Showing papers on "Continuum mechanics published in 2003"

Microstructure of martensite : why it forms and how it gives rise to the shape-memory effect

Abstract: 1. Introduction 2. Review of Continuum Mechanics 3. Continuum Theory of Crystalline Solids 4. Martensitic Phase Transformation 5. Twinning in Martensite 6. Origin of Microstructure 7. Special Microstructures 8. Analysis of Microstructure 9. The Shape-Memory Effect 10. Thin Films 11. Geometrically Linear Theory 12. Piece-wise Linear Elasticity 13. Polycrystals

Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics

Abstract: A model, based on the theory of nonlocal continuum mechanics, on the column buckling of multiwalled carbon nanotubes is presented. The present analysis considers that each of the nested concentric tubes is an individual column and that the deflection of all the columns is coupled together through the van der Waals interactions between adjacent tubes. Based on this description, a condition is derived in terms of the parameters that describe the van der Waals forces and the small internal length scale effects. In particular, an explicit expression is derived for the critical axial strain of a double walled carbon nanotube which clearly demonstrates that small scale effects contribute significantly to the mechanical behavior of multiwalled carbon nanotubes and cannot be ignored.

Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model

Abstract: An analytical model based on a molecular mechanics approach is presented to relate the elastic properties of a single-walled carbon nanotube to its atomic structure. We derive closed-form expressions for elastic modulus and Poisson's ratio as a function of the nanotube diameter. Properties at different length scales are directly connected via these expressions. The analytically calculated elastic properties for achiral nanotubes using force constants obtained from experimental data of graphite are compared to those based on tight binding numerical calculations. This study represents a preliminary effort to develop analytical methods of molecular mechanics for applications in nanostructure modeling.

Molecular scale contact line hydrodynamics of immiscible flows.

Abstract: From extensive molecular dynamics simulations on immiscible two-phase flows, we find the relative slipping between the fluids and the solid wall everywhere to follow the generalized Navier boundary condition, in which the amount of slipping is proportional to the sum of tangential viscous stress and the uncompensated Young stress. The latter arises from the deviation of the fluid-fluid interface from its static configuration. We give a continuum formulation of the immiscible flow hydrodynamics, comprising the generalized Navier boundary condition, the Navier-Stokes equation, and the Cahn-Hilliard interfacial free energy. Our hydrodynamic model yields interfacial and velocity profiles matching those from the molecular dynamics simulations at the molecular-scale vicinity of the contact line. In particular, the behavior at high capillary numbers, leading to the breakup of the fluid-fluid interface, is accurately predicted.

Description of Elastic Forces in Absolute Nodal Coordinate Formulation

Abstract: 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 cross-section 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.

Multiple-choice test of energy and momentum concepts

Abstract: We investigate student understanding of energy and momentum concepts at the level of introductory physics by designing and administering a 25-item multiple choice test and conducting individual interviews. We find that most students have difficulty in qualitatively interpreting basic principles related to energy and momentum and in applying them in physical situations.

The Mode III Crack Problem in Microstructured Solids Governed by Dipolar Gradient Elasticity: Static and Dynamic Analysis

