Formulation Of Continuum Mechanics Research Articles

Viscoelastic material models in peridynamics

In this paper we propose a new viscoelastic material model within the framework of the nonlocal peridynamic formulation of continuum mechanics. Using Fourier- and Laplace-transforms we derive integral-representation formulas using a Green’s function approach for both local and nonlocal viscoelasticity. As an example we calculate the local and nonlocal response of an infinite viscoelastic bar impacted by a point load. We show both analytically and numerically that local viscoelasticity is recovered in the limit as the peridynamic lengthscales become small.

Damage and Failure Process of Concrete Structure under Uniaxial Compression Based on Peridynamics Modeling

Peridynamics is a nonlocal formulation of continuum mechanics, which uses integral formulation rather than the spatial partial differential equations. The peridynamic approach avoids using any spatial derivatives, which arise naturally in the classical local theory. It has shown effectiveness and advantage in solving discontinuous problems at both macro- and microscales. In this paper, the peridynamic theory is used to analyze damage and progressive failure of concrete structures. A nonlocal peridynamic model for concrete columns under uniaxial compression is developed. Numerical example illustrates that cracks in a peridynamic body of concrete form spontaneously. The result of the example clarifies the unique advantage of modeling damage accumulation and progressive failure of concrete based on peridynamic theory. This study provides a new promising alternative for analyzing complicated discontinuity problems. Finally, some open problems and future research trends in peridynamics are discussed.

A new approach to model cross-linked actin networks: Multi-scale continuum formulation and computational analysis

The mechanical properties of a cell are defined mainly by the cytoskeleton. One contributor within this three-dimensional structure is the actin cortex which is located underneath the lipid bilayer. It forms a nearly isotropic and densely cross-linked protein network. We present a continuum mechanical formulation for describing the mechanical properties of in vitro model systems based on their micro-structure, i.e. the behavior of a single filament and its spatial arrangement. The network is considered elastic, viscous effects being neglected. Filamentous actin is a biopolymer with a highly nonlinear force–stretch relationship. This can be well described by a worm-like chain model that includes extensibility of the filament, which we call the β-model. A comparison with experimental data shows good agreement with values for the physically interpretable parameters. To make these properties applicable to three dimensions we used a non-affine micro-sphere network, which accounts for filaments, equally distributed in space. The assembled model results in a strain-energy density which is a function of the deformation gradient, and it is validated with experimental data from rheological experiments of in vitro reconstituted actin networks. The Cauchy stress and elasticity tensors are obtained within the continuum mechanics framework and implemented into a finite element program to solve boundary-value problems.

A strain space framework for numerical hyperplasticity

Numerical simulations of high strain rate plastic flow have historically been built in a hypoelastic framework and use radial return (Wilkins’ method) as the solution algorithm. We show how each of these choices can lead to inaccurate and possibly nonconvergent results. We describe an alternative solution procedure based on a simple multiple time scale perturbation theory that is stable, accurate, computationally efficient and simple to implement. Further extension of these results then leads to a strain space formulation that has additional computational advantages. We illustrate our development with numerical experiments.This paper is dedicated to my friend and colleague Christo Christov on the occasion of his 60th birthday, in recognition of his many important and creative contributions to the formulation of continuum mechanics.

Nonlocal continuum mechanics formulation for axial, flexural, shear and contraction coupled wave propagation in single walled carbon nanotubes

This paper presents the effect of nonlocal scaling parameter on the coupled i.e., axial, flexural, shear and contraction, wave propagation in single-walled carbon nanotubes (SWCNTs). The axial and transverse motion of SWCNT is modeled based on first order shear deformation theory (FSDT) and thickness contraction. The governing equations are derived based on nonlocal constitutive relations and the wave dispersion analysis is also carried out. The studies shows that the nonlocal scale parameter introduces certain band gap region in all wave modes where no wave propagation occurs. This is manifested in the wavenumber plots as the region where the wavenumber tends to infinite or wave speed tends to zero. The frequency at which this phenomenon occurs is called the escape frequency. Explicit expressions are derived for cut-off and escape frequencies of all waves in SWCNT. It is also shown that the cut-off frequencies of shear and contraction mode are independent of the nonlocal scale parameter. The results provided in this article are new and are useful guidance for the study and design of the next generation of nanodevices that make use of the coupled wave propagation properties of single-walled carbon nanotubes.

