Showing papers on "Continuum mechanics published in 1993"

8 – Continuum Models for Layered and Blocky Rock

Abstract: Publisher Summary This chapter presents the framework of Cosserat continuum theory (CT) to model the behavior of granular, layered, and blocky rock. The materials of geomechanics are distinguished from other engineering materials such as steel and concrete, primarily by their visible inhomogeneity. The mechanical modeling of such materials is complex but is needed for the numerical and analytical prediction or back-analysis of forces and displacements within rock bodies around engineering structures. Nonhomogeneous materials can be modeled by conventional continuum theories only if the characteristic fabric length of the material is vanishingly small, as compared to some characteristic structural length. The chapter discusses the most important kinematic and static relationships for plane, infinitesimal deformations of Cosserat's continuum. The absence of any characteristic length for the localized strain in conventional continuum models is the reason for the critical dependence of finite element calculations in related boundary value problems on the employed finite element grid. Analytical solutions of boundary value problems of Cosserat theory require approximately the same mathematical effort as corresponding boundary value problems of conventional continuum mechanics.

The Continuum Mechanics of Coherent Two-Phase Elastic Solids with Mass Transport

Abstract: We develop a theory for the dynamics of an interface in a two-phase elastic solid with kinetics driven by mass transport and stress. We consider a two-phase system consisting of bulk regions separated by a sharp interface endowed with energy and capable of supporting force. Our discussion is based on balance laws for mass and force in conjunction with a version of the second law-appropriate to a mechanical system out of equilibrium-which we use to develop a suitable constitutive theory for the interface. It is assumed that mass transport is characterized by the bulk diffusion of a single independent species; we neglect mass diffusion within the interface; limit our discussion to a continuous chemical potential and to a coherent interface; neglect the elasticity of the interface; and consider only infinitesimal deformations, neglecting inertia. We show that the field equations and free-boundary conditions can be developed in a simple manner in terms of the diffusion potential and its time derivatives, as opposed to the usual formulation in terms of concentration. Natural consequences of the thermodynamic framework are Lyapunov functions for the resulting evolution problems. This leads to a hierarchy of variational principles that should describe the equilibrium shapes of misfitting particles as well as possible microstructures that might form; these principles are applicable both in the absence and presence of an applied stress.

Engineering mechanics of composite structures

Abstract: Continuum mechanics - basic concepts and problem formulations asymptotic homogenization of regular structures elasticity of regular composite structures coupled field and mechanics of periodic composites general homogenization models for composite plates and shells with rapidly varying thickness structurally nonhomogeneous periodic plates and shells network and framework reinforced shells and plates with regular structure.

The effect of block rotations on the global seismic moment tensor and the patterns of seismic P and T axes

Abstract: Distributed brittle deformation of the Earth's crust involving block rotations is comparable to the deformation of a granular material, with fault blocks acting like the grains. The deformation of a granular material is not adequately described using classical continuum mechanics because the individual grains within the material rotate in a manner that is not uniquely determined by the large-scale average deformation. Thus a theoretical link has not existed between the kinematics of deformation involving block rotation and the associated effects on the seismic moment tensor and focal mechanism solutions. We establish this link using micropolar continuum theory (Eringen, 1964, 1966 a, b; Eringen and Suhubi, 1964) and the analysis of the effect of block rotations on fault slickenline patterns by Twiss et al. (1991). This theory takes into account two separate scales of motion: a large-scale average motion of the material, the macromotion, composed of a macrodeformation rate (i.e., a macrostrain rate) and a macrospin, and a local motion, the microspin, that describes the average rotation rate of grains in the material. The micropolar kinematic theory allows us to predict the orientations of coseismic slip directions V on local shear planes of any orientation in a large-scale shear zone. We define a local and a global asymmetric micropolar seismic moment tensor in terms of these slip directions. For a restricted kinematic model, the theory shows that two scalar parameters, D and W, determine the symmetry of the global micropolar seismic moment tensor and the pattern of seismic P (shortening) and T (lengthening) axes. The deformation rate parameter D is defined in terms of the principal values of the deformation rate tensor . The deformation is transtensional (constrictional) if 0 ≤ D ≤ 0.5, plane strain if D = 0.5, and transpressional (flattening) if 0.5 < D ≤ 1. The net vorticity parameter W is a normalized value of the difference between microspin and macrospin . It is an objective variable. W = 0 implies that the global micropolar seismic moment tensor is symmetric and that P and T axis patterns have orthorhombic or higher symmetry. W ≠ implies that the global micropolar seismic moment tensor is asymmetric and that P and T axis patterns have monoclinic symmetry. The antisymmetric part of the global micropolar seismic moment tensor is associated with the net vorticity that characterizes the deformation. W has different values for different models of rigid block rotation and thus could serve to identify the rotation mechanism.