Abstract: This study aims at determining the elastic stress and displacement fields around a crack in a microstructured body under a remotely applied loading of the antiplane shear (mode III) type. The material microstructure is modeled through the Mindlin-Green-Rivlin dipolar gradient theory (or strain-gradient theory of grade two). A simple but yet rigorous version of this generalized continuum theory is taken here by considering an isotropic linear expression of the elastic strain-energy density in antiplane shearing that involves only two material constants (the shear modulus and the so-called gradient coefficient). In particular, the strain-energy density function, besides its dependence upon the standard strain terms, depends also on strain gradients. This expression derives from form Il of Mindlin's theory, a form that is appropriate for a gradient formulation with no couple-stress effects (in this case the strain-energy density function does not contain any rotation gradients). Here, both the formulation of the problem and the solution method are exact and lead to results for the near-tip field showing significant departure from the predictions of the classical fracture mechanics. In view of these results, it seems that the conventional fracture mechanics is inadequate to analyze crack problems in microstructured materials. Indeed, the present results suggest that the stress distribution ahead of the tip exhibits a local maximum that is bounded. Therefore, this maximum value may serve as a measure of the critical stress level at which further advancement of the crack may occur. Also, in the vicinity of the crack tip, the crack-face displacement closes more smoothly as compared to the classical results. The latter can be explained physically since materials with microstructure behave in a more rigid way (having increased stiffness) as compared to materials without microstructure (i.e., materials governed by classical continuum mechanics). The new formulation of the crack problem required also new extended definitions for the J-integral and the energy release rate. It is shown that these quantities can be determined through the use of distribution (generalized function) theory. The boundary value problem was attacked by both the asymptotic Williams technique and the exact Wiener-Hopf technique. Both static and time-harmonic dynamic analyses are provided.

Dynamic fracture modeling with a meshfree peridynamic code

Abstract: Publisher Summary The peridynamic model is an alternate theory of continuum mechanics that is specifically oriented toward modeling problems, in which cracks or other discontinuities emerge spontaneously as a body deforms under load. In this study, a code that implements this theory is applied to the Kalthoff-Winkler dynamic single-fracture experiment in a tough steel specimen. Many problems of fundamental importance in mechanics involve the spontaneous emergence of discontinuities, such as cracks, in the interior of a body. The classical theory of continuum mechanics is in some ways suited to modeling this type of problem because the theory uses partial differential equations as a mathematical description. Although much work has been devoted to special techniques aimed at working around this problem—particularly in the theory of fracture mechanics—these techniques are not fully satisfactory either in principle or in practice as general descriptions of fracture. This difficulty is inherited by numerical methods that implement the classical theory, including almost all finite-element and finite-difference codes in common usage.

Deriving a particle system from continuum mechanics for the animation of deformable objects

Abstract: Mass-spring and particle systems have been widely employed in computer graphics to model deformable objects because they allow fast numerical solutions In this work, we establish a link between these discrete models and classical mathematical elasticity It turns out that discrete systems can be derived from a continuum model by a finite difference formulation and approximate classical continuum models unless the deformations are large In this work, we present the derivation of a particle system from a continuum model, compare it to the models of classical elasticity theory, and assess its accuracy In this way, we gain insight into the way discrete systems work and we are able to specify the correct scaling when the discretization is changed Physical material parameters that describe materials in continuum mechanics are also used in the derived particle system

Identification of mechanical properties by displacement field measurement: A variational approach

Abstract: We study a parameter identification problem associated with a two-dimensional mechanical problem. In the first part, the experimental technique of determining the displacement field is briefly presented. The variational method proposed herein is based on the minimization of either a separately convex functional or a convex functional that leads to the reconstruction of the elastic tensor and the stress field. These two reconstructed fields are continuous and piecewise linear on a triangulation of the two-dimensional domain. Some numerical and experimental examples are presented to test the performance of the algorithms.

Constitutive relations for the interaction force in multicomponent particulate flows

Abstract: 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 diffusive) force. We propose a constitutive relation which is general and frame-indifferent and thus suitable for use in many flow conditions. At the end, we provide an alternative approach for finding the drag force on a particle in a particulate mixture. This approach has been used in the non-Newtonian fluid mechanics to find the drag force on surfaces.

