Showing papers on "Continuum mechanics published in 2015"

Analytical continuum mechanics à la Hamilton-Piola least action principle for second gradient continua and capillary fluids

Abstract: In this paper a stationary action principle is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments. We remark that these fluids are sometimes also called Korteweg–de Vries or Cahn–Allen fluids. In general, continua whose deformation energy depends on the second gradient of placement are called second gradient (or Piola–Toupin, Mindlin, Green–Rivlin, Germain or second grade) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both the material and spatial descriptions and the corresponding Euler–Lagrange equations and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and ∇C or on C−1 and ∇C−1, where C is the Cauchy–Green deformation tensor. When particularized to energies which characterize fluid materia...

The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results

Abstract: In order to found continuum mechanics, two different postulations have been used. The first, introduced by Lagrange and Piola, starts by postulating how the work expended by internal interactions in a body depends on the virtual velocity field and its gradients. Then, by using the divergence theorem, a representation theorem is found for the volume and contact interactions which can be exerted at the boundary of the considered body. This method assumes an a priori notion of internal work, regards stress tensors as dual of virtual displacements and their gradients, deduces the concept of contact interactions and produces their representation in terms of stresses using integration by parts. The second method, conceived by Cauchy and based on the celebrated tetrahedron argument, starts by postulating the type of contact interactions which can be exerted on the boundary of every (suitably) regular part of a body. Then it proceeds by proving the existence of stress tensors from a balance-type postulate. In this paper, we review some relevant literature on the subject, discussing how the two postulations can be reconciled in the case of higher gradient theories. Finally, we underline the importance of the concept of contact surface, edge and wedge s-order forces.

Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof

Abstract: Since the works by Gabrio Piola, it has been debated the relevance of higher-gradient continuum models in mechanics. Some authors even questioned the logical consistency of higher-gradient theories, and the applicability of generalized continuum theories seems still open. The present paper considers a pantographic plate constituted by Euler beams suitably interconnected and proves that Piola’s heuristic homogenization method does produce an approximating continuum in which deformation energy depends only on second gradients of displacements. The Γ-convergence argument presented herein shows indeed that Piola’s conjecture can be rigorously proven in a Banach space whose norm is physically dictated by energetic considerations.

Bubble dynamics in a viscoelastic medium with nonlinear elasticity

Abstract: In a variety of recently developed medical procedures, bubbles are formed directly in soft tissue and may cause damage. While cavitation in Newtonian liquids has received significant attention, bubble dynamics in tissue, a viscoelastic medium, remains poorly understood. To model tissue, most previous studies have focused on Maxwell-based viscoelastic fluids. However, soft tissue generally possesses an original configuration to which it relaxes after deformation. Thus, a Kelvin–Voigt-based viscoelastic model is expected to be a more appropriate representation. Furthermore, large oscillations may occur, thus violating the infinitesimal strain assumption and requiring a nonlinear/finite-strain elasticity description. In this article, we develop a theoretical framework to simulate spherical bubble dynamics in a viscoelastic medium with nonlinear elasticity. Following modern continuum mechanics formalism, we derive the form of the elastic forces acting on a bubble for common strain-energy functions (e.g. neo-Hookean, Mooney–Rivlin) and incorporate them into Rayleigh–Plesset-like equations. The main effects of nonlinear elasticity are to reduce the violence of the collapse and rebound for large departures from the equilibrium radius, and increase the oscillation frequency. The present approach can readily be extended to other strain-energy functions and used to compute the stress/deformation fields in the surrounding medium.

A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations

Abstract: We describe a model based on continuum mechanics that reduces the study of a significant class of problems of discrete dislocation dynamics to questions of the modern theory of continuum plasticity As applications, we explore the questions of the existence of a Peierls stress in a continuum theory, dislocation annihilation, dislocation dissociation, finite-speed-of-propagation effects of elastic waves vis-a-vis dynamic dislocation fields, supersonic dislocation motion, and short-slip duration in rupture dynamics

Nonhydrostatic granular flow over 3‐D terrain: New Boussinesq‐type gravity waves?