Modeling of penetrant diffusion in glassy polymers with an integral sorption deborah number

Abstract: A mathematical model was developed to explain the anomalous penetrant diffusion behavior in glassy polymers The model equations were derived by using the linear irreversible thermodynamics theory and the kinematic relations in continuum mechanics, showing the coupling between the polymer mechanical behavior and penetrant transport The Maxwell model was used as the stress–strain constitutive equation, from which the polymer relaxation time was defined An integral sorption Deborah number was proposed as the ratio of the characteristic relaxation time in the glassy region to the characteristic diffusion time in the swollen region With this definition, an integral sorption process was characterized by a single Deborah number and the controlling mechanism was identified in terms of the value of the Deborah number The model equations were two coupled nonlinear differential equations A finite difference method was developed for solving the model equations Numerical simulation of integral sorption of penetrants in glassy polymers was performed The simulation results show that (1) the present model can predict Case II transport behavior as well as the transition from Case II to Fickian diffusion and (2) the integral sorption Deborah number is a major parameter affecting the transition © 1993 John Wiley & Sons, Inc

A Dynamical Theory of Structured Solids. I Basic Developments

Abstract: Based on the well-accepted notion of a Bravais lattice of a crystal at the atomic scale and with particular reference to inelastic behaviour of materials, this paper is concerned with the construction of a macroscopic dynamical theory of solids which incorporates the effect of the presence of the atoms and their arrangements. The theory incorporates a wide variety of microstructural processes occurring at various physical scales and has a range approaching the atomic scale. These processes include the effect of the motion of individual dislocations, which are modeled here as continuous distributions at the macroscopic scale. The formulation of the basic theory, apart from the kinematical and kinetical variables employed in classical continuum mechanics, utilizes a triad of independent vector-valued variables - called directors - (or an equivalent tensor-valued variable) which represent the lattice vectors and are determined by additional momentum-like balance laws associated with the rate of change of lattice deformation in the spirit of a Cosserat (or directed) continuum. A suitable composition of the triad of directors and the ordinary deformation gradient is identified as a measure of permanent or plastic deformation, the referential gradient of which plays a significant role in the kinematics of lattice defects. In particular, a uniquely defined skew-symmetric part of the gradient of plastic deformation is identified as a measure of the density of dislocations in the crystal. The additional momentum-like balance laws associated with the rate of lattice deformation include the effect of forces necessary to maintain the motion of dislocations, as well as the inertia effects on the microscopic and submicroscopic scales arising from the director fields. The basic theoretical developments also provide important clarifications pertaining to the structure of the constitutive response functions for both viscoplasticity and (the more usual) rate-independent plasticity.

