convergence between India and southern Tibet along the Himalayan arc. Measurements of ten campaign-style and six permanent sites with global positioning system (GPS) precise geodesy provide these bounds on the style and rates of the large-scale deformation in the Tibet-Himalaya region. Divergence between sites at Leh, Ladakh, India, and Shiquanhe, western Tibet, as well as slow relative motion among sites within the Ladakh network, limit right-lateral slip parallel to the Karakorum fault to only 3.4 ± 5 mm/yr. This low rate concurs with a recent estimate of 3-4 mm/yr for Late Holocene time, but disagrees with the much higher rate of 30-35 mm/yr that has been used to argue for plate-like behavior of the Tibetan Plateau. Convergence between Ladakh and the Indian subcontinent at 18.8 ± 3 mm/yr at 224° ± 17° (1σ) differs little from estimates of convergence across the central segment of the Himalaya. Finally, lengthening of the baseline between Leh, Ladakh, and Lhasa (in southeastern Tibet) at 17.8 ± 1 mm/yr or between Leh and Bayi (farther to the southeast) at 18 ± 3 mm/yr, is consistent with an extrapolation of rates of east-west extension of the Tibetan Plateau based both on shorter GPS baselines (e.g., Lhasa-Simikot) and on diverging slip vectors of earthquakes in the Himalaya. We interpret these results to indicate that Tibet behaves more like a fl uid than like a plate.

Recent mechanical models for cloth simulation have evolved toward accurate representation of elastic stiffness based on continuum mechanics, converging to formulations that are largely analogous to fast finite element methods. In the context of tensile deformations, these formulations usually involve the linearization of tensors, so as to express linear elasticity in a simple way. However, this approach needs significant adaptations and approximations for dealing with the nonlinearities resulting from large cloth deformations. Toward our objective of accurately simulating the nonlinear properties of cloth, we show that this linearization can indeed be avoided and replaced by adapted strain-stress laws that precisely describe the nonlinear behavior of the material. This leads to highly streamlined computations that are particularly efficient for simulating the nonlinear anisotropic tensile elasticity of highly deformable surfaces. We demonstrate the efficiency of this method with exa...

Peridynamics is a formulation of continuum mechanics based on integral equations. It is a nonlocal model, accounting for the effects of long-range forces. Correspondingly, classical molecular dynamics is also a nonlocal model. Peridynamics and molecular dynamics have similar discrete computational structures, as peridynamics computes the force on a particle by summing the forces from surrounding particles, similarly to molecular dynamics. We demonstrate that the peridynamics model can be cast as an upscaling of molecular dynamics. Specifically, we address the extent to which the solutions of molecular dynamics simulations can be recovered by peridynamics. An analytical comparison of equations of motion and dispersion relations for molecular dynamics and peridynamics is presented along with supporting computational results.

This is a comprehensive review of local direct measurement shear stress transducers. Transducers are first classified by their movement, measuring mode, and mechanism. These categories are then subclassified into active or passive movement, static or dynamic measuring mode, and rotational or translational mechanisms. Over 80 transducers are reviewed and tabulated. Finally, sources of transducer error are analyzed. Primary sources of error are transducer and housing misalignment, material ingress around the active face, ..

Two complementary methodologies are described to quantify the effects of crack-tip stress triaxiality (constraint) on the macroscopic measures of elastic-plastic fracture toughness J and Crack-Tip Opening Displacement (CTOD). In the continuum mechanics methodology, two parameters J and Q suffice to characterize the full range of near-tip environments at the onset of fracture. J sets the size scale of the zone of high stresses and large deformations while Q scales the near-tip stress level relative to a high triaxiality reference stress state. The material's fracture resistance is characterized by a toughness locus Jc(Q) which defines the sequence of J-Q values at fracture determined by experiment from high constraint conditions (Q ~0) to low constraint conditions (Q < 0). A micromechanics methodology is described which predicts the toughness locus using crack-tip stress fields and critical J-values from a few fracture toughness tests. A robust micromechanics model for cleavage fracture has evolved from the observations of a strong, spatial self-similarity of crack-tip principal stresses under increased loading and across different fracture specimens. We explore the fundamental concepts of the J-Q description of crack-tip fields, the fracture toughness locus and micromechanics approaches to predict the variability of macroscopic fracture toughness with constraint under elastic-plastic conditions. Computational results are presented for a surface cracked plate containing a 6:1 semielliptical, a = t/4 flaw subjected to remote uniaxial and biaxial tension. Crack-tip stress fields consistent with the J-Q theory are demonstrated to exist at each location along the crack front. The micromechanics model employs the J-Q description of crack-front stresses to interpret fracture toughness values measured on laboratory specimens for fracture assessment of the surface cracked plate. Jo = ~/ hFB(0.1/0.0)"

Assume that the physics on the microscale (interactions between atoms,
molecules, defects in crystals . . . ) is understood.What is the appropriate mesoscale
continuum theory for the problem? What are the assumptions involved and how do
they define the limitations of the continuum model? To answer these questions, we
begin with the definition of mesoscale continuum kinematics from the microscale
kinematics. The geometry of micro-structure (e.g., order vs. disorder) has a decisive
role in defining the continuum kinematics. We thus arrive at three kinematic
formulations: mass continuum, lattice continuum and granular continuum. Then,
upon formulating the power balance, we use the principle of virtual power to
arrive at a variety of mathematical formulations: simple continuum with moving
boundaries, phase field formulation, and, a higher order, size-dependent continuum.
The problems considered include: mixing of fluids and capillary flows, granular
flow/deformation, and, polycrystalline diffusional/dislocation creep accompanied
by dislocation plasticity.

The objective of this paper is to investigate the accuracy of the elastic force models that can be used in the absolute nodal coordinate finite element formulation. This study focuses on the description of the elastic forces in three-dimensional beams. The elastic forces of the absolute nodal coordinate formulation can be derived using a continuum mechanics approach. This study investigates the accuracy and usability of such an approach for a three-dimensional absolute nodal coordinate beam element. This study also presents an improvement proposal for the use of a continuum mechanics approach in deriving the expression of the elastic forces in the beam element. The improvement proposal is verified using several numerical examples that show that the proposed elastic force model of the beam element agrees with the analytical results as well as with the solutions obtained using existing finite element formulation. In the beam element under investigation, global displacements and slopes are used as the nodal coordinates, which resulted in a large number of nodal degrees of freedom. This study provides a physical interpretation of the nodal coordinates used in the absolute nodal coordinate beam element. It is shown that a beam element based on the absolute nodal coordinate formulation relaxes the assumption of a rigid crosssection and is capable of representing a distortional deformation of the cross-section. The numerical results also imply that the beam element does not suffer from the phenomenon called shear locking.

With the eventual aim of describing flowing elasto-plastic materials, we focus on the elementary brick of such a flow, a plastic event, and compute the long-range perturbation it elastically induces in a medium submitted to a global shear strain. We characterize the effect of a nearby wall on this perturbation, and quantify the importance of finite size effects. Although for the sake of simplicity most of our explicit formulae deal with a 2D situation, our statements hold for 3D situations as well.