Editing Continuum mechanics (section)

==Governing equations==
Continuum mechanics deals with the behavior of materials that can be approximated as continuous for certain length and time scales.  The equations that govern the mechanics of such materials include the balance laws for [[conservation of mass|mass]], [[conservation of momentum|momentum]], and [[conservation of energy|energy]]. [[Kinematics|Kinematic]] relations and [[constitutive equations]] are needed to complete the system of governing equations.  Physical restrictions on the form of the constitutive relations can be applied by requiring that the [[second law of thermodynamics]] be satisfied under all conditions.  In the continuum mechanics of solids, the second law of thermodynamics is satisfied if the [[Clausius–Duhem inequality|Clausius–Duhem]] form of the entropy inequality is satisfied.

The balance laws express the idea that the rate of change of a quantity (mass, momentum, energy) in a volume must arise from three causes:

#the physical quantity itself flows through the surface that bounds the volume,
#there is a source of the physical quantity on the surface of the volume, or/and,
#there is a source of the physical quantity inside the volume.

Let <math>\Omega</math> be the body (an open subset of Euclidean space) and let   <math>\partial \Omega </math> be its surface (the boundary of <math>\Omega</math>).

Let the motion of material points in the body be described by the map
:<math>\mathbf{x} = \boldsymbol{\chi}(\mathbf{X}) = \mathbf{x}(\mathbf{X})</math>
where <math>\mathbf{X}</math> is the position of a point in the initial configuration and <math>\mathbf{x}</math> is the location of the same point in the deformed configuration.

The deformation gradient is given by
:<math>\boldsymbol{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \nabla \mathbf{x}  ~.</math>

===Balance laws===
Let <math>f(\mathbf{x},t)</math> be a physical quantity that is flowing through the body.  Let <math>g(\mathbf{x},t)</math> be sources on the surface of the body and let <math>h(\mathbf{x},t)</math> be sources inside the body.  Let <math>\mathbf{n}(\mathbf{x},t)</math> be the outward unit normal to the surface <math>\partial \Omega </math>.  Let <math>\mathbf{v}(\mathbf{x},t)</math> be the flow velocity of the physical particles that carry the physical quantity that is flowing.  Also, let the speed at which the bounding surface <math>\partial \Omega </math> is moving be <math>u_n</math> (in the direction <math>\mathbf{n}</math>).

Then, balance laws can be expressed in the general form
:<math>
    \cfrac{d}{dt}\left[\int_{\Omega} f(\mathbf{x},t)~\text{dV}\right] = 
      \int_{\partial \Omega } f(\mathbf{x},t)[u_n(\mathbf{x},t) - \mathbf{v}(\mathbf{x},t)\cdot\mathbf{n}(\mathbf{x},t)]~\text{dA} + 
      \int_{\partial \Omega } g(\mathbf{x},t)~\text{dA} + \int_{\Omega} h(\mathbf{x},t)~\text{dV} ~.
  </math>
The functions <math>f(\mathbf{x},t)</math>, <math>g(\mathbf{x},t)</math>, and <math>h(\mathbf{x},t)</math> can be scalar valued, vector valued, or tensor valued - depending on the physical quantity that the balance equation deals with.  If there are internal boundaries in the body, jump discontinuities also need to be specified in the balance laws.

If we take the [[Lagrangian and Eulerian specification of the flow field|Eulerian]] point of view, it can be shown that the balance laws of mass, momentum, and energy for a solid can be written as (assuming the source term is zero for the mass and angular momentum equations)
:<math>
    {
    \begin{align}
      \dot{\rho} + \rho (\boldsymbol{\nabla} \cdot \mathbf{v}) & = 0 
          & & \qquad\text{Balance of Mass} \\
      \rho~\dot{\mathbf{v}} - \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} - \rho~\mathbf{b} & = 0 
          & & \qquad\text{Balance of Linear Momentum (Cauchy's first law of motion)} \\
      \boldsymbol{\sigma} & = \boldsymbol{\sigma}^T
          & & \qquad\text{Balance of Angular Momentum (Cauchy's second law of motion)} \\
      \rho~\dot{e} - \boldsymbol{\sigma}:(\boldsymbol{\nabla}\mathbf{v}) + \boldsymbol{\nabla} \cdot \mathbf{q} - \rho~s & = 0
          & & \qquad\text{Balance of Energy.}
    \end{align}
    }
  </math>

