Showing papers on "Continuum mechanics published in 1983"

Shape Design Sensitivity Analysis of Elastic Structures

Abstract: Design problems in which the shape of two- or three-dimensional elastic bodies plays the role of design are studied. Five prototype problems are formulated in a unified variational form, with performance measures involving natural frequency, displacement, and stress in the structure. The material derivative of continuum mechanics and an adjoint variable method of design sensitivity analysis are used to develop an explicit formula for variation of performance measures in terms of normal movement of the boundary of the physical domain. Examples are presented for beams, membranes, shafts, and three-dimensional elastic solids.

A nonlinear fracture mechanics approach to the growth of small cracks

Abstract: An analytical model of crack closure is used to study the crack growth and closure behavior of small cracks in plates and at notches. The calculated crack opening stresses for small and large cracks, together with elastic and elastic plastic fracture mechanics analyses, are used to correlate crack growth rate data. At equivalent elastic stress intensity factor levels, calculations predict that small cracks in plates and at notches should grow faster than large cracks because the applied stress needed to open a small crack is less than that needed to open a large crack. These predictions agree with observed trends in test data. The calculations from the model also imply that many of the stress intensity factor thresholds that are developed in tests with large cracks and with load reduction schemes do not apply to the growth of small cracks. The current calculations are based upon continuum mechanics principles and, thus, some crack size and grain structure exist where the underlying fracture mechanics assumptions become invalid because of material inhomogeneity (grains, inclusions, etc.). Admittedly, much more effort is needed to develop the mechanics of a noncontinuum. Nevertheless, these results indicate the importance of crack closure in predicting the growth of small cracks from large crack data.

Solar System Magnetohydrodynamics

Abstract: Continuum mechanics is that branch of physics that treats the motions of infinitely deformable matter It embraces hydrodynamics, aerodynamics, magnetohydrodynamics (MHD), and magnetogasdynamics The first two differ in that the former is incompressible and the latter compressible fluid dynamics The prefix, magneto, signifies the addition of the ponderemotive force (colloquially called the J-cross-B force) to the usual pressure gradient, gravitational and viscous forces of fluid dynamics Magnetofluid mechanics applies to fluids that can carry electrical currents, such as liquid metals and plasmas Our interest in Solar System MHD is confined to the latter

Nonlinear Elastic Shell Theory

Abstract: Publisher Summary This chapter provides an overview of nonlinear elastic shell theory The chapter starts with the integral equations of motion and thermodynamics of a three-dimensional body taken over its reference shape Specializing to shell-like bodies, chapter discusses definitions of a two-dimensional stress resultant, stress couple, displacement, spin (angular velocity), entropy resultant, temperature mean, temperature drop, and other fields These definitions provide a concrete, unambiguous tie between two- and three-dimensional continuum mechanics The two-dimensional integral equations, for sufficiently smooth fields, imply differential equations Those of conservation of linear and rotational momentum yield in a natural way a mechanical work identity that provides, automatically, two-dimensional strains conjugate to the stress resultants and couples This leads to a mechanical theory of shells in which a strain energy density, depending on these strains only, is assumed to exist In the chapter, a thermodynamic theory is formulated from a postulated first law in which the internal energy is a function of the aforementioned two-dimensional strains, plus an entropy resultant and couple Invariance requirements and dimensional analysis are used to simplify the form of the strain energy density (or the free-energy density in the thermodynamic theory)

Analysis of Some Mixed Finite Element Methods for Plane Elasticity Equations

Abstract: We analyze some mixed finite element methods, based on rectangular elements, for solving the two-dimensional elasticity equations. We prove error estimates for a method proposed by Taylor and Zienkiewicz and for some new variants of the known equilibrium methods. A numerical example is given demonstrating the performance of the various algorithms considered. 1. Introduction. In the numerical solution of problems of continuum mechanics, the stresses are normally of primary interest in the elastic region. It is therefore natural to design the numerical algorithms so that the stresses can be obtained directly without first computing the displacements. Such methods can be derived from the dual variational formulation of the elasticity problem. The corresponding finite element algorithms are usually formulated as mixed methods where both the displacements and the stresses are first approximated, and the displacements are then eliminated from the discrete equations. In many cases the elimination can be rather effectively done using penalty/perturbation techniques or their iterative variants; cf. (3), (11), (12). The best known finite element methods of the above type are the so-called equilibrium methods, first proposed by Fraejis de Veubeke (17) (cf. also (14), (16), (18)) and analyzed theoretically by Johnson and Mercier (9) (cf. also (8)). In these methods, one uses specific composite elements which allow the equilibrium condi- tion between the stresses and the volume load to be satisfied exactly in the case where the volume load is zero.

