Showing papers on "Continuum mechanics published in 1990"

Mechanics of Solid Materials

Abstract: 1. Elements of the physical mechanisms of deformation and fracture 2. Elements of continuum mechanics and thermodynamics 3. Identification and theological classification of real solids 4. Linear elasticity, thermoelasticity and viscoelasticity 5. Plasticity 6. Viscoplasticity 7. Damage mechanics 8. Crack mechanics.

Isotropic and anisotropic damage variables in continuum damage mechanics

Abstract: The present work analyzes the implication and limitation of some scalar damage models. In particular, the elastic‐damage thermodynamic potential and the effective stress concept are reexamined. It is demonstrated that isotropic damage does not necessarily imply scalar damage representation in general. The notion of isotropic and anisotropic damage variables in continuum damage mechanics is then discussed. In addition, some results from micromechanical analyses are applied to show the direct relationship between the fourth‐order damage tensor and the damage‐induced compliance tensor characteristic of microcrack‐weakened brittle materials. It is shown that even for isotropic damage one should employ an isotropic fourth‐order damage tensor (not a scalar damage variable) to characterize the state of damage in materials, in accordance with the effective stress concept. In general, however, a damage tensor is anisotropic and should be derived from micromechanical analysis when possible.

Effective elastic moduli of porous solids

Abstract: The principles of continuum mechanics can be extended to porous solids only if the effective moduli are known. Although the effective bulk modulus has already been determined by approximating the geometry of a porous solid to be a hollow sphere, bounds could only be established for the other moduli. This problem of indeterminacy of the moduli is solved in this study using a particular model from the variation of the effective Poisson's ratio. In addition to this, the results are extended for the hollow sphere to real geometry by introducing a porositydependent factor. These results are compared with experimental data and the agreement is found to be good. As the effective Poisson's ratio cannot be determined accurately using experiments, the derived equation is verified using finite element analysis.

Nonlinear analysis in soil mechanics : theory and implementation

Abstract: Part I. FUNDAMENTALS. 1. Introduction. Characteristics of soil behavior.Idealizations and material modeling. Historical review of plasticity in soil mechanics. Nonlinear stress analyses in geotechnical engineering. Need, objectives and scope. References. 2. Basic Concept of Continuum Mechanics. Introduction. Notations. Stresses in three dimensions. Definitions and notations. Cauchy's formulas, index notation, and summation convention. Principal axes of stresses. Deviatoric stress. Geometrical representation of stresses. Strains in three dimensions. Definitions and notations. Deviatoric strain. Octahedral strains and principal shear strains. Equations of solid mechanics. Equations of equilibrium (or motion). Geometric (compatibility) conditions. Constitutive relations. Summary. References. Part II. MATERIAL MODELING-BASIC CONCEPTS. 3. Elasticity and Modeling . Introduction. Elastic models in geotechnical engineering. Linear elastic model (generalized Hooke's law). Cauchy elastic model. Hyperelastic model. Hypoelastic model. Uniqueness, stability, normality, and convexity for elastic materials. Uniqueness. Drucker's stability postulate. Existence of W and v. Restrictions - normality and convexity. Linear elastic stress-strain relations. Generalized Hooke's law. A plane of symmetry. Two planes of symmetry (orthotropic symmetry). Transverse and cubic isotropies. Full isotropy. Isotropic linear elastic stress-strain relations. Tensor forms. Three-dimensional matrix forms. Plane stress case. Plane strain case. Axisymmetric case. Isotropic nonlinear elastic stress-strain relations based on total formulation. Nonlinear elastic model with secant moduli. Cauchy elastic model. Hyperelastic (green) model. Isotropic nonlinear elastic stress-strain relations based on incremental formulation. Nonlinear elastic model with secant muduli. Cauchy elastic model. Hyerelastic model. Hypoelastic model. Summary. References. 4. Perfect Plasticity and Modeling. Introduction. Deformation theory. An illustrative example. Variable moduli models. Flow theory. Yield criteria. Flow rule. Basic requirements. Perfect plasticity models. Tresca and von Mises models. Coulomb model. Drucker-Prager model. Prandtl-Reuss stress-strain relations. Generalized stress-strain relations. Stiffness formulation. General description. Stiffness coefficients. Summary. References. 5. Hardening Plasticity and Modeling. Introduction. Flow theory. Loading function. Hardening rule. Flow rule. Drucker's postulate. Hardening plasticity models. Lade-Duncan model. Lade model. Nested yield surface models. Generalized multi-surface models. Bounding surface models. Prandtl-Reuss stress-strain relations. Prandtl-Reuss equations. Matrix form of Prandtl-Reuss equations. Generalized stress-strain relations. Incremental stress-strain relations. Isotropic hardening. Kinematic hardening. Mixed hardening. Stiffness formulation. General description. Stiffness coefficients. Summary. References. PART III.

