Continuum mechanics deals with the behavior of abstracts that can be approximated as connected for assertive breadth and time scales. The equations that administer the mechanics of such abstracts cover the antithesis laws for mass, momentum, and energy. Kinematic relations and basal equations are bare to complete the arrangement of administering equations. Concrete restrictions on the anatomy of the basal relations can be activated by acute that the additional law of thermodynamics be annoyed beneath all conditions. In the continuum mechanics of solids, the additional law of thermodynamics is annoyed if the Clausius–Duhem anatomy of the anarchy asperity is satisfied.
The antithesis laws accurate the abstraction that the amount of change of a abundance (mass, momentum, energy) in a aggregate accept to appear from three causes:
the concrete abundance itself flows through the apparent that bound the volume,
there is a antecedent of the concrete abundance on the apparent of the volume, or/and,
there is a antecedent of the concrete abundance central the volume.
Let Ω be the physique (an accessible subset of Euclidean space) and let \partial \Omega be its apparent (the abuttals of Ω).
Let the motion of actual credibility in the physique be declared by the map
\mathbf{x} = \boldsymbol{\chi}(\mathbf{X}) = \mathbf{x}(\mathbf{X})
where \mathbf{X} is the position of a point in the antecedent agreement and \mathbf{x} is the area of the aforementioned point in the askew configuration.
The anamorphosis acclivity is accustomed by
\boldsymbol{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \boldsymbol{\mathbf{x}} \cdot \nabla ~.
edit Antithesis laws
Let f(\mathbf{x},t) be a concrete abundance that is abounding through the body. Let g(\mathbf{x},t) be sources on the apparent of the physique and let h(\mathbf{x},t) be sources central the body. Let \mathbf{n}(\mathbf{x},t) be the apparent assemblage accustomed to the apparent \partial \Omega . Let \mathbf{v}(\mathbf{x},t) be the acceleration of the concrete particles that backpack the concrete abundance that is flowing. Also, let the acceleration at which the bonds apparent \partial \Omega is affective be un (in the administration \mathbf{n}).
Then, antithesis laws can be bidding in the accepted form
\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} ~.
Note that the functions f(\mathbf{x},t), g(\mathbf{x},t), and h(\mathbf{x},t) can be scalar valued, agent valued, or tensor admired - depending on the concrete abundance that the antithesis blueprint deals with. If there are centralized boundaries in the body, jump discontinuities aswell charge to be authentic in the antithesis laws.
If we yield the Lagrangian point of view, it can be apparent that the antithesis laws of mass, momentum, and activity for a solid can be accounting as
{ \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} \\ \boldsymbol{\sigma} & = \boldsymbol{\sigma}^T & & \qquad\text{Balance of Angular Momentum} \\ \rho~\dot{e} - \boldsymbol{\sigma}:(\boldsymbol{\nabla}\mathbf{v}) + \boldsymbol{\nabla} \cdot \mathbf{q} - \rho~s & = 0 & & \qquad\text{Balance of Energy.} \end{align} }
In the aloft equations \rho(\mathbf{x},t) is the accumulation physique (current), \dot{\rho} is the actual time acquired of ρ, \mathbf{v}(\mathbf{x},t) is the atom velocity, \dot{\mathbf{v}} is the actual time acquired of \mathbf{v}, \boldsymbol{\sigma}(\mathbf{x},t) is the Cauchy accent tensor, \mathbf{b}(\mathbf{x},t) is the physique force density, e(\mathbf{x},t) is the centralized activity per assemblage mass, \dot{e} is the actual time acquired of e, \mathbf{q}(\mathbf{x},t) is the calefaction alteration vector, and s(\mathbf{x},t) is an activity antecedent per assemblage mass.
With account to the advertence configuration, the antithesis laws can be accounting as
{ \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}^T -\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} }
In the above, \boldsymbol{P} is the aboriginal Piola-Kirchhoff accent tensor, and ρ0 is the accumulation physique in the advertence configuration. The aboriginal Piola-Kirchhoff accent tensor is accompanying to the Cauchy accent tensor by
\boldsymbol{P} = J~\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T} ~\text{where}~ J = \det(\boldsymbol{F})
We can alternatively ascertain the nominal accent tensor \boldsymbol{N} which is the alter of the aboriginal Piola-Kirchhoff accent tensor such that
\boldsymbol{N} = \boldsymbol{P}^T = J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma} ~.
Then the antithesis laws become
{ \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} -\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} }
The operators in the aloft equations are authentic as such that
\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 ~;~~ \boldsymbol{\nabla} \cdot \mathbf{v} = \sum_{i=1}^3 \frac{\partial v_i}{\partial x_i} = v_{i,i} ~;~~ \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 ~.
where \mathbf{v} is a agent field, \boldsymbol{S} is a second-order tensor field, and \mathbf{e}_i are the apparatus of an orthonormal base in the accepted configuration. Also,
\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 ~;~~ \boldsymbol{\nabla}_{\circ}\cdot\mathbf{v} = \sum_{i=1}^3 \frac{\partial v_i}{\partial X_i} = v_{i,i} ~;~~ \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
where \mathbf{v} is a agent field, \boldsymbol{S} is a second-order tensor field, and \mathbf{E}_i are the apparatus of an orthonormal base in the advertence configuration.
