Continuum mechanics is a branch of
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
that deals with the mechanical behavior of material
s modeled as a continuous
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
rather than as discrete particles
. The French mathematician
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
was the first to formulate such models in the 19th century.
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 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
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the ...
(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 is
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
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form
where -\infty < a(x), b(x) < \infty and the integral are
This integral being zero holds for all groups, that is, for all intervals