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Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous medium (also called a continuum) rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

Continuum mechanics deals with deformable bodies, as opposed to rigid bodies. A continuum model assumes that the substance of the object completely fills the space it occupies. While ignoring the fact that matter is made of atoms, this provides a sufficiently accurate description of matter on length scales much greater than that of inter-atomic distances. The concept of a continuous medium allows for intuitive analysis of bulk matter by using differential equations that describe the behavior of such matter according to physical laws, such as mass conservation, momentum conservation, and energy conservation. Information about the specific material is expressed in constitutive relationships.

Continuum mechanics treats the physical properties of solids and fluids independently of any particular coordinate system in which they are observed. These properties are represented by tensors, which are mathematical objects with the salient property of being independent of coordinate systems. This permits definition of physical properties at any point in the continuum, according to mathematically convenient continuous functions. The theories of elasticity, plasticity and fluid mechanics are based on the concepts of continuum mechanics.

Concept of a continuum

The concept of a continuum underlies the mathematical framework for studying large-scale forces and deformations in materials. Although materials are composed of discrete atoms and molecules, separated by empty space or microscopic cracks and crystallographic defects, physical phenomena can often be modeled by considering a substance distributed throughout some region of space. A continuum is a body that can be continually sub-divided into infinitesimal elements with local material properties defined at any particular point. Properties of the bulk material can therefore be described by continuous functions, and their evolution can be studied using the mathematics of calculus.

Apart from the assumption of continuity, two other independent assumptions are often employed in the study of continuum mechanics. These are homogeneity (assumption of identical properties at all locations) and isotropy (assumption of directionally invariant vector properties).^[1] If these auxiliary assumptions are not globally applicable, the material may be segregated into sections where they are applicable in order to simplify the analysis. For more complex cases, one or both of these assumptions can be dropped. In these cases, computational methods are often used to solve the differential equations describing the evolution of material properties.

Major areas

An additional area of continuum mechanics comprises elastomeric foams, which exhibit a curious hyperbolic stress-strain relationship. The elastomer is a true continuum, but a homogeneous distribution of voids gives it unusual properties.^[2]

Formulation of models

Continuum mechanics models begin by assigning a region in three-dimensional Euclidean space to the material body ${\displaystyle {\mathcal {B}}}$ being modeled. 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 ${\displaystyle t}$ is labeled ${\displaystyle \kappa _{t}({\mathcal {B}})}$.

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

${\displaystyle \mathbf {x} =\sum _{i=1}^{3}x_{i}\mathbf {e} _{i},}$

where ${\displaystyle \mathbf {e} _{i}}$ are the coordinate vectors in some frame of reference chosen for the problem (See figure 1). This vector can be expressed as a function of the particle position ${\displaystyle \mathbf {X} }$ in some reference configuration, for example the configuration at the initial time, so that

${\displaystyle \mathbf {x} =\kappa _{t}(\mathbf {X} ).}$

This function needs to have various properties so that the model makes physical sense. ${\displaystyle \kappa _{t}(\cdot )}$ needs to be:

• continuous in time, so that the body changes in a way which is realistic,
• globally invertible at all times, so that the body cannot intersect itself,
• orientation-preserving, as transformations which produce mirror reflections are not possible in nature.

For the mathematical formulation of the model, ${\displaystyle \kappa _{t}(\cdot )}$ is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated.

Forces in a continuum

A solid is a deformable body that possesses shear strength, sc. a solid can support shear forces (forces parallel to the material surface on which they act). Fluids, on the other hand, do not sustain shear forces.

Following the classical dynamics of Newton and Euler, the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces ${\displaystyle \mathbf {F} _{C}}$ and body forces ${\displaystyle \mathbf {F} _{B}}$.^[3] Thus, the total force ${\displaystyle {\mathcal {F}}}$ applied to a body or to a portion of the body can be expressed as:

${\displaystyle {\mathcal {F}}=\mathbf {F} _{C}+\mathbf {F} _{B}}$

Surface forces

Surface forces or contact forces, expressed as force per unit area, can act either on the bounding surface of the body, as a result of mechanical contact with other bodies, or on imaginary internal surfaces that bound portions of the body, as a result of the mechanical interaction between the parts of the body to either side of the surface (Euler-Cauchy's stress principle). When a body is acted upon by external contact forces, internal contact forces are then transmitted from point to point inside the body to balance their action, according to Newton's third law of motion of conservation of linear momentum and angular momentum (for continuous bodies these laws are called the Euler's equations of motion). The internal contact forces are related to the body's deformation through constitutive equations. The internal contact forces may be mathematically described by how they relate to the motion of the body, independent of the body's material makeup.^{[citation needed]}

The distribution of internal contact forces throughout the volume of the body is assumed to be continuous. Therefore, there exists a contact force density or Cauchy traction field^[4] ${\displaystyle \mathbf {T} (\mathbf {n} ,\mathbf {x} ,t)}$ that represents this distribution in a particular configuration of the body at a given time ${\displaystyle t\,\!}$. It is not a vector field because it depends not only on the position ${\displaystyle \mathbf {x} }$ of a particular material point, but also on the local orientation of the surface element as defined by its normal vector ${\displaystyle \mathbf {n} }$.^{[5][page needed]}

Any differential area ${\displaystyle dS\,\!}$ with normal vector ${\displaystyle \mathbf {n} }$ of a given internal surface area ${\displaystyle S\,\!}$, bounding a portion of the body, experiences a contact force ${\displaystyle d\mathbf {F} _{C}\,\!}$ arising from the contact between both portions of the body on each side of ${\displaystyle S\,\!}$, and it is given by

${\displaystyle d\mathbf {F} _{C}=\mathbf {T} ^{(\mathbf {n} )}\,dS}$

where ${\displaystyle \mathbf {T} ^{(\mathbf {n} )}}$ is the surface traction,^[6] also called stress vector,^[7] traction,^{[8][page needed]} or traction vector.^[9] The stress vector is a frame-indifferent vector (see Euler-Cauchy's stress principle). Zdroj:https://en.wikipedia.org?pojem=Continuum_(physics)
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