Geometric factors affecting the internal stress distribution and high temperature creep rate of discontinuous fiber reinforced metals

Abstract: The creep deformation behavior of metal-matrix composites has been studied by a continuum mechanics treatment utilizing finite element techniques. The objective of the work has been to understand the underlying mechanisms of fiber reinforcement at high temperatures and to quantify the importance of reinforcement phase geometry on the overall deformation rate. Internal stress distributions are presented for a material that consists of stiff elastic fibers in an elastic, power law creeping matrix. Results indicate that large triaxial stresses develop in the matrix, and that these stresses have a strong effect on reducing the creep rate of the composite. Reinforcement phase geometry, as measured by the fiber volume fraction, aspect ratio, separation, and overlap, greatly influences the degree of constraint on the flowing matrix material and the overall deformation rate. Theoretical predictions from this modeling are compared to experimental results of creep deformation in metal-matrix composite systems with varying degrees of agreement.

Constitutive relation and creep-fatigue life model for eutectic tin-lead solder

Abstract: A constitutive equation for a typical SnPb eutectic or near-eutectic solder joint is developed, based on empirical data in shear and generalized to three dimensions. Three strain components are considered: elastic, time independent plastic, and steady-state creep. A continuum mechanics rather than a metallurgical approach is taken with emphasis on the formulation of an equation useful for predicting solder behavior under a variety of conditions. Solutions of the constitutive equation predict the experimental hysteresis data for the loading histories available. Solutions of the equation for the test conditions of three independent sets of solder fatigue data show that the equation, together with the matrix creep failure indicator, can give an estimate of fatigue life. >

Electrodynamics of continua. Volume 1 - Foundations and solid media. Volume 2 - Fluids and complex media

Abstract: A unified approach is presented to the nonlinear continuum theory of deformable and fluent media subject to electromagnetic and thermal loads. Basic laws are used to establish the macroscopic electromagnetic theory are treated from first principles and nonlinear constitutive equations for large fields are developed. Many solutions of linear and nonlinear problems in the field of rigid media, elastic dielectrics, piezoelectricity, magnetoelasticity, ferromagnets, and magnetohydrodynamics are discussed. Applications are extended to ferrofluids, electrodynamics, memory-dependent materials, nonlocal theories, and relativistic continua.

Generalized constitutive equation for polymeric liquid crystals Part 1. Model formulation using the Hamiltonian (poisson bracket) formulation

Abstract: The Hamiltonian formulation of equations in continuum mechanics through Poisson brackets, developed in Z.R. Iwinski and L.A. Turski, Lett. Appl. Eng. Sci., 4 (1976), 179–191, P.J. Morrison and J.M. Greene, Phys. Rev. Lett., 45 (1980) 790–794, I.E. Dzyaloshinskii and G.E. Volovick, Ann. Phys., 125 (1980) 67–97, D.D. Holm, J.E. Marsden, T. Ratiu and A. Weinstein, Phys. Rep., 123 (1985) 1–116, M. Grmela, Phys. Lett. A, 130 (1988) 81–86, and A.N. Beris and B.J. Edwards, J. Rheol., 34 (1990) 55–78, for a class of incompressible fluids, is used here in order to generate a constitutive equation for the stress and the order parametr tensor for a polymeric liquid crystal. A free energy expression, of the type used by Doi in his theory for concentrated solutions of rigid rods, is used in addition to the Frank elasticity expression employed in the Leslie—Ericksen—Parodi (LEP) theory to model the effect of spatial gradients in the liquid crystalline microstructure. For homogeneous systems, the analysis leads to a model which is equivalent to a generalization of Doi theory out to fourth-order terms in S . Truncating this model at second-order terms gives the Doi equations exactly. To evaluate the expanded model, results for steady simple shear and extensional flows are compared against the Doi model predictions. The constitutive equation resulting from the expanded model is compared against the LEP constitutive equation and the parameters between the two are correlated. The additional stress terms for non-homogeneous systems reduce to a recently presented (B.J. Edwards and A.N. Beris, J. Rheol., 33 (1989) 1189–1193; M. Grmela, Phys. Lett. A, 137 (1989) 342–348) generalization of the Ericksen stress expression in terms of the second-order parameter tensor. The model presented is a generalization and extension of the order-parameter-based theory of Doi which allows a greater flexibility in describing the rheological properties of polymeric liquid crystalline systems

