Showing papers on "Continuum mechanics published in 1984"

A continuum mechanics based four‐node shell element for general non‐linear analysis

Abstract: A new four‐node (non‐flat) general quadrilateral shell element for geometric and material non‐linear analysis is presented. The element is formulated using three‐dimensional continuum mechanics theory and it is applicable to the analysis of thin and thick shells. The formulation of the element and the solutions to various test and demonstrative example problems are presented and discussed.

Continuum Theory for Strain‐Softening

Abstract: In heterogeneous materials such as concretes or rocks, failure occurs by progressive distributed damage during which the material exhibits strain‐softening, i.e., a gradual decline of stress at increasing strain. It is shown that strain‐softening which is stable within finite‐size regions and leads to a nonzero energy dissipation by failure can be achieved by a new type of nonlocal continuum called the imbricate continuum. Its theory is based on the hypothesis that the stress depends on the change of distance between two points lying a finite distance apart. This continuum is a limit of a discrete system of imbricated (regularly overlapping) elements which have a fixed length, l, and a cross‐section area that tends to zero as the discretization is refined. The principal difference from the existing nonlocal continuum theory is that the equation of motion involves not only the averaging of strains but also the averaging of stress gradients. This assures that the finite element stiffness matrices are symmet...

Imbricate continuum and its variational derivation

Abstract: The one-dimensional imbricate nonlocal continuum, which was developed in another paper in order to model strain-softening within zones of finite size, is extended here to two or three dimensions. The continuum represents a limit of a system of imbricated (overlapping) elements that have a fixed size and a diminishing cross section as the mesh is refined. The proper variational method for the imbricate continuum is developed, and the continuum equations of motion are derived from the principle of virtual work. They are of difference-differential type and involve not only strain averaging but also stress gradient averaging for the so-called broad-range stresses characterizing the forces within the characteristic volume of heterogeneous material. The gradient averaging may be defined by a difference operator, or an averaging integral, or by least-square fitting of a homogeneous strain field. A differential approximation with higher order displacement derivatives is also shown. The theory implies a boundary layer which requires special treatment. The blunt crack band model, previously used in finite element analysis of progressive fracturing, is extended by the present theory into the range of mesh sizes much smaller than the characteristic width of the crack band front. Thus, the crack band model is made part of a convergent discretization scheme. The nonlocal continuum aspects are captured by an imbricated arrangement of finite elements, which are of the usual type.

On the Rotated Stress Tensor and the Material Version of the Doyle-Ericksen Formula

Abstract: Doyle & Ericksen [1956, p. 77] observed that the Cauchy stress tensor σ can be derived by varying the internal free energy ψ with respect to the Rie- mannian metric g on the ambient space: σ=2ϱ ∂ψ/∂g. Their formula has gone virtually unnoticed in the literature of continuum mechanics. In this paper we shall establish the material version of this formula: \( \sum { = 2\varrho \partial \bar{\Psi }} /\partial G \) for the rotated stress tensor Σ, and shall address some of the reasons why these formulae are of fundamental significance. Making use of these formulae one can derive elasticity tensors and establish rate forms of the hyperelastic constitutive equations for the Cauchy stress tensor and the rotated stress tensor, as discussed in Sections 4 and 5. The role of the rotated stress tensor in the formulation of continuum theories has been noted by Green & Naghdi [1965]. Generalizations of hypoelasticity based on the use of the rotated stress tensor have been considered by Green & McInnis [1967].

Instability of Nonlocal Continuum and Strain Averaging

Abstract: Nonlocal continuum, in which the (macroscopic smoothed‐out) stress at a point is a function of a weighted average of (macroscopic smoothed‐out) strains in the vicinity of the point, are of interest for modeling of heterogeneous materials, especially in finite element analysis. However, the choice of the weighting function is not entirely empirical but must satisfy two stability conditions for the elastic case: (1) No eigenstates of nonzero strain at zero stress, called unresisted deformation, may exist; and (2) the wave propagation speed must be real and positive if the material is elastic. It is shown that some weighting functions, including one used in the past, do not meet these conditions, and modifications to meet them are shown. Similar restrictions are deduced for discrete weighting functions for finite element analysis. For some cases, they are found to differ substantially from the restriction for the case of a continuum if the averaging extends only over a few finite elements.

