Mechanical models of residually stressed fibre-reinforced solids,which do not resist bending,have been developed in the literature.However,in some residually stressed fibre-reinforced elastic solids,resistance to fibr...Mechanical models of residually stressed fibre-reinforced solids,which do not resist bending,have been developed in the literature.However,in some residually stressed fibre-reinforced elastic solids,resistance to fibre bending is significant,and the mechanical behavior of such solids should be investigated.Hence,in this paper,we model the mechanical aspect of residually stressed elastic solids with bending stiffness due to fibre curvature,which up to the authors’knowledge has not been mechanically modeled in the past.The proposed constitutive equation involves a nonsymmetric stress and a couple-stress tensor.Spectral invariants are used in the constitutive equation,where each spectral invariant has an intelligible physical meaning,and hence they are useful in experiment and analysis.A prototype strain energy function is proposed.Moreover,we use this prototype to give results for some cylindrical boundary value problems.展开更多
We introduce the notion of property(RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S~l(G) of rapidly decreasing functions on G with respect to a continuous length function l is a...We introduce the notion of property(RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S~l(G) of rapidly decreasing functions on G with respect to a continuous length function l is a dense spectral invariant and Fréchet *-subalgebra of the reduced groupoid C~*-algebra C~*(G) of G when G has property(RD) with respect to l, so the K-theories of both algebras are isomorphic under inclusion. Each normalized cocycle c on G, together with an invariant probability measure on the unit space G~0 of G, gives rise to a canonical map τon the algebra C(G) of complex continuous functions with compact support on G. We show that the map τcan be extended continuously to S~l(G) and plays the same role as an n-trace on C~*(G) when G has property(RD) and c is of polynomial growth with respect to l, so the Connes’ fundament paring between the K-theory and the cyclic cohomology gives us the K-theory invariants on C~*(G).展开更多
文摘Mechanical models of residually stressed fibre-reinforced solids,which do not resist bending,have been developed in the literature.However,in some residually stressed fibre-reinforced elastic solids,resistance to fibre bending is significant,and the mechanical behavior of such solids should be investigated.Hence,in this paper,we model the mechanical aspect of residually stressed elastic solids with bending stiffness due to fibre curvature,which up to the authors’knowledge has not been mechanically modeled in the past.The proposed constitutive equation involves a nonsymmetric stress and a couple-stress tensor.Spectral invariants are used in the constitutive equation,where each spectral invariant has an intelligible physical meaning,and hence they are useful in experiment and analysis.A prototype strain energy function is proposed.Moreover,we use this prototype to give results for some cylindrical boundary value problems.
基金Supported by the NNSF of China(Grant Nos.11271224 and 11371290)
文摘We introduce the notion of property(RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S~l(G) of rapidly decreasing functions on G with respect to a continuous length function l is a dense spectral invariant and Fréchet *-subalgebra of the reduced groupoid C~*-algebra C~*(G) of G when G has property(RD) with respect to l, so the K-theories of both algebras are isomorphic under inclusion. Each normalized cocycle c on G, together with an invariant probability measure on the unit space G~0 of G, gives rise to a canonical map τon the algebra C(G) of complex continuous functions with compact support on G. We show that the map τcan be extended continuously to S~l(G) and plays the same role as an n-trace on C~*(G) when G has property(RD) and c is of polynomial growth with respect to l, so the Connes’ fundament paring between the K-theory and the cyclic cohomology gives us the K-theory invariants on C~*(G).
