By C. Dascalu, Gérard A. Maugin, Claude Stolz
This quantity provides fresh advancements within the thought of defects and the mechanics of fabric forces. many of the contributions have been offered on the overseas Symposium on disorder and fabric Forces (ISDMM2007), held in Aussois, France, March 2007.
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Additional resources for Defect and Material Mechanics: Proceedings of the International Symposium on Defect and Material Mechanics (ISDMM), held in Aussois, France, March 25–29, 2007
5) for all virtual velocities. Now, in this classical case there is an understood extra compatibility condition between the virtual ordinary velocities (v) and the virtual velo˙ This condition cities of the deformation gradient (F). establishes that the deformation gradient velocities must be derived from the (globally smooth) ordinary velocity field. 5) is enforced only under the condition that: F˙ = ∇v. 7) and the natural boundary condition: TN = t R , f R v + tr (BP˙ T ) d + t R vd S. 8) where N is the unit exterior normal to the (unsupported part of the) boundary in the reference configuration.
Oleaga Fig. 10 The mapping properties of the extended Gt g t(z) ζ z [ a(t) . ] b(t) ξ(t) ξ(t) ’ Γt −1 g0 (ζ) a’ b’ . ut We want now to relate the coefficients cn (t) with the ones of the initial field cn (0). Using the expansion for h 0 (z) we have that: Ut (θ ) = Re h 0 G t eiθ ∞ = Re ck (0) G t e iθ k=0 The expression for the first coefficient (notice that Ut (−θ ) = Ut (θ )) should read (we drop for a while the dependence on t of the ai ’s and bi ’s): ∞ c0 (t) = Re 1 π ck (0) j=0 π j G t eiθ dθ 0 = c0 (0) + c1 (0) a0 + c2 (0) × a02 + 2a1 b1 + 2a2 b2 + · · · (44) Similarly, we obtain that for c1 (t), ∞ c1 (t) = c j (0) j=0 2 π π Re G t eiθ j cos(θ )dθ 0 = c1 (0) (a1 + b1 ) + 2c2 (0) (a0 (a1 + b1 ) + a1 b2 + a2 (b1 + b3 )) + · · · (45) For c2 (t) we have: ∞ c2 (t) = c j (0) j=0 2 π We can now apply the complex version of the energy release formula (cf.
J Elast 6:313–326 Di Carlo A (2005) Surface and bulk growth unified. In: Steinmann P, Maugin GA (eds) Mechanics of material forces. Advances in mechanics and mathematics, vol 11. Springer, pp 53–64 Di Carlo A, Quiligotti S (2002) Growth and balance. Mech Res Commun 29:449–456 Epstein M (2005) Self-driven continuous dislocations and growth. In: Steinmann P, Maugin GA (eds) Mechanics of material forces. Springer, pp 129–139 Epstein M, Maugin GA (1990) The energy-momentum tensor and material uniformity in finite elasticity.