Showing papers on "Continuum mechanics published in 2009"

Nonlinear Finite Element Methods

Abstract: Nonlinear Phenomena.- Basic Equations of Continuum Mechanics.- Spatial Discretization Techniques.- Solution Methods for Time Independent Problems.- Solution Methods for Time Dependent Problems.- Stability Problems.- Adaptive Methods.- Special Structural Elements.- Special Finite Elements for Continua.- Contact Problems.- Automation of the Finite Element Method by J. Korelc.

Friction laws at the nanoscale

Abstract: Macroscopic laws of friction do not generally apply to nanoscale contacts. Although continuum mechanics models have been predicted to break down at the nanoscale, they continue to be applied for lack of a better theory. An understanding of how friction force depends on applied load and contact area at these scales is essential for the design of miniaturized devices with optimal mechanical performance. Here we use large-scale molecular dynamics simulations with realistic force fields to establish friction laws in dry nanoscale contacts. We show that friction force depends linearly on the number of atoms that chemically interact across the contact. By defining the contact area as being proportional to this number of interacting atoms, we show that the macroscopically observed linear relationship between friction force and contact area can be extended to the nanoscale. Our model predicts that as the adhesion between the contacting surfaces is reduced, a transition takes place from nonlinear to linear dependence of friction force on load. This transition is consistent with the results of several nanoscale friction experiments. We demonstrate that the breakdown of continuum mechanics can be understood as a result of the rough (multi-asperity) nature of the contact, and show that roughness theories of friction can be applied at the nanoscale.

Nonlinear mechanics of single-atomic-layer graphene sheets

Abstract: The unique lattice structure and properties of graphene have drawn tremendous interests recently. By combining continuum and atomistic approaches, this paper investigates the mechanical properties of single-atomic-layer graphene sheets. A theoretical framework of nonlinear continuum mechanics is developed for graphene under both in-plane and bending deformation. Atomistic simulations are carried out to deduce the effective mechanical properties. It is found that graphene becomes highly nonlinear and anisotropic under finite-strain uniaxial stretch, and coupling between stretch and shear occurs except for stretching in the zigzag and armchair directions. The theoretical strength (fracture strain and fracture stress) of perfect graphene lattice also varies with the chiral direction of uniaxial stretch. By rolling graphene sheets into cylindrical tubes of various radii, the bending modulus of graphene is obtained. Buckling of graphene ribbons under uniaxial compression is simulated and the critical strain for the onset of buckling is compared to a linear buckling analysis.

Linearized theory of peridynamic states.

Abstract: A state-based peridynamic material model describes internal forces acting on a point in terms of the collective deformation of all the material within a neighborhood of the point. In this paper, the response of a state-based peridynamic material is investigated for a small deformation superposed on a large deformation. The appropriate notion of a small deformation restricts the relative displacement between points, but it does not involve the deformation gradient (which would be undefined on a crack). The material properties that govern the linearized material response are expressed in terms of a new quantity called the modulus state. This determines the force in each bond resulting from an incremental deformation of itself or of other bonds. Conditions are derived for a linearized material model to be elastic, objective, and to satisfy balance of angular momentum. If the material is elastic, then the modulus state is obtainable from the second Frechet derivative of the strain energy density function. The equation of equilibrium with a linearized material model is a linear Fredholm integral equation of the second kind. An analogue of Poincare’s theorem is proved that applies to the infinite dimensional space of all peridynamic vector states, providing a condition similar to irrotationality in vector calculus.

Variational Principles of Continuum Mechanics

Abstract: Some Applications of Variational Methods to Development.- Theory of Elastic Plates and Shells.- Elastic Beams.- Some Stochastic Variational Problems.- Homogenization.- Homogenization of Random Structures: a Closer View.- Some Other Applications.

Peridynamics as an Upscaling of Molecular Dynamics

Abstract: 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.

