Showing papers on "Continuum mechanics published in 1989"

Continuum Mechanics of Electromagnetic Solids

Abstract: 1. Essential Properties of Electromagnetic Solids. 2. Elements of Continuum Mechanics. 3. General Equations of Nonlinear Electromagnetic Continua. 4. Elastic Dielectrics and Piezoelectricity. 5. Elastic Conductors. 6. Elastic Ferromagnets. 7. Elastic Ionic Crystals, Ferroelectrics and Ceramics. Appendices. Index.

Boundary Conditions at the Cartilage-Synovial Fluid Interface for Joint Lubrication and Theoretical Verifications

Abstract: The objective of this study is to establish and verify the set of boundary conditions at the interface between a biphasic mixture (articular cartilage) and a Newtonian or non-Newtonian fluid (synovial fluid) such that a set of well-posed mathematical problems may be formulated to investigate joint lubrication problems. A "pseudo-no-slip" kinematic boundary condition is proposed based upon the principle that the conditions at the interface between mixtures or mixtures and fluids must reduce to those boundary conditions in single phase continuum mechanics. From this proposed kinematic boundary condition, and balances of mass, momentum and energy, the boundary conditions at the interface between a biphasic mixture and a Newtonian or non-Newtonian fluid are mathematically derived. Based upon these general results, the appropriate boundary conditions needed in modeling the cartilage-synovial fluid-cartilage lubrication problem are deduced. For two simple cases where a Newtonian viscous fluid is forced to flow (with imposed Couette or Poiseuille flow conditions) over a porous-permeable biphasic material of relatively low permeability, the well known empirical Taylor slip condition may be derived using matched asymptotic analysis of the boundary layer at the interface.

On the mechanics of crack closing and bonding in linear viscoelastic media

Abstract: The mechanics of quasi-static crack closing and bonding of surfaces of the same or different linear viscoelastic materials is described. Included is a study of time-dependent joining of initially curved surfaces under the action of surface forces of attraction and external loading. Emphasis is on the use of continuum mechanics to develop equations for predicting crack length or contact size as a function of time for relatively general geometries; atomic and molecular processes associated with the healing or bonding process are taken into account using a crack tip idealization which is similar to that used in the Barenblatt method for fracture. Starting with a previously developed correspondence principle, an expression is derived for the rate of movement of the edge of the bonded area. The effects of material time-dependence and the stress intensity factor are quite different from those for crack growth. A comparison of intrinsic and apparent energies of fracture and bonding is made, and criteria are given for determining whether or not bonding can occur. Examples are given to illustrate use of the basic theory for predicting healing of cracks and growth of contact area of initially curved surfaces. Finally, the effect of bonding time on joint strength is estimated from the examples on contact area growth.

On the application of dual variables in continuum mechanics

Abstract: Stress and strain tensors that arise in the expression of the stress power are called “conjugate variables”. More special is the term “dual variables” which has been introduced in connection with incremental constitutive relations of hypoelasticity and plasticity, where the rates of both tensors arise. We propose a rational rule from which the most natural form of tensor-valued kinematic and dynamic variables (strain and stress tensors) including their corresponding time rates can be deduced. Dual variables and their associated dual derivatives are characterized by the property that apart from the stress power also the incremental stress power is invariant under a group of transformations that corresponds to a set of physically reasonable intermediate configurations. We outline the precursory history of these concepts and then discuss in detail how the invariance properties can be realized in the various stress and strain measures. We finally demonstrate the concept in three different applications: The rate form of the principle of virtual work, the formulation of constitutive relations in viscoelasticity and the formulation of incremental constitutive assumptions of rate-independent plasticity.

3‐D shape optimal design and automatic finite element regridding

Abstract: A unified method for continuum shape design sensitivity analysis and optimal design of mechanical components is developed. A domain method of shape design sensitivity analysis that uses the material derivative concept of continuum mechanics is employed. For numerical implementation of shape optimal design, parameterization of the boundary shape of mechanical components is defined and illustrated using a Bezier surface. In shape design problems, nodal points of the finite element model move as the shape changes. A method of automatic regridding to account for shape change has been developed using a design velocity field in the physical domain that obeys the governing equilibrium equations of the elastic solid. For numerical implementation of the continuum shape design sensitivity analysis and automatic regridding, an established finite element analysis code is used. To demonstrate the feasibility of the method developed, shape design optimization of a main engine bearing cap is carried out as an example.