Quantum continuum mechanics in a strong magnetic field

We extend a recent formulation of quantum continuum mechanics [J. Tao et. al, Phys. Rev. Lett. {\bf 103}, 086401 (2009)] to many-body systems subjected to a magnetic field. To accomplish this, we propose a modified Lagrangian approach, in which motion of infinitesimal volume elements of the system is referred to the "quantum convective motion" that the magnetic field produces already in the ground-state of the system. In the linear approximation, this approach results in a redefinition of the elastic displacement field $\uv$, such that the particle current $\jv$ contains both an electric displacement and a magnetization contribution: $\jv=\jv_0+n_0\partial_t \uv+\nabla \times (\jv_0\times \uv)$, where $n_0$ and $\jv_0$ are the particle density and the current density of the ground-state and $\partial_t$ is the partial derivative with respect to time. In terms of this displacement, we formulate an "elastic approximation" analogous to the one proposed in the absence of magnetic field. The resulting equation of motion for $\uv$ is expressed in terms of ground-state properties -- the one-particle density matrix and the two-particle pair correlation function -- and in this form it neatly generalizes the equation obtained for vanishing magnetic field.

A Two‐Dimensional Computational Droplet Contact Model

AbstractThe interaction between capillary fluid films and micro‐structural rough surfaces is one of the main challenges in studying self‐cleaning mechanisms. The surface behavior of the deformable fluid film is governed by the Young‐Laplace equation, which is highly non‐linear. Therefore, a numerical solution is introduced using the finite element method, based on a continuum mechanical formulation. Surface and line contact at the fluid‐structure interface are modeled by enforcing a contact constraint, and a contact angle, respectively. The numerical solution is validated against the analytical solution of a test case. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Entropic formulation of relativistic continuum mechanics

An entropic formulation of relativistic continuum mechanics is developed in the Landau-Lifshitz frame. We introduce two spatial scales, one being the small scale representing the linear size of each material particle and the other the large scale representing the linear size of a large system which consists of material particles and is to linearly regress to the equilibrium. We propose a local functional which is expected to represent the total entropy of the larger system and require the entropy functional to be maximized in the process of linear regression. We show that Onsager's original idea on linear regression can then be realized explicitly as current conservations with dissipative currents in the desired form. We demonstrate the effectiveness of this formulation by showing that one can treat a wide class of relativistic continuum materials, including standard relativistic viscous fluids and relativistic viscoelastic materials.

Sparse meshless models of complex deformable solids

A new method to simulate deformable objects with heterogeneous material properties and complex geometries is presented. Given a volumetric map of the material properties and an arbitrary number of control nodes, a distribution of the nodes is computed automatically, as well as the associated shape functions. Reference frames attached to the nodes are used to apply skeleton subspace deformation across the volume of the objects. A continuum mechanics formulation is derived from the displacements and the material properties. We introduce novel material-aware shape functions in place of the traditional radial basis functions used in meshless frameworks. In contrast with previous approaches, these allow coarse deformation functions to efficiently resolve non-uniform stiffnesses. Complex models can thus be simulated at high frame rates using a small number of control nodes.