Convection and diffusion of polymer network strands

Abstract: A review of several important constitutive equations is herein conducted with an eye towards determining those most suitable for use in modelling polymer melt processing. General principles are invoked for a priori screening of the equations without needing detailed comparison of the model predictions with experimental data. These principles, which are derived from continuum mechanics, thermodynamics and molecular kinetic theory, and dela with convection and diffusion of entangled polymer strands during flow, are: (1) During sudden deformations, the stress is a unique function of the total strain. (2) The second law of thermodynamics holds for all deformations. (3) The constitutive equation can be derived from a plausible molecular model which describes the convection and diffusion. (4) The model parameters can be determined by a reasonable number of rheometric experiments. Based on these principles, it is concluded that separable free energy models are the most promising. These are either BKZ integral models with a kernel factorable into a time-dependent and a strain-dependent part. or sets of Maxwell-type differential equations that employ a generalized convected derivative, and that are linear in stress in the absence of flow.

Dislocation Kinetics and the Formation of Deformation Bands

Abstract: By viewing material states as superpositions of perfect lattice and dislocated substates, we can adopt the approach of continuum mechanics to arrive at a system of differential equations describing the interaction effects between stress, dislocations and deformation. The various arguments that we utilize are broadly related to ideas earlier advanced in theories of continuous distribution of dislocations and in models of dislocation dynamics. Following a brief account of the general theoretical structure in the introduction, we discuss in the first part of this paper the ability of the present method to recover previous phenomenological models of material behavior, such as macroscopic theories of plasticity. In the second part of the paper, we illustrate the ability of the present method to predict dislocation phenomena associated with the formation of plastic zones, deformation bands and dislocation patterns.

The boundary integral equation (boundary element) method in engineering stress analysis

Abstract: The principles of the boundary integral equation (BIE) or boundary element method (BEM) are discussed in a non-mathematical way The technique is compared with other numerical methods, part

Mechanics and Physics of the Growth of Small Cracks

Abstract: : The mechanics and physics of the sub-critical propagation of small fatigue cracks are reviewed in terms of reported differences in behavior between long and short flaws based on fracture mechanics, microstructural and environmental viewpoints. Cracks are considered short when their length is small compared to relevant microstructural dimensions (a continuum mechanics limitation), when their length is small compared to the scale of local plasticity (a linear elastic fracture mechanics limitation), and when they are merely physically-small (e.g., less than 0.5-1 mm). For all three cases, it is shown that, at the same nominal driving force, the growth rates of the short flaws are likely to be greater than (or at least equal to) the corresponding growth rates of long flaws; a situation which can lead to non-conservative defect-tolerant lifetime predictions where existing (long crack) data are utilized. Reasons for this problem of similitude between long and short flaw behavior are discussed in terms of the roles of crack driving force, local plasticity, microstructure, crack shape, crack extension mechanism, premature closure of the crack, and local crack tip environment.

Non-Newtonian behavior of polymeric liquids

Abstract: A survey is given of the three main approaches to the study of the non-linear constitutive equations for polymer solutions and polymer melts: rheometric measurements in shear and elongational flows; continuum mechanics results for special flows and useful empiricisms; and molecular theories, including single-molecule theories for dilute solutions and network and reptational theories for melts.

Application of plasticity theory to soil behavior : a new sand model

Abstract: The representation of rheological soil behavior by constitutive equations is a now branch of soil mechanics which has been expanding for 30 years. Based on continuum mechanics, numerical methods (finite elements) and experimental techniques, this now discipline allows practicing engineers to solve complex geotechnical problems. Although all soils are constituted of discrete mineral particles, forces and displacements within them are represented by continuous stresses and strains. Most stress-strain relationships, which describe the soil behavior, are derived from plasticity theory. Originated for metals, the conventional plasticity is presented and illustrated simultaneously with a metal and a soil model. Each plasticity concept may be criticized when applied to soil. A recent theory, called "bounding surface plasticity," generalizes the conventional plasticity and describes more accurately the cyclic responses of metals and clays. This new theory is first presented and linked with the conventional plasticity, then applied to a new material, sand. Step by step a new sand model is constructed, mainly from data analysis with an interactive computer code. In its present development, only monotonic loading s are investigated. In order to verify the model ability to describe sand responses, isotropic, drained and undrained tests on the dense Sacramento River sand are simulated numerically and compared with real test results and predictions with another model. Finally the new constitutive equation, which was formulated in the p-q space for axisymmetric loadings, is generalized in the six-dimensional stress state with the assumption of isotropy and a particular Lode's angle contribution. This new model is ready to be used in finite element codes to represent a sand behavior.