Small-scale effect on the free in-plane vibration of nanoplates by nonlocal continuum model

Abstract: In the present paper, small-scale effects on the free in-plane vibration (FIV) of nanoplates are investigated employing nonlocal continuum mechanics. Equations of motion of the nonlocal plate model for the aforementioned study are derived and presented. Explicit relations for natural frequencies are obtained through direct separation of variables. It has been shown that nonlocal effects are quite significant in in-plane vibration studies and need to be included in the continuum model of nanoplates such as in graphene sheets.

Morphogenesis of thin hyperelastic plates: A constitutive theory of biological growth in the Föppl-von Kármán limit

Abstract: The shape of plants and other living organisms is a crucial element of their biological functioning. Morphogenesis is the result of complex growth processes involving biological, chemical and physical factors at different temporal and spatial scales. This study aims at describing stresses and strains induced by the production and reorganization of the material. The mechanical properties of soft tissues are modeled within the framework of continuum mechanics in finite elasticity. The kinematical description is based on the multiplicative decomposition of the deformation gradient tensor into an elastic and a growth term. Using this formalism, the authors have studied the growth of thin hyperelastic samples. Under appropriate assumptions, the dimensionality of the problem can be reduced, and the behavior of the plate is described by a two-dimensional surface. The results of this theory demonstrate that the corresponding equilibrium equations are of the Foppl–von Karman type where growth acts as a source of mean and Gaussian curvatures. Finally, the cockling of paper and the rippling of a grass blade are considered as two examples of growth-induced pattern formation.

Defect evolution and pore collapse in crystalline energetic materials

Abstract: This work examines the use of crystal based continuum mechanics in the context of dynamic loading. In particular, we examine model forms and simulations which are relevant to pore collapse in crystalline energetic materials. Strain localization and the associated generation of heat are important for the initiation of chemical reactions in this context. The crystal mechanics based model serves as a convenient testbed for the interactions among wave motion, slip kinetics, defect generation kinetics and physical length scale. After calibration to available molecular dynamics and single crystal gas gun data for HMX (octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine), the model is used to predict behaviors for the collapse of pores under various conditions. Implications for experimental observations are discussed.

Dynamical density functional theory for molecular and colloidal fluids: a microscopic approach to fluid mechanics.

Abstract: In recent years, a number of dynamical density functional theories (DDFTs) have been developed for describing the dynamics of the one-body density of both colloidal and atomic fluids. In the colloidal case, the particles are assumed to have stochastic equations of motion and theories exist for both the case when the particle motion is overdamped and also in the regime where inertial effects are relevant. In this paper, we extend the theory and explore the connections between the microscopic DDFT and the equations of motion from continuum fluid mechanics. In particular, starting from the Kramers equation, which governs the dynamics of the phase space probability distribution function for the system, we show that one may obtain an approximate DDFT that is a generalization of the Euler equation. This DDFT is capable of describing the dynamics of the fluid density profile down to the scale of the individual particles. As with previous DDFTs, the dynamical equations require as input the Helmholtz free energy functional from equilibrium density functional theory (DFT). For an equilibrium system, the theory predicts the same fluid one-body density profile as one would obtain from DFT. Making further approximations, we show that the theory may be used to obtain the mode coupling theory that is widely used for describing the transition from a liquid to a glassy state.

On the method of virtual power in continuum mechanics

Abstract: The method of virtual power is generally used to produce balance equations for nontraditional continua such as continua with various types of microstructure. Here I show that the expression of the internal power can be deduced from that of the external power using a general invariance requirement due to Noll and a generalized version of Cauchy’s tetrahedron theorem. In other words, the measures of deformation and stress, as well as the balance equations, are determined by the expression chosen for the external power and by the invariance assumptions. A pair of examples taken from the literature shows that both ingredients are essential for defining a specific class of continua.

