continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

Explanation

A continuum model assumes that the substance of the object fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atoms, and so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate. These models can be used to derive differential equations that describe the behavior of such objects using physical laws, such as mass conservation, momentum conservation, and energy conservation, and some information about the material is provided by constitutive relationships.
Continuum mechanics deals with the physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed. The physical properties are then represented by tensors, which are mathematical objects with the property of being independent of coordinate systems. Coordinate systems allow these tensors to be expressed computationally.

Concept of a continuum

Space separates molecules that make up solids, liquids, and gases. Materials have cracks and discontinuities on a microscopic level. Physical phenomena can, however, be modeled if the materials exist as a continuum, meaning the matter in the body is continuously distributed and fills the entire 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 validity of the continuum assumption may be verified by a theoretical analysis, in which either some clear periodicity is identified or statistical homogeneity and ergodicity of the microstructure exist. More specifically, the continuum hypothesis/assumption hinges on the concepts of a representative elementary volume and separation of scales based on the Hill–Mandel condition. This condition provides a link between an experimentalist's and a theoretician's viewpoint on constitutive equations (linear and nonlinear elastic/inelastic or coupled fields) as well as a way of spatial and statistical averaging of the microstructure.
When the separation of scales does not hold, or when one wants to establish a continuum of a finer resolution than the size of the representative volume element (RVE), a statistical volume element (SVE) is employed, which results in random continuum fields. The latter then provide a micromechanics basis for stochastic finite elements (SFE). The levels of SVE and RVE link continuum mechanics to statistical mechanics. Experimentally, the RVE can only be evaluated when the constitutive response is spatially homogenous

Car traffic as an introductory example

Consider car traffic on a highway, with just one lane for