Advanced mechanics of solids

Abstract: 1 Basic Concepts of Continuum Mechanics- 2 Elastic Material- 3 The Theory of Simple Beams I- 4 Torsion of Prismatic Bars- 5 Curved Beams- 6 Simple Beams II: Energy Principles- 7 Two-dimensional Problems- 8 Plates and Shells- 9 Stability of Equilibrium- 10 Some Basic Concepts of Dynamics- 11 Oscillators with One Degree of Freedom- 12 Systems of Several Degrees of Freedom- 13 Answers to the Exercises

Non-local coupling of viscoplasticity and anisotropic viscodamage for impact problems using the gradient theory

Abstract: A general framework for the analysis of heterogeneous media that assesses a strong coupling between viscoplasticity and anisotropic viscodamage evolution is formulated for-impact related problems within the framework of thermodynamic laws and nonlinear continuum mechanics. The proposed formulations include thermo-elasto-viscoplastici- ty with anisotropic thermo-elasto-viscodamage, a dynamic yield criterion of a von Mises type and a dynamic viscodamage criterion, the associated flow rules, non-linear strain hardening, strain-rate hardening, and temperature softening. The constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material. That is, the damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses are replaced by the effective stresses. The evolution laws are impeded in a finite deformation framework based on the multiplicative decomposition of the deformation gradient into elastic, viscoplastic, and viscodamage parts. Since the material macroscopic thermomechanical response under high-impact loading is governed by different physical mechanisms on the macroscale level, the proposed three-dimensional kinematical model is introduced with manifold structure accounting for discontinuous fields of dislocation interactions (plastic hardening), and crack and void interactions (damage hardening). The non-local theory of viscoplasticity and viscodamage that incorporates macroscale interstate variables and their higher-order gradients is used here to describe the change in the internal structure and in order to investigate the size effect of statistical inhomogeneity of the evolution-related viscoplasticity and viscodamage hardening variables. The gradients are introduced here in the hardening internal state variables and are considered to be independent of their local counterparts. It also incorporates the thermomechanical coupling effects as well as the internal dissipative effects through the rate-type covariance constitutive structure with a finite set of internal state variables. The model presented in this paper can be considered as a framework, which enables one to derive various non-local and gradient viscoplasticity and viscodamage theories by introducing simplifying assumptions.

Computer Simulation of the Mechanical Properties of Amorphous Polymer Nanostructures

Abstract: The mechanical properties of polymeric nanostructures have been investigated by means of Monte Carlo and molecular dynamics simulations. A continuum mechanics model was applied to determine an apparent modulus for such structures from results of virtual deformation simulations. A recently proposed method based on strain fluctuations was also used to calculate the elasticity tensor for the structures. The effect of size on the apparent modulus of ultra-small structures was explored. Our results indicate that, for a given system at a specified temperature, the modulus of a small structure can be significantly smaller than that of the bulk material. Furthermore, in small systems the elastic constants are shown to become anisotropic.

Tensor Analysis and Continuum Mechanics

Abstract: Preface. 1: Tensors. 1. First steps with tensors. 2. Operations of tensors. 3. Euclidean vector space. 4. Exterior algebra. 5. Point spaces. Exercises. 2 : Lagrangian and Eulerian Descriptions. 1. Lagrangian description. 2. Eulerian description. Exercises. 3 : Deformations. 1. Homogeneous transformation. 2. Tangential homogeneous transformation. 3. Infinitesimal transformation. Exercises. 4: Kinematics of Continua. 1. Lagrangian kinematics. 2. Eulerian kinematics. 3. Material derivatives of circulation, flux, and volume. Exercises. 5: Fundamental Laws: Principle of Virtual Work. 1. Conservation of mass and continuity equation. 2. Fundamental laws of dynamics. 3. Theorem of kinetic energy. 4. Study of stresses. 5. Principle of virtual work. 6. Thermomechanics and balance equations. Exercises. 6: Linear Elasticity. 1. Elasticity and tests. 2. Generalized Hooke's law in linear elasticity. 3. Equations and principles in elastostatics. 4. Classical problems. Exercises. Summary of Formulae. Bibliography. Glossary of Symbols. Index.