In the above equations <math>\rho(\mathbf{x},t)</math> is the mass density (current),   <math>\dot{\rho}</math> is the material time derivative of <math>\rho</math>, <math>\mathbf{v}(\mathbf{x},t)</math> is the particle velocity, <math>\dot{\mathbf{v}}</math> is the material time derivative of <math>\mathbf{v}</math>, <math>\boldsymbol{\sigma}(\mathbf{x},t)</math> is the [[Cauchy stress tensor]], <math>\mathbf{b}(\mathbf{x},t)</math> is the body force density, <math>e(\mathbf{x},t)</math> is the internal energy per unit mass, <math>\dot{e}</math> is the material time derivative of <math>e</math>, <math>\mathbf{q}(\mathbf{x},t)</math> is the heat flux vector, and <math>s(\mathbf{x},t)</math> is an energy source per unit mass. The operators used are defined [[#Operators|below]].
  
With respect to the reference configuration (the Lagrangian point of view), the balance laws can be written as
:<math>
    {
    \begin{align}
      \rho~\det(\boldsymbol{F}) - \rho_0 &= 0 & &  \qquad \text{Balance of Mass} \\
      \rho_0~\ddot{\mathbf{x}} - \boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{P} -\rho_0~\mathbf{b} & = 0  & & 
        \qquad \text{Balance of Linear Momentum} \\
      \boldsymbol{F}\cdot\boldsymbol{P}^T & = \boldsymbol{P}\cdot\boldsymbol{F}^T  & & 
        \qquad \text{Balance of Angular Momentum} \\ 
      \rho_0~\dot{e} - \boldsymbol{P}^T:\dot{\boldsymbol{F}} + \boldsymbol{\nabla}_{\circ}\cdot\mathbf{q} - \rho_0~s & = 0
          & & \qquad\text{Balance of Energy.} 
    \end{align}
    }
  </math>

In the above, <math>\boldsymbol{P}</math> is the first [[Piola-Kirchhoff stress tensor]], and   <math>\rho_0</math> is the mass density in the reference configuration.  The first Piola-Kirchhoff stress tensor is related to the Cauchy stress tensor by
:<math>
    \boldsymbol{P} = J~\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T}
   ~\text{where}~ J = \det(\boldsymbol{F})
  </math>

We can alternatively define the nominal stress tensor <math>\boldsymbol{N}</math> which is the transpose of the first Piola-Kirchhoff stress tensor such that
:<math>
    \boldsymbol{N} = \boldsymbol{P}^T = J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma} ~.
  </math>
Then the balance laws become
:<math>
    {
    \begin{align}
      \rho~\det(\boldsymbol{F}) - \rho_0 &= 0 & &  \qquad \text{Balance of Mass} \\
      \rho_0~\ddot{\mathbf{x}} - \boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{N}^T -\rho_0~\mathbf{b} & = 0  & & 
        \qquad \text{Balance of Linear Momentum} \\
      \boldsymbol{F}\cdot\boldsymbol{N} & = \boldsymbol{N}^T\cdot\boldsymbol{F}^T  & & 
        \qquad \text{Balance of Angular Momentum} \\ 
      \rho_0~\dot{e} - \boldsymbol{N}:\dot{\boldsymbol{F}} + \boldsymbol{\nabla}_{\circ}\cdot\mathbf{q} - \rho_0~s & = 0
          & & \qquad\text{Balance of Energy.} 
    \end{align}
    }
  </math>

====Operators====

The operators in the above equations are defined as
:<math>
    \begin{align}
\boldsymbol{\nabla} \mathbf{v}
  &= \sum_{i,j = 1}^3 \frac{\partial v_i}{\partial x_j}\mathbf{e}_i\otimes\mathbf{e}_j
  = v_{i,j}\mathbf{e}_i\otimes\mathbf{e}_j ~; \\[1ex]
\boldsymbol{\nabla} \cdot \mathbf{v}
  &= \sum_{i=1}^3 \frac{\partial v_i}{\partial x_i}
  = v_{i,i} ~; \\[1ex]
\boldsymbol{\nabla} \cdot \boldsymbol{S}
  &= \sum_{i,j=1}^3 \frac{\partial S_{ij}}{\partial x_j}~\mathbf{e}_i 
  = \sigma_{ij,j}~\mathbf{e}_i ~.
\end{align}
  </math>
where <math>\mathbf{v}</math> is a vector field, <math>\boldsymbol{S}</math> is a second-order tensor field, and <math>\mathbf{e}_i</math> are the components of an orthonormal basis in the current configuration.  Also,
:<math>
    \begin{align}
\boldsymbol{\nabla}_{\circ} \mathbf{v}
  &= \sum_{i,j = 1}^3 \frac{\partial v_i}{\partial X_j}\mathbf{E}_i\otimes\mathbf{E}_j
  = v_{i,j}\mathbf{E}_i\otimes\mathbf{E}_j ~; \\[1ex]
\boldsymbol{\nabla}_{\circ}\cdot\mathbf{v}
  &=  \sum_{i=1}^3 \frac{\partial v_i}{\partial X_i} = v_{i,i} ~; \\[1ex]
\boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{S}
  &= \sum_{i,j=1}^3 \frac{\partial S_{ij}}{\partial X_j}~\mathbf{E}_i
  = S_{ij,j}~\mathbf{E}_i
\end{align} 
  </math>
where <math>\mathbf{v}</math> is a vector field, <math>\boldsymbol{S}</math> is a second-order tensor field, and <math>\mathbf{E}_i</math> are the components of an orthonormal basis in the reference configuration.

