Clausius-Duhem inequality

Rational thermodynamics is formulated based on the following hypotheses (i) absolute temperature and entropy are not limited to near-equilibrium situations, (ii) it is assumed that systems have memories, their behavior at a given instant of time is determined by the history of the variables, and (iii) the second law of thermodynamics is expressed in mathematical terms by means of the Clausius-Duhem inequality. The balance equations were combined with the Clausius-Duhem inequality by means of arbitrary source terms, or by an approach based on Lagrange multipliers. 

The Clausius-Duhem equation is the fundamental inequality for a single-component system. The selection of the independent constitutive variables depends on the type of system being considered. A process is then described by solving the balance equations with the constitutive relations and the Clausius-Duhem inequality. 

The dependence of 4> is expressed by ordinary functions instead of functionals. WithEq. (14.57) the Clausius-Duhem inequality becomes... 

As discussed in Ref [39] the following results [Eqs. (15)-(17)] are obtained from the evaluation of the Clausius-Duhem inequality if additionally pk is assumed to be constant ... 

By Gauss theorem and (3.68) we obtain entropy inequality in the local form called the Clausius-Duhem inequality... 

Finally, we can eliminate q and Q from energy balance (3.107) and Clausius-Duhem inequality (3.109) and use the following definition of the specific free energy f and the temperature gradient g... 

Edelen, D.G.B. The Clausius-Duhem inequality, time reversal, and the onsager reciprocity relations. Rec. Adv. Engng. Sci. 6, 129-133 (1975)... 

When a singular surface involves field variables that are affected by the motion and deformation of the medium, the geometrical and kinematical compatibility conditions should be supplemented by restrictions originating from the local balance equations. These conditions are called the dynamical conditions of compatibility. The dynamical conditions of compatibility are due to the local conservation of mass, balance of linear and angular momenta, balance of energy, and the local Clausius-Duhem inequality on ait). 

Where U(Vi ) represents the trapped energy due to hardening, depending only on the internal state parameters Vj. Substituting this into the Clausius-Duhem inequality and simplifying leads to ... 

The relation (3.11), which is equivalent to the classical Clausius-Duhem inequality, implies that the dissipative part of the internal work due to stress is converted into heat. This suggests that the irreversible process of classical thermodynamics gives a conversion of energy from one form to another. The result is evident, because its framework is built on a potential theory. 

In fact, as mentioned in Sect. 3.1.3, the CIT theory introduces this entropy inequality (3.63), which is known as the Clausius-Duhem inequality. A similar situation to that described for the case of a viscous fluid occurs, for example, in heat engines, where the energy loss due to friction between metals is difficult to quantify. 

As we did for the case of the First Law of Thermodynamics we can obtain the Second Law of Thermodynamics (Clausius-Duhem inequality) in a Lagrangian description. The entropy density in a Lagrangian description can be written as... 

The essentials of the constitutive theory are that the experimental results are reproduced after satisfying the requirements of thermodynamics such as the Clausius-Duhem inequality and the Gibbs-Duhem relation. The constitutive theory that includes chemical processes is complex because all the conservation laws for mass, linear momentum/moment of momentum and energy are involved, and experiments to determine the parameters are extremely difficult to conduct. 

Note D.3 (Internal dissipation and Clausius-Duhem inequality). The Second Law of Thermodynamics is considered in terms of the dissipation of internai structure. Referring to the definition of entropy (D.IO) and its effects for a reversible process... 

It should be noted that the Clausius-Duhem inequality, giving the condition of internal dissipation, is exclusively satisfied for the equality part by the reversible process and for the pure inequality part by the irreversible process. The equality of the non-negative condition is not satisfied for the pure irreversible process . ... 

This gives the Eulerian form of the Second Law of Thermodynamics for the continuum, also referred to as the Clausius-Duhem inequality. Note that the first term of the r.h.s. of (D.76) is a consequence of the internal dissipation in mechanical energy. It should also be noted that the equality is provided for reversible processes whereas the pure inequality is applicable to irreversible processes. 

Constitutive laws for viscoelastic materials are however more complex than the Hooke s law and consequently the number of parameters to identify increases. Frameworks could also be used to ensure that all parameters of the proposed laws are physically admissible. To do so, it is possible to use the thermodynamic of irreversible processes as the framework. Based on the concepts of continuum mechanics and irreversible thermodynamics, the Clausius-Duhem inequality is obtained for given problems where dissipation mechanisms are of importance, e.g., viscous deformation. Fundamental equations leading to a generic form of the Clausius-Duhem inequality have been well covered by many authors (Bazarov, 1964 Coussy, 2010 Lemaitre and Chaboche, 1990 Mase and Mase, 1999) and thus will be only summarized later in section 3. Based on the generic form of the Qausius-Duhem inequality, models or constitutive laws are further developed considering various assumptions closely related to materials of interest. 

Finally, combining the two equations obtained with the first and second principles, i.e., equations (17) and (21), and the Helmholtz free energy (9), a general form of the Clausius-Duhem inequality is obtained ... 

The inequality (24) must be satisfied for any thermodynamic process. According to the local state law (Lemaitre and Chaboche, 1990), the Clausius-Duhem inequality leads to the following state equation ... 

The parameters of this tensor must be identified, as well. Based on the thermodynamics of irreversible processes (TIP) and on the choice of the internal state variables (ISV), the Clausius-Duhem inequality can be written as ... 

The Clausius-Duhem inequality asserts that the rate of the entropy production is not negative ... 

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