By Dario Graffi
R. Bouc, G. Geymonat: Periodic difficulties in thermoviscoelasticity.- A. Day: The thermodynamics of fabrics with memory.- M. Fabrizio: Sulla nozione di stato nella termomeccanica dei continui.- G. Fichera: Analytic difficulties of hereditary phenomena.- M. Ribaric: The black field process and platforms with memory.- R.S. Rivlin: the idea of constitutive equations.-
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Extra resources for Materials with Memory (C.I.M.E. Summer Schools, 74)
Sample text
We have f i r s t l y : Lemma 4- Let the assumption A1 be satisfied and l e t eo,-s&L; (5). e. t 15. e. ) : Find a E I , q E L i (E) such t h a t In order t o solve t h i s l a s t problem the following l e m a i s basic. ) has a solution a - # L2 (J) t o the equation # E I , q € L2 (E) given by . where c I is arbitrary. 16) O + q, are We can give now the main theorem. 16) has one and only one solution z E L$ ( J ) such t h a t i E L$n- (J). 6 t e r ) has a solution (2 , and a l l the solutions A 4 are E = E + c 5 = 5 + c where c E I is arbitrary'.
7- The periodic bilateral problem f o r a Maxwell body. 1- We consider in t h i s section some questions concerning existence. 12) i n the case of a17 the data are T-periodic in time and with the constitutive equation that we have studied i n 5 6. t) a surfacic 0 density of forces on a0 la,Q with only a tangential component on a2Q(see 55). e. ap . A l l the data are assumed T-periodic in time. 12). 5). 2). 4). 5). we have now t o solve He assume here that the nonlinear heat equation is uncoupled from the motion motion equation.
20) $& -1 c uc. 21) 11 uC(t)ll ( E (x) I t can be proved t h a t meaning. uC, - gc + v 1 exp -k ( 0C ' 11 oC(0)ll E exists and - 7K - ic k L ~ K t a-e. i (E) ] U =~ 0 t b 0 0 and so uc(0) has a 8- The periodic unilateral Signorini problem. 1- We are dealing now with a signorini(*) periodic viscoelastic problem i n the quasi s t a t i c case. e. "contact without friction". 2- Example . Let u s consider a plane medium with a crack (on a4n ). This medium i s submitted t o T-periodic forces or stresses of mechanical o r thermal nature.