Terahertz wave propagation in uniform nanorods: A nonlocal continuum mechanics formulation including the effect of lateral inertia

The dynamic testing of materials and components often involves predicting the propagation of stress waves in slender rods. The present work deals with the analysis of the wave propagation characteristics of nanorods. The nonlocal elasticity theory and also the lateral inertia are incorporated into classical/local rod model to capture unique features of the nanorods under the umbrella of continuum mechanics theory. The strong effect of the nonlocal scale has been obtained which leads to substantially different wave behaviors of nanorods from those of macroscopic rods. Nonlocal rod/bar model is developed for nanorods including the lateral inertia effects. The analysis shows that the wave characteristics are highly over estimated by the classical rod model, which ignores the effect of small-length scale. The wave propagation properties of the nanorod obtained from the present formulations are compared with the continuum rod model, nonlocal second and fourth order strain gradient models, Born-K a ´ rm a ´ n model and the nonlocal stress gradient model. It has also been shown that, the unstable second order strain gradient model can be replaced by considering the inertia gradient terms in the formulations. The effects of both the nonlocal scale and the diameter of the nanorod on spectrum curves are highlighted in the present manuscript. The results provided in this article are useful guidance for the study and design of the next generation of nanodevices that make use of the wave propagation properties of single-walled carbon nanotubes.

Mechano-kinetic coupling approach for materials with dynamic internal structure

Coupled multiscale approaches for the analysis and simulation of complex multiphysics phenomena in solids have been of great interest during the past decade. These include concurrent MD/FEM models, atomistic-to-continua homogenization techniques and deterministic multiple time scale approaches. In this article we discuss a generic approach to concurrently couple the Monte-Carlo master equation of microscopic kinetic processes, not accessible by the direct particle dynamics, with the continuum mechanics formulation. This approach can be adequate for the modeling and validation of advanced contemporary materials with dynamic internal structure, as well as evolutionary and degradation processes in materials. The approach is illustrated in the application to the models of interstitial and vacancy diffusion in fuel cell catalysts and binary material systems.

Collisions between two nonlinear deformable bodies stated within Continuum Mechanics

Fil: Buezas, Fernando Salvador. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina. Universidad Nacional del Sur. Departamento de Fisica; Argentina

Hamilton-type principles applied to ice-sheet dynamics: new approximations for large-scale ice-sheet flow

AbstractIce-sheet modelers tend to be more familiar with the Newtonian, vectorial formulation of continuum mechanics, in which the motion of an ice sheet or glacier is determined by the balance of stresses acting on the ice at any instant in time. However, there is also an equivalent and alternative formulation of mechanics where the equations of motion are instead found by invoking a variational principle, often called Hamilton’s principle. In this study, we show that a slightly modified version of Hamilton’s principle can be used to derive the equations of ice-sheet motion. Moreover, Hamilton’s principle provides a pathway in which analytic and numeric approximations can be made directly to the variational principle using the Rayleigh–Ritz method. To this end, we use the Rayleigh–Ritz method to derive a variational principle describing the large-scale flow of ice sheets that stitches the shallow-ice and shallow-shelf approximations together. Numerical examples show that the approximation yields realistic steady-state ice-sheet configurations for a variety of basal tractions and sliding laws. Small parameter expansions show that the approximation reduces to the appropriate asymptotic limits of shallow ice and shallow stream for large and small values of the basal traction number.

Peridynamics as an Upscaling of Molecular Dynamics

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.

Modeling deflagration-to-detonation transition in granular explosive pentaerythritol tetranitrate