On the mathematical modelling of material behavior in continuum mechanics

Abstract: The classical theories of continuum mechanics — linear elasticity, viscoelasticity, plasticity and hydrodynamics — are defined by special constitutive equations. These can be understood to be asymptotic approximations of a quite general constitutive model, valid under restrictive assumptions for the stress functional or the input processes. The general theory of material behavior develops systematic methods to represent material properties in a context of physical evidence and mathematical consistency. According to experimental observations material behavior may be rate independent or rate dependent with or without equilibrium hysteresis. This motivates four different constitutive theories, namely elasticity, plasticity, viscoelasticity and viscoplasticity. Constitutive equations can be formulated explicitly as functionals. Then, the particular constitutive models correspond to continuity properties of these functionals, related to convenient function spaces. On the other hand, a system of differential equations may lead to an implicit definition of a stress functional. In this case additional variables are introduced, which are called internal variables. For these variables additional evolution equations must be formulated, specifying the rate of change of the internal variables in dependence on their present values and the strain (or stress) input. In the context of different models of inelastic material behavior the evolution equations have different mathematical characteristics. These concern the existence of equilibrium solutions and their stability properties. Rate independent material behavior is modelled by means of evolution equations, which are related to an arclength instead of the time as independent variable. It can be shown that the rate independent constitutive equations of elastoplasticity are the asymptotic limit of rate dependent viscoplasticity for slow deformation processes.

Boundary interactions for two-dimensional granular flows. Part 1. Flat boundaries, asymmetric stresses and couple stresses

Abstract: The behaviour of a granular flow at a boundary cannot be specified independently of what is happening in the rest of the flow field. This paper describes a study of two fictitious, but instructive, flat boundary types using a computer simulation of a two-dimensional granular flow with the goal of trying to understand the possible effects of the boundary on the flow. The two boundary conditions, Type A and Type B, differ largely in the way that they apply torques to the flow particles. During a particle–wall collision, the Type A boundary applies the force at the particle surface, thus applying the largest mechanistically possible torque to the particle, while the Type B boundary applies the force directly to the particle centre, resulting in the application of zero torque. Though a small change on continuum scales (i.e. the point at which the force is applied has only been moved by a particle radius) it makes a huge difference to the macroscopic behaviour of the system. Generally, it was found that, near boundaries, large variations in continuum properties occur over distances of a particle diameter, a non-continuum scale, throwing into doubt whether boundaries may be accurately modelled via continuum mechanics. Finally, the large torques applied to the particles by the Type A boundary induce asymmetries in the stress tensor, which, in these steady flows, are balanced by gradients in a couple stress tensor. Thus, near boundaries, a frictional granular material must be modelled as a polar fluid.

Continuum mechanics in environmental sciences and geophysics

Abstract: Modern continuum mechanics is the topic of this book. After its introduction it will be applied to a few typical systems arising in the environmental sciences and in geophysics. In large lake/ocean dynamics peculiar effects of the rotation of the Earth will be analyzed in linear/nonlinear processes of a homogenous and inhomogenous water body. Strong thermomechanical coupling paired with nonlinear rheology affects the flow of large ice sheets (such as Antarctica and Greenland) and ice shelves. Its response to the climatic forcing in an environmental of greenhouse warming may significantly affect the life of future generations. The mechanical behavior of granular materials under quasistatic loadings requires non-classical mixture concepts and encounters generally complicated elastic-plastic-type constitutive behavior. Creeping flow of soils, consolidation processes and ground water flow are described by such theories. Rapid shearing flow of granular materials lead to constitutive relations for the stresses which incorporate rate independent behavior of Mohr-Coulomb type together with dispersive stress contributions due to particle collisions. Rockfalls, sturzstroms, snow and ice avalanches, but also debris flow and sea ice drifting can be described with such formulations.

An equilibrium theory of dislocation continua

Abstract: A homogenised model for elastic media containing large numbers of dislocations is described. First, discrete dislocations are discussed from the mathematical and crystallographical points of view, and their stress fields are calculated. These building blocks are averaged to construct a homogenised model in which the dislocation distribution is averaged to a number density tensor. Equilibrium configurations are then considered and it is proved that, in the absence of externally applied stresses, the only possible finite, simply connected distributions are ones in which all components of stress vanish everywhere. Some examples are given of these zero-stress everywhere (ZSE) distributions, and their geometrical interpretation is considered in terms of the plastic distortion tensor, which shows that they are equivalent to local rotations of the crystal lattice. Finally, some conjectures are made about the response of a cellular ZSE distribution to an applied stress, introducing the idea of “polarization” by a...

