Showing papers on "Continuum mechanics published in 2013"

Coarse-grained local and objective continuum description of three-dimensional granular flows down an inclined surface

Abstract: Dry, frictional, steady-state granular flows down an inclined, rough surface are studied with discrete particle simulations. From this exemplary flow situation, macroscopic fields, consistent with the conservation laws of continuum theory, are obtained from microscopic data by time-averaging and spatial smoothing (coarse-graining). Two distinct coarse-graining length scale ranges are identified, where the fields are almost independent of the smoothing length w. The smaller, sub-particle length scale, w ≪ d, resolves layers in the flow near the base boundary that cause oscillations in the macroscopic fields. The larger, particle length scale, w ≈ d, leads to smooth stress and density fields, but the kinetic stress becomes scale-dependent; however, this scale-dependence can be quantified and removed. The macroscopic fields involve density, velocity, granular temperature, as well as strain-rate, stress, and fabric (structure) tensors. Due to the plane strain flow, each tensor can be expressed in an inherently anisotropic form with only four objective, coordinate frame invariant variables. For example, the stress is decomposed as: (i) the isotropic pressure, (ii) the “anisotropy” of the deviatoric stress, i.e., the ratio of deviatoric stress (norm) and pressure, (iii) the anisotropic stress distribution between the principal directions, and (iv) the orientation of its eigensystem. The strain rate tensor sets the reference system, and each objective stress (and fabric) variable can then be related, via discrete particle simulations, to the inertial number, I. This represents the plane strain special case of a general, local, and objective constitutive model. The resulting model is compared to existing theories and clearly displays small, but significant deviations from more simplified theories in all variables – on both the different length scales.

An Introduction to Continuum Mechanics, Second Edition

Abstract: 1. Introduction 2. Vectors and tensors 3. Kinematics of continua 4. Stress measures 5. Conservation and balance laws 6. Constitutive equations 7. Linearized elasticity 8. Fluid mechanics and heat transfer 9. Linearized viscoelasticity.

Mechanical properties of magneto-sensitive elastomers: unification of the continuum-mechanics and microscopic theoretical approaches

Abstract: A new theoretical formalism is developed for the study of the mechanical behaviour of magneto-sensitive elastomers (MSEs) under a uniform external magnetic field This formalism allows us to combine macroscopic continuum-mechanics and microscopic approaches for complex analysis of MSEs with different shapes and with different particle distributions It is shown that starting from a model based on an explicit discrete particle distribution one can separate the magnetic field inside the MSE into two contributions: one which depends on the shape of the sample with finite size and the other, which depends on the local spatial particle distribution The magneto-induced deformation and the change of elastic modulus are found to be either positive or negative, their dependences on the magnetic field being determined by a non-trivial interplay between these two contributions Mechanical properties are studied for two opposite types of coupling between the particle distribution and the magneto-induced deformation: absence of elastic coupling and presence of strong affine coupling Predictions of a new formalism are in a qualitative agreement with existing experimental data

Generalized Continua from the Theory to Engineering Applications

Abstract: On the roots of continuum mechanics in differential geometry -- a review.- Cosserat media.- Cosserat-type shells.- Cosserat-type rods.- Micromorphic media.- Electromagnetism and generalized continua.- Computational methods for generalized continua.

Analytical closed-form solutions for size-dependent static pull-in behavior in electrostatic micro-actuators via Fredholm integral equation

Abstract: In this paper, a novel method is proposed for the first time to obtain static pull-in voltages with fringing field effects in electrostatically actuated cantilever and clamped-clamped micro-beams where the mid-plane stretching and the residual axial load are taken into account for clamped-clamped boundary conditions. The non-classical Euler–Bernoulli beam model containing one material length scale parameter is adopted to effectively capture the size effect. In the solution procedure, the governing fourth-order differential equation of variable coefficients is converted into a Fredholm integral equation. By adopting the first natural mode of the cantilever and clamped-clamped micro-beams as a deflection shape function, the resulting equation is solved for the static pull-in voltages. The accuracy of the present analytical closed-form solution is verified through comparing with the experimentally measured and numerical data conducted in the published works. From the experimental data available in the literature, the value of the material length scale parameter for the (poly)silicon is estimated to be in the order of magnitude of 10 −1 μm. Then, the effect of the material length scale parameter on the pull-in voltages of the cantilever and clamped-clamped micro-beams is investigated. The results indicate that the tensile residual stress can extend the validity range of the classical continuum mechanics to lower beam thicknesses. It is also found that microcantilever beams are more sensitive to the size effect than their corresponding clamped-clamped micro-beams.