Based on an approach suggested by Stewart et al. [Phys. Fluids 6, 2515 (1994)] we develop a model to simulate deflagration-to-detonation transition (DDT) in pentaerythritol tetranitrate (PETN) powders. The model uses a continuum mechanics formulation of conservation laws for a mixture of solid reactants and gas products, written in terms of mixture quantities plus two independent variables used to account for exothermic conversion of solid reactants into gas products, and compaction associated with pore collapse and grain rearrangement. We propose a simple empirical dependence of the reaction rate on the initial bed compaction that allows us to calibrate the model for a wide range of initial conditions. For the solid reactants we use a wide-ranging equation of state (EOS) developed by Davis and co-workers in a series of papers [Proceedings of the Tenth International Symposium on Detonation, 1993, pp. 369–376; Explosive Effects and Applications (Springer, New York, 1998), Chap. 1, Combust. Flame 120, 399 (2000); Proceedings of the 12th International Symposium Detonation, San Diego, CA, 2002, pp. 624–631; . ONR 333-05-2; Proceedings of the Eighth Detonation Symposium, 1985, pp. 785–795; Proceedings of the 11th International Symposium on Detonation, 1998, pp. 303–308]. The EOS for powder uses the P-α model of Herrmann [J. Appl. Phys. 40, 2490 (1969)] and Carrol and Holt [J. Appl. Phys. 43, 759 (1972)]. To close the system, we suggest phenomenological closure relations, consistent with the limit of a compressible inert material and of a solid fully reactive material, such that the EOS can be posed only in terms of mixture quantities and the reaction and compaction variables. We demonstrate the model’s ability to capture DDT in PETN powders by matching transients typically observed in experiments through simulation. We show that for flows calculated using nonideal EOSs and complex reaction kinetics such as those formulated in our model, it is possible to define a separatrix, i.e., the C+ characteristic that separates the C+ characteristics that evolve into the detonation front from those that evolve away from it. We comment on the effects that the variability in the grain microstructure in PETN explosive powder beds can have on the overall mechanics of DDT and discuss possible ways to model this.

A continuum mechanical theory for turbulence: a generalized Navier–Stokes-α equation with boundary conditions

We develop a continuum-mechanical formulation and generalization of the Navier–Stokes-α equation based on a recently developed framework for fluid-dynamical theories involving higher-order gradient dependencies. Our flow equation involves two length scales α and β. The first of these enters the theory through the specific free-energy α2|D|2, where D is the symmetric part of the gradient of the filtered velocity, and contributes a dispersive term to the flow equation. The remaining scale is associated with a dissipative hyperstress which depends linearly on the gradient of the filtered vorticity and which contributes a viscous term, with coefficient proportional to β2, to the flow equation. In contrast to Lagrangian averaging, our formulation delivers boundary conditions and a complete structure based on thermodynamics applied to an isothermal system. For a fixed surface without slip, the standard no-slip condition is augmented by a wall-eddy condition involving another length scale l characteristic of eddies shed at the boundary and referred to as the wall-eddy length. As an application, we consider the classical problem of turbulent flow in a plane, rectangular channel of gap 2h with fixed, impermeable, slip-free walls and make comparisons with results obtained from direct numerical simulations. We find that α/β ~ Re0.470 and l/h ~ Re−0.772, where Re is the Reynolds number. The first result, which arises as a consequence of identifying the specific free-energy with the specific turbulent kinetic energy, indicates that the choice β = α required to reduce our flow equation to the Navier–Stokes-α equation is likely to be problematic. The second result evinces the classical scaling relation η/L ~ Re−3/4 for the ratio of the Kolmogorov microscale η to the integral length scale L.

Magnetoelastic buckling of a rectangular block in plane strain

Of interest here is the stability of a rectangular block subjected to a uniform magnetic field perpendicular to its longitudinal axis. The two ends of the block are frictionless and kept parallel to each other. This boundary value problem is motivated by the classical problem of magnetoelastic buckling in which a cantilever beam subjected to a transverse magnetic field buckles when the applied field reaches a critical value. This work presents a finite strain continuum mechanics formulation of the stability problem of a homogeneous, compressible, magnetoelastic rectangular block in plane strain subjected to a uniform transverse magnetic field. The applied variational approach employs an unconstrained energy minimization recently proposed by the authors. The analytical solution for the critical buckling fields for both the antisymmetric and symmetric modes are obtained for three different constitutive laws. The corresponding result for thin beams is extracted asymptotically for a special material and the solution is compared to previously published results. The critical magnetic field is shown to increase monotonically with the block's aspect ratio for each material and mode type. Antisymmetric modes are always the critical buckling modes for stress saturated and neo-Hookean materials, except for a narrow range of moderate aspect ratios (about 0.25 ) where symmetric modes become critical. For strain-saturated solids no buckling is possible above a maximum aspect ratio.