The inner product is defined as
:<math>
    \boldsymbol{A}:\boldsymbol{B} = \sum_{i,j=1}^3 A_{ij}~B_{ij} = \operatorname{trace}(\boldsymbol{A}\boldsymbol{B}^T) ~.
  </math>

===Clausius–Duhem inequality===
The [[Clausius–Duhem inequality]] can be used to express the second law of thermodynamics for elastic-plastic materials.  This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved.

Just like in the balance laws in the previous section, we assume that there is a flux of a quantity, a source of the quantity, and an internal density of the quantity per unit mass.  The quantity of interest in this case is the entropy.  Thus, we assume that there is an entropy flux, an entropy source, an internal mass density <math>\rho</math> and an internal specific entropy (i.e. entropy per unit mass) <math>\eta</math> in the region of interest.

Let <math>\Omega</math> be such a region and let <math>\partial \Omega </math> be its boundary.  Then the second law of thermodynamics states that the rate of increase of  <math>\eta</math> in this region is greater than or equal to the sum of that supplied to <math>\Omega</math> (as a flux or from internal sources) and the change of the internal entropy density <math>\rho\eta</math> due to material flowing in and out of the region.
  
Let <math>\partial \Omega </math> move with a flow velocity <math>u_n</math> and let particles inside <math>\Omega</math> have velocities <math>\mathbf{v}</math>.  Let <math>\mathbf{n}</math> be the unit outward normal to the surface <math>\partial \Omega </math>.  Let <math>\rho</math> be the density of matter in the region, <math>\bar{q}</math> be the entropy flux at the surface, and <math>r</math> be the entropy source per unit mass.  
Then the entropy inequality may be written as
:<math>
    \cfrac{d}{dt}\left(\int_{\Omega} \rho~\eta~\text{dV}\right) \ge
    \int_{\partial \Omega} \rho~\eta~(u_n - \mathbf{v}\cdot\mathbf{n}) ~\text{dA} + 
    \int_{\partial \Omega} \bar{q}~\text{dA} + \int_{\Omega} \rho~r~\text{dV}.
  </math>
The scalar entropy flux can be related to the vector flux at the surface by the relation <math>\bar{q} = -\boldsymbol{\psi}(\mathbf{x})\cdot\mathbf{n}</math>.  Under the assumption  of incrementally isothermal conditions, we have
:<math>
    \boldsymbol{\psi}(\mathbf{x}) = \cfrac{\mathbf{q}(\mathbf{x})}{T} ~;~~ r = \cfrac{s}{T}
  </math>
where <math>\mathbf{q}</math> is the heat flux vector, <math>s</math> is an energy source per unit mass, and <math>T</math> is the absolute temperature of a material point at <math>\mathbf{x}</math> at time <math>t</math>.

We then have the Clausius–Duhem inequality in integral form:
:<math>
    {
    \cfrac{d}{dt}\left(\int_{\Omega} \rho~\eta~\text{dV}\right) \ge
    \int_{\partial \Omega} \rho~\eta~(u_n - \mathbf{v}\cdot\mathbf{n}) ~\text{dA} - 
    \int_{\partial \Omega} \cfrac{\mathbf{q}\cdot\mathbf{n}}{T}~\text{dA} + \int_\Omega \cfrac{\rho~s}{T}~\text{dV}.
    }
  </math>
We can show that the entropy inequality may be written in differential form as
:<math>
    {
    \rho~\dot{\eta} \ge - \boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right)
       + \cfrac{\rho~s}{T}.
    }
  </math>
In terms of the Cauchy stress and the internal energy, the Clausius–Duhem inequality may be written as 
:<math>
    {
      \rho~(\dot{e} - T~\dot{\eta}) - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le 
           - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T}.
    }
  </math>