Notes on Continuum Mechanics

Abstract: Preface.- Abbreviations.- Operators And Symbols.- Si-Units.- Introduction.- 1 Mechanics.- 2 What Is Continuum Mechanics.- 3 Scales Of Material Studies.- 4 The Initial Boundary Value Problem (Ibvp).- 1 Tensors.- 1.1 Introduction.- 1.2 Algebraic Operations With Vectors.- 1.3 Coordinate Systems.- 1.4 Indicial Notation.- 1.5 Algebraic Operations With Tensors.- 1.6 The Tensor-Valued Tensor Function.- 1.7 The Voigt Notation.- 1.8 Tensor Fields.- 1.9 Theorems Involving Integrals.- Appendix A: A Graphical Representation Of A Second-Order Tensor.- A.1 Projecting A Second-Order Tensor Onto A Particular Direction.- A.2 Graphical Representation Of An Arbitrary Second-Order Tensor.- A.3 The Tensor Ellipsoid.- A.4 Graphical Representation Of The Spherical And Deviatoric Parts.- 2 Continuum Kinematics.- 2.1 Introduction.- 2.2 The Continuous Medium.- 2.3 Description Of Motion.- 2.4 The Material Time Derivative.- 2.5 The Deformation Gradient.- 2.6 Finite Strain Tensors.- 2.7 Particular Cases Of Motion.- 2.8 Polar Decomposition Of F.- 2.9 Area And Volume Elements Deformation.- 2.10 Material And Control Domains.- 2.11 Transport Equations.- 2.12 Circulation And Vorticity.- 2.13 Motion Decomposition: Volumetric And Isochoric Motions.- 2.14 The Small Deformation Regime.- 2.15 Other Ways To Define Strain.- 3 Stress.- 3.1 Introduction.- 3.2 Forces.- 3.3 Stress Tensors.- 4 Objectivity Of Tensors.- 4.1 Introduction.- 4.2 The Objectivity Of Tensors.- 4.3 Tensor Rates.- 5 The Fundamental Equations Of Continuum Mechanics.- 5.1 Introduction.- 5.2 Density.- 5.3 Flux.- 5.4 The Reynolds Transport Theorem.- 5.5 Conservation Law.- 5.6 The Principle Of Conservation Of Mass. The Mass Continuity Equation.- 5.7 The Principle Of Conservation Of Linear Momentum. The Equations Of Motion.- 5.8 The Principle Of Conservation Of Angular Momentum. Symmetry Of The Cauchy Stress Tensor.- 5.9 The Principle Of Conservation Of Energy. The Energy Equation.- 5.10 The Principle Of Irreversibility. Entropy Inequality.- 5.11 Fundamental Equations Of Continuum Mechanics.- 5.12 Flux Problems.- 5.13 Fluid Flow In Porous Media (Filtration).- 5.14 The Convection-Diffusion Equation.- 5.15 Initial Boundary Value Problem (Ibvp) And Computational Mechanics.- 6 Introduction To Constitutive Equations.- 6.1 Introduction.- 6.2 The Constitutive Principles.- 6.3 Characterization Of Constitutive Equations For Simple Thermoelastic Materials.- 6.4 Characterization Of The Constitutive Equations For A Thermoviscoelastic Material.- 6.5 Some Experimental Evidence.- 7 Linear Elasticity.- 7.1 Introduction.- 7.2 Initial Boundary Value Problem Of Linear Elasticity.- 7.3 Generalized Hooke's Law.- 7.4 The Elasticity Tensor.- 7.5 Isotropic Materials.- 7.6 Strain Energy Density.- 7.7 The Constitutive Law For Orthotropic Material.- 7.8 Transversely Isotropic Materials.- 7.9 The Saint-Venant's And Superposition Principles.- 7.10 Initial Stress/Strain.- 7.11 The Navier-Lame Equations.- 7.12 Two-Dimensional Elasticity.- 7.13 The Unidimensional Approach.- 8 Hyperelasticity.- 8.1 Introduction.- 8.2 Constitutive Equations.- 8.3 Isotropic Hyperelastic Materials.- 8.4 Compressible Materials.- 8.5 Incompressible Materials.- 8.6 Examples Of Hyperelastic Models.- 8.7 Anisotropic Hyperelasticity.- 9 Plasticity.- 9.1 Introduction.- 9.2 The Yield Criterion.- 9.3 Plasticity Models In Small Deformation Regime (Uniaxial Cases).- 9.4 Plasticity In Small Deformation Regime (The Classical Plasticity Theory).- 9.5 Plastic Potential Theory.- 9.6 Plasticity In Large Deformation Regime.- 9.7 Large-Deformation Plasticity Based On The Multiplicative Decomposition Of The Deformation Gradient.- 10 Thermoelasticity.- 10.1 Thermodynamic Potentials.- 10.2 Thermomechanical Parameters.- 10.3 Linear Thermoelasticity.- 10.4 The Decoupled Thermo-Mechanical Problem In A Small Deformation Regime.- 10.5 The Classical Theory Of Thermoelasticity In Finite Strain (Large Deformation Regime).- 10.6 Thermoelasticity Based On The Multiplicative Decomposition Of The Deformation Gradient..- 10.7 Thermoplasticity In A Small Deformation Regime.- 11 Damage Mechanics.- 11.1 Introduction.- 11.2 The Isotropic Damage Model In A Small Deformation Regime.- 11.3 The Generalized Isotropic Damage Model.- 11.4 The Elastoplastic-Damage Model In A Small Deformation Regime.- 11.5 The Tensile-Compressive Plastic-Damage Model.- 11.6 Damage In A Large Deformation Regime.- 12 Introduction To Fluids.- 12.1 Introduction.- 12.2 Fluids At Rest And In Motion.- 12.3 Viscous And Non-Viscous Fluids.- 12.4 Laminar Turbulent Flow.- 12.5 Particular Cases.- 12.6 Newtonian Fluids.- 12.7 Stress, Dissipated And Recoverable Powers.- 12.8 The Fundamental Equations For Newtonian Fluids.- Bibliography.- Index.

