Showing papers on "Continuum mechanics published in 1970"

Delayed failure - The Griffith problem for linearly viscoelastic materials

Abstract: The unstable growth of a crack in a large viscoelastic plate is considered, within the framework of continuum mechanics. Starting from the local stress and deformation fields at the tip of the crack, a non-linear, first order differential equation is found to describe the time history of the crack size if the stress applied far from the crack is constant. The differential equation contains the creep compliance and the intrinsic surface energy of the material. The surface energy concept for viscoelastic materials is clarified. Inertial effects are not considered, but the influence of temperature is included for thermorheologically simple materials. Initial crack velocities are given as a function of applied load in closed form, as well as a comparison of calculated crack growth history with experiments. Above a certain high stress, crack propagation ensues at high speeds controlled by material inertia while at a lower limit infinite time is required to produce crack growth. Thus an upper and lower limit criterion of the Griffith type exists. For rate insensitive (elastic) materials the two limits coalesce and only the brittle fracture criterion of Griffith exists. The implications of these results for creep fracture in metals and inorganic glasses are examined.

Schaum's outline of theory and problems of continuum mechanics

Abstract: Mathematical Foundations Analysis of Stress Deformation and Strain Motion and Flow Fundamental Laws of Continuum Mechanics Linear Elasticity Fluids Plasticity Viscoelasticity

Numerical continuum approaches to analysis of nonlinear rock deformation

Abstract: In the continuum approach to problems of geologic mechanics, the ground surrounding a deformation zone is replaced mathematically by an idealized material that deforms in accordance with principles of continuum mechanics. Studies published to date have been rather few in number and have often followed a methodology characterized by restrictive material property and boundary condition assumptions. In order to avoid these difficulties, emphasis in this paper is given to modern numerical methods. Discrete-element formulations of matrix structural analysis are shown to be particularly useful, inasmuch as they are adaptable to solutions of systems characterized by nonlinear anisotropic materials, heterogeneously distributed, containing internal discontinuities, and of arbitrary topographic or internal boundary configuration.

On 'thermo-rheologically simple' solids.

Abstract: Starting with the results for a non-isothermal finite linear theory of viscoelasticity, a systematic derivation of a linearized theory with infinitesimal deformation is presented for ‘thermo-rheologically simple’ solids. A comparison of the resulting constitutive equations and the internal dissipation with those given previously is included.

Homogeneous constitutive equations for materials with permanent memory

Abstract: : Nonlinear homogeneous constitutive equations are developed in this thesis for highly filled polymeric materials such as solid propellants. In the range of strains below vacuole dilatation these materials obey the homogeneity rule of linearity but do not obey the superposition rule. Such materials typically exhibit an irreversible 'stress softening' called the 'Mullins' Effect.' The development in this dissertation stems from attempting to mathematically describe the failing microstructure of these composite materials in terms of a linear cumulative damage model. It is demonstrated that pth order Lebesgue norms of the strain history can be used to describe the state of damage in these materials and can also be used in the constitutive equation to characterize their time dependent mechanical response to strain disturbances. Stress analysis procedures for materials having nonlinear homogeneous constitutive equations are developed for two and three dimensional proportional boundary value problems. A series of correspondence principles are derived wherein half of the solution, either the stresses or the strains, can be obtained by solving an equivalent linear elastic problem. The remaining half of the solution can be obtained by substituting the linear elastic solution into the nonlinear homogeneous constitutive equation.

Lagrangian Formulation of Statics of Shells

Abstract: A Lagrangian formulation of the equations of equilibrium of nonlinear thin elastic shell theory referred to nonorthogonal midsurface coordinates is presented. In analogy with the definition of the Lagrangian stress tensor of nonlinear continuum mechanics, stress and moment resultants are introduced and the equations of equilibrium with reference to the undeformed state derived. All results are “exact” within the Kirchhoff-Love hypothesis and are stated both in tensorial as well as in physical component form.

On the formal structure of continuum mechanics Part I: Deformation theory

Abstract: The purpose of this and of following papers is to give an operator form to the equations of statics of continuous media in the case of small deformations (linear case) and of large deformations (non linear), aiming to obtain a unitary view of many known results in linear and nonlinear mechanics. In this paper an operator version of compatibility equations for large deformations, is given introducing a nonlinear operator that will be shown to be of the potential type. Some new results follow, namely a variational formulation of compatibility conditions extending a similar result obtained by the author in the small deformation theory. This formulation in three-dimensional space is similar to that of Einstein describing the gravitational field in space-time. An operational version of Bianchi's identities and a construction of boundary compatibility conditions are also given.

Mach's principle and Newtonian mechanics

Abstract: The union of Mach's principle and Newtonian mechanics gives rise to Relational Mechanics. We find that the characteristics of the revised mechanics are: (1) freedom from any reference to absolute space; (2) the identity of inertial and gravitational mass; (3) the relative acceleration of a body in a gravitational field dependent on the mass of the body. All these results are valid in the context of a Newtonian mechanics which is being developed in the center-of-mass system of all the particles. The conservation of linear momentum, energy, angular momentum are expressed in relational terms, i.e., no reference is made to absolute space. Relational Mechanics is a classical relativistic theory which can be formulated to satisfy Einsteinian relativistic requirements. The Hamiltonian formalism for Relational Mechanics is discussed.

Artificial Intelligence in Continuum Mechanics

Abstract: An exploratory software system for the area of continuum mechanics is described. The system, named CONFORM, is capable of recognizing, retrieving, and associating formulas, performing algebraic manipulation at various levels (including differentiation and integration, vector and tensor analysis, and matrix algebra), and of performing embryonic problem-solving and theorem-proving activities in continuum mechanics.

Continuum theories for composite materials

Abstract: Continuum mechanics application to elastic composite materials, investigating plane deformation of reinforced medium

Some boundary effects in continuum mechanics

Abstract: This thesis is concerned with the estimation of the effects of an external boundary on some dynamical system of continuum mechanics which have previously been analysed for infinite media. The problems considered are:- (i) The forced torsional oscillations of a rigid spheroidal inclusion in a bounded axisymmetric elastic solid. (ii) The diffraction of torsional stress waves travelling along the axis of an infinite elastic cylinder by a) a fixed rigid inclusion and b) a penny shaped crack. (iii) The diffraction of harmonic sound waves in a circular tube by a) a soft spheroidal obstacle and b) a rigid disc. (iv) The steady swirling flow of an inviscid fluid past a rigid spheroidal body in a coaxial tube (v) The forced torsional oscillations of a) a rigid sphere and b) a rigid disc in an axisymmetric container of viscous fluid. These are particular examples of a general class of boundary value problems for the reduced wave equations which may be formulated by means of Green's theorem and appropriate Green's functions as Fredholm integral equations of the first kind. By perturbing on the static solution low frequency" approximations for quantities of physical interest exhibiting explicitly the first order effects of the external boundaries are obtained. The advantage of this procedure is that it may be used in a large variety of situations where the geometry of the problem prohibits the use of exact processes such as the method of separation of variables. Integral equation formulations may also facilitate the use of a direct boundary perturbation on the infinite medium solution; this technique is used here in a few particular examples.