By Gianfranco Capriz
This e-book proposes a brand new normal surroundings for theories of our bodies with microstructure after they are defined in the scheme of the con tinuum: along with the standard fields of classical thermomechanics (dis placement, pressure, temperature, etc.) a few new fields input the image (order parameters, microstress, etc.). The e-book can be utilized in a semester direction for college kids who've already lectures at the classical idea of continua and is meant as an creation to big themes: fabrics with voids, liquid crystals, meromorphic con tinua. actually, the content material is largely that of a chain of lectures given in 1986 on the Scuola Estiva di Fisica Matematica in Ravello (Italy). i need to thank the clinical Committee of the Gruppo di Fisica Matematica of the Italian nationwide Council of analysis (CNR) for the invitation to educate within the tuition. I additionally thank the Committee for arithmetic of CNR and the nationwide technology starting place: they've got supported my examine over a long time and given me the chance to check the subjects provided during this booklet, particularly via a USA-Italy software initiated by means of Professor Clifford A. Truesdell. My curiosity within the box dates again to a interval of collaboration with Paolo Podio-Guidugli and a few of the elemental rules got here up in the course of our discussions.
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Additional resources for Continua with Microstructure
H. §11. The Conservative Case in Statics In the static and conservative case, the balance equations proposed in Sections 8 and 9 can be deduced also from a variational principle. E through F, v, Grad v only. E let us assume, as already done in earlier Sections, that body actions f, JJ and contact actions t, (1 through the boundary are present. 3). E under the given body and surface actions. H. We define the global virtual work of the external actions as the quantity r f p*(f' 15x + JJ. 15 v) + (t* .
12) can be written a a div(sym T + dived ® S». 13) We pass on now to some developments which are a necessary premise to the study of uniaxial liquid crystals; precisely we examine the consequences of the introduction of the constraint Idl = 1 for d. The constraint implies that micromotions belong locally to the class of rigid rotations. Hence we can adapt to the special case certain §19. 7). 3), in our case the Cauchy stress is expressed again by the sum T = sym a T a a + skw(d ® z + (grad d)ST).
Then recourse must be made to results of Section 14, preserving, however, for the active components of stress and micro~ the hypothesis of their dependence on a potential. The stress constraint we want to introduce is the coincidence of the values of v and 1, so that the function W(l) of Section 14 is the trivial one W(l) = 1. 16), we recognize that it must be t" T = sym Ta - a] P- (ax)" aV + ax) aV + div sa - , 1 [P( 1. 3) We come now to the constitutive prescription of the active components. The potential qJ is a function of 1 and grad 1 (which is exactly the property assumed in Section 12, when we adapt it to the present circumstances) and thus the following relation must apply pcp = p(~~ )i + p(a(g~:d I)} (grad It = T· D + (i + ~·(grad i).