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Download Continuum Mechanics - Progress in Funds., Engrg. Applns. by Y. Gan PDF

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Extra info for Continuum Mechanics - Progress in Funds., Engrg. Applns.

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On the contrary, if L dry  1 , the surface energy is larger than the strain energy, and then the adhesion of the two beams is possible. Equation (18) also requires that the contact angle  Y must satisfy 0  Y   2 , that is, the beams must be hydrophilic. In other words, capillary adhesion cannot happen between two hydrophobic hairs. The critical length Ldry for capillary adhesion of two beams increases with the decreasing of the surface tension and the Young’s contact angle  Y in the range of 0  Y   2 , and with the increasing of the end distance d of the two beams.

It follows from a chase that the kernel of the k − τ k with ( ρk ) ∈ Z2 ( g ) ⊂ canonical projection Riemann → Weyl is defined by ρkl,ij = τli,j 1 lj,i l,ij k ) ∈ T ∗ ⊗ gˆ . Accordingly ( n − 2) τ = nρ − ( n/2( n − 1)) ω ρ provides the Z2 ( gˆ1 ) and (τli,j 2 ij ij ij S2 T ∗ and we get nρkl,ij = δik τlj − δjk τli + ωlj ω ks τsi − ωli ω ks τsj , that is: isomorphism Ricci ρkl,ij = 1 1 (δk ρ − δjk ρli + ωlj ω ks ρsi − ωli ω ks ρsj ) − (δk ω − δjk ωli )ρ (n − 2) i lj (n − 1)(n − 2) i lj We check that ρri,rj = ρij , obtaining therefore a splitting of the right vertical sequence in the last diagram that allows to define the Weyl tensor by difference.

8. Conclusion In continuum mechanics, the classical approach is based on differential invariants and only involves derivatives of finite transformations. Accordingly, the corresponding variational calculus can only describe forces as it only involves translations. It has been the idea of E. and F. Cosserat to change drastically this point of view by considering a new differential geometric tool, now called Spencer sequence, and a corresponding variational calculus involving both translations and rotations in order to describe torsors, that is both forces and couples.

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