Turbulent kinetic energy and a possible hierarchy of length scales in a generalization of the Navier-Stokesαtheory

We present a continuum-mechanical formulation and generalization of the Navier-Stokes alpha theory based on a general framework for fluid-dynamical theories with gradient dependencies. Our flow equation involves two additional problem-dependent length scales alpha and beta. The first of these scales enters the theory through the internal kinetic energy, per unit mass, alpha2|D|2, where D is the symmetric part of the gradient of the filtered velocity. The remaining scale is associated with a dissipative hyperstress which depends linearly on the gradient of the filtered vorticity. When alpha and beta are equal, our flow equation reduces to the Navier-Stokes alpha equation. In contrast to the original derivation of the Navier-Stokes alpha equation, which relies on Lagrangian averaging, our formulation delivers boundary conditions. For a confined flow, our boundary conditions involve an additional length scale l characteristic of the eddies found near walls. Based on a comparison with direct numerical simulations for fully developed turbulent flow in a rectangular channel of height 2h, we find that alphabeta approximately Re(0.470) and lh approximately Re(-0.772), where Re is the Reynolds number. The first result, which arises as a consequence of identifying the internal kinetic energy with the turbulent kinetic energy, indicates that the choice alpha=beta required to reduce our flow equation to the Navier-Stokes alpha equation is likely to be problematic. The second result evinces the classical scaling relation eta/L approximately Re(-3/4) for the ratio of the Kolmogorov microscale eta to the integral length scale L . The numerical data also suggests that l < or = beta . We are therefore led to conjecture a tentative hierarchy, l < or = beta < alpha , involving the three length scales entering our theory.

Modeling UHMWPE wear debris generation

It is widely recognized that polyethylene wear debris is one of the main causes of long-term prosthesis loosening. The noxious bioreactivity associated with this debris is determined by its size, shape, and quantity. The aim of this study was to develop a numerical tool that can be used to investigate the primary polyethylene wear mechanisms involved. This model illustrates the formation of varying flow of polyethylene debris with various shapes and sizes caused by elementary mechanical processes. Instead of using the classical continuum mechanics formulation for this purpose, we used a divided materials approach to simulate debris production and release. This approach involves complex nonlinear bulk behaviors, frictional adhesive contact, and characterizes material damage as a loss of adhesion. All the associated models were validated with various benchmark tests. The examples given show the ability of the numerical model to generate debris of various shapes and sizes such as those observed in implant retrieval studies. Most of wear mechanisms such as abrasion, adhesion, and the shearing off of micro-asperities can be described using this approach. Furthermore, it could be applied to study the effects of friction couples, macroscopic geometries, and material processing (e.g. irradiation) on wear.

Kinetics of phase transformations in the peridynamic formulation of continuum mechanics

We study the kinetics of phase transformations in solids using the peridynamic formulation of continuum mechanics. The peridynamic theory is a nonlocal formulation that does not involve spatial derivatives, and is a powerful tool to study defects such as cracks and interfaces. We apply the peridynamic formulation to the motion of phase boundaries in one dimension. We show that unlike the classical continuum theory, the peridynamic formulation does not require any extraneous constitutive laws such as the kinetic relation (the relation between the velocity of the interface and the thermodynamic driving force acting across it) or the nucleation criterion (the criterion that determines whether a new phase arises from a single phase). Instead this information is obtained from inside the theory simply by specifying the inter-particle interaction. We derive a nucleation criterion by examining nucleation as a dynamic instability. We find the induced kinetic relation by analyzing the solutions of impact and release problems, and also directly by viewing phase boundaries as traveling waves. We also study the interaction of a phase boundary with an elastic non-transforming inclusion in two dimensions. We find that phase boundaries remain essentially planar with little bowing. Further, we find a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase boundaries at the edges of the inclusion while the original phase boundary slows down or stops. Transformation proceeds as the freshly nucleated phase boundaries propagate leaving behind some untransformed martensite around the inclusion.