Peridynamics: a nonlocal continuum theory.

Abstract: The peridynamic theory is a nonlocal theory of continuum mechanics based on an integro-differential equation without spatial derivatives, which can be easily applied in the vicinity of cracks, where discontinuities in the displacement field occur. In this paper we give a survey on important analytical and numerical results and applications of the peridynamic theory.

Symmetry classes for even-order tensors

Abstract: The purpose of this article is to give a complete and general answer to the recurrent problem in continuum mechanics of the determination of the number and the type of symmetry classes of an even-order tensor space. This kind of investigation was initiated for the space of elasticity tensors. Since then, different authors solved this problem for other kinds of physics such as photoelectricity, piezoelectricity, flexoelectricity, and strain-gradient elasticity. All the aforementioned problems were treated by the same computational method. Although being effective, this method suffers the drawback not to provide general results. And, furthermore, its complexity increases with the tensorial order. In the present contribution, we provide general theorems that directly give the sought results for any even-order constitutive tensor. As an illustration of this method, and for the first time, the symmetry classes of all even-order tensors of Mindlin second strain-gradient elasticity are provided.

Analytical continuum mechanics \`a la Hamilton-Piola: least action principle for second gradient continua and capillary fluids

Abstract: In this paper a stationary action principle is proven to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments, for instance by Cahn and Hilliard. Remark that these fluids are sometimes also called Korteweg-de Vries or Cahn-Allen. In general continua whose deformation energy depend on the second gradient of placement are called second gradient (or Piola-Toupin or Mindlin or Green-Rivlin or Germain or second gradient) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both material and spatial description and the corresponding Euler-Lagrange bulk 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 grad C or on C^-1 and grad C^-1 ; where C is the Cauchy-Green deformation tensor. When particularized to energies which characterize fluid materials, the capillary fluid evolution conditions (see e.g. Casal or Seppecher for an alternative deduction based on thermodynamic arguments) are recovered. A version of Bernoulli law valid for capillary fluids is found and, in the Appendix B, useful kinematic formulas for the present variational formulation are proposed. Historical comments about Gabrio Piola's contribution to continuum analytical mechanics are also presented. In this context the reader is also referred to Capecchi and Ruta.

A continuum model for the static pull-in behavior of graphene nanoribbon electrostatic actuators with interlayer shear and surface energy effects

Abstract: Based on multi-beam shear model theory, a continuum mechanics model is developed to investigate the pull-in instability of wedged/curved multilayer graphene nanoribbon (MLGNR) cantilever nanobeams subjected to electrostatic and Casimir forces. The first-order fringing-field correction, the interlayer shear between neighboring graphene nanoribbons (GNRs), surface elasticity, and residual surface tension are incorporated into the analytical model. An explicit closed-form analytical solution to the governing fourth-order nonlinear differential equation of variable coefficients is introduced for the static pull-in behavior of electrostatic nanoactuators using a Fredholm integral equation of the first kind. A comparison study for a [001] silver electrostatic nanoactuator indicates that the proposed analytical closed-form solution yields an improved accuracy over other analytical and numerical methods existing in literature. The results indicate that the interfacial slip between GNRs and the surface material pa...

