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Continuum mechanics introduction

"Continuum mechanics combines mathematical operations and physical processes in continuous media for various subjects of engineering, physics, and applied mathematics." Viewed on a microscopic scale, matter is composed of individual atoms separated by empty space: it is bumpy, granular, discontinuous and requires a quantum mechanical description. In the macroscopic world of solids, fluids, and everyday experience, matter resembles a continuous distribution of stuff: it can be mathematically divided and subdivided into infinitesimally small volumes, each having a definite location and smooth trajectory. This classical macroscopic model is the domain of continuum mechanics.

While media like sand may appear too granular for continuous modeling, if the characteristic length $S$ of physical interest exceeds the average distance $\lambda$ between molecular collisions (or between bumping sand grains) a continuous description may still be valid:

${\rm {Kn}}={\lambda \over S}<1\qquad (macroscopic),$

${\rm {Kn}}={\lambda \over S}\geq 1\qquad (microscopic),$

${\rm {u}}=x(X,t)-X$

The Knudsen number ${\rm {Kn}}$ shown here is helpful in making this determination; ultimately, the validity of the continuum assumption needs to be verified with experimental testing and measurements on the real material under consideration and under similar loading conditions.

mathematical operations and physical processes include .... These are further described below.

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that deals with the analysis of the kinematics and mechanical behavior of materials modeled as a continuum, i.e., solids and fluids (e.g., liquids and gases). A continuum concept assumes that the substance of the body is distributed throughout — and completely fills — the space it occupies.

The continuum concept ignores the fact that matter is made of atoms, is not continuous, and that it commonly has some sort of heterogeneous microstructure, allowing the approximation of physical quantities, such as energy and momentum, at the infinitesimal limit. Differential equations can thus be employed in solving problems in continuum mechanics. Some of these differential equations are specific to the materials being investigated and are called constitutive equations, while others capture fundamental physical laws, such as the conservation of mass (the continuity equation), the conservation of momentum (the equations of motion and equilibrium), and the conservation of energy (the first law of thermodynamics).

Continuum mechanics deals with physical quantities of solids and fluids which are independent of any particular coordinate system in which they are observed. These physical quantities are then represented by tensors, which are mathematical objects that are independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience.

In fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made.

The macroscopic view

On a microscopic scale, matter is composed of individual atoms separated by empty space: it is bumpy, granular, discontinuous and requires a [[Quantum mechanics|quantum mechanical] description. On the macroscopic scale of daily experience, matter is best modeled as a continuous distribution of infinitesimal volumes and their surfaces.

However, certain physical phenomena can be modeled assuming the materials exist as a continuum, meaning the matter in the body is continuously distributed and fills the entire region of space it occupies. A continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material.

The concept of continuum is a macroscopic physical model, and its validity depends on the type of problem and the scale of the physical phenomena under consideration. A material may be assumed to be a continuum when the distance between the physical particles is very small compared to the dimension of the problem. For example, such is the case when analyzing the deformation behavior of soil deposits in soil mechanics. A given volume of soil is composed of discrete solid particles (grains) of minerals that are packed in a certain manner with voids between them. In this sense, soils evade the definition of a continuum. To simplify the deformation analysis of the soil, the volume of soil can be assumed to be a continuum because the grain particles are very small compared to the scale of the problem.

The validity of the continuum assumption needs to be verified with experimental testing and measurements on the real material under consideration and under similar loading conditions.

Mathematical modeling of a continuum

Figure 1. Configuration of a continuum body

In continuum mechanics, a material body ${\mathcal {B}}$ is a set of infinitesimal volumetric elements $\ X$, called particles or material points.

Presentation of continuum model

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$u=x(X,t)-X$

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Notes

• Chung, T. J. (2007). General Continuum Mechanics. Cambridge University Press. p. 3. ISBN 978-0-521-87406-9.