Modeling the Behavior of Materials

Abstract: A first requirement in the calculation of problems in mechanics is a formulation of the material behavior. The material description should include elastic, and elastic-plastic flow. Appropriate yield criteria must be employed. The literature includes many complicated forms to describe material behavior, some of which have been developed to aid the mathematics in the analytic solution of the equation of motion. Numerical techniques can solve the equation of continuum mechanics in two or three dimensions and time with second order accuracy. With these techniques the equations of motion are completely independent of equations that describe material behavior and any mathematical form may be used. The objective of the material models is to provide a theoretical description applicable to a wide class of practical problems, but using simple idealizations of the outstanding features of the real phenomena.

On the response and symmetry of elastic and hyperelastic membrane points

Abstract: In continuum mechanics constitutive equations for elastic and hyperelastic (3-dim.) material points are important mathematical models with many applications. In this paper we present constitutive equations for elastic and hyperelastic (2-dim.) membrane points. As in the theory of (3-dim.) material points, the central topic for the theory of (2-dim.) membrane points are conditions of material frame-indifference, conditions of material symmetry, and representations for the response functions and for the stored energy functions.

36—finite-element analysis of yarns part i: yarn model and energy formulation

Abstract: The finite-element method is applied to derive alternative governing equations for Carnaby and Grosberg's continuous-filament-yarn model. This is shown to give a reasonable simulation of the stress–strain curves of wool yarns when tested at a short gauge length. The finite-element analysis used here is based on the principle of virtual work, and, because of the large deformations occurring, non-linear continuum mechanics are used to describe the configuration of the model.

On the Mechanics of Modulated Structures

Abstract: The purpose of this lecture is to illustrate the appropriateness and potential of the methods of continuum mechanics in modeling modulated structures. Modulations are viewed, in general, as occurrences which may involve one or more properties of a system and extend from a submicroscopic to a macroscopic scale. They are also viewed as capable of possessing wave lengths and amplitudes which may vary from very small to very large values.

Surface integral finite element hybrid method for localized problems in continuum mechanics

Abstract: Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1984.

A unified constitutive theory for polymeric liquids I. Basic considerations and a simplified model

Abstract: By combining a continuum mechanical approach with considerations of network theory and thermodynamics of irreversible processes, a set of differentialtype constitutive equations for polymeric liquids are obtained which provide expressions for the stress tensor, evolution equations of the effective Finger strain and Cauchy strain for the network deformation, and a first order differential equation governing the rigidity modulus. Unlike Giesekus' recent “unified approach” that starts from the bead-spring model, the theory lends itself more readily to a better understanding of most of the current theories based on continuum mechanics and molecular network concepts. Different recent models such as those due to Leonov, Dashner—Van Arsdale, Phan Thien—Tanner, and Acierno et al. (or “Marrucci”) can be unambiguously interpreted as resulting from specific approximations or additional assumptions.

Shape optimization of elastic bars in torsion

Abstract: The problem of shape optimal design for multiply-connected elastic bars in torsion is formulated and solved numerically. A variational formulation for the equation is presented in a Sobolev space setting and the material derivative idea of Continuum Mechanics is used for the shape design sensitivity analysis. The finite element method is used for a numerical solution of the variational state equation and is integrated into an iterative optimization algorithm. Numerical results are presented for both simply- and doubly-connected bars, with prescribed bounds on admissible location of both inner and outer boundaries.

Statics in the momentum current picture

Abstract: Newton’s Second Law is equivalent to the continuity equation for momentum in integral form. This insight leads to an alternative picture of forces as momentum currents. The purpose of the present paper is to introduce a new approach to statics problems and their solutions in terms of momentum currents. In particular, the relation between the distribution of momentum currents and the elastic stresses within a medium will be considered and a simple way to pictorially represent those stresses with the help of momentum flow diagrams will be discussed. Handling statics problems in the momentum current picture immediately displays their relationship to analogous problems in electrical network theory. This uncovers a structural relationship between the role of electric charge in the theory of electricity and the role of momentum in mechanics. For example, it will be shown that the familiar method for the solution of statics problems in terms of free‐body diagrams is equivalent to the use of a junction rule for momentum currents.