Abstract: Understanding granular mass flow is a basic step in the prediction and control of natural or man-made disasters related to avalanches on the Earth. Savage and Hutter (1989) pioneered the mathematical modeling of these geophysical flows introducing Saint-Venant-type mass and momentum depth-averaged hydrostatic equations using the continuum mechanics approach. However, Denlinger and Iverson (2004) found that vertical accelerations in granular mass flows are of the same order as the gravity acceleration, requiring the consideration of nonhydrostatic modeling of granular mass flows. Although free surface water flow simulations based on nonhydrostatic depth-averaged models are commonly used since the works of Boussinesq (1872, 1877), they have not yet been applied to the modeling of debris flow. Can granular mass flow be described by Boussinesq-type gravity waves? This is a fundamental question to which an answer is required, given the potential to expand the successful Boussinesq-type water theory to granular flow over 3-D terrain. This issue is explored in this work by generalizing the basic Boussinesq-type theory used in civil and coastal engineering for more than a century to an arbitrary granular mass flow using the continuum mechanics approach. Using simple test cases, it is demonstrated that the above question can be answered in the affirmative way, thereby opening a new framework for the physical and mathematical modeling of granular mass flow in geophysics, whereby the effect of vertical motion is mathematically included without the need of ad hoc assumptions.

Numerical Simulation of Wetting Phenomena with a Phase‐Field Method Using OpenFOAM®

Abstract: The phase-field method coupled with the Navier-Stokes equations is a rather new approach for scale-resolving numerical simulation of interfacial two-phase flows. The intention is to implement it as finite-volume method in the open source library for computational continuum mechanics OpenFOAM® and make it freely available. An overview on the governing equations is given and the numerical method is shortly discussed. The focus is on application and validation of the code for some fundamental wetting phenomena, namely the capillary rise in a narrow channel and the spreading of a droplet on a flat surface, which is chemically homogeneous or regularly patterned. The numerical results on static meshes agree well with analytical solutions and experimental/numerical results from literature. Also, first 3D finite-volume simulations with adaptive mesh refinement near the interface are presented as a key element to achieve CPU-time efficient simulations.

Average balance equations, scale dependence, and energy cascade for granular materials.

Abstract: A new averaging method linking discrete to continuum variables of granular materials is developed and used to derive average balance equations. Its novelty lies in the choice of the decomposition between mean values and fluctuations of properties which takes into account the effect of gradients. Thanks to a local homogeneity hypothesis, whose validity is discussed, simplified balance equations are obtained. This original approach solves the problem of dependence of some variables on the size of the averaging domain obtained in previous approaches which can lead to huge relative errors (several hundred percentages). It also clearly separates affine and nonaffine fields in the balance equations. The resulting energy cascade picture is discussed, with a particular focus on unidirectional steady and fully developed flows for which it appears that the contact terms are dissipated locally unlike the kinetic terms which contribute to a nonlocal balance. Application of the method is demonstrated in the determination of the macroscopic properties such as volume fraction, velocity, stress, and energy of a simple shear flow, where the discrete results are generated by means of discrete particle simulation.

Introduction to Practical Peridynamics: Computational Solid Mechanics Without Stress and Strain

Abstract: Deformable Solids Beginnings of the Theory of Elasticity Continuum Mechanics Fracture Mechanics Bond-Based Continuum Peridynamics Particle Lattice Model for Solids Elastic Bond-Based Peridynamic Lattice Model State-Based Peridynamic Lattice Model (SPLM) Elastic SPLM Plasticity Damage Particle Dynamics Computational Implementation Simulation of Reinforced Concrete

Stress and Strain

Abstract: In this chapter, the fundamentals of continuum mechanics are presented, in particular the concepts of stress and strain, and the general equations of motion for a continuum medium. Finally, it is introduced the discipline of rheology and the basic rheological models, which are widely used in geodynamics.

Relaxation mechanism of plastic deformation and its justification using the example of the sharp yield point phenomenon in whiskers

Abstract: The generality of the dynamic approach to a wide range of problems of the continuum mechanics, including deformation at the rates determining quasi-static deformation conditions has been demonstrated using the example of deformation of cadmium and copper whiskers. The sharp yield point phenomenon has been analyzed in terms of the theory of dislocations and phenomenological integrated criteria of plasticity. It has been shown that the characteristic relaxation times used in these criteria, regardless of the applied model of plasticity, reflect essential properties of the very deformation process itself.

Modeling and simulation of the bending behavior of electrically-stimulated cantilevered hydrogels

Abstract: A systematic development of a chemo–electro–mechanical continuum model—for the application of electrically-stimulated cantilevered hydrogels—and its numerical implementation are presented in this work. The governing equations are derived within the framework of the continuum mechanics of mixtures. The finite element method is then utilized for the numerical treatment of the model. For the numerical simulation a cantilevered strip of an anionic hydrogel immersed in a NaCl solution bath is considered. An electric field is applied to electrically stimulate the aforementioned hydrogel. The application of the electric field alters the initial concentrations of the ionic species due to the chemo–electrical coupling. The gradual increase in the applied electric field leads to the bending movement of the hydrogel. Concluding, the presented multi-field continuum model is capable of simulating hydrogel bending actuators and also more complex systems e.g. gel finger grippers.