Introduction to Continuum Mechanics for Engineers

Abstract: 1. One-Dimensional Continuum Mechanics.- 1.1. Kinematics of Motion and Strain.- 1.2. Balance of Mass.- 1.3. Balance of Linear Momentum.- 1.4. Balance of Energy.- 1.5. General Balance.- 1.6. The Entropy Inequality.- 1.7. Example Constitutive Equations.- 1.8. Thermodynamic Restrictions.- 1.9. Small Departures from Thermodynamic Equilibrium.- 1.10. Small Departures from Static Equilibrium.- 1.11. Some Features of the Linear Model.- 2. Kinematics of Motion.- 2.1. Bodies and Deformations.- 2.2. Velocity, Acceleration, and Deformation Gradients.- 2.3. Transformation of Linear, Surface, and Volume Elements.- 2.4. Strain Kinematics.- 2.5. Infinitesimal Strain Kinematics.- References.- 3. Equations of Balance.- 3.1. Balance of Mass.- 3.2. Balance of Linear Momentum.- 3.3. Balance of Angular Momentum.- 3.4. Balance of Energy.- 3.5. The Entropy Inequality.- 3.6. Jump Equations of Balance-Material Versions.- References.- 4. Models of Material Behavior.- 4.1. Examples.- 4.2. Isothermal Elasticity-Thermodynamic Restrictions.- 4.3. Isothermal Elasticity-Material Frame Indifference.- 4.4. Isothermal Elasticity-Material Symmetry.- 4.5. Incompressible Isothermal Elasticity.- 4.6. Thermoelastic Material with Heat Conduction and Viscous Dissipation-Constitutive Assumptions.- 4.7. Thermoelastic Material with Heat Conduction and Viscous Dissipation-General Thermodynamic Restrictions.- 4.8. Thermoelastic Material with Heat Conduction and Viscous Dissipation-Equilibrium Thermodynamic Restrictions.- 4.9. Thermoelastic Material with Heat Conduction and Viscous Dissipation-Material Frame Indifference.- 4.10. Thermoelastic Material with Heat Conduction and Viscous Dissipation-Material Symmetry.- 4.11. Constitutive Equations for a Compressible, Conducting, Viscous Fluid.- 4.12. Constitutive Equations for an Isotropic Linear Thermoelastic Solid with Heat Conduction.- References.- 5. Materials with Internal State Variables.- 5.1. Constitutive Assumptions and Thermodynamic Results.- 5.2. Maxwell-Cattaneo Heat Conductor.- 5.3. Maxwellian Materials.- 5.4. Closing Remarks-Alternate Forms of the Entropy Inequality.- References.- Appendix A. Mathematical Preliminaries.- A.1. Vector Spaces.- A.2. Linear Transformations.- A.3. Inner Product Spaces.- A.4. Components of Vectors and Linear Transformations.- A.5. Cross Products, Determinants, and the Polar Decomposition Theorem.- A.6. Multilinear Functionals and Tensor Algebra.- A.7. Euclidean Point Spaces, Coordinate Systems.- A.8. Vector Analysis.- Appendix B. Representation Theorems.

Squeezing flow of fibre-reinforced viscous fluids

Abstract: A relatively simple continuum model is described for the viscous flow of highly anisotropic materials such as fibre-reinforced resins. The theory is applied to the flow of such a fluid when squeezed between two rigid platens.

A transport theorem for moving interfaces

Abstract: where S*(t) is a surface which evolves with time t, f(x, t), defined for all x e and all t, is the density (per unit area) of a superficial quantity such as energy, and da(x) is the area measure on surfaces in R-\\ The evaluation of (1) is nontrivial when S?{t) evolves within a fixed region QcR3 and dS^{t) C is nonempty, for then a portion of (1) must balance an outflow of / due to the transport of portions of S^(t) across <9£2. We assume that S*(t) is smooth and oriented by n(x, t), a particular choice of continuous unit-normal field, and we write V(x, t) and k(x, t) for the normal velocity and total curvature. (Total curvature is twice the normal curvature.) It is the purpose of this note to prove the transport theorem:1

Energy‐Based Coupled Elastoplastic Damage Models at Finite Strains

Abstract: Energy‐based continuum damage‐elastoplasticity theories at finite strains are proposed within the framework of damage mechanics. The proposed damage models are based on the effective stress concept, damage threshold loading/unloading conditions, and the multiplicative split of finite kinematics. The models are linked to the history of “damage energy release rate” within representative volumes. The elastoplastic damage constitutive theories feature a thermodynamic basis, characterization of damage, coupling of damage and plasticity, as well as an anisotropic microcrack opening/closing mechanism. Both spatial and material descriptions are discussed. A simple and efficient computational integration algorithm is also given. In particular, a three‐step operator split algorithm is developed within the present framework. A numerical experiment of a notched specimen involving damage coupled with plastic flow is presented to illustrate the capability of the proposed method.

