Summary
The theory of thermomechanical behavior of a linear viscoelastic Cosserat continuum is developed by making use of the fundamental concepts of continuum mechanics and irreversible thermodynamics. It is shown that, under conditions of arbitrary temporal and spatial variation of the temperature field, the stress tensor, the couple stress tensor and entropy density are derivable from a potential which is the free energy density Ψ.
Under isothermal conditions, it is found that six relaxation functions are needed to describe the linear mechanical behavior of a centro-symmetric isotropic viscoelastic Cosserat materials in a small deformation field, as opposed to two relaxation functions which describe the mechanical behavior of a classical isotropic viscoelastic materials. If a material element is intrinsically constrained such that the geometrically independent rotation vector is identified as the deformation induced rotation vector, then the independent relaxation functions are reduced from six to four. Finally, the explicit form of the linear thermo-mechanical constitutive equations under time varying, spatially inhomogeneous temperature field are derived.
Zusammenfassung
Die Theorie des thermomechanischen Verhaltens eines linear-viskoelastischen Cosserat-Kontinuums wird unter Verwendung der grundlegenden Begriffe der Kontinuumsmechanik und der irreversiblen Thermodynamik entwickelt. Es wird gezeigt, daß für eine beliebige zeitliche und örtliche Variation des Temperaturfeldes der Spannungstensor, der Momentenspannungstensor und die Entropiedichte von einem Potential, der Dichte Ψ der freien Energie, abgeleitet werden können.
Für isotherme Bedingungen zeigt sich, daß sechs Relaxationsfunktionen notwendig sind, um das lineare mechanische Verhalten eines vollisotropen viskoelastischen Cosserat-Materials für kleine Deformationen zu beschreiben; im Gegensatz zu zwei das mechanische Verhalten beschreibenden Relaxationsfunktionen des klassischen isotropen viskoelastischen Materials. Ist die Bewegungsfreiheit des materiellen Elements derart eingeschränkt, daß der geometrisch unabhängige Rotationsvektor dem durch die Deformation bedingten Rotationsvektor gleichgesetzt wird, so reduziert sich die Zahl der unabhängigen Relaxationsfunktionen von sechs auf vier. Abschließend werden die linearen thermomechanischen Grundgleichungen für ein zeitlich veränderliches, örtlich inhomogenes Temperaturfeld in expliziter Form hergeleitet.
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Peng, T.J., Valanis, K.C. Thermomechanical behavior of viscoelastic polar materials. Acta Mechanica 11, 203–216 (1971). https://doi.org/10.1007/BF01176556
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DOI: https://doi.org/10.1007/BF01176556