# Continuum mechanics

In physics, continuous body mechanics, or simply continuum mechanics, is the branch of classical mechanics and statistical mechanics that studies the behavior of continuous bodies, i.e., macroscopic physical systems in which the size of the observed phenomena is such that they are not affected by the molecular structure of the matter, and for which it is assumed that the matter is uniformly distributed and fills the space occupied by the body. More formally, a continuous body is defined as a body whose material points are identifiable with the geometric points of a regular region of physical space, and which is endowed with mass for which there exists a mass density function that can represent its measure.

The continuous body is a phenomenological model that includes both solids and fluids, which is why it is specifically referred to as solid mechanics and fluid mechanics, and is associated with the concept of a deformable body in that its parts undergo changes in shape and volume during motion. A limiting case of a continuous body is the rigid body, whose study, developed by rational mechanics, is defined on the basis of a finite number of degrees of freedom and leads to systems of ordinary differential equations. Deformable continua, on the other hand, can be thought of as systems with infinite degrees of freedom, and their mechanical equations take the form of partial differential equations.

A classification of continuous body models can be made on the basis of the size of the region of space they occupy. The three-dimensional models include the Cauchy continuum, which is the best-known and most important continuous body model in the discipline, so much so that the term continuum mechanics is often used synonymously with Cauchy continuum mechanics. The three-dimensional models also include the polar Cosserat continuum model, which has a richer local structure than the point structure of the Cauchy model, also expressed in terms of the orientation of its material points. Both two-dimensional continua, such as plates, slabs, and shells, and one-dimensional continua, such as the beam model studied in structural engineering, are widely used in structural mechanics because of their greater simplicity.

## Continuum Mechanics Relationships

The study of the mechanical behavior of continuous bodies is based on the kinematic characterization of the continuous body (configuration, deformation, motion) and relates such notions of the body to the mass assigned to it and the forces acting on it. Such relations are of two kinds:

1. of a general nature, or fundamental equations, common to all continuous bodies;
2. of a particular kind, or constitutive laws, which distinguish one class of continuous bodies from another.

The former include the fundamental equations of balance, such as the conservation of mass, the balance of momentum, the balance of internal energy, and the balance of mechanical energy, which embody the laws of physics that the body must obey, regardless of the material of which it is made. These relationships give rise to the theories of statics and dynamics.

In the latter, the focus is on developing the so-called constitutive laws that characterize the behavior of certain ideal materials that make up the body: the perfectly elastic solid and the viscous fluid are well-known examples.

Mathematically, the fundamental equations of continuum mechanics can be developed in two different but equivalent formulations. The first, in integral or global form, is derived by applying the basic principles to a finite portion of the body’s volume. The other, in differential or field form, leads to equations (partial derivatives) resulting from the application of the basic principles to elements of very small volumes (infinitesimals).

Continuum mechanics deals with physical quantities of solids and fluids that do not depend on the coordinate system in which they are observed. These quantities are conveniently represented by tensors, which are mathematical objects that are independent of the coordinate system. Thus, the relations of continuum mechanics have a tensorial character. For computational purposes, these tensors can be expressed in certain coordinate systems.

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