Continuum Mechanics of Anisotropic Materials

Abstract: Chapter 1. Introduction Chapter 2. Mechanical modeling of materials 2.1 Introduction 2.2 Models and the real physical world 2.3 Guidelines for modeling objects and solving mechanics problems 2.4 The types of models used in mechanics 2.5 The particle model 2.6 The rigid object model 2.7 The deformable continuum model 2.8 Lumped parameter models 2.9 Statistical models 2.10 Cellular automata 2.11 The limits of reductionism 2.12 References Appendix 2A Laplace transform refresher Appendix 2B First order differential equations Appendix 2C Electrical analogs of the spring and dashpot models Chapter 3. Basic continuum kinematics 3.1 The deformable material model, the continuum 3.2 Rates of change and the spatial representation of motion 3.3 Infinitesimal motions 3.4 The strain conditions of compatibility Chapter 4. Continuum formulations of conservation laws 4.1 The conservation principles 4.2 The conservation of mass 4.3 The state of stress at a point 4.4 The stress equations of motion 4.5 The conservation of energy Chapter 5. Formulation of constitutive equations 5.1 Guidelines for the formulation of constitutive equations 5.2 Constitutive ideas 5.3 Localization 5.4 Invariance under rigid object motions 5.5 Determinism 5.6 Linearization 5.7 Coordinate invariance 5.8 Homogeneous versus inhomogeneous constitutive models 5.9 Restrictions due to material symmetry 5.10 The symmetry of the material coefficient tensors 5.11 Restrictions on the coefficients representing material properties 5.12 Summary of results 5.13 Relevant literature Chapter 6 Modeling material symmetry 6.1 Introduction 6.2 The representative volume element (RVE) 6.3 Crystalline materials and textured materials 6.4 Planes of mirror symmetry 6.5 Characterization of material symmetries by planes of symmetry 6.6 The forms of the 3D symmetric linear transformation A 6.7 The forms of the 6D symmetric linear transformation 6.8 Curvilinear anisotropy 6.9 Symmetries that permit chirality 6.10 Relevant literature Chapter 7. Four linear continuum theories 7.1 Formation of continuum theories 7.2 The theory of fluid flow through rigid porous media 7.3 The theory of elastic solids 7.4 The theory of viscous fluids 7.5 The theory of viscoelastic materials 7.6 Relevant literature Chapter 8 Modeling material microstructure 8.1 Introduction 8.2 The representative volume element (RVE) 8.3 Effective material parameters 8.4 Effective elastic constants 8.5 Effective permeability 8.6 Structural gradients 8.7 Tensorial representations of microstructure 8.8 Relevant literature Chapter 9. Poroelasticity 9.1 Poroelastic materials 9.2 The stress-strain-pore pressure constitutive relation 9.3 The fluid content-stress-pore pressure constitutive relation 9.4 Darcy's Law 9.5 Matrix material and pore fluid incompressibility constraints 9.6 The undrained elastic coefficients 9.7 Expressions of mass and momentum conservation 9.8 The basic equations of poroelasticity 9.9 The basic equations of incompressible poroelasticity 9.10 Some example isotropic poroelastic problems 9.11 An example: the unconfined compression of an anisotropic disc 9.12 Relevant literature Chapter 10 Mixture 10.1 Introduction 