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The Clausius–Duhem inequality is a way of expressing the
second law of thermodynamics The second law of thermodynamics is a physical law based on Universal (metaphysics), universal empirical observation concerning heat and Energy transformation, energy interconversions. A simple statement of the law is that heat always flows spont ...
that is used in
continuum mechanics 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. Continuum mec ...
. This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable.. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved. It was named after the German physicist
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Sadi Carnot's principle ...
and French physicist
Pierre Duhem Pierre Maurice Marie Duhem (; 9 June 1861 – 14 September 1916) was a French theoretical physicist who made significant contributions to thermodynamics, hydrodynamics, and the theory of Elasticity (physics), elasticity. Duhem was also a prolif ...
.


Clausius–Duhem inequality in terms of the specific entropy

The Clausius–Duhem inequality can be expressed in
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
form as : \frac\left(\int_\Omega \rho \eta \, dV\right) \ge \int_ \rho \eta \left(u_n - \mathbf\cdot\mathbf\right) dA - \int_ \frac~ dA + \int_\Omega \frac~dV. In this equation t is the time, \Omega represents a body and the integration is over the volume of the body, \partial \Omega represents the surface of the body, \rho is the
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
of the body, \eta is the specific
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
(entropy per unit mass), u_n is the normal velocity of \partial \Omega, \mathbf is the
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
of particles inside \Omega, \mathbf is the unit normal to the surface, \mathbf is the
heat In thermodynamics, heat is energy in transfer between a thermodynamic system and its surroundings by such mechanisms as thermal conduction, electromagnetic radiation, and friction, which are microscopic in nature, involving sub-atomic, ato ...
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phe ...
vector, s is an
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
source per unit mass, and T is the absolute
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
. All the variables are functions of a material point at \mathbf at time t. In differential form the Clausius–Duhem inequality can be written as : \rho \dot \ge - \boldsymbol \cdot \left(\frac\right) + \frac where \dot is the time derivative of \eta and \boldsymbol \cdot (\mathbf) is the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
of the
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
\mathbf.


Clausius–Duhem inequality in terms of specific internal energy

The inequality can be expressed in terms of the
internal energy The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...
as : \rho~(\dot - T~\dot) - \boldsymbol:\boldsymbol\mathbf \le - \cfrac where \dot is the time derivative of the specific internal energy e (the internal energy per unit mass), \boldsymbol is the Cauchy stress, and \boldsymbol\mathbf is the
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
of the velocity. This inequality incorporates the balance of energy and the balance of linear and angular momentum into the expression for the Clausius–Duhem inequality.


Dissipation

The quantity : \mathcal = \rho~(T~\dot-\dot) + \boldsymbol:\boldsymbol\mathbf - \cfrac \ge 0 is called the
dissipation In thermodynamics, dissipation is the result of an irreversible process that affects a thermodynamic system. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to a final form, wh ...
which is defined as the rate of internal
entropy production Entropy production (or generation) is the amount of entropy which is produced during heat process to evaluate the efficiency of the process. Short history Entropy is produced in irreversible processes. The importance of avoiding irreversible p ...
per unit volume times the
absolute temperature Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion. Thermodynamic temperature is typically expres ...
. Hence the Clausius–Duhem inequality is also called the dissipation inequality. In a real material, the dissipation is always greater than zero.


See also

*
Entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
*
Second law of thermodynamics The second law of thermodynamics is a physical law based on Universal (metaphysics), universal empirical observation concerning heat and Energy transformation, energy interconversions. A simple statement of the law is that heat always flows spont ...


References


External links


Memories of Clifford Truesdell
by Bernard D. Coleman, Journal of Elasticity, 2003.
Thoughts on Thermomechanics
by
Walter Noll Walter Noll (January 7, 1925 June 6, 2017) was a mathematician, and Professor Emeritus at Carnegie Mellon University. He is best known for developing mathematical tools of classical mechanics, thermodynamics, and continuum mechanics. Biography B ...
, 2008. {{DEFAULTSORT:Clausius-Duhem inequality Continuum mechanics