Strain-displacement relations and strain engineering in 2d materials

Abstract: We investigate the electromechanical coupling in 2d materials. For non-Bravais lattices, we find important corrections to the standard macroscopic strain - microscopic atomic-displacement theory. We put forward a general and systematic approach to calculate strain-displacement relations for several classes of 2d materials. We apply our findings to graphene as a study case, by combining a tight binding and a valence force-field model to calculate electronic and mechanical properties of graphene nanoribbons under strain. The results show good agreement with the predictions of the Dirac equation coupled to continuum mechanics. For this long wave-limit effective theory, we find that the strain-displacement relations lead to a renormalization correction to the strain-induced pseudo-magnetic fields. Implications for nanomechanical properties and electromechanical coupling in 2d materials are discussed.

Coupled Axisymmetric Thermo-Poro-Mechanical Finite Element Analysis of Energy Foundation Centrifuge Experiments in Partially Saturated Silt

Abstract: The paper presents an axisymmetric, small strain, fully-coupled, thermo-poro-mechanical (TPM) finite element analysis (FEA) of soil–structure interaction (SSI) between energy foundations and partially saturated silt. To account for the coupled processes involving the mechanical response, gas flow, water species flow, and heat flow, nonlinear governing equations are obtained from the fundamental laws of continuum mechanics, based on mixture theory of porous media at small strain. Constitutive relations consist of the effective stress concept, Fourier’s law for heat conduction, Darcy’s law and Fick’s law for pore liquid and gas flow, and an elasto-plastic constitutive model for the soil solid skeleton based on a critical state soil mechanics framework. The constitutive parameters employed in the thermo-poro-mechanical FEA are mostly fitted with experimental data. To validate the TPM model, the modeling results are compared with the observations of centrifuge-scale tests on semi-floating energy foundations in compacted silt. Variables measured include the thermal axial strains and temperature in the foundations, surface settlements, and volumetric water contents in the surrounding soil. Good agreement is obtained between the experimental and modeling results. Thermally-induced liquid water and water vapor flow inside the soil were found to have an impact on SSI. With further improvements (including interface elements at the foundation-soil interface), FEA with the validated TPM model can be used to predict performance and SSI mechanisms for energy foundations.

Hybrid molecular-continuum simulations using smoothed dissipative particle dynamics

Abstract: We present a new multiscale simulation methodology for coupling a region with atomistic detail simulated via molecular dynamics (MD) to a numerical solution of the fluctuating Navier-Stokes equations obtained from smoothed dissipative particle dynamics (SDPD). In this approach, chemical potential gradients emerge due to differences in resolution within the total system and are reduced by introducing a pairwise thermodynamic force inside the buffer region between the two domains where particles change from MD to SDPD types. When combined with a multi-resolution SDPD approach, such as the one proposed by Kulkarni et al. [J. Chem. Phys. 138, 234105 (2013)], this method makes it possible to systematically couple atomistic models to arbitrarily coarse continuum domains modeled as SDPD fluids with varying resolution. We test this technique by showing that it correctly reproduces thermodynamic properties across the entire simulation domain for a simple Lennard-Jones fluid. Furthermore, we demonstrate that this approach is also suitable for non-equilibrium problems by applying it to simulations of the start up of shear flow. The robustness of the method is illustrated with two different flow scenarios in which shear forces act in directions parallel and perpendicular to the interface separating the continuum and atomistic domains. In both cases, we obtain the correct transient velocity profile. We also perform a triple-scale shear flow simulation where we include two SDPD regions with different resolutions in addition to a MD domain, illustrating the feasibility of a three-scale coupling.

