Abstract
Thermodynamics includes a theoretical and an applied part. The applied thermodynamics aims for calculating the temperature distribution in a continuum body. We will study this approach for macroscopic and microscopic systems. The difference between macroscopic and microscopic systems relies on the used constitutive equation. The theoretical thermodynamics has the goal of defining constitutive (material) equations that close the balance equations. By using thermodynamics we will derive the constitutive equations necessary in the computational reality. In the preceding chapter we have employed many constitutive equations with an ad-hoc method. In this chapter we will answer the question of how to derive these equations in a thermodynamically consistent manner. We will analyze such an approach and derive the Navier–Stokes–Fourier equations for a viscous fluid. We will employ the same method for viscoelastic materials and then for plastic deformations. Much use of the method will be made in the next chapter, too.
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- 1.
Density means per volume.
- 2.
Specific means per mass.
- 3.
A laser (Light Amplification by Stimulated Emission of Radiation) generates a focused beam of photons in the same wavelength (coherent).
- 4.
It is named for Carl Friedrich Gauß.
- 5.
The constitutive equation is named after Jean-Baptiste Joseph Fourier.
- 6.
Constant temperature of the surroundings is a warm bath idealization. Suppose that there is so much water in a bath; a heat exchange with the body within the bath does not affect the temperature of the bath, at all. The water on the surface of the body remains at the same constant temperature all the time.
- 7.
It is named after Victor Gustave Robin.
- 8.
We apply a Dirichlet condition strongly by exchanging the solution with the given solution by using “DirichletBC” in the code. Instead of this method, we can satisfy the condition weakly by writing it under the boundary integral.
- 9.
An adiabatic boundary prevents heat transfer across boundaries.
- 10.
See [16] for some interesting explanations on the characteristics of the heat propagation.
- 11.
See, for example, [18].
- 12.
It is named for Carlo Cattáneo and Pierre Vernotte.
- 13.
For some intuitive explanations and examples of the thermal radiation, see [2, Chap. II, Sect. 9–4].
- 14.
The law is named after Josef Stefan and Ludwig Boltzmann.
- 15.
- 16.
- 17.
Under the assumption that no phase changes occur.
- 18.
Of course they are coupled, however, independent. We can hold the temperature fixed and move the body or restrict any motion and change the temperature.
- 19.
The formulation holds for fluids with elasticity, too. Therefore, we need to introduce,
, for a fixed frame, \(w_i=0\). The proof of this identity is out of scope, therefore, we explain it in Appendix A.4 on p. 301. - 20.
For another, more conventional derivation, see [1, Sect. 3].
- 21.
Since we use a Euclidean transformation to test the objectivity, a constant velocity is accepted, too. Consider a rigid body moving with a constant velocity, it actually rests in a coordinate system moving with this velocity. Hence, we can always introduce a constantly moving coordinate system, which is allowed in the Euclidean transformation, where the body rests.
- 22.
For instance in [14] the internal energy is introduced as a full recoverable quantity such that the first integral is automatically justified. Either we can accept the axiom of existence of \(\, {\mathrm d}u\) as the 1st law, or the assumption that the internal energy is fully recoverable as the 1st law.
- 23.
For a brief explanation of this deficiency, we refer to [13, Chap. 2].
- 24.
Since state variables are derived from the primitive variables we can also name them as primary variables.
- 25.
This assumption is another weak point in the methodology, the 1st law of thermodynamics only states that the internal energy has a perfect differential, \(\, {\mathrm d}u\), but not the free energy. Concretely, we have to make sure that \(\, {\mathrm d}(T\eta )\) exists.
- 26.
Although this is mathematically obvious that the dual variables have to depend on the same set of arguments of the free energy, namely on the state variables, this condition is called the equipresence principle, see [17, Sect. 293.\(\eta \)].
- 27.
See [10] for a detailed explanation about the Muller-calorimeter named after F. Horst Müller.
- 28.
It is named after James Clerk Maxwell.
- 29.
Nematic fluids used in LCD (Liquid Crystal Display) is a prominent polar material.
- 30.
It is named after Pierre Curie.
- 31.
For the sake of clarity, the coefficients are functions of the invariants of thermodynamical forces.
- 32.
- 33.
In a fixed domain we simply write \(\frac{\, {\mathrm d}(\cdot )}{\, {\mathrm d}t}\) instead of
and obtain the balance equations for open systems. - 34.
A material system is a closed system possessing the same particles over time. In a material system no (mass) convection is allowed.
- 35.
For an alternative derivation of Gibbs’s equation we refer to [12, Chap. 8].
- 36.
Mathematicians call this transformation a Legendre transformation named after Adrien-Marie Legendre.
- 37.
We present a monolithic approach; however, many commercial codes still use a staggered schema. A staggered schema solves the field equations subsequently such that the results from each solution are used in the subsequent field equation. Such an approach is used in Sect. 1.9, where the balance equations are solved subsequently.
- 38.
We also use a linear strain measure, \(\varepsilon _{ij}\), instead of \(E_{ij}\) in order to attain an identical formulation for plasticity as given in the literature.
- 39.
It is named after Jean-Marie Constant Duhamel and Franz Ernst Neumann.
- 40.
For the inverse of a tensor of rank four we need an identity tensor of rank four. This method can be challenging. Instead of that, the inverse is found by using the Voigt notation. For linear materials we can always rewrite the compliance tensor in the Voigt notation, which is a \(6\times 6\) matrix and its inverse is easy to determine. From the resulting \(6\times 6\) matrix in the Voigt notation, the stiffness tensor is obtained.
- 41.
We use H for the plastic modulus instead of h as in Sect. 1.6 since we have started to use h for the convective heat transfer coefficient in the mixed boundary conditions for temperature.
, for a fixed frame, 
and obtain the balance equations for open systems.References
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Abali, B.E. (2017). Thermodynamics. In: Computational Reality. Advanced Structured Materials, vol 55. Springer, Singapore. https://doi.org/10.1007/978-981-10-2444-3_2
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