Showing papers on "Continuum mechanics published in 1998"

A tensorial approach to computational continuum mechanics using object-oriented techniques

Abstract: In this article the principles of the field operation and manipulation (FOAM) C++ class library for continuum mechanics are outlined. Our intention is to make it as easy as possible to develop reliable and efficient computational continuum-mechanics codes: this is achieved by making the top-level syntax of the code as close as possible to conventional mathematical notation for tensors and partial differential equations. Object-orientation techniques enable the creation of data types that closely mimic those of continuum mechanics, and the operator overloading possible in C++ allows normal mathematical symbols to be used for the basic operations. As an example, the implementation of various types of turbulence modeling in a FOAM computational-fluid-dynamics code is discussed, and calculations performed on a standard test case, that of flow around a square prism, are presented. To demonstrate the flexibility of the FOAM library, codes for solving structures and magnetohydrodynamics are also presented with appropriate test case results given. © 1998 American Institute of Physics.

Theory of Multicomponent Fluids

Abstract: Preliminaries.- Physical Reality, Corpuscular Models, Continuum Models.- Classical Continuum Theory.- Viscous and Inviscid Fluids and Elastic Solids.- Kinetic Theory.- Classical Theory of Solutions.- Continuum Theory.- Continuum Balance Equations for Multicomponent Fluids.- Mixture Equations.- Averaging Theory.- Ensemble Averaging.- Other Averages.- Averaged Equations.- Postulational and Averaging Approaches.- Modeling Multicomponent flows.- Closure Framework.- Relation of Microstructure to Constitutive Equations.- Maxwell-Boltzmann Dynamics.- Interfacial Area.- Equations of Motion for Dilute Flow.- Consequences.- Nature of the Equations.- Well-Posedness.- Solutions for Shearing Flows.- Wave Dynamics.

Theory of sintering: from discrete to continuum

Abstract: Theoretical concepts of sintering were originally based upon ideas of the discrete nature of particulate media. However, the actual sintering kinetics of particulate bodies are determined not only by the properties of the particles themselves and the nature of their local interaction with each other, but also by macroscopic factors. Among them are externally applied forces, kinematic constraints (e.g. adhesion of the sample's end face and furnace surface), and inhomogeneity of properties in the volume under investigation (e.g. inhomogeneity of initial density distribution created during preliminary forming operations). Insufficient treatment of the questions enumerated above was one of the basic reasons hindering the use of sintering theory. A promising approach is connected with the use of continuum mechanics, which has been successfully applied to the analysis of compaction of porous bodies. This approach is based upon the theories of plastic and nonlinear-viscous deformation of porous bodies. Similar ideas have recently been embodied in a continuum theory of sintering. The main results of the application of this theory for the solution of certain technological problems of sintering are introduced including their thermo–mechanical aspects.

A Constitutive Law for Mitral Valve Tissue

Abstract: Biaxial mechanical testing and theoretical continuum mechanics analysis are employed to formulate a constitutive law for cardiac mitral valve anterior and posterior leaflets. A strain energy description is formulated based on the fibrous architecture of the tissue, accurately describing the large deformation, highly nonlinear transversely isotropic material behavior. The results show that a simple three-coefficient exponential constitutive law provides an accurate prediction of stress-stretch behavior over a wide range of deformations. Regional heterogenity may be accommodated by spatially varying a single coefficient and incorporating collagen fiber angle. The application of this quantitative information to mechanical models and bioprosthetic development could provide substantial improvement in the evaluation and treatment of valvular disease, surgery, and replacement.