Poisson bracket formulation of incompressible flow equations in continuum mechanics

Abstract: The Hamiltonian formulation of equations in continuum mechanics through Poisson brackets is presented for a number of incompressible fluids, including the Euler inviscid fluid, the Newtonian viscous fluid, a perfectly elastic Poisson brackets is presented for a number of incompressible fluids, including the Euler inviscid fluid, the Newtonian viscous fluid, a perfectly elastic medium, the upper‐convected Maxwell, and the Oldroyd‐B viscoelastic fluids. The analysis, expanding previous results reported by Grmela, leads to a generalized Poisson bracket formalism from which all of the above‐mentioned cases can be recovered. Furthermore, the Poisson bracket formulation can easily incorporate model changes, as shown in the application of the hydrodynamic interaction correction to the Hookean dumbbell (upper‐convected Maxwell) model. The Hamiltonian formulation is fully explained here through a novel interpretation of the functional derivative through which the constraints of the flow are incorporated into the c...

Poisson bracket formulation of viscoelastic flow equations of differential type: A unified approach

Abstract: The Hamiltonian formulation of equations in continuum mechanics through a generalized bracket operation is shown here to reproduce a variety of incompressible viscoelastic fluid models, including the Giesekus model (with particular cases the upper‐convected Maxwell and the Oldroyd‐B models), the FENE–P dumbbell, the Phan‐Thien/Tanner, the Leonov, the Bird/DeAguiar, and the bead–spring chain models. The analysis allows comparison of the differential models on a more fundamental level than previously possible by reformulating the equations in terms of the Hamiltonian (system energy) and the dissipation of the system expressed as functionals involving the velocity vector and structural parameter(s). In fact, all of these models involve only slight variations of the same general Hamiltonian and the dissipation tensor. An advantage of this formulation is the establishment of thermodynamic admissibility criteria which in complex flows can shed light on the range of validity and/or faithfulness of the numerical calculations involving the above models. The usefulness of the generalized bracket formulation lies in the systematic approach that it provides in addressing one of the fundamental problems that the engineer working with complex materials has to deal with: how to transfer information that has been painstakingly provided by the physical chemist, addressing fundamental problems on a molecular level, from the microscopic scale to the macroscopic level where the engineer actually needs the model in dealing with everyday industrial problems. It is hoped that this new formulation can be used in the future to systematically generate continuum constitutive models, which are thermodynamically consistent, and based on microscopic analysis. Thus, it is the purpose here to narrow the gap between detailed (molecular) microscopic descriptions of the motions of polymer chains and (macroscopic phenomenological) continuum approaches. We believe that the generalized bracket formulation, due to its inherent simplicity and symmetry, has the potential to provide an answer to very complex situations, such as multicomponent structured media and coupled transport phenomena.

Micro-mechanics of crack initiation

Abstract: Prior to the crack initiation, damage is most often localized at a scale below the size of the classical representative volume element of the continuum mechanics. This allows the stress and strain analyses in a component to neglect the strain-damage coupling at macro-scale. At the micro-scale, this coupling plays a very important role which can be emphasized by a two scale element of an elastoplastic damaged micro-element embedded in an elastic or elastoplastic macro-element. The Lin-Taylor hypothesis of strain compatibility allows the determination of the damage at micro-scale by solving the coupled constitutive equations for a given macro-strain history. It is shown how this model may be cast in the form of a post-processor of a finite element code and how a simple damage law coupled with strain constitutive equations replicates the main features of ductile or creep crack initiation, low cycle and high cycle fatigue for the case of a three-dimensional state of stress.

Matrix-Tensor Methods in Continuum Mechanics

Abstract: The purposes of the text are: To introduce the engineer to the very important discipline in applied mathematics-tensor methods as well as to show the fundamental unity of the different fields in continuum mechanics-with the unifying material formed by the matrix-tensor theory and to present to the engineer modern engineering problems.

Variational Theory of Mixtures in Continuum Mechanics

Abstract: In continuum mechanics, the equations of motion for mixtures are derived through the use of Hamilton's extended principle which regards the mixture as a collection of distinct continua. The internal energy is assumed to be a function of densities, entropies and successive spatial gradients of each constituent. We first write the equations of motion for each constituent of an inviscid miscible mixture of fluids without chemical reactions or diffusion. Our work leads to the equations of motion in an universal thermodynamic form in which interaction terms subject to constitutive laws, difficult to interpret physically, do not occur. For an internal energy function of densities, entropies and spatial gradients, an equation describing the barycentric motion of the constituents is obtained. The result is extended for dissipative mixtures and an equation of energy is obtained. A form of Clausius-Duhem's inequality which represents the second law of thermodynamics is deduced. In the particular case of compressible mixtures, the equations reproduce the classical results. Far from critical conditions, the interfaces between different phases in a mixture of fluids are layers with strong gradients of density and entropy. The surface tension of such interfaces is interpreted.

