Geodesic
Thermodynamics of monopolar continuum of grade $n$
Applications of Mathematics, Tome 16 (1971) no. 5, pp. 370-383
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Contrary to the theory of the simple material it is assumed that the values of physical quantities at a point are arrected by the deformation history of a finite neighborhood of the point. In the case of the monopolar continuum of grade $n$, the physical quantities are in a functional dependence on the temperature and, moreover, on 4n$ deformation gradients which are found from a single shift function. Equations of equilibrium and the boundary values for all $n$ stress tensors are evaluated on the basis of the First Law of Thermodynamics. Introducing the Hilbert space with the norm which expresses the fading of the memory it is possible to derive the system of constitutive equations from the Second Law of Thermodynamics. These equations enable us to evaluate the entropy as well as all stress tensors provided the functional dependence of the free energy on the history of $n$ deformation gradients and on the history of temperature is given.
Contrary to the theory of the simple material it is assumed that the values of physical quantities at a point are arrected by the deformation history of a finite neighborhood of the point. In the case of the monopolar continuum of grade $n$, the physical quantities are in a functional dependence on the temperature and, moreover, on 4n$ deformation gradients which are found from a single shift function. Equations of equilibrium and the boundary values for all $n$ stress tensors are evaluated on the basis of the First Law of Thermodynamics. Introducing the Hilbert space with the norm which expresses the fading of the memory it is possible to derive the system of constitutive equations from the Second Law of Thermodynamics. These equations enable us to evaluate the entropy as well as all stress tensors provided the functional dependence of the free energy on the history of $n$ deformation gradients and on the history of temperature is given.
DOI : 10.21136/AM.1971.103368
Classification : 74A15 
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Bucháček, Karel. Thermodynamics of monopolar continuum of grade $n$. Applications of Mathematics, Tome 16 (1971) no. 5, pp. 370-383. doi: 10.21136/AM.1971.103368

[1] C. Truesdell R. A. Toupin: The Classical Field Theories, Handbuch der Physik. III/1 (1960), Springer Verlag, Berlin. | MR

[2] C. Truesdell W. Noll: The Non-Linear Field Theories of Mechanics, Handbuch der Physik. III/3 (1965), Springer Verlag, Berlin. | MR

[3] B. Coleman: Thermodynamics of Materials with Memory. Arch. Rational Mech. Anal., 17 (1964), 1 - 46. | DOI | MR

[4] B. Coleman: On Thermodynamics, Strain Impulses and Viscoelasticity. Arch. Rational Mech. Anal., 17 (1964), 230-254. | DOI | MR | Zbl

[5] A. E. Green R. S. Rivlin: Multipolar Continuum Mechanics. Arch. Rational Mech. Anal., 17 (1964), 113-147. | DOI | MR

[6] B. T. Koйтep: Моментные напряжения в теории упругости. Механика, 91 (1965), 89-112.

[7] A. E. Green R. S. Rivlin: Simple Force and Stress Multipoles. Arch. Rational Mech. Anal., 16 (1964), 325-353. | DOI | MR

[8] A. E. Green P. M. Naghdi: A General Theory of an Elastic Plastic Continuum. Arch. Rational Mech. Anal., 18 (1965), 251 - 281. | DOI | MR

[9] P. Perzyna: Teoria lepkoplastyczności. Panstwowe wydawnictwo naukowe, Warszawa 1966.

[10] R. D. Midlin: Micro-structure in Linear Elasticity. Arch. Rational Mech. Anal., 16 (1964), 15-78. | MR

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