By Gianfranco Capriz

This e-book proposes a brand new normal environment 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, rigidity, temperature, etc.) a few new fields input the image (order parameters, microstress, etc.). The ebook can be utilized in a semester path for college students who've already lectures at the classical thought of continua and is meant as an advent to important issues: fabrics with voids, liquid crystals, meromorphic con­ tinua. in truth, the content material is basically 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 study (CNR) for the invitation to coach within the college. I additionally thank the Committee for arithmetic of CNR and the nationwide technological know-how origin: they've got supported my learn over a long time and given me the chance to review the themes provided during this ebook, specifically via a USA-Italy application 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 fundamental principles got here up in the course of our discussions.

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11) is the only relevant equation to be added to the balance equation of Cauchy and in fact it can take up the role of equation of balance of moment of momentum. 12) In this case it is enough that d symmetric. T 9" have null divergence for T to be §1O. 1) they do not involve ~ at all. 2) where 'I is the coefficient of surface expansion '1:= lIF-Tn*1 = 1/IFTnl. 1) are Pn* = 'It, ~n* = 'I (1. 4) 26 I. 1) seems often hard to impose. The experiments in special cases (on liquid crystals, for instance) suggest that "conditions of place" are more significant; even more appropriate at times are consequences of constitutive boundary conditions.

X 2+ K») = 1 p(f' x + P' v) -1 (T' grad + t~ (t· x + 0"' v) x + ~'V + y. grad v). 1) The identification of terms in this equation is as follows. On the left-hand side the time-rate of change of the total kinetic energy appears; on the right-hand side the power of external body actions and the power of external surface actions appear in succession. Thus the last integral must be interpreted as the power of internal actions; its density is -(T' grad x + ~. v + y. grad v). 2) identically zero along any rigid virtual motion.

2) For the other constitutive choices it is necessary to explore in detail, case by case, the behaviour of the material element:- We show below how one can express X(v, v) in the case one wants to study continua with voids, using v to measure the void fraction, when the lacunae are so finely dispersed that the continuum model still makes sense. Then v is a real number belonging to a subinterval of [0, 1J and its measure is obviously not influenced by rotations of the body. The element of a continum is imagined as a small ball with a concentric spherical cavity as in the first example in Section 7; as remarked there each element has volume equal to the reciprocal of the number density v of the lacunae, and the volume of each lacuna is vivo No diffusion of lacunae through the matrix is envisaged and the only motion allowed within the element is an expansion/contraction of the lacuna and corresponding radial motions of the spherical crust.

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