A finite deformation continuum\discrete model for the description of fragmentation and damage in brittle materials

Abstract: A dynamic finite element analysis of large displacements, high strain rate deformation behavior of brittle materials is presented in total Lagrangian coordinates. A continuum\discrete damage model capable of capturing fragmentation at two size scales is derived by combining a continuum damage model and a discrete damage model for brittle failure. It is assumed that size and distribution of potential fragments are known a priori, through either experimental findings or materials properties, and that macrocracks can nucleate and propagate along the boundaries of these potential fragments. The finite deformation continuum multiple-plane microcracking damage model accounts for microcracks within fragments. Interface elements, with cohesive strength and reversible unloading before debonding, between potential fragments describe the initiation of macrocracks, their propagation, and coalescence leading to the formation of discrete fragments. A surface-defined multibody contact algorithm with velocity dependent friction is used to describe the interaction between fragments and large relative sliding between them. The finite element equations of motion are integrated explicitly using a variable time step. Outputs are taken at discrete time intervals to study material failure in detail. The continuum\discrete damage model and the discrete fragmentation model, employing interface elements alone, are used to simulate a ceramic rod on rod impact. Stress wave attenuation, fragmentation pattern, and overall failure behavior, obtained from the analyses using the two models, are compared with the experimental results and photographs of the failing rod. The results show that the continuum\discrete model captures the stress attenuation and rod pulverization in agreement with the experimental observations while the pure discrete model underpredicts stress attenuation when the same potential fragment size is utilized. Further analyses are carried out to study the effect of potential fragment size and friction between sliding fragments. It is found that compared with the continuum\discrete damage model, the discrete fragmentation model is more sensitive to the multi-body discretization.

Macroscopic equations of motion for two-phase flow in porous media

Abstract: The usual macroscopic equations of motion for two-phase immiscible displacement in porous media are known to be physically incomplete because they do not contain the surface tension and surface areas governing capillary phenomena. Therefore, a more general system of macroscopic equations is derived here that incorporates the spatiotemporal variation of interfacial energies. These equations are based on the theory of mixtures in macroscopic continuum mechanics. They include wetting phenomena through surface tensions instead of the traditional use of capillary pressure functions. Relative permeabilities can be identified in this approach that exhibit a complex dependence on the state variables. A capillary pressure function can be identified in equilibrium that shows the qualitative saturation dependence known from experiment. In addition, the proposed equations include a description of the spatiotemporal changes of residual saturations during immiscible displacement.

Vibrations of elasto-plastic bodies

Abstract: 1 Foundations and equations of continuum mechanics.- 2 Plasticity theory and internal friction in materials.- 3 Three-dimensional cyclic deformations of elastoplastic materials.- 4 Single-frequency vibrations in elastoplastic bodies.- 5 Random deformation of elastoplastic materials.- 6 Random vibrations of elastoplastic bodies.- 7 Propagation of vibration in a nonlinear dissipative medium.- 8 Propagation of vibration in media with complex structure.- References.

A continuum model of layered rock masses with non-associative joint plasticity

Abstract: Layered rock masses can be modelled either as standard, orthotropic continua if the layer bending can be neglected or as Cosserat continua if the influence of layer bending is essential. This paper presents a finite element smeared joint model based on the Cosserat theory. The layers are assumed to be elastic with equal thickness and equal mechanical properties. All the cosserat parameters are expressed through the elastic properties of layers, layer thickness and joint stiffness. Plastic-slip as well as tensile-opening of layer interface (joint) are accounted for in a manner similar to the conventional non-associative plasticity theory. As an application, the behaviour of an excavation in a layered rock mass is examined. The displacement and stress fields given by smeared joint models based on the Cosserat continuum and the conventional anisotropic continuum approaches are compared with those obtained from the discrete joint model. The conventional anisotropic continuum model is found to break-down completely when the effective shear modulus in the direction parallel to layering is low in comparison to the shear modulus of the intact layer, whereas the Cosserat model is found to be capable of accurately reproducing complex load–deflection patterns irrespective of the differences in shear moduli. © 1998 John Wiley & Sons, Ltd.