A hierarchical multi-physics model for design of high toughness steels

Abstract: In support of the computational design of high toughness steels as hierarchically structured materials, a multiscale, multiphysics methodology is developed for a `ductile fracture simulator.' At the nanometer scale, the method unites continuum mechanics with quantum physics, using first-principles calculations to predict the force-distance laws for interfacial separation with both normal and plastic sliding components. The predicted adhesion behavior is applied to the description of interfacial decohesion for both micron-scale primary inclusions governing primary void formation and submicron-scale secondary particles governing microvoid-based shear localization that accelerates primary void coalescence. Fine scale deformation is described by a `Particle Dynamics' method that extends the framework of molecular dynamics to multi-atom aggregates. This is combined with other meshfree and finite-element methods in two-level cell modeling to provide a hierarchical constitutive model for crack advance, combining conventional plasticity, microstructural damage, strain gradient effects and transformation plasticity from dispersed metastable austenite. Detailed results of a parallel experimental study of a commercial steel are used to calibrate the model at multiple scales. An initial application provides a Toughness-Strength-Adhesion diagram defining the relation among alloy strength, inclusion adhesion energy and fracture toughness as an aid to microstructural design. The analysis of this paper introduces an approach of creative steel design that can be stated as the exploration of the effective connections among the five key-components: elements selection, process design, micro/nanostructure optimization, desirable properties and industrial performance by virtue of innovations and inventions.

Plastic slip distribution in two-phase laminate microstructures: Dislocation-based versus generalized-continuum approaches

Abstract: The size-dependent mechanical response of a simple model microstructure is investigated using continuum dislocation-based, Cosserat and strain-gradient models of crystal plasticity. The governing equations and closed-form analytical solutions for plastic slip and lattice rotation are directly compared. The microstructure consists of a periodic succession of hard (elastic) and soft (elastoplastic single-crystal) layers, subjected to single glide perpendicular to the layers. In the dislocation-based approach, inhomogeneous plastic deformation and lattice rotation are shown to develop in the soft channels, either because of bowing of dislocations or owing to pile-up formation. The generalized continuum non-local models are found to be able to reproduce the plastic slip and lattice rotation distribution. In particular, a correspondence was found between the generalized-continuum results and line tension effects; the additional or higher- order balance equations introduced in the non-local models turn out to b...

The effects of wall and rolling resistance on the couple stress of granular materials in vertical flow

Abstract: The couple stress of a three-dimensional vertical granular flow is investigated by means of a combined approach of discrete element method and averaging method. The velocity, mass density and couple stress are quantified under various flow conditions. The velocity and mass density profiles are illustrated to be consistent with those obtained by the previous experiments and numerical simulations, confirming the validity of the proposed approach. The couple stress profiles are shown to be significantly affected by the wall supporting the vertical flow, and be contributed by the rolling resistance due to the asymmetrical normal traction distributions in the contact areas between particles and between particle and wall. For mono-sized particles, the couple stress far from the wall can be ignored although it may vary slightly causing the fluctuation of flow behavior; however, the couple stress in the region close to a wall must be taken into account to properly describe the flow behavior of particles. For multi-sized particle, the couple stress is mainly contributed by the sliding resistance and to a less degree by the rolling resistance; the transport of particle plays a limited role. Implication of the present numerical results to continuum modeling is also discussed.