On the Invariance of Constitutive Equations According to the Kinetic Theory of Gases

Abstract: Iterative techniques for solving the Boltzmann equation in the kinetic theory of gases yield expressions for the stress tensor and heat flux vector that are analogous to constitutive equations in continuum mechanics. However, these expressions are not generally invariant under the Euclidean group of transformations, whereas constitutive equations in continuum mechanics are usually required to be by the principle of material frame indifference. This disparity in invariance properties has led some previous investigators to argue that Euclidean invariance should be discarded as a contraint on constitutive equations. It is proven mathematically in this paper that the results of the Chapman-Enskog iterative procedure have no direct bearing on this issue. In order to settle this question, it is necessary to examine mathematically the effect of superimposed rigid body rotations on solutions of the Boltzmann equation. A preliminary investigation along these lines is presented which suggests that the kinetic theory is consistent with material frame indifference in at least a strong approximate sense provided that the disparity in the time scales of the microscopic and macroscopic motions is extremely large—a condition which is usually a prerequisite for the existence of constitutive equations.

Thermostatics of non-simple materials

Abstract: Etude de la thermostatique de materiaux dont l'energie libre et les contraintes a l'equilibre dependent des gradients de deformation. Dans ce cadre on etudie les materiaux de grade r a variables internes

Truesdell invariance in relativistic electromagnetic fields

Abstract: The Truesdell derivative of a contravariant tensor fieldXabis defined with respect to a null congruencelaanalogous to the Truesdell stress rate in classical continuum mechanics. The dynamical consequences of the Truesdell invariance with respect to a timelike vectoruaof the stress-energy tensor characterizing a charged perfect fluid with null conductivity are the conservation of pressure (p), charged density (e) an expansion-free flow, constancy of the Maxwell scalars, and vanishing spin coefficientsα+¯β = ¯σ − λ = τ = 0 (assuming freedom conditionsk = λ = e ψ + ¯γ = 0). The electromagnetic energy momentum tensor for the special subcases of Ruse-Synge classification for typesA andB are described in terms of the spin coefficients introduced by Newman-Penrose.

On the work hardening behaviour of metallic materials with spherical second-phase particles at high temperatures

Abstract: The work hardening behaviour of metallic materials with spherical second-phase particles at elevated temperatures has been discussed theoretically, based on the continuum mechanics model which incorporated the effect of the dynamic recovery by diffusion of atoms. It was found that the theoretical model developed in this study could satisfactorily explain the experimental results of tensile tests in an austenitic heat-resisting steel with M23C6 carbides. The model was also applied to the understanding of the internal stress during high-temperature creep in this steel.

Accurate Nonlinear Equations and a Perturbation Solution for the Free Nonplanar Vibrations of Inextensional Beams

Abstract: : In the present work free nonplanar motions of an inextensional elastic beam with no warping and shear deformation, supported in an arbitrary manner but no axially restrained, are studied without introducing simplifying assumptions. The kinematically nonlinear beam model is derived within the theory of rods from a continuum mechanics point of view. Although the usefulness of such model is not recognized, it has been followed here because its major generality; indeed it permits an unitary approach for both cases of inextensional and deformable beam.

Review of Continuum Mechanics Factors in Adhesive Fracture

Abstract: The possibility of treating adhesive fracture as an engineering analysis problem in continuum mechanics has been emphasized in previous reviews, with applications to designs which utilize bonded surfaces and/or various composite materials Cost-effective use of this analysis ability will ultimately depend, of course, upon the accuracy of fracture and structural life prediction, as well as the availability of methods to conduct non-destructive examination

Nonlinear Response of Coupled Flow-Stress Problems

Abstract: Equations governing the nonlinear phenomenon of fluid flow through deformable porous media are presented in this paper. The proposed formulation is based on a continuum theory of mixtures using a total Lagrangian approach, where large displacement/large strain cases are considered. The incremental finite element equations are obtained using a direct Galerkian approach. Numerical results are presented for several problems assuming infinitesimal strains. Comparisons with known solutions demonstrate the validity of this approach. The effect of nonlinear solid phase behavior is discussed and shown to be very substantial.