Nonlinear Continuum Mechanics for Finite Element Analysis

Abstract: Nonlinear continuum mechanics of solids is a fascinating subject. All the assumptions inherited from an overexposure to linear behaviour and analysis must be re-examined. The standard definitions of strain designed for small deformation linear problems may be totally misleading when finite motion or large deformations are considered. Nonlinear behaviour includes phenomena like `snap-through', where bifurcation theory is applied to engineering design. Capabilities in this field are growing at a fantastic speed; for example, modern automobiles are presently being designed to crumple in the most energy absorbing manner in order to protect the occupants. The combination of nonlinear mechanics and the finite element method is a very important field. Most engineering designs encountered in the fusion effort are strictly limited to small deformation linear theory. In fact, fusion devices are usually kept in the low stress, long life regime that avoids large deformations, nonlinearity and any plastic behaviour. The only aspect of nonlinear continuum solid mechanics about which the fusion community now worries is that rare case where details of the metal forming process must be considered. This text is divided into nine sections: introduction, mathematical preliminaries, kinematics, stress and equilibrium, hyperelasticity, linearized equilibrium equations, discretization and solution, computer implementation and an appendix covering an introduction to large inelastic deformations. The authors have decided to use vector and tensor notation almost exclusively. This means that the usual maze of indicial equations is avoided, but most readers will therefore be stretched considerably to follow the presentation, which quickly proceeds to the heart of nonlinear behaviour in solids. With great speed the reader is led through the material (Lagrangian) and spatial (Eulerian) co-ordinates, the deformation gradient tensor (an example of a two point tensor), the right and left Cauchy-Green tensors, the Eulerian or Almansi strain tensor, distortional components, strain rate tensors, rate of deformation tensors, spin tensors and objectivity. The standard Cauchy stress tensor is mentioned in passing, and then virtual work and work conjugacy lead to alternative stress representations such as the Piola-Kirchoff representation. Chapter 5 concentrates on hyperelasticity (where stresses are derived from a stored energy function) and its subvarieties. Chapter 6 proceeds by linearizing the virtual work statement prior to discretization and Chapter 7 deals with approaches to solving the formulation. In Chapter 8 the FORTRAN finite element code written by Bonet (available via the world wide web) is described. In summary this book is written by experts, for future experts, and provides a very fast review of the field for people who already know the topic. The authors assume the reader is familiar with `elementary stress analysis' and has had some exposure to `the principle of the finite element method'. Their goals are summarized by the statement, `If the reader is prepared not to get too hung up on details, it is possible to use the book to obtain a reasonable overview of the subject'. This is a very nice summary of what is going on in the field but as a stand-alone text it is much too terse. The total bibliography is a page and a half. It would be an improvement if there were that much reference material for each chapter.

Finite element analysis for composite structures

Abstract: Preface. 1. Some Results from Continuum Mechanics. 2. A Brief History of FEM. 3. Natural Modes for Finite Elements. 4. Composites. 5. Composite Beam Element. 6. Composite Plate and Shell Element. 7. Computational Statistics. 8. Nonlinear Analysis of Anisotropic Shells. 9. Programming Aspects. Appendices: A. Geometry of the Bema Element in Space. B. Contents of the Floppy Disk. Bibliography. Index.

A continuum model with microstructure for materials with flaws and inclusions

Abstract: A continuum endowed with affine microstructure is adopted for the macroscopic description of fiber composite materials. The microstructure is made of a rigid and of a deformable local structure. The former represents the fibers of the composite, perceived as rigid inclusions. The latter accounts for the presence of distributed flaws, considered as slit microcracks. In the framework of a degree one theory, a formula for the mechanical power is derived from a discrete microscopic model using an integral procedure of equivalence. Constitutive elastic stress-strain relationships, accounting for the geometry of the internal phases, are identified The balance equations for both the continuum macro and micro-actions are derived from the axiom of vanishing power and of invariance of power under change of observer. It is also shown that the material symmetries are preserved in the transition from fine to gross description.

