Editing Continuum mechanics (section)

==Kinematics: motion and deformation==
[[Image:Displacement of a continuum.svg|class=skin-invert-image|400px|right|thumb|Figure 2. Motion of a continuum body.]]
A change in the configuration of a continuum body results in a [[displacement field (mechanics)|displacement]]. The displacement of a body has two components: a rigid-body displacement and a [[Deformation (mechanics)|deformation]]. A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration <math>\kappa_0(\mathcal B)</math> to a current or deformed configuration <math>\kappa_t(\mathcal B)</math> (Figure 2).

The motion of a continuum body is a continuous time sequence of displacements. Thus, the material body will occupy different configurations at different times so that a particle occupies a series of points in space which describe a path line.

There is continuity during motion or deformation of a continuum body in the sense that:
* The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.
* The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.

It is convenient to identify a reference configuration or initial condition which all subsequent configurations are referenced from. The reference configuration need not be one that the body will ever occupy. Often, the configuration at <math>t=0</math> is considered the reference configuration, <math>\kappa_0 (\mathcal B)</math>. The components <math>X_i</math> of the position vector <math>\mathbf X</math> of a particle, taken with respect to the reference configuration, are called the material or reference coordinates.

When analyzing the motion or [[Deformation (mechanics)|deformation]] of solids, or the [[fluid mechanics|flow]] of fluids, it is necessary to describe the sequence or evolution of configurations throughout time. One description for motion is made in terms of the material or referential coordinates, called material description or Lagrangian description.

===Lagrangian description===
In the Lagrangian description the position and physical properties of the particles are described in terms of the material or referential coordinates and time. In this case '''the reference configuration is the configuration at <math>t=0</math>'''. An observer standing in the frame of reference observes the changes in the position and physical properties as the material body moves in space as time progresses. The results obtained are independent of the choice of initial time and reference configuration, <math>\kappa_0(\mathcal B)</math>. This description is normally used in [[solid mechanics]].

In the Lagrangian description, the motion of a continuum body is expressed by the mapping function <math>\chi(\cdot)</math> (Figure 2),

:<math>\mathbf x=\chi(\mathbf X, t)</math>

which is a mapping of the initial configuration <math>\kappa_0(\mathcal B)</math> onto the current configuration <math>\kappa_t(\mathcal B)</math>, giving a geometrical correspondence between them, i.e. giving the position vector <math>\mathbf{x}=x_i\mathbf e_i</math> that a particle <math>X</math>, with a position vector <math>\mathbf X</math> in the undeformed or reference configuration <math>\kappa_0(\mathcal B)</math>, will occupy in the current or deformed configuration <math>\kappa_t(\mathcal B)</math> at time <math>t</math>. The components <math>x_i</math> are called the spatial coordinates.

Physical and kinematic properties <math>P_{ij\ldots}</math>, i.e. thermodynamic properties and flow velocity, which describe or characterize features of the material body, are expressed as continuous functions of position and time, i.e. <math>P_{ij\ldots}=P_{ij\ldots}(\mathbf X,t)</math>.

The material derivative of any property <math>P_{ij\ldots}</math> of a continuum, which may be a scalar, vector, or tensor, is the time rate of change of that property for a specific group of particles of the moving continuum body. The material derivative is also known as the ''substantial derivative'', or ''comoving derivative'', or ''convective derivative''. It can be thought as the rate at which the property changes when measured by an observer traveling with that group of particles.

In the Lagrangian description, the material derivative of <math>P_{ij\ldots}</math> is simply the partial derivative with respect to time, and the position vector <math>\mathbf X</math> is held constant as it does not change with time. Thus, we have

:<math>\frac{d}{dt}[P_{ij\ldots}(\mathbf X,t)]=\frac{\partial}{\partial t}[P_{ij\ldots}(\mathbf X,t)]</math>

The instantaneous position <math>\mathbf x</math> is a property of a particle, and its material derivative is the ''instantaneous flow velocity'' <math>\mathbf v</math> of the particle. Therefore, the flow velocity field of the continuum is given by

:<math>\mathbf v = \dot{\mathbf x} =\frac{d\mathbf x}{dt}=\frac{\partial \chi(\mathbf X,t)}{\partial t} </math>

Similarly, the acceleration field is given by

:<math>\mathbf a= \dot{\mathbf v} = \ddot{\mathbf x} =\frac{d^2\mathbf x}{dt^2}=\frac{\partial^2 \chi(\mathbf X,t)}{\partial t^2} </math>

Continuity in the Lagrangian description is expressed by the spatial and temporal continuity of the mapping from the reference configuration to the current configuration of the material points. All physical quantities characterizing the continuum are described this way. In this sense, the function <math>\chi(\cdot)</math> and <math>P_{ij\ldots}(\cdot)</math> are single-valued and continuous, with continuous derivatives with respect to space and time to whatever order is required, usually to the second or third.

===Eulerian description===
Continuity allows for the inverse of <math>\chi(\cdot)</math> to trace backwards where the particle currently located at <math>\mathbf x</math> was located in the initial or referenced configuration <math>\kappa_0(\mathcal B)</math>. In this case the description of motion is made in terms of the spatial coordinates, in which case is called the spatial description or Eulerian description, i.e. '''the current configuration is taken as the reference configuration'''.