On the Lagrangian formulation of continuum mechanics

Abstract: The aim of this paper is to show that a Lagrangian formulation of continuum mechanics (in the spirit of field theory) can deliver, not only equations of motion, but the conservation laws related to the material symmetries in the perfect continuum. Those conservation laws in the presence of defects lead to the path-independent integrals broadly used in fracture mechanics. They are basically related to the (material) forces on a defect in a continuum and can be interpreted as the equations of motion for a defect with respect to the material. The quantity playing the role of the stress tensor in this formulation is the material momentum tensor. A simple example of a material force in the form of a path-independent integral in the fluid is considered.

Global existence of solutions of the equations of one-dimensional thermoviscoelasticity with initial data in $BV$ and $L^1$

Abstract: : This paper discusses the Cauchy problem associated with a particular system of equations of one-dimensional nonlinear thermoviscoelasticity with the initial data given in the class of functions of bounded variation (denoted by BV). It has been known that the class of BV is a suitable function space for the study of evolution equations which arise in continuum mechanics in order to admit solutions possessing shocks. This fact has been exploited in the analysis of hyperbolic conservation laws which describe the motion of a continuum when mechanical and thermal dissipations are neglected. On the other hand, only the smooth (classical) solutions have been studied for the equations which include such dissipative terms. Our goal is to study the global existence of weaker solutions of systems which include such dissipative terms. Our main result shows that when the initial data are sufficiently small in the Lagrangian form and BV norms, the system (1) of the abstract has global solutions in time possessing specific regularity properties. (Author)

Numerical simulation of forming processes : the use of the Arbitrary-Eulerian-Lagrangian (AEL) formulation and the finite element method

Abstract: Symbols and notatien

A general field theory of Cauchy continuum - Classical mechanics

Abstract: The fundamentals of Cauchy continuum mechanics are set into discussion in view of unifying the methods of classical and relativistic field theory. From a general variational principle—also valid for history-dependent materials—a set of four Euler-Lagrange equations are obtained. Noether's Erster Satz supplies, ten (actually reducing to seven) further equalities which, combined with the previous ones, yield the conservation laws of the field theory, expressing the properties of invariance of physical laws with respect to infinitesimal transformations of the reference frame. Some topics are discussed, concerning conservation laws in classical field theory.

Recent investigations on strain and stress rates in nonlinear continuum mechanics

Abstract: Presenting some recent considerations and results, the present paper deals with two basic concepts of the continuum mechanics: strain-and stress-rates. Upon a brief systematic survey of concepts of strain and stress, a so far unknown explicit expression for the rate of the right stretch tensor is offered in absolute notation. This paper then suggests to distinguish two ways of defining objective stress-rates. Following the second procedure, after analyzing several particular cases, the author proposes a generalized Jaumann flux, which contains the majority of the existing definitions for stress-rate and the Hill's result as well.

A small strain, large rotation theory and finite element formulation of thin curved beams

Abstract: In this thesis a theory for the geometrically nonlinear analysis of thin curved beam-type structures is proposed and an associated displacement finite element formulation developed. An exact two-dimensional large rotation theory, which is based on an intrinsic coordinate system, has been developed. Four alternative Lagrangian formulations of the theory have been presented for comparison. A family of two-dimensional thin curved beam elements has been developed by using the constraint technique to include the convective coordinate system. The elements are relatively simple and the minimum number of degrees of freedom necessary has been used. The Total Lagrangian formulation has been shown to be numerically more effective than the Updated Lagrangian formulation. A new Total Lagrangian formulation that includes the effect of curvature change on axial force in the incremental equilibrium equations has been developed. The formulation is based on the geometric strains and has the capability of predicting true axial force values in large rotation and curvature problems. This approach can be used in the general continuum mechanics largz; d3formation formulations. A large rotation theory for three-dimensional beams and Total Lagrangian formulations of the theory, which are based on the Green strains and the geometric strains, have been developed. The theory correctly describes the large rotation elastic response of a thin eccentric curved beam of rectangular cross-section. Material nonlinearity, which is based an the von-Mises yield function and the Prandtl-Reuss flow rule and in which isotropic hardening is assumed, has been included in the formulation. A family of three-diinensional beam elements, that can accurately accommodate the theory, has been developed by the constraint technique. The elements are suitable for use as stiffeners in the analysis of stiffened shell structures. The elements, which have been developed, have been implemented in the LUSAS finite element system. The accuracy of the results obtained has been demonstrated by comparison with*published results.

On large strain theories in sheet metal forming : general theories and numerical considerations with particular emphasis on strain analysis, non-conservative loading and plastic anisotropy

Abstract: The fundamentals of continuum mechanics in large strain problems are treated and analysed. Special emphasis is directed to strain analysis in large deformation problems and a comparison between the ...

On a micromechanical fracture model for cracked reinforced composites

Abstract: The fracture process of reinforced composite materials is examined. In the outer region of the crack tip anisotropic continuum mechanics is employed, while for the crack tip region a heterogeneous micromechanical model is proposed. A solution is obtained using combined boundary layer — non-linear finite elements.