Micromechanics versus macromechanics: a combined approach for metal matrix composite constitutive modelling

Abstract: The cyclic constitutive behaviour description of long fibre metal matrix composites needs to take into account viscoplasticity of the matrix, damage of the constituents and interfaces, manufacture residual stresses and damage deactivation effects. In order to incorporate in the model the main constituent characteristics and the composite parameters (volume fractions, fibre shape and arrangements) a combined approach is proposed, i.e. that of using a micromechanics-based analysis for the thermo-elastoviscoplasticity of the composite with damaging effects when the damage is active (i.e. when the microcracks are open). The developed model is based on transformation field analysis and on the effective stress-effective strain space within the continuum damage mechanics of the constituents. The particularization to a two-phase material permits an explanation of the macroscopic constitutive operators of the composite. The obtained macroscopic model is then formulated in such a way as to describe the damage deactivation effects that take place under cyclic conditions for compressive-like loadings. The formalism is extended from that developed previously for elastic brittle ceramic matrix composites, taking into account the possible deactivation for a given (or varying) non-zero strain. The deactivation criterion ensures the continuity of the stress-strain response for any multiaxial loadings. The proposed model is then applied to a SiC/Ti metal matrix composite, with unidirectional long fibres.

Homogenization of discrete media

Abstract: Material such as granular media, beam assembly are easily seen as discrete media. They look like geometrical points linked together thanks to energetic expressions. Our purpose is to extand discrete kinematics to the one of an equivalent continuous material. First we explain how we build the localisation tool for periodic materials according to estimated continuum medium type (classical Cauchy, and Cosserat media). Once the bridge built between discrete and continuum media, we exhibit its application over two bidimensional beam assembly structures: the honey comb and a structural reinforced variation. The new behavior is then applied for the simple plan shear problem in a Cosserat continuum and compared with the real discrete solution. By the mean of this example, we establish the agreement of our new model with real structures. The exposed method has a longer range than mechanics and can be applied to every discrete problems like electromagnetism in which relationship between geometrical points can be summed up by an energetic function.

Nonlocal stress field of interface dislocations

Abstract: The basic theory of nonlocal elasticity is stated with emphasis on the difference between the nonlocal theory and classical continuum mechanics. The concept of Nonlocal Interface Residual (NIR) is introduced in nonlocal theory. With the concept of NIR and the nonlocal constitutive equation, we calculate nonlocal stresses due to an edge dislocation on the interface of bi-materials. The nonlocal stress distribution along an interface is quite different from the classical one. Instead of the singularity in the dislocation core, nonlocal stress gives a finite value in the core. A maximum of the stress is also found near the dislocation core.

Serrated Flow and Surface Markings in Aluminum Alloys

Abstract: Serrated flow and associated progressive surface markings severely restrict the application of some aluminum sheet alloys for automotive body exteriors. This paper attempts to approach the phenomenon from the localization theory of continuum mechanics as well as from the classical atomistic and dislocation considerations. Plane strain tension tests were conducted for a commercial Al-Mg alloy (5182-O) at different strain rates and temperatures, and the local temperature changes were measured by an infrared thermal imaging system. Continuum mechanics analysis provided the insight into the myth that band surface markings never appear under biaxial tension strain states. In addition, continuum mechanics analysis shed light on the observation that PLC bands were not seen on the surface of plane strain tension specimens even though the stress-strain curves exhibited serrations. Finally, it is emphasized that only by combining the efforts of continuum mechanics at the macroscale and materials science at the microscale, can a complete understanding of the phenomenon be reached.

Computation of coarse grain structures using a homogeneous equivalent medium

Abstract: This paper deals with the improvement of concentration relations when some basic hypothesis of homogenization techniques are no longer valid inducing important errors in the deduced local fields. In the first part of this work we propose a new formalism of the standard concentration rules developed in the homogenization of periodic media framework. In a second part, we insist on the fact that a homogeneous equivalent medium replacing a coarse grain material is in fact expected to be a generalized continuum. Additional stiffnesses must be attributed to the unit cell and special non-homogeneous boundary conditions and their periodic counterparts are proposed. The resulting HEM turns out to be a Cosserat continuum.

Contribution à la modélisation géométrique et thermodynamique d'une classe de milieux faiblement continus

Abstract: Abstract.