Skew-symmetric couple-stress fluid mechanics

Abstract: There can be no doubt as to the importance of vortical motion in fluid mechanics. Yet, very little attention is given typically to the balance law of angular momentum and to its role in defining the fundamental character of stress, which as a result is usually assumed as a symmetric tensor. Here, we allow for the possibility of couple-stresses, along with general non-symmetric force–stresses, and develop a self-consistent size-dependent theory within the context of classical continuum mechanics. This development relies upon the identification of the following key components for the dynamic response of three-dimensional fluid continua: (i) fundamental, uniquely defined kinematical measures of flow, (ii) an independent set of energy conjugate variables, (iii) the corresponding permissible natural and essential boundary conditions, and (iv) a non-redundant set of body-force and inertial contributions. Based upon this formulation, one can recognize that the previous couple-stress theory for fluids suffers from some inconsistencies, which may have restricted its applicability in the study of viscous flows. After presenting the general formulation of the new consistent theory, we specialize for incompressible viscous flow and consider the problem of generalized Poiseuille flow within this size-dependent fluid mechanics. Finally, we conclude that the theory presented here may provide a basis for a broad range of fluid mechanics applications and for fundamental studies of flows at the finest scales for which a continuum representation is valid.

Fractional Calculus for Continuum Mechanics - anisotropic non-locality

Abstract: In this paper the generalisation of previous author's formulation of fractional continuum mechanics to the case of anisotropic non-locality is presented. The considerations include the review of competitive formulations available in literature. The overall concept bases on the fractional deformation gradient which is non-local, as a consequence of fractional derivative definition. The main advantage of the proposed formulation is its analogical structure to the general framework of classical continuum mechanics. In this sense, it allows, to give similar physical and geometrical meaning of introduced objects.

A novel two-parameter linear elastic constitutive model for bond based peridynamics

Abstract: A novel linear elastic constitutive model is developed for bond based peridynamics. For a peridynamic constitutive model, the relevant material parameters are derived from energy equivalence to a classical linear elastic continuum mechanics model. The commonly used microelastic model is a central force model characterized by a single micromodulus constant, and hence the effective Poisson’s ratio of the isotropic peridynamic material is found to always be 1/3 in 2D and 1/4 in 3D. The elastic modulus of the peridynamic material is also dependant on the input Poisson’s ratio and as a result, the strain energy for simple loading conditions, for eg. a uniaxial tension test, is not correctly estimated by the microelastic bond model. This originates from the fact that typically a bulk expansion test is used to calibrate the peridynamic model which relates the micromodulus to the bulk modulus of the material only. In the present work, a novel linear elastic constitutive model incorporating two peridynamic material parameters is proposed, analogous to the idea of two material constants used to describe linear isotropic materials in classical theory. Numerical results are presented to demonstrate that after initial calibration using a simple biaxial test, the material model shows correct Poisson’s contraction and strain energies for a range of Poisson’s ratios. A damage model based on an energy criterion is implemented, and a dynamic crack propagation and crack branching problem is considered. Results are compared with both the microelastic model and other numerical methods from published literature. It is observed that the two parameter model shows a significant improvement in predicting displacements as a result of the inclusion of a tangential stiffness parameter.

A dissipative force between colliding viscoelastic bodies: Rigorous approach

Abstract: A collision of viscoelastic bodies is analysed within a mathematically rigorous approach. We develop a perturbation scheme to solve continuum mechanics equation, which deals simultaneously with strain and strain rate in the bulk of the bodies' material. We derive dissipative force that acts between particles and express it in terms of particles' deformation, deformation rate and material parameters. It differs noticeably from the currently used dissipative force, found within the quasi-static approximation and does not suffer from inconsistencies of this approximation. The proposed approach may be used for other continuum mechanics problems where the bulk dissipation is addressed.

Elements of Crustal Geomechanics

Abstract: Preface 1. Geomaterials and crustal geomechanics 2. Elements of rheology 3. Forces and stresses 4. Elements of kinematics 5. Elements of linear elasticity 6. From continuum mechanics to fluid mechanics 7. Elements of linear fracture mechanics 8. Laboratory investigations on geomaterials under compression 9. Homogenized geomaterials 10. Fractures and faults 11. Elements of seismology 12. Elements of solid-fluid interactions 13. Methods for stress fields evaluation from in situ observations 14. Elements of stress fields and crustal rheology References Appendix A. Elements of tensors in rectangular coordinates Index.

Simulation and control of chaotic nonequilibrium systems

Abstract: Overview of Atomistic Mechanics Formulating Atomistic Simulations Thermodynamics, Statistical Mechanics, and Temperature Continuum Mechanics: Continuity, Stress, Heat Flux, Applications Numerical Molecular Dynamics and Chaos Time-Reversible Atomistic Thermostats Key Results from Nonequilibrium Simulations Second Law, Reversibility, Instability Outlook for Progress and Life on Earth