Prediction of steady state creep behavior of two phase composites

Abstract: A simple continuum mechanics-based model has been developed to predict the steady state creep rates of composites containing coarse and rigid reinforcements from the matrix creep behavior. The model has been derived on the basis of a unit cell, representative of the composite microstructure, which is idealized to a pattern of periodic, cubic inclusions distributed uniformly in a continuous creeping matrix. Comparisons of the predicted creep rates are made with the experimental data of a number of two phase systems as well as transversely loaded continuous fiber reinforced composites. A good agreement between the predicted and measured creep rates is seen for most of the systems. However, for some composites, the calculations overestimate the creep rates significantly at intermediate volume fractions, typically, 0.3–0.4. It is suggested that factors such as the differences between the microstructure of the matrix in the composite and that of the monolithic matrix could be responsible for the differences in the predicted and experimentally measured creep behavior. Finally, an assessment of the predictions of the model proposed in this study with rigorous, self-consistent calculations as well as finite element simulations has been made.

Analysis of multiple-bay frames using continuum model

Abstract: An approximate method that can be used to determine displacements and member forces of multiple-bay frames is presented. The method utilizes simple continuum models. Finite-element representations of building frameworks resulting from continuum methodology require significantly fewer degrees of freedom than classical discrete finite-element models, which individually model each of the beam and column elements. As a result, the use of continuum models to analyze structures can result in considerable savings in terms of computational effort. This makes the method attractive for use in the analysis of large frames, especially during the preliminary design stage. This paper reviews the continuum methodology and presents a method for computing discrete element forces from the continuum element displacements. In determining element forces, the effects of deep-beam-type cross-sectional warping are approximated. Examples including the analysis of fixed- and pinned-base single- and multiple-bay plane frames illustrate the accuracy of the continuum analyses procedure.

Continuum Mechanics, Nonlinear Finite Element Techniques and Computational Stability

Abstract: This three lectures course will give a modern concept of finite-element- analysis in nonlinear solid mechanics using material (Lagrangian) and spatial (Eulerian) coordinates. Elastic response of solids is treated as an essential example for the geometrically and material nonlinear behavior. Furthermore a brief introduction in stability analysis and the associated numerical algorithms will be given.

Stresses in Rock and Rock Masses

Abstract: Publisher Summary In continuum mechanics, some aspects of rock mechanics may be dealt with very efficiently without referring to stresses. Rocks, and even more so, rock masses, are neither continuous nor purely solid and one may question the validity of applying concepts of continuum mechanics to such materials. Because rocks are heterogeneous and, most of the time, polyphasic, the concept of stress refers, in rock mechanics, to mean forces per unit area. When the rock is assimilated to a continuum material, the elementary representative volume is the smallest volume for which there is equivalence between the idealized continuum material and the real rock. This equivalence principle may be illustrated by considering a simple porous material. This material is made up of a single material matrix and a single fluid completely filling the pore space assumed to be fully interconnected.

Finite elements analysis of viscoelastic fracture

Abstract: A numerical procedure based on the finite elements method and Schapery's formulation is proposed to determine the critical condition of cracks in viscoelastic structures. Some initial results trying to couple fracture mechanics with continuum damage mechanics also are presented.

Foundation of Continuum Mechanics

Abstract: The following lectures give a short indroduction into modern continuum mechanics. The presentation omits many details and should motivate the reader to study the references.

An introduction to continuum mechanics -after Truesdell and Noll

Abstract: Preface. 0: Preliminary Results. 1: Framings. 2: Bodies and Motion. 3: Kinematics. 4: Cauchy Stress Tensor. 5: Examples on Stress Constitutive Relations. 6: Noll's Simple Material. 7: Internally Constrained Materials. 8: Material Classification from Symmetry. 9: Canonical Stress Functions for Isotropic Materials. 10: Classical Infinitesimal Theory of Elasticity. 11: Shear of an Isotropic Elastic Rectangular Block. 12: Torsion of an Isotropic Elastic Circular Cylinder. References. Quotation References. Name Index. Subject Index.

