Continuum mechanics is a branch of mechanics
that deals with the mechanical behavior of material
s 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.
Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atom
s, and so is not continuous; however, on length scale
s 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 behavior of such objects, and some information about the material under investigation is added through 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 tensor
s, 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 a continuum
Materials, such as solids, liquids and gases, are composed of molecule
s separated by space. On a microscopic scale, materials have cracks and discontinuities. 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 validity of the continuum assumption may be verified by a theoretical analysis, in which either some clear periodicity is identified or statistical homogeneity
of the microstructure
exists. 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 that of the representative volume element (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 fluid
s, the Knudsen number
is used to assess to what extent the approximation of continuity can be made.
Car traffic as an introductory example
Consider car traffic on a highway, with just one lane for simplicity.
Somewhat surprisingly, and in a tribute to its effectiveness, continuum mechanics effectively models the movement of cars via a partial differential equation
(PDE) for the density of cars.
The familiarity of this situation empowers us to understand a little of the continuum-discrete dichotomy underlying continuum modelling in general.
To start modelling define that:
measures distance (in km) along the highway;
is time (in minutes);
is the density of cars on the highway (in cars/km in the lane); and
is the flow velocity
(average velocity) of those cars 'at' position
Conservation derives a PDE (Partial differential equation
Cars do not appear and disappear.
Consider any group of cars: from the particular car at the back of the group located at
to the particular car at the front located at
The total number of cars in this group
Since cars are conserved (if there is overtaking, then the `car at the front \ back' may become a different car)
But via the Leibniz integral rule
This integral being zero holds for all groups, that is, for all intervals