10.2 Kinematics of mixtures 10.3 The conservation laws for mixtures 10.4 A statement of irreversibility in mixture processes 10.5 Donnan equilibrium and osmotic pressure 10.6 Continuum model for a charged porous medium the governing equations 10.7 Linear irreversible thermodynamics and the four constituent mixture 10.8 Modeling swelling and compression experiments on the intervertebral disc 10.9 Relevant literature Chapter 11. Kinematics and mechanics of large deformations 11.1 Large deformations 11.2 Large homogeneous deformations 11.3 Polar decomposition of the deformation gradients 11.4 The strain measures for large deformations 11.5 Measures of volume and surface change in large deformations 11.6 Stress measures 11.7 Finite deformation elasticity 11.8 The isotropic finite deformation stress-strain relation 11.9 Finite deformation hyperelasticity 11.10 Incompressible elasticity 11.11 Relevant literature Chapter 12. Plasticity Theory 12.1 Extension of von Mises criterion to anisotropic materials 12.2 Yield criteria for pressure sensitive anisotropic materials 12.3 Some particular deformation characteristics exhibited by granular materials (dilatancy/contractancy, anisotropy, hardening/softening, and shear localization). 12.4 Dilatant double shearing kinematics 12.5 Evolution equations for the material parameters 12.6 Numerical biaxial compression test of anisotropic granular materials 12.6 Numerical triaxial compression test of anisotropic granular materials 12.7 Plasticity theories for crystalline materials Appendix A. Matrices and tensors A.1 Introduction and rationale A.2 Definition of square, column and row matrices A.3 The types and algebra of square matrices A.4 The algebra of n-tuples A.5 Linear transformations A.6 Vector spaces A.7 Second rank tensors A.8 The moment of inertia tensor A.9 The alternator and vector cross products A.10 Connection to Mohr's circles A.11 Special vectors and tensors in six dimensions A.12 The gradient operator and the divergence theorem A.13 Tensor components in cylindrical coordinates

Chemo-thermomechanical ageing of elastomers based on multiphase continuum mechanics

Abstract: This article concerns the constitutive representation of one of the chemical ageing processes that occur in elastomers, chemo-thermomechanical ageing, which takes place as an irreversible, time-delayed chemical reaction when a medium diffuses into an unlike solid. This process is inhomogeneous in component parts of finite thickness, and as it can be thermally activated, ageing is accelerated on an increase in temperature. The application of multiphase continuum mechanics to these basic characteristics enables a thermodynamically coupled material model to be formulated, which is able to describe not only the viscoelasticity, but also the chemical decomposition and reformation processes that occur in the polymer network. The evolution principle of Liu-Muller is used to evaluate the thermomechanical consistency of the model obtained. Subsequent to this, the finite element method is applied to solve the resulting set of partial equations, which corresponds to a coupled multifield problem. The article closes with convincing simulations of illustrative examples.

