The Micropolar Thermoelasticity
Thermoelasticity investigates the. interaction of the field of deformation with the field of temperature and combines, on the basis of the thermodynamics of the irreversible processes, two separately developing branches of science, namely the theory of el
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1. Introduction
Thermoelasticity investigates the .interaction of the field of deformation with the field of temperature and c~mbines,
on the basis of the thermodynamics of the irrevers-
ible proces ses, two separately developing branches of science, namely the theory of elasticity and the theory of heat conduction. At the present moment, after 20 years of the development, the thermoelasticity of the Hooke's continuum is fully formed. The fundamental assumptions have been worked out [1] - [5], the fundamental relations and different equations have been elaborated, the fundamental energy and variational theorems obtained. The entire classical thermoelasticity has been formulated in a number of monographs. On the background of the development of the classical thermoelasticity the achievements of Cosserat's continuum thermo elasticity
[6 - 9J , are still modest. Though all
more important theorems have been derived, the domain of
W. Nowacki et al. (eds.), Micropolar Elasticity © Springer-Verlag Wien 1972
106
W. Nowacki "The Micropo1ar Thermoe1asticity"
the particular solutions is incomparably smaller. The fundamentals of the micropo1ar, Cosserats' thermoe1asticity were formulated in 1966 by the author of the present study [10], [11]. We present, in a concise form, the fundamental relations and the fundamental equations of Cosserats' continuum thermo elasticity. The principle of the energy conservation and the entropy balance are our point of departure
!!.J[..!..."" 0'0', 'IT, + Iww) + UJdN =fe x.v: +'('UJ'.) d.V + dt 2,""-Y"· "" ~" ~ " (1.1)
v
+
jc p~""~ + m~'UTi,) d,A - 1q ~ni.d.A v
~
A
and
(1.2) In eq. (1.1) U denotes the internal energy referred to the unit of volume,
X~, y~
are the components of the body forces and mo-
ments acting on the surface A bounding the body,
U,L'
CPL denote
the components of the displacement vector and rotation vector, respectively, u.~=",,~~ cp~=ur~ are their time derivatives,
'1
is the
flux of heat vector, ~ the density, I the rotational inertia. The term on the left-hand side of eq. (1. 1) represents the time change of the internal and kinetic energies. The first term on the right hand side of the equation is the power of body forces and body moments, the second term is the ~ower
of traction and surface moments. The last term expres-
ses the amount of heat transmitted into the volume V by the
The Principle of the Energy Conservation. Entropy Balance 107 heat conduction. The left-hand side of the balance of entropy equation (1.2) represents the increase of entropy. The first term on the right-hand side is the increase of entropy due to the exchange of the entropy with environment, the second term expresses the production of entropy generated by heat conduction. Here
e n·T \!:!I=-¥" > 0,
according to the postulate of the therm£.
dynamics of irreversible processes. In eq. (1.2) S denotes the entropy referred to the unit of volume, T
is the absolute temperature,
e
is the
source of entropy. Transforming eqs. (1.1) and (1.2) by means of the equation of mo
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