Dynamics of saturated rocks. I: Equations of motion

Abstract: The field and constitutive equations expressing the dynamic behavior of fully saturated elastic rocks with two degrees of porosity—one due to the pores and the other due to the fissures—are developed. The corresponding linearized governing equations of motion are then constructed to form a system of 11 partial differential equations with 11 unknowns. The various phenomenological coefficients of the theory are identified and expressed in terms of measurable quantities. The quasi-static case with two degrees of porosity and the dynamic case with one degree of porosity are easily obtained as special cases of the present formulation and compared with those due to Aifantis and Biot, respectively. A comparison of the present equations of motion against those due to Wilson and Aifantis is also made.

Phenomenological aspects of continuum damage mechanics

Abstract: The objective of the paper is to review the general principles and the main possibilities of CDM, considering both the experimental aspects, the theoretical concepts and some applications to structures. CDM is considered as a general method to treat the progressive deterioration of materials and structures in the framework of Continuum Mechanics.

Irreducible tensor description. III. Thermodynamics of a low-temperature phonon gas

Abstract: Let one assume that the interacting phonon gas, whose behavior is governed by the Boltzmann–Peierls equation, inhabits an insulating crystal at sufficiently low temperature. Then, within the framework of a single acoustic phonon branch and of an isotropic long‐wavelength approximation to the dispersion relation, the simplest acceptable version of extended irreversible thermodynamics, based upon the nine‐moment system of differential equations for the slow and fast gas‐state variables, is carefully investigated. It is clearly demonstrated that, in virtue of the structure simplifications just mentioned, the conceptually different (macroscopic, kinetic, and variational) procedures, which are discussed in this paper, appear complementary to each other. Finally, with the help of a suitable contraction of the nine‐moment system of field equations, for Callaway’s relaxation model a slightly generalized nonlinear variant of ordinary low‐temperature phonon hydrodynamics is explicitly derived.

Basic equations for shocks in nonlinear electroelastic materials

Abstract: The complete set of balance equations and jump relations useful in the study of the propagation of strong discontinuities of the shock‐wave type is deduced both in its Eulerian form and material form (the latter being the most useful one) for nonlinear anisotropic electroelastic materials such as piezoelectric ceramics. In particular, the relevant form of the so‐called Hugoniot equation is obtained, and all expressions are given a useful form in terms of the displacement gradient and the material electric field (or the gradient of electric potential in the quasielectrostatic approximation). The equations obtained are based on a strict use of nonlinear continuum mechanics and the electrodynamics of continua with electromechanical interactions.

Three-dimensional vibrations of Tethered Satellite System

Abstract: HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Three-dimensional vibrations of Tethered Satellite System Monica Pasca, Marcello Pignataro, Angelo Luongo

Time dependent reliability model incorporating continuum damage mechanics for high-temperature ceramics

Abstract: Presently there are many opportunities for the application of ceramic materials at elevated temperatures. In the near future ceramic materials are expected to supplant high temperature metal alloys in a number of applications. It thus becomes essential to develop a capability to predict the time-dependent response of these materials. The creep rupture phenomenon is discussed, and a time-dependent reliability model is outlined that integrates continuum damage mechanics principles and Weibull analysis. Several features of the model are presented in a qualitative fashion, including predictions of both reliability and hazard rate. In addition, a comparison of the continuum and the microstructural kinetic equations highlights a strong resemblance in the two approaches.

The order of a damage tensor

Abstract: In this paper, the stress deformation constitutive relations for continua are discussed and a stress deformation constitutive relation expressed by functional tensorial expression is found. When we study the anisotropic damage of anisotropic materials either from a macroscopic continuum mechanics model or from a microdefect model, there exists a limit to the order of a damage tensor, and the condition under which the damage variable may be described by a tensor lower than those of the highest order is found.

Nonlocal continuum damage/plasticity model for impulse-loaded rc beams

Abstract: In this paper, a rate‐independent constitutive model for plain concrete is proposed for application to the analysis of impulse‐loaded structural members. The model combines a continuum damage approach, using a scalar damage variable, with a pressure‐sensitive plasticity model. The plasticity model incorporates a nonassociated flow rule in regions of low compressive or tensile hydrostatic pressures and an associated flow rule elsewhere; the possibility of energy generation through use of a nonassociated flow rule is avoided through modification of the flow rule relations. Strain softening is also included and the deleterious side effects (i.e., mesh dependency and unreasonable energy dissipation) are avoided through use of a nonlocal definition of the damage parameter. The concrete model is combined with a uniaxial steel model and a layered, large‐strain, Timoshenko beam element to perform the analysis of impulse‐loaded, simply supported, reinforced‐concrete beams.