Intrinsic material length, Theory of Critical Distances and Gradient Mechanics: analogies and differences in processing linear-elastic crack tip stress fields

Abstract: The Theory of Critical Distances (TCD) is a bi-parametrical approach suitable for predicting, under both static and high-cycle fatigue loading, the non-propagation of cracks by directly post-processing the linear-elastic stress fields, calculated according to continuum mechanics, acting on the material in the vicinity of the geometrical features being assessed. In other words, the TCD estimates static and high-cycle fatigue strength of cracked bodies by making use of a critical distance and a reference strength which are assumed to be material constants whose values change as the material microstructural features vary. Similarly, Gradient Mechanics postulates that the relevant stress fields in the vicinity of crack tips have to be determined by directly incorporating into the material constitutive law an intrinsic scale length. The main advantage of such a method is that stress fields become non-singular also in the presence of cracks and sharp notches. The above idea can be formalized in different ways allowing, under both static and high-cycle fatigue loading, the static and high-cycle fatigue assessment of cracked/notched components to be performed without the need for defining the position of the failure locations a priori. The present paper investigates the existing analogies and differences between the TCD and Gradient Mechanics, the latter formalized according to the so-called Implicit Gradient Method, when such theories are used to process linear-elastic crack tip stress fields.

Notes on Metric Independent Analysis of Classical Fields

Abstract: A metric independent geometric framework for some fundamental objects of continuum mechanics is presented. In the geometric setting of general differentiable manifolds, balance principles for extensive properties are formulated and Cauchy's theorem for fluxes is proved. Fluxes in an n-dimensional space are represented as differential (n − 1)-forms. In an analogous formulation of stress theory, a distinction is made between the traction stress, enabling the evaluation of the traction on the boundaries of the various regions, and the variational stress, which acts on the derivative of a virtual velocity field to produce the virtual power density. The relation between the two stress fields is examined as well as the resulting differential balance law. As an application, metric-invariant aspects of electromagnetic theory are presented within the framework of the foregoing flux and stress theory.

Three-dimensional free vibration analysis of multi-layered graphene sheets embedded in elastic matrix

Abstract: Three dimensional vibration analysis of multi-layered graphene sheets embedded in polymer matrix is carried out employing nonlocal continuum mechanics. By using the Fourier series expansion in the in-plane axis, the governing equations in term of displacements can be obtained. By exerting the traction free surface boundary conditions to the eigenfrequency equation, natural frequencies are obtained. Accuracy of the present work is validated by comparing the numerical results with those obtained in the open literature. The effect of nonlocal parameter, length of square plate, aspect ratio, plate thickness and half wave numbers in the frequency behavior are examined.

Fractional Gradient Elasticity from Spatial Dispersion Law

Abstract: Non-local elasticity models in continuum mechanics can be treated with two different approaches: the gradient elasticity models (weak non-locality) and the integral non-local models (strong non-locality). This article focuses on the fractional generalization of gradient elasticity that allows us to describe a weak non-locality of power-law type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. We prove that the continuous limit maps the equations for lattice with this spatial dispersion into the continuum equations with fractional Laplacians in the Riesz form. A weak non-locality of power-law type in the non-local elasticity theory is derived from the fractional weak spatial dispersion in the lattice model. The suggested continuum equations, which are obtained from the lattice model, describe a fractional generalization of the gradient elasticity. These equations of fractional elasticity are solved for some special cases: sub-gradient elasticity and super-gradient elasticity.

Symmetry classes for even-order tensors

Abstract: The purpose of this article is to give a complete and general answer to the recurrent problem in continuum mechanics of the determination of the number and the type of symmetry classes of an even-order tensor space. This kind of investigation was initiated for the space of elasticity tensors. Since then, different authors solved this problem for other kinds of physics such as photoelectricity, piezoelectricity, flexoelectricity, and strain-gradient elasticity. All the aforementioned problems were treated by the same computational method. Although being effective, this method suffers the drawback not to provide general results. And, furthermore, its complexity increases with the tensorial order. In the present contribution, we provide general theorems that directly give the sought results for any even-order constitutive tensor. As an illustration of this method, and for the first time, the symmetry classes of all even-order tensors of Mindlin second strain-gradient elasticity are provided.