The close artefact is authentic as
\boldsymbol{A}:\boldsymbol{B} = \sum_{i,j=1}^3 A_{ij}~B_{ij} = trace(\boldsymbol{A}\boldsymbol{B}^T) ~.
edit Clausius–Duhem inequality
The Clausius–Duhem asperity can be acclimated to accurate the additional law of thermodynamics for elastic-plastic materials. This asperity is a account apropos the irreversibility of accustomed processes, abnormally if activity amusement is involved.
Just like in the antithesis laws in the antecedent section, we accept that there is a alteration of a quantity, a antecedent of the quantity, and an centralized physique of the abundance per assemblage mass. The abundance of absorption in this case is the entropy. Thus, we accept that there is an anarchy flux, an anarchy source, and an centralized anarchy physique per assemblage accumulation (η) in the arena of interest.
Let Ω be such a arena and let \partial \Omega be its boundary. Again the additional law of thermodynamics states that the amount of access of η in this arena is greater than or according to the sum of that supplied to Ω (as a alteration or from centralized sources) and the change of the centralized anarchy physique due to actual abounding in and out of the region.
Let \partial \Omega move with a acceleration un and let particles central Ω accept velocities \mathbf{v}. Let \mathbf{n} be the assemblage apparent accustomed to the apparent \partial \Omega . Let ρ be the physique of amount in the region, \bar{q} be the anarchy alteration at the surface, and r be the anarchy antecedent per assemblage mass. Again the anarchy asperity may be accounting as
\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}.
The scalar anarchy alteration can be accompanying to the agent alteration at the apparent by the affiliation \bar{q} = -\boldsymbol{\psi}(\mathbf{x})\cdot\mathbf{n}. Beneath the acceptance of incrementally isothermal conditions, we have
\boldsymbol{\psi}(\mathbf{x}) = \cfrac{\mathbf{q}(\mathbf{x})}{T} ~;~~ r = \cfrac{s}{T}
where \mathbf{q} is the calefaction alteration vector, s is an activity antecedent per assemblage mass, and T is the complete temperature of a actual point at \mathbf{x} at time t.
We again accept the Clausius–Duhem asperity in basic form:
{ \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}. }
We can appearance that the anarchy asperity may be accounting in cogwheel anatomy as
{ \rho~\dot{\eta} \ge - \boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) + \cfrac{\rho~s}{T}. }
In agreement of the Cauchy accent and the centralized energy, the Clausius–Duhem asperity may be accounting as
{ \rho~(\dot{e} - T~\dot{\eta}) - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T}. }
The antithesis laws accurate the abstraction that the amount of change of a abundance (mass, momentum, energy) in a aggregate accept to appear from three causes:
the concrete abundance itself flows through the apparent that bound the volume,
there is a antecedent of the concrete abundance on the apparent of the volume, or/and,
there is a antecedent of the concrete abundance central the volume.
Let Ω be the physique (an accessible subset of Euclidean space) and let \partial \Omega be its apparent (the abuttals of Ω).
Let the motion of actual credibility in the physique be declared by the map
\mathbf{x} = \boldsymbol{\chi}(\mathbf{X}) = \mathbf{x}(\mathbf{X})
where \mathbf{X} is the position of a point in the antecedent agreement and \mathbf{x} is the area of the aforementioned point in the askew configuration.
The anamorphosis acclivity is accustomed by
\boldsymbol{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \boldsymbol{\mathbf{x}} \cdot \nabla ~.
edit Antithesis laws
Let f(\mathbf{x},t) be a concrete abundance that is abounding through the body. Let g(\mathbf{x},t) be sources on the apparent of the physique and let h(\mathbf{x},t) be sources central the body. Let \mathbf{n}(\mathbf{x},t) be the apparent assemblage accustomed to the apparent \partial \Omega . Let \mathbf{v}(\mathbf{x},t) be the acceleration of the concrete particles that backpack the concrete abundance that is flowing. Also, let the acceleration at which the bonds apparent \partial \Omega is affective be un (in the administration \mathbf{n}).
Then, antithesis laws can be bidding in the accepted form
\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} ~.
Note that the functions f(\mathbf{x},t), g(\mathbf{x},t), and h(\mathbf{x},t) can be scalar valued, agent valued, or tensor admired - depending on the concrete abundance that the antithesis blueprint deals with. If there are centralized boundaries in the body, jump discontinuities aswell charge to be authentic in the antithesis laws.