Fractal solids, product measures and fractional wave equations

Abstract: This paper builds on the recently begun extension of continuum thermomechanics to fractal porous media that are specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d and a resolution length scale R. The focus is on pre-fractal media (i.e. those with lower and upper cut-offs) through a theory based on a dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D, d and R. This procedure allows a specification of a geometry configuration of continua by ‘fractal metric’ coefficients, on which the continuum mechanics is subsequently constructed. While all the derived relations depend explicitly on D, d and R, upon setting D = 3 and d = 2, they reduce to conventional forms of governing equations for continuous media with Euclidean geometries. Whereas the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure, making it capable of grasping local fractal anisotropy. Finally, the one-, two- and three-dimensional wave equations are developed, showing that the continuum mechanics approach is consistent with that obtained via variational energy principles.

A continuum mechanics based derivation of Reissner’s large-displacement finite-strain beam theory: the case of plane deformations of originally straight Bernoulli–Euler beams

Abstract: In the present paper, we present a continuum mechanics based derivation of Reissner’s equations for large-displacements and finite-strains of beams, where we restrict ourselves to the case of plane deformations of originally straight Bernoulli–Euler beams. For the latter case of extensible elastica, we succeed in attaching a continuum mechanics meaning to the stress resultants and to all of the generalized strains, which were originally introduced by Reissner at the beam-theory level. Our derivations thus circumvent the problem of needing to determine constitutive relations between stress resultants and generalized strains by physical experiments. Instead, constitutive relations at the stress–strain level can be utilized. Subsequently, this is exemplarily shown for a linear relation between Biot stress and Biot strain, which leads to linear constitutive relations at the beam-theory level, and for a linear relation between the second Piola–Kirchhoff stress and the Green strain, which gives non-linear constitutive relations at the beam theory level. A simple inverse method for analytically constructing solutions of Reissner’s non-linear relations is shortly pointed out in Appendix I.

Mechanics of Continua

Abstract: So far we have treated only point-like particles; an exception was the treatment of the rigid body in Chap. 8. In this chapter we will investigate deformable bodies, for a twofold reason: (1) If a body is not rigid but deformable, it has its own internal dynamics, which is independent of its state of motion (disregarding inertial forces), as was investigated, e.g., in Chap. 8. (2) The continuum mechanics is a first example of a field theory. A more detailed treatment of a field theory is the subject of the Maxwell theory of the electromagnetic fields.2

Elements of Continuum Mechanics

Abstract: Continuum mechanics is concerned with the motion and deformation of continuous bodies (for instance, a glacier). A body consists of an infinite number of material elements, called particles. For any time t, each particle is identified by a position vector x (relative to a prescribed origin O) in the physical space e, and the continuous set of position vectors for all particles of the body is called a configuration κ of the body. If t is the actual time, the corresponding configuration is called the present configuration κ t . In addition, we define a reference configuration κ r which refers to a fixed (or initial) time t 0.

Assessing continuum postulates in simulations of granular flow

Abstract: Continuum mechanics relies on the fundamental notion of a mesoscopic volume"element" in which properties averaged over discrete particles obey deterministic relationships. Recent work on granular materials suggests a continuum law may be inapplicable, revealing inhomogeneities at the particle level, such as force chains and slow cage breaking. Here, we analyze large-scale three-dimensional Discrete-Element Method (DEM) simulations of different granular flows and show that an approximate"granular element" defined at the scale of observed dynamical correlations (roughly three to five particle diameters) has a reasonable continuum interpretation. By viewing all the simulations as an ensemble of granular elements which deform and move with the flow, we can track material evolution at a local level. Our results confirm some of the hypotheses of classical plasticity theory while contradicting others and suggest a subtle physical picture of granular failure, combining liquid-like dependence on deformation rate and solid-like dependence on strain. Our computational methods and results can be used to guide the development of more realistic continuum models, based on observed local relationships betweenaverage variables.