Generalization of the method of perturbation of the form of the boundary in the mechanics of deformable bodies

Abstract: Introduction. Under the title "A Method of Pertubation of the Form of the Boundary" there,appeared in the scientific literature an approximate analytical method of solution of boundary value problems of continuum mechanics for noncanonical domains (domains not allowing a solution by the method of direct separation of variables). Its conceptual basis was contained in a paper by Guz' [4], published in 1962, in which, for the first time, an effective, approximate analytical method was proposed for studying stress concentration around curvilinear holes in shells. This idea proved to be sufficiently fruitful, and the developed method so general, that with no principal difficulties it was extended not only to threedimensional boundary value problems of elasticity theory for bodies of revolution [6, 13] and noncircular cylinders [7, 15], but also to broad classes of three-dimensional boundary value problems of the mechanics of deformable b6dies [12] and related problems of continuum mechanics [5, 9] for noncanonical domains. The recursion relationships and differential operators constructed in an arbitrary approximation [16] make it possible to solve stated problems in principle with specified accuracy. A broadening of the possibilities of applying the method proposed in [4] (subsequently referred to in [8] as a first variant of the method of perturbation of the form of the boundary) was enhanced through its application, along with other analytical methods, to the solution of boundary value problems for media with complicated elastic properties. Thus, along with the methods of successive approximations and the perturbation of elastic properties, it was applied for the first time to the solution of planar [I0] and spatial [14] nonlinear physical problems of elasticity theory and also to the solution of boundary value problems for curvilinear-orthotropic bodies of noncanonical form [18]. The indicated first version of the method of perturbation of the form of the boundary, along with an integral Laplace transform with respect to the time, was also applied to the solution of spatial [27] and planar [23] related problems of the mechanics of saturated porous media with noncanonical processing.

The principle of material frame indifference and the covarianee principle

Abstract: A comparison between the formulations of the principle of material frame indifference in continuum mechanics, which principle refers to stress-strain relations without inertial forces, and the covariance principle in the theory of general relativity indicates that a relationship between them can be established. It is shown that the principle of material frame indifference follows from the covariance principle in the nonrelativistic limit when inertia is considered to be absent. As a result, the principle of material frame indifference has received further justification but it cannot retain its status of a fundamental principle.

Numerical Simulation of Filling and Discharging Processes in Silos

Abstract: In this paper a numerical method for the calculation of stress- and velocity- fields in silos during storage and during mass- or core flow is presented based upon a consistent continuum mechanics approach. For the granular bulk material a constitutive highly nonlinear model is used that covers the solid-like behaviour of the material during at rest conditions as well as the fluid-like behaviour during discharging. The resulting set of differential equations are solved by an Eulerian Finite Element Method spatially and by Finite Difference Techniques in time. On the basis of this consistent approach and the developed computer programm a great variety of silo problems can be studied. In contrast to other methods no restrictions as to the start of discharging are necessary. Computations can be done over a long period until a constant state of stresses and velocities is reached.

Constitutive equations in viscoelasticity through the Legendre-Fenchel transformation

Abstract: A thermodynamic scheme is investigated which allows interesting applications in continuum mechanics. Then the Legendre-Fenchel transformation is considered and it is shown how it can be applied to determine solutions to the Clausius-Duhem inequality as the statement of the second law. Next the attention is confined to linear viscoelastic solids, viewed as materials with internal variables, and a general expression for the stress-strain relation is derived.

Nonlinear Continuum Mechanics and Applied Analysis

Abstract: : Ropes in Equilibrium is concerned with the equilibria of ropes or strings that are lying on surfaces or intertwined with other strings Stability and folds ; A Model for Disclinations In Nematic Liquid Crystals; On the Maneuvering of Vehicles addressed here is the kinematics governing the motion of vehicles that move on rolling wheels Restricted Quadriatic Forms, Inertia Theorems, and the Schur Complement; On the Kinematics of Wheeled Mobile Robots; On second-order conditions in constrained variational principles; Optimal Design of Columns Against Buckling; Director theories of rods reduces the equilibrium conditions for a uniform isotropic rod to a phase-plane for the curvature of the centerline of the rod; and 'The Stability of Relative Equilibria Many important Hamiltonian systems have periodic solutions that are associated with symmetries of the equations The equations governing the rotation of heavy rigid bodies comprise one such system in which the special solutions are known as permanent rotations