Variational time integrators in computational solid mechanics

Abstract: This thesis develops the theory and implementation of variational integrators for computational solid mechanics problems, and to some extent, for fluid mechanics problems as well. Variational integrators for finite dimensional mechanical systems are succinctly reviewed, and used as the foundations for the extension to continuum systems. The latter is accomplished by way of a space-time formulation for Lagrangian continuum mechanics that unifies the derivation of the balance of linear momentum, energy and configurational forces, all of them as Euler-Lagrange equations of an extended Hamilton's principle. In this formulation, energy conservation and the path independence of the J- and L-integrals are conserved quantities emanating from Noether's theorem. Variational integrators for continuum mechanics are constructed by mimicking this variational structure, and a discrete Noether's theorem for rather general space-time discretizations is presented. Additionally, the algorithms are automatically (multi)symplectic, and the (multi)symplectic form is uniquely defined by the theory. For instance, in nonlinear elastodynamics the algorithms exactly preserve linear and angular momenta, whenever the continuous system does. A class of variational algorithms is constructed, termed asynchronous variational integrators (AVI), which permit the selection of independent time steps in each element of a finite element mesh, and the local time steps need not bear an integral relation to each other. The conservation properties of both synchronous and asynchronous variational integrators are discussed in detail. In particular, AVI are found to nearly conserve energy both locally and globally, a distinguishing feature of variational integrators. The possibility of adapting the elemental time step to exactly satisfy the local energy balance equation, obtained from the extended variational principle, is analyzed. The AVI are also extended to include dissipative systems. The excellent accuracy, conservation and convergence characteristics of AVI are demonstrated via selected numerical examples, both for conservative and dissipative systems. In these tests AVI are found to result in substantial speedups, at equal accuracy, relative to explicit Newmark. In elastostatics, the variational structure leads to the formulation of discrete path-independent integrals and a characterization of the configurational forces acting in discrete systems. A notable example is a discrete, path-independent J-integral at the tip of a crack in a finite element mesh.

A method for linking thermally activated dislocation mechanisms of yielding with continuum plasticity theory

Abstract: This work combines classical continuum mechanics, the continuously dislocated continuum theory as developed by Kroner and Bilby with discrete dislocation theory to develop quantities that permit models involving interactions between individual dislocations to be incorporated into a description of multiaxial yielding of a material. Two quantities distinguish this approach from earlier efforts: firstly, a dislocation mobility tensor relating the velocity of a dislocation configuration to the net Peach–Koehler force on the configuration and, secondly, a vector quantity representing the dislocation content of the materials. The theory of thermally activated motion of dislocations past obstacles is employed to relate the dislocation velocity to stress by a stress-dependent mobility tensor whose components are determined by the nature of the interaction of the moving dislocation with the obstacle. An example is presented in which the obstacle is a forest dislocation that affects a gliding dislocation through mu...

On `best\' shell models – From classical shells, degenerated and multi-layered concepts to 3D

Abstract: Problems of solid mechanics are most generally formulated within 3D continuum mechanics. However, engineering models favor reduced dimensions, in order to portray mechanical properties by surface or curvilinear approximations. Such attempts for dimensional reduction constitute interactions between theoretical formulations and numerical techniques. A classical reduced model for thin bodies is represented by shell theory, an approximation in terms of resultants and first-order moments. If the shell theory, with its inherent errors, is considered as qualitatively insufficient for a particular problem, a further improvement is given by solid shell models, which are gained by direct linear interpolation of the 3D kinematic relations. They improve considerably the analytic capabilities for shells, especially when their congenital locking effects are handled by variational `convergence tricks'. The next step towards 3D quality are layered shells or solid shell elements. The present paper compares these three approximation stages from the point of view of multi-director (integral) transformations of classical continuum mechanics. It offers physical convergence requirements for each of the treated models.

Geometrically non‐linear anisotropic inelasticity based on fictitious configurations: Application to the coupling of continuum damage and multiplicative elasto‐plasticity

Abstract: The objective of this contribution is the formulation and algorithmic treatment of a phenomenological framework to capture anisotropic, geometrically non-linear inelasticity. In addition to the intermediate configuration of multiplicative elasto-plasticity, we further introduce two microscopic configurations of Lagrangian and Eulerian type which characterize the so-called fictitious undamaged material. This kinematical framework enables us to apply two well-established postulates based on standard terminology in non-linear continuum mechanics. Concerning the free energy function, the postulate of strain energy equivalence is adopted and in view of the plastic dissipation potential the concept of effective stress is a natural outcome of the underlying kinematical assumptions. Finally, we focus on the integration technique for the class of obtained evolution equations and present numerical examples for a prototype model to underline the applicability of the proposed framework. Copyright © 2003 John Wiley & Sons, Ltd.