Generalized constitutive equation for polymeric liquid crystals: Part 2. non-homogeneous systems

Abstract: The Hamiltonian formulation of equations in continuum mechanics through Poisson brackets was used in Ref. 1 to develop a constitutive equation for the stress and the order parameter tensor for a polymeric liquid crystal. These equations were shown to reduce to the homogeneous Doi equations as well as to the Leslie-Ericksen-Parodi (LEP) constitutive equations under small deformations [1]. In this paper, these equations are fitted against the non-homogeneous Doi equations through the simulation of the spinodal decomposition of the isotropic state when it is suddenly brought into a parameter region in which it is thermodynamically unstable. Linear stability analysis reveals the wavelength of the most unstable fluctuation as well as its initial growth rate. Results predicted from this theory compare well with the predictions of Doi for the spinodal decomposition using an extended molecular rigid-rod theory in terms of the distribution function. This completes the development of a generalized constitutive equation for polymeric liquid crystals initiated in Part 1.

Dynamic continuum modeling of beamlike space structures using finite element matrices

Abstract: Subscripts A rational and straight-forward method is introduced for developing equivalent continuum models of large beam-like periodic lattice structures based on energy equivalence Extended Timoshenko bean model is chosen to take account of the effects due to couplings between extension, transverse shear and bending deformations The procedure for developing continuum models involves utilizing well-defined existing finite element matrices directly in caluclating strain and kinetic energies from which equivalent continuum structural and dynamic properties are induced The numerical results of free vibration analysis show that the method developed in this paper gives very reliable dynamic characteristics compared to other methods Nomenclature

Probabilistic formulation of damage-evolution law of cementitious composites

Abstract: A model is proposed with a probabilistic evolution of damage in the material. The constitutive law is built through the development of a two-level approach. The micro-level is assumed to be constituted of elastic-brittle springs whose strength follows a probabilistic distributin. The representative macro-volume of material contains a given number of these elementary defects and its damage (loss of elastic properties) is computed from the knowledge of the local states. The macro-behavior results from the interactions between all the micro-defects. The model may be considered as representative of the behavior of brittle and almost brittle materials. It exhibits scattering and size effect. A Weibull distribution law is assumed for the local probabilities of failure and a parallel loose bundle connects the micro-defects. These simple hypotheses lead to analytical expressions of the probabilistic constitutive law. The approach developed appears as an intermediate model between continuous damage mechanics and probabilistic brittle fracture. The knowledge of a single parameter \IN\N\dt, number of defects in a given volume, provides the degree of ductility of the material.

Asymptotic models for a long, elastic cylinder

Abstract: We study asymptotic expansions for the displacement field of a long elastic cylinder under various constitutive assumptions. We show that under simple hypotheses it is possible to derive from the equations of continuum mechanics two known beam equations and several different string models. Some of the string models correspond to those studied by S. Antman and R. Dickey. We also show that under our assumptions the problem of asymptotic expansion can be reduced to that of algebraic geometry.

Kinematics of media with continuously changing topology

Abstract: The fundamental postulate of continuum mechanics states that a body is a three-dimensional differentiable manifold and its motions are diffeomorphisms. Simple thought experiments with cyclic motions of dislocations show that they do not preserve topology (set of neighborhoods). The same is valid for chaotic and turbulent motions with coarse-graining. To describe such motions, kinematics of a generalized continuum mechanics is suggested. Observables are defined operationally in the laboratory system which is not anymore equivalent to the Lagrangian picture. The body is a submanifold of a higher-dimensional space and generalized motions are its diffeomorphisms. In a gauge-theoretic interpretation, the motion is a translational connection with the curvature identified as a “dislocation” density-flux.

Metric-torsion gauge theory of continuum line defects

Abstract: Basic points underlying the geometrization of continuum defects are discussed. Following an analogy with gravitational gauge theories, a metric-torsion gauge theory of continuum line defects is developed. Gauge-invariant action integrals are constructed and their equations of motion are obtained. A Lagrangian containing curvature terms up to second power has constant-curvature solutions. In linear approximation these solutions correspond to line defects which form closed loops separately.