If we yield the Lagrangian point of view, it can be apparent that the antithesis laws of mass, momentum, and activity for a solid can be accounting as
{ \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} \\ \boldsymbol{\sigma} & = \boldsymbol{\sigma}^T & & \qquad\text{Balance of Angular Momentum} \\ \rho~\dot{e} - \boldsymbol{\sigma}:(\boldsymbol{\nabla}\mathbf{v}) + \boldsymbol{\nabla} \cdot \mathbf{q} - \rho~s & = 0 & & \qquad\text{Balance of Energy.} \end{align} }
In the aloft equations \rho(\mathbf{x},t) is the accumulation physique (current), \dot{\rho} is the actual time acquired of ρ, \mathbf{v}(\mathbf{x},t) is the atom velocity, \dot{\mathbf{v}} is the actual time acquired of \mathbf{v}, \boldsymbol{\sigma}(\mathbf{x},t) is the Cauchy accent tensor, \mathbf{b}(\mathbf{x},t) is the physique force density, e(\mathbf{x},t) is the centralized activity per assemblage mass, \dot{e} is the actual time acquired of e, \mathbf{q}(\mathbf{x},t) is the calefaction alteration vector, and s(\mathbf{x},t) is an activity antecedent per assemblage mass.
With account to the advertence configuration, the antithesis laws can be accounting as
{ \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}^T -\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} }
In the above, \boldsymbol{P} is the aboriginal Piola-Kirchhoff accent tensor, and ρ0 is the accumulation physique in the advertence configuration. The aboriginal Piola-Kirchhoff accent tensor is accompanying to the Cauchy accent tensor by
\boldsymbol{P} = J~\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T} ~\text{where}~ J = \det(\boldsymbol{F})
We can alternatively ascertain the nominal accent tensor \boldsymbol{N} which is the alter of the aboriginal Piola-Kirchhoff accent tensor such that
\boldsymbol{N} = \boldsymbol{P}^T = J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma} ~.
Then the antithesis laws become
{ \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} -\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} }
The operators in the aloft equations are authentic as such that
\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 ~;~~ \boldsymbol{\nabla} \cdot \mathbf{v} = \sum_{i=1}^3 \frac{\partial v_i}{\partial x_i} = v_{i,i} ~;~~ \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 ~.
where \mathbf{v} is a agent field, \boldsymbol{S} is a second-order tensor field, and \mathbf{e}_i are the apparatus of an orthonormal base in the accepted configuration. Also,
\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 ~;~~ \boldsymbol{\nabla}_{\circ}\cdot\mathbf{v} = \sum_{i=1}^3 \frac{\partial v_i}{\partial X_i} = v_{i,i} ~;~~ \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
where \mathbf{v} is a agent field, \boldsymbol{S} is a second-order tensor field, and \mathbf{E}_i are the apparatus of an orthonormal base in the advertence configuration.
The close artefact is authentic as
\boldsymbol{A}:\boldsymbol{B} = \sum_{i,j=1}^3 A_{ij}~B_{ij} = trace(\boldsymbol{A}\boldsymbol{B}^T) ~.
edit Clausius–Duhem inequality
The Clausius–Duhem asperity can be acclimated to accurate the additional law of thermodynamics for elastic-plastic materials. This asperity is a account apropos the irreversibility of accustomed processes, abnormally if activity amusement is involved.
Just like in the antithesis laws in the antecedent section, we accept that there is a alteration of a quantity, a antecedent of the quantity, and an centralized physique of the abundance per assemblage mass. The abundance of absorption in this case is the entropy. Thus, we accept that there is an anarchy flux, an anarchy source, and an centralized anarchy physique per assemblage accumulation (η) in the arena of interest.
Let Ω be such a arena and let \partial \Omega be its boundary. Again the additional law of thermodynamics states that the amount of access of η in this arena is greater than or according to the sum of that supplied to Ω (as a alteration or from centralized sources) and the change of the centralized anarchy physique due to actual abounding in and out of the region.
Let \partial \Omega move with a acceleration un and let particles central Ω accept velocities \mathbf{v}. Let \mathbf{n} be the assemblage apparent accustomed to the apparent \partial \Omega . Let ρ be the physique of amount in the region, \bar{q} be the anarchy alteration at the surface, and r be the anarchy antecedent per assemblage mass. Again the anarchy asperity may be accounting as
\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}.
The scalar anarchy alteration can be accompanying to the agent alteration at the apparent by the affiliation \bar{q} = -\boldsymbol{\psi}(\mathbf{x})\cdot\mathbf{n}. Beneath the acceptance of incrementally isothermal conditions, we have
\boldsymbol{\psi}(\mathbf{x}) = \cfrac{\mathbf{q}(\mathbf{x})}{T} ~;~~ r = \cfrac{s}{T}
where \mathbf{q} is the calefaction alteration vector, s is an activity antecedent per assemblage mass, and T is the complete temperature of a actual point at \mathbf{x} at time t.
We again accept the Clausius–Duhem asperity in basic form:
{ \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}. }
We can appearance that the anarchy asperity may be accounting in cogwheel anatomy as
{ \rho~\dot{\eta} \ge - \boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) + \cfrac{\rho~s}{T}. }
In agreement of the Cauchy accent and the centralized energy, the Clausius–Duhem asperity may be accounting as
{ \rho~(\dot{e} - T~\dot{\eta}) - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T}. }
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