The Eulerian description, introduced by [[d'Alembert]], focuses on the current configuration <math>\kappa_t(\mathcal B)</math>, giving attention to what is occurring at a fixed point in space as time progresses, instead of giving attention to individual particles as they move through space and time. This approach is conveniently applied in the study of [[fluid mechanics|fluid flow]] where the kinematic property of greatest interest is the rate at which change is taking place rather than the shape of the body of fluid at a reference time.{{sfn|Spencer|1980|p=83}}

Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function

:<math>\mathbf X=\chi^{-1}(\mathbf x, t)</math>

which provides a tracing of the particle which now occupies the position <math>\mathbf x</math> in the current configuration <math>\kappa_t(\mathcal B)</math> to its original position <math>\mathbf X</math> in the initial configuration <math>\kappa_0(\mathcal B)</math>.

A necessary and sufficient condition for this inverse function to exist is that the determinant of the [[Jacobian matrix and determinant|Jacobian matrix]], often referred to simply as the Jacobian, should be different from zero. Thus,

:<math>J = \left| \frac{\partial \chi_i}{\partial X_J} \right| = \left| \frac{\partial x_i}{\partial X_J} \right| \neq 0</math>

In the Eulerian description, the physical properties <math>P_{ij\ldots}</math> are expressed as

:<math>P_{ij \ldots}=P_{ij\ldots}(\mathbf X,t)=P_{ij\ldots}[\chi^{-1}(\mathbf x,t),t]=p_{ij\ldots}(\mathbf x,t)</math>

where the functional form of <math>P_{ij \ldots}</math> in the Lagrangian description is not the same as the form of <math>p_{ij \ldots}</math> in the Eulerian description.

The material derivative of <math>p_{ij\ldots}(\mathbf x,t)</math>, using the chain rule, is then

:<math>\frac{d}{dt}[p_{ij\ldots}(\mathbf x,t)]=\frac{\partial}{\partial t}[p_{ij\ldots}(\mathbf x,t)]+ \frac{\partial}{\partial x_k}[p_{ij\ldots}(\mathbf x,t)]\frac{dx_k}{dt}</math>

The first term on the right-hand side of this equation gives the ''local rate of change'' of the property <math>p_{ij\ldots}(\mathbf x,t)</math> occurring at position <math>\mathbf x</math>. The second term of the right-hand side is the ''convective rate of change'' and expresses the contribution of the particle changing position in space (motion).

Continuity in the Eulerian description is expressed by the spatial and temporal continuity and continuous differentiability of the flow velocity field. All physical quantities are defined this way at each instant of time, in the current configuration, as a function of the vector position <math>\mathbf x</math>.

===Displacement field===
{{main|Displacement field (mechanics)}}

The vector joining the positions of a particle <math>P</math> in the undeformed configuration and deformed configuration is called the [[displacement (vector)|displacement vector]] <math>\mathbf u(\mathbf X,t)=u_i\mathbf e_i</math>, in the Lagrangian description, or <math>\mathbf U(\mathbf x,t)=U_J\mathbf E_J</math>, in the Eulerian description.

A ''displacement field'' is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field,  In general, the displacement field is expressed in terms of the material coordinates as

:<math>\mathbf u(\mathbf X,t) = \mathbf b+\mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = \alpha_{iJ}b_J + x_i - \alpha_{iJ}X_J</math>

or in terms of the spatial coordinates as

:<math>\mathbf U(\mathbf x,t) = \mathbf b+\mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = b_J + \alpha_{Ji}x_i - X_J \,</math>

where <math>\alpha_{Ji}</math> are the direction cosines between the material and spatial coordinate systems with unit vectors <math>\mathbf E_J</math> and <math>\mathbf e_i</math>, respectively. Thus

:<math>\mathbf E_J \cdot \mathbf e_i = \alpha_{Ji}=\alpha_{iJ}</math>

and the relationship between <math>u_i</math> and <math>U_J</math> is then given by

:<math>u_i=\alpha_{iJ}U_J \qquad \text{or} \qquad U_J=\alpha_{Ji}u_i</math>

Knowing that
:<math>\mathbf e_i = \alpha_{iJ}\mathbf E_J</math>
then
:<math>\mathbf u(\mathbf X,t)=u_i\mathbf e_i=u_i(\alpha_{iJ}\mathbf E_J)=U_J\mathbf E_J=\mathbf U(\mathbf x,t)</math>

It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in <math>\mathbf b=0</math>,  and the direction cosines become [[Kronecker delta]]s, i.e.

:<math>\mathbf E_J \cdot \mathbf e_i = \delta_{Ji}=\delta_{iJ}</math>

Thus, we have

:<math>\mathbf u(\mathbf X,t) = \mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = x_i - \delta_{iJ}X_J</math>

or in terms of the spatial coordinates as

:<math>\mathbf U(\mathbf x,t) = \mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = \delta_{Ji}x_i - X_J </math>
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==Fundamental laws==
===Conservation of mass===
===Conservation of momentum===
P<sub>i</sub>=P<sub>f</sub>

===Conservation of energy===
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