More clarity on the concept of material frame-indifference in classical continuum mechanics

Abstract: There was and still is a considerable amount of confusion in the community of classical continuum mechanics on the concept of material frame-indifference. An extensive review is presented which will point out and try to resolve various misconceptions that still accompany the literature of material frame-indifference. With the tools of differential geometry a precise terminology is developed ending in a consequent mathematical framework, in which not only the concept of material frame-indifference can be formulated naturally, but showing advantages that go beyond all conventional considerations on invariance used so far in classical continuum mechanics. As an exemplification the Navier-Stokes equations and the corresponding Reynolds averaged equations are written in a general covariant form within Newtonian mechanics.

Elastic waves propagation in 1D fractional non-local continuum

Abstract: Aim of this paper is the study of waves propagation in a fractional, non-local 1D elastic continuum. The non-local effects are modeled introducing long-range central body interactions applied to the centroids of the infinitesimal volume elements of the continuum. These non-local interactions are proportional to a proper attenuation function and to the relative displacements between non-adjacent elements. It is shown that, assuming a power-law attenuation function, the governing equation of the elastic waves in the unbounded domain, is ruled by a Marchaud-type fractional differential equation. Wave propagation in bounded domain instead involves only the integral part of the Marchaud fractional derivative. The dispersion of elastic waves, as well as waves propagation in unbounded and bounded domains are discussed in detail.

Quantifying the size-dependent effect of the residual surface stress on the resonant frequencies of silicon nanowires if finite deformation kinematics are considered

Abstract: There are two major objectives to the present work. The first objective is to demonstrate that, in contrast to predictions from linear surface elastic theory, when nonlinear, finite deformation kinematics are considered, the residual surface stress does impact the resonant frequencies of silicon nanowires. The second objective of this work is to delineate, as a function of nanowire size, the relative contributions of both the residual (strain-independent) and the surface elastic (strain-dependent) parts of the surface stress to the nanowire resonant frequencies. Both goals are accomplished by using the recently developed surface Cauchy-Born model, which accounts for nanoscale surface stresses through a nonlinear, finite deformation continuum mechanics model that leads to the solution of a standard finite element eigenvalue problem for the nanowire resonant frequencies. In addition to demonstrating that the residual surface stress does impact the resonant frequencies of silicon nanowires, we further show that there is a strong size dependence to its effect; in particular, we find that consideration of the residual surface stress alone leads to significant errors in predictions of the nanowire resonant frequency, with an increase in error with decreasing nanowire size. Correspondingly, the strain-dependent part of the surface stress is found to have an increasingly important effect on the resonant frequencies of the nanowires with decreasing nanowire size.

Analysis of micropolar elastic beams

Abstract: In this paper, a linear theory for the analysis of beams based on the micropolar continuum mechanics is developed. Power series expansions for the axial displacement and micro-rotation fields are assumed. The governing equations are derived by integrating the momentum and moment of momentum equations in the micropolar continuum theory. Body couples and couple stresses can be supported in this theory. After some simplifications, this theory can be reduced to the well-known Timoshenko and Euler–Bernoulli beam theories. The nature of flexural and longitudinal waves in the infinite length micropolar beam has been investigated. This theory predicts the existence of micro-rotational waves which are not present in any of the known beam theories based on the classical continuum mechanics. Also, the deformation of a cantilever beam with transverse concentrated tip loading has been studied. The pattern of deflection of the beam is similar to the classical beam theories, but couple stress and micro-rotation show an oscillatory behavior along the beam for various loadings.

Extremum and variational principles for elastic and inelastic media with fractal geometries

Abstract: This paper further continues the recently begun extension of continuum mechanics and thermodynamics to fractal porous media which are specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d, and a resolution lengthscale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through a theory based on dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. In effect, the global forms of governing equations may be cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D, d and R. Here we first generalize the principles of virtual work, virtual displacement and virtual stresses, which in turn allow us to extend the minimum energy theorems of elasticity theory. Next, we generalize the extremum principles of elasto-plastic and rigid-plastic bodies. In all the cases, the derived relations depend explicitly on D, d and R, and, upon setting D = 3 and d = 2, they reduce to conventional forms of governing equations for continuous media with Euclidean geometries.