degrees of freedom. That means, in contrast to classicalThat means, in - PDF document

degrees of freedom. That means, in contrast to classicalThat means, in contrast to classical)micromorphic material particles undergo an additional. The indices of therespectively, of the material body.micromorphic continuabut, due to its complexity, the practical usefulness ofthis theory is inversely proportional to its generality.Aslightly more special case is that of microstretch con-microstretch continua. That means, particles of micro-micropolar continuais the Kronecker delta. That means, onlyThat means, onlyIn micropolar theory the director may be envisaged asan orthogonal tripod circumscribed by a unit sphere,centered within an anisometric particle, e.g. a triaxialellipsoid. For rotationally symmetric material particles,e.g. spheroids (biaxial ellipsoids) or cylinders, one-index quantities, e.g. Ericksen's orientational vectors[17] would suffice to determine the orientation.With the deformation function , (3)respectively, where the referential gradient of the defor-. (4)derivatives. The quantity dettion (1). This is in complete analogy with the fact that, (5)micromotions. (7)deformation tensors (strain measures)the wryness tensor WW)(8)and, (9)respectively, where (third-order permutation pseudotensor). These deforma-measures)es))(10)and , (11)crogyration vectorits gradient. These deforma-Micropolar materials Ceramics 49(3) 170-180 (2005)171 tions of objectivity the reader should consult Eringen'sbooks [1,2].Since for micropolar continua (in contrast to micro-morphic and microstretch continua) the directors arerigid, it is possible to represent their micromotion as arigid body rotation (with an angle of rotation ) withrespect to an axis nk. According to Eringen's theoremAccording to Eringen's theorem)()microrotation tensorangle of rota-, (12)become Kronecker deltas. Similarly, the microgyra-rate of rotationotationp. 28 and [2], p. 8)k= nk+ sinnk+ (1 - cos)(n×n)k. (13). (14)BALANCE LAWS OF MICROPOLAR THEORYbalance laws applying to micropolar continua= 0 (15)= 0 (16)) = 0 (17)) = 0 (18)5. Balance of energy (first law of thermodynamics):= 0 (19). (20)ty, the velocity vector, microinertia tensorthe microgyration vector, the stress tensor, couple stress tensor, . (21). Further, internal energy per unit mass, the heat flux vector, the absolute temperature. These balancesolid, fluid or other. The balances of mass, linearangular momentum and energy extra terms appear inry (rational thermomechanics). The balance of microin-not appear in the classical theory. Preliminary versions. Preliminary versions(1909) work of the Cosserat brothers [20]. The latterwork, however, does not define a specific spin inertiamicroinertia tensor. For the balance laws at discontinu-. For the balance laws at discontinu-concrete engineering problems appropriate constitutiveequations (corresponding to the material model chosen)should be inserted into the balance laws. As in classicalcontinuum theory, this results in field equations whichprocess and geometry in question. The boundary condi-[1,2]. The main task for materials scientists and engi-neers, however, is the proper selection and implementa-CONSTITUTIVE EQUATIONS OF MICROPOLARfree energy (Helmholtz potential)Pabst W. 172Ceramics 49(3) 170-180 (2005) { isotropic solids= 0 (44)= 0 . (48)linear constitutive equations of isotropicmicropolar thermoelastic solids. (49). (50). (51). (52)The free energy (Helmholtz potential) is in thissimply due to reasons of symmetry. Thus, linear isotrop-ature coefficient (thermal expansion coefficient) (gradient) materials. They are constants only for homo-, (54)microisotropic or spin-isotropic= 0 . (55)= 0 . (55), (56), (57), (58), (59), (60), (61), (62). (63)Moreover, the requirement of non-negative kineticenergy [1],, (64), (65) . (66)Pabst W. 174Ceramics 49(3) 170-180 (2005) = 1). Their value for the micropolar tensile modulus(Young's modulus),, (90)bone. Yang and Lakes' investigation showed, inter alia,determined stiffness than the prediction of Toupin's [28]and Mindlin's and Thiersten's [29] couple stress theoryThiersten's [29] couple stress theoryolar theory to bone can be found e.g. in [31,32]. For fur-ther applications of micropolar theory to solids, includ-ing the modelling of grain size effects and wave propa-fects and wave propa-ences cited therein.The theory of micropolar fluidsfluid mechanics. Among these are liquid crystals and. While the first is beyondthe scope of this paper, we mention a few facts about thelatter. In classical fluid mechanics, the generation ofvortices is attributed to viscous friction. When there isthe fluid has no vortices. The process of vortex creationThe process of vortex creationsource for the creation of vortices. For the completemicropolar description of turbulence and many otherapplications to problems of micropolar fluid mechanicsthe reader should consult [2,11,39-41] and many papers1,39-41] and many paperstions such as flow in a non-coaxial plate-plate rheome-ter, lubrication problem (generalized Reynolds equa-tion), Stokes flow about a sphere, stagnation flow, Tay-lor-Bénard instability, boundary layer flow over a plate,, boundary layer flow over a plate,lar fluid lubrication with reference to human joints isdiscussed in [47,48]. Trivially, the theory of micropolaras the classical theory of Newtonian fluids (Navier-as mentioned above. Clearly, it must remain a desidera-, it must remain a desidera-done in [49]. It is evident, however, that Eringen's theo-than any other present theory in this field. Therefore itis highly probable that micropolar theory, when formu-lated in sufficient generality, can account for the essen-, can account for the essen-importance would be a rational answer to the question,in what way the microstructure tensors (orientationaltensors) occurring in the latter theories are related to thetensors occurring in the micropolar theory.attention and cannot be treated in a standard way. For. For, (91)order tensor, related to the evolution of fluid (suspen-sion) microstructure during flow. In the case of aniso-tation can change during flow, leading to changes in themetry. Eringen's argument for introducing this funda-velocity difference between microstructure (suspendedbalance law.) , (92)) , (92)kl]nl= 0 , (93)= 0 , (93)kl]are the symmetric and antisymmetric parts of aij,respectively.rigid sphere suspensions, (94)Micropolar materials Ceramics 49(3) 170-180 (2005)177 trast to usual tensors, but similar to the deformation gradient,directors correspond to a mapping between two differentferent2In order to retain as far as possible the structure of classicalthermodynamics, the temperature change rate is omitted apriori from the set of independent variables of the dissipationfunction here, in contrast to [1,2].3This enumeration combines two different logical categories,materials (such as "concrete with sand"). Further, some cate-not in another.This work was part of the project "Mechanics andthermomechanics of disperse systems, porous materialsand composites", (Grant Agency of the Czech Republicacknowledged. 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