Micromechanical Considerations in Shock Compression of Solids

Abstract: The application of what is commonly known as solid continuum mechanics has been very successful in descriptions of the shock-compression process. It has given us the jump conditions, useful concepts of average quantities such as density and specific internal energy (for example), and constitutive descriptions (including equations of state) involving these average quantities and their time rates of change. Even as we profitably use these ideas, we always have in mind micromechanical concepts such as the crystal lattice, the electronic structure of the crystallographic system, and lattice defects which give rise to important physical phenomena.

On objective surface rates

Abstract: In this paper we derive objective, in the sense of surface, rates of tensors and we give a correct formulation of a two-dimensional continuum. Furthermore we present objective rates of tensors on the boundary line enclosing the surface under consideration. It can be considered as a first step to derive objective rates for generalized continua as Cosserat continua and Kirchhoff-Love type nonlinear shell theories.

Non-Equilibrium Thermodynamics of Electromagnetic Solids

Abstract: The thermomechanics of electromagnetic continua is a branch of energetics which deals with a unification of continuum mechanics and electrodynamics of material media under the umbrella of general thermodynamics. This obviously goes in the direction indicated by the great P.Duhem early in this century (Duhem, 1911, 1914/1954). This ambitious, somewhat Aristotelian-like, scheme also adds one difficulty to the other. In effect, in addition to the cumbersome and rather heavy framework of nonlinear continuum mechanics (such as exposed in modern treatises, e.g., Truesdell and Toupin, I960, Truesdell and Noll, 1965; Eringen 1980, Eringen 1971–1976), one has to consider electromagnet ism (e.g., Jackson, 1962) and then combine them (in an nonlinear manner; this is not a linear superposition) in the harmonious frame of thermodynamics. Some of the difficulties met have to do with the electrodynamics of moving bodies (writing of fields and equations in appropriate frames), while others relate to the introduction of a general deformation field (“material” writing of fields). Finally, there are difficulties connected with the inherent complexity of some of the behaviors (e.g., hysteresis), and even more so, the non-unique thermodynamical framework at the time of writing!

7 – Thermal–Hydraulic–Mechanical Coupling Analysis of Rock Mass

Abstract: Publisher Summary This chapter describes methods for the analysis of coupled thermal, hydraulic, and mechanical phenomena of rock mass. The prediction of the phenomena is most important for the plan and design of a structure constructed in a rock mass. For coupled thermal and mechanical problems, there are many examples in which classical continuum mechanics are applied with numerical calculation techniques. In such cases, the medium is often considered as elastic. When plastic deformation because of thermal stress is considered, the yield function is needed as a function of temperature in addition to stress. When a repository is constructed in saturated deep crystalline rock, convection of the groundwater induced by increasing temperatures is the most important matter for assessment. As groundwater does not homogeneously fill the volume of a fractured rock, the effect of groundwater on the deformation behavior can be anisotropic. When fracture flow is considered as a phenomenon restricted to a plane, the effect of the deformation on the permeability is considered as a change in the area of the channels in the fracture plane.

Counterexample to some shape equations for axisymmetric vesicles

Abstract: Three different shape equations for axisymmetric vesicles have been derived from the same spontaneous-curvature model in literature. The validity of the equations has been examined by means of a rigorous analytical solution for axisymmetric vesicles. A counterexample is given to show the invalidity of two of the equations

One-dimensional Models for Pulsating Combustion

Abstract: The flow field in a pulse combustor can over the major part of the combustor be approximated with an oscillatory “plug” flow, indicating that the flow can be simplified to a one-dimensional flow. Hence, a natural way to start developing theoretical models for pulsating combustion would be to formulate and examine one-dimensional models; such simplified models can serve as a theoretical instrument for designers of pulse combustors. Applying the methods and principles of modern continuum mechanics, a one-dimensional model has been derived and formulated. The model is constituted by a system of non-linear, partial differential equations representing the conservation of mass, the balance of momentum and the balance of energy. In order to complete and close the system of differential equations a set of suitable constitutive relations is suggested and discussed.