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Continuum Mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behaviour of materials modelled as a continuous mass rather than as discrete particles. The French mathematician Augustin Louis Cauchy was the first to formulate such models in the 19^th century, but research in the area continues today.

Modelling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Modelling 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. Fundamental physical laws such as the conservation of mass, the conservation of momentum, and the conservation of energy may be applied to such models to derive differential equations describing the behaviour of such objects, and some information about the particular material studied is added through a constitutive relation.

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

Concept of Continuum

Materials, such as solids, liquids and gases, are composed of molecules separated by empty space. On a macroscopic scale, materials have cracks and discontinuities. However, certain physical phenomena can be modelled 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 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 exists. More specifically, the continuum hypothesis/assumption hinges on the concepts of a representative volume element (RVE) (sometimes called "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 that of the RVE size, one employs a statistical volume element (SVE), which, in turn, leads to 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. The RVE may be assessed only in a limited way via experimental testing: when the constitutive response becomes spatially homogeneous.

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

Formulation of continuum models

Continuum mechanics models begin by assigning a region in three dimensional Euclidean spaces to the material body B being modelled. The points within this region are called particles or material points. Different configurations or states of the body correspond to different regions in Euclidean space. The region corresponding to the body's configuration at time t is labelled K_t(B).

A particular particle within the body in a particular configuration is characterized by a position vector.

where e_i are the coordinate vectors in some frame of reference chosen for the problem. This vector can be expressed as a function of the particle position X in some reference configuration, for example the configuration at the initial time, so that

x = K_t (X)

This function needs to have various properties so that the model makes physical sense. K_t (.) needs to be:

1. Continuous in time, so that the body changes in a way which is realistic,

2. Globally invertible at all times, so that the body cannot intersect itself,

3. Orientation-preserving, as transformations which produce mirror reflections are not possible in nature.

For the mathematical formulation of the model, K_t (.) is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated.