Basic Equations Of Continuum Mechanics Research Articles

Fundamentals of continuum mechanics – classical approaches and new trends11The paper is dedicated to Alexander Manzhirov’s 60th birthday.

Continuum mechanics is a branch of mechanics that deals with the analysis of the mechanical behavior of materials modeled as a continuous manifold. Continuum mechanics models begin mostly by introducing of three-dimensional Euclidean space. The points within this region are defined as material points with prescribed properties. Each material point is characterized by a position vector which is continuous in time. Thus, the body changes in a way which is realistic, globally invertible at all times and orientation-preserving, so that the body cannot intersect itself and as transformations which produce mirror reflections are not possible in nature. For the mathematical formulation of the model it is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated. Finally, the kinematical relations, the balance equations, the constitutive and evolution equations and the boundary and/or initial conditions should be defined. If the physical fields are non-smooth jump conditions must be taken into account.The basic equations of continuum mechanics are presented following a short introduction. Additionally, some examples of solid deformable continua will be discussed within the presentation. Finally, advanced models of continuum mechanics will be introduced.

Energy flow relations from quadratic quantities in three-dimensional isotropic medium and exact formulation for one-dimensional waves

From the basic equations of continuum mechanics in a three-dimensional (3D) isotropic and damped medium excited by harmonic body forces, exact expressions are obtained from quadratic variables for time-averaged energy quantities: kinetic- and strain-energy densities, structural intensity, structural intensity divergence and curl. These energy quantities are split into four components: longitudinal, shear and two mixed ones. Each component is governed by similar relations of different quadratic variables. For 1D wave fields, an exact formulation based on quadratic variables is derived. The fundamental solutions of this formulation are analyzed for unloaded, and for concentrated loaded systems. The energy-models reported in the literature consider only some components of these solutions. The energy density and structural intensity components obtained from the quadratic formulation and from the usual displacement formulation are equivalent; this is illustrated for the energy transfers modeled by the quadratic formulation, in comparison with the displacement formulation, for a one-dimensional, longitudinal- and shear-wave field with wave conversion at one end.

Formulation of a Non‐linear Higher‐Order Shell Theory Using the Convective Description

AbstractBy formulating a higher‐order non‐linear shell theory, we start with the basic equations of continuum mechanics of a three‐dimensional body. By assuming a director set on a reference surface of a shell, the description shifts into the shell space. The position vector for the body is introduced as a power series with respect to the out‐of‐surface variable. By considering more than the usual linear part of the series, the complete three‐dimensional strain tensor is available. The motion of the continuum can be represented through the usage of the convective description which combines properties from both the spatial and the material description. Furthermore, the rate‐type formulation gives the opportunity to deal with geometrical nonlinearities in a three‐dimensional field problem efficiently. Following these two concepts results as well in a position‐dependent as in a time‐dependent base vector set; applying them leads to an initial boundary value problem (IBVP).

Reformulation of elasticity theory for discontinuities and long-range forces

Some materials may naturally form discontinuities such as cracks as a result of deformation. As an aid to the modeling of such materials, a new framework for the basic equations of continuum mechanics, called the ‘peridynamic’ formulation, is proposed. The propagation of linear stress waves in the new theory is discussed, and wave dispersion relations are derived. Material stability and its connection with wave propagation is investigated. It is demonstrated by an example that the reformulated approach permits the solution of fracture problems using the same equations either on or off the crack surface or crack tip. This is an advantage for modeling problems in which the location of a crack is not known in advance.

Numerical heat conductivity in smooth particle applied mechanics.

Smooth particle applied mechanics provides a method for solving the basic equations of continuum mechanics, interpolating these equations onto a grid made up of moving particles. The moving particle grid gives rise to a thoroughly artificial numerical heat conductivity, analogous to the numerical viscosities associated with finite-difference schemes. We exploit an isomorphism linking the smooth-particle method to conventional molecular dynamics, and evaluate this numerical heat conductivity. We find that the corresponding thermal diffusivity is comparable in value to the numerical kinematic viscosity, and that neither is described very well by the Enskog theory. {copyright} {ital 1996 The American Physical Society.}

Some applications of geometry is continuum mechanics

Some contemporary ideas from differential geometry are applied to continuum mechanics. The Lie derivative is used to clarify the notion of “objective rates”, an intrinsic treatment of Piola transformations is described, a simplified proof of Vainberg's theorem for potential operators is given by way of the Poincaré lemma on infinite dimensional manifolds, and a new derivation of the basic equations of continuum mechanics is presented which is valid in a general Riemannian manifold setting.

Heat and mass transport in cylindrical gas-filled incandescent lamps

The heat and mass transport in a gas-filled incandescent lamp is governed by convection and heat conduction or mass diffusion, respectively. Proceeding from the basic equations of continuum mechanics a numerical method has been developed to calculate the temperature distribution, the flow field of the filling gas, and the flow of the wire material in a horizontal cylindrical lamp. The results are compared to interferometric temperature measurements and to flow patterns obtained by a tracer technique.

Allgemeine rahmengleichungen der kontinuumsdynamik

An extension of the basic equations of thermodynamics is given for systems undergoing general deformation together with a new formulation for the basic equations of continuum mechanics.