Continuum mechanics

About: Continuum mechanics is a research topic. Over the lifetime, 5042 publications have been published within this topic receiving 181027 citations.

A micromechanical model for a viscoelastic cohesive zone

Abstract: A micromechanical model for a viscoelastic cohesive zone is formulated herein. Care has been taken in the construction of a physically-based continuum mechanics model of the damaged region ahead of the crack tip. The homogenization of the cohesive forces encountered in this region results in a damage dependent traction-displacement law which is both single integral and internal variable-type. An incrementalized form of this traction-displacement law has been integrated numerically and placed within an implicit finite element program designed to predict crack propagation in viscoelastic media. This research concludes with several example problems on the response of this model for various displacement boundary conditions.

Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics

Abstract: Introduction Continuum Mechanics in the Twentieth Century The Objective of This Book The Contents of This Book Historical Note Standard Continuum Mechanics Theory of Motion and Deformation Basic Thermomechanics of Continua Examples of Thermomechanical Behaviors Eshelbian Mechanics for Elastic Bodies The Notion of Eshelby Material Stress Eshelby Stress in Small Strains in Elasticity Classical Introduction of the Eshelby Stress by Eshelby's Original Reasoning Another Example Due to Eshelby: Material Force on an Elastic Inhomogeneity Gradient Elastic Materials Interface in a Composite The Case of a Dislocation Line (Peach-Koehler Force) Four Formulations of the Balance of Linear Momentum Variational Formulations in Elasticity More Material Balance Laws Eshelby Stress and Kroner's Theory of Incompatibility Field Theory Introduction Elements of Field Theory: Variational Formulation Application to Elasticity Conclusive Remarks Canonical Thermomechanics of Complex Continua Introduction Reminder Canonical Balance Laws of Momentum and Energy Examples without Body Force Variable alpha as an Additional Degree of Freedom Comparison with the Diffusive Internal-Variable Theory Example: Homogeneous Dissipative Solid Material Described by Means of a Scalar Diffusive Internal Variable Conclusion and Comments Local Structural Rearrangements of Matter and Eshelby Stress Changes in the Reference Configuration Material Force of Inhomogeneity Some Geometric Considerations Continuous Distributions of Dislocations Pseudo-Inhomogeneity and Pseudo-Plastic Effects A Variational Principle in Nonlinear Dislocation Theory Eshelby Stress as a Resolved Shear Stress Second-Gradient Theory Continuous Distributions of Disclinations Discontinuities and Eshelby Stresses Introduction General Jump Conditions at a Moving Discontinuity Surface Thermomechanical Shock Waves Thermal Conditions at Interfaces in Thermoelastic Composites Propagation of Phase-Transformation Fronts On Internal and Free Energies The Case of Complex Media Applications to Problems of Materials Science (Metallurgy) Singularities and Eshelby Stresses The Notion of Singularity Set The Basic Problem of Fracture and Its Singularity Global Dissipation Analysis of Brittle Fracture The Analytical Theory of Brittle Fracture Singularities and Generalized Functions Variational Inequality: Fracture Criterion Dual I-Integral of Fracture Other Material Balance Laws and Path-Independent Integrals Generalization to Inhomogeneous Bodies Generalization to Dissipative Bodies A Curiosity: "Nondissipative" Heat Conductors Generalized Continua Introduction Field Equations of Polar Elasticity Small-Strain and Small-Microrotation Approximation Discontinuity Surfaces in Polar Materials Fracture of Solid Polar Materials Other Microstructure Modelings Systems with Mass Changes and/or Diffusion Introduction Volumetric Growth First-Order Constitutive Theory of Growth Application: Anisotropic Growth and Self-Adaptation Illustrations: Finite-Element Implementation Intervention of Nutriments Eshelbian Approach to Solid-Fluid Mixtures Single-Phase Transforming Crystal and Diffusion Electromagnetic Materials Maxwell Could Not Know Noether's Theorem but... Electromagnetic Fields in Deformable Continuous Matter Variational Principle Based on the Direct Motion Variational Principle Based on the Inverse Motion Geometrical Aspects and Material Uniformity Remark on Electromagnetic Momenta Balance of Canonical Momentum and Material Forces Electroelastic Bodies and Fracture Transition Fronts in Thermoelectroelastic Crystals The Case of Magnetized Elastic Materials Application to Nonlinear Waves Wave Momentum in Crystal Mechanics Conservation Laws in Soliton Theory Examples of Solitonic Systems and Associated Quasiparticles Sine Gordon Equation and Associated Equations Nonlinear Schrodinger Equation and Allied Systems Driving Forces Acting on Solitons A Basic Problem of Materials Science: Phase-Transition Front Propagation Numerical Applications Introduction Finite-Difference Method Finite-Volume Method-Continuous Cellular Automata Finite-Element Method Conclusive Remarks More on Eshelby-Like Problems and Solutions Introduction Analogy: Path-Independent Integrals in Heat and Electricity Conductions The Eshelbian Nature of Aerodynamic Forces The World of Configurational Forces Bibliography Index

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.

Theory of Mixtures

Abstract: In developing the balance equations for a multicomponent flow, there are two approaches available. Some authors (Whitaker 1967, Slattery 1967) start from the equations of fluid mechanics, valid on the particle scale, and then integrate these equations in regions sufficiently large to contain a representative mass of all components. In case of multiphase particulate flow, those regions must be much greater than the size of the particles contained in the system. The spatially averaged properties then become field variables and the new balance equations constitute a set of local equations describing the flow of a multicomponent mixture. The second approach is the theory of mixtures which uses the concepts of continuum mechanics, considering all components as superimposed interacting continua (Bowen 1976, Truesdell 1984, Dobran 1985). The macroscopic balances are established as the fundamental equations and, from them, the local balances and jump conditions are deduced. The field variables in the continuum approach are equivalent to the averaged variables in the first approach, so that both methods give the same results (Drew 1983). In both cases the local variables cannot be experimentally measured and are not to be confused with the experimental variables of fluid mechanics. In the work that follows, we use the continuum approach of the theory of mixtures.

Reduced symmetry elements in linear elasticity

Abstract: In continuum mechanics problems, we have to work in most cases with symmetric tensors, symmetry expressing the conservation of angular momentum. Discretization of symmetric tensors is however difficult and a classical solution is to employ some form of reduced symmetry. We present two ways of introducing elements with reduced symmetry. The first one is based on Stokes problems, and in the two-dimensional case allows to recover practically all interesting elements on the market. This however is (definitely) not true in three dimensions. On the other hand the second approach (based on a very nice property of several interpolation operators) works for three-dimensional problems as well, and allows, in particular, to prove the convergence of the Arnold-Falk-Winther element with simple and standard arguments, without the use of the Berstein-Gelfand-Gelfand resolution.