Download Nonlinear Mechanics of Soft Fibrous Materials by Luis Dorfmann, Raymond W. Ogden PDF

By Luis Dorfmann, Raymond W. Ogden

The e-book offers a cutting-edge evaluation of the basic theories, confirmed versions and ongoing learn relating to the modeling of those fabrics. techniques are conventionally used to enhance constitutive family for hugely deformable fibrous fabrics. in line with the phenomenological method, a pressure strength density functionality should be outlined by way of pressure invariants. the opposite strategy is predicated on kinetic theories, which treats a fibrous fabric as a randomly orientated inter-tangled community of lengthy molecular chains bridged via everlasting and transitority junctions. on the micro-level, those are linked to chemical crosslinks and energetic entanglements, respectively.

The papers comprise rigorously crafted overviews of the basic formula of the three-d concept from a number of issues of view, and tackle their equivalences and adjustments. additionally integrated are ideas to boundary-value difficulties that are amenable to experimental verification. an extra point is the pliability of filaments, balance of equilibrium and thermodynamics of the molecular community theory.

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The latter are obtained from the axial stress σzz , which is given by (97)2 and clearly depends on r. From it we construct the resultant axial load, denoted N , on a tube cross section, which is given by b N = 2π σzz rdr, (107) a and is independent of z. By using (97) and the equilibrium equation this can be written b N =π a b ˆ λ )r dr + π ˆ λ − λW (2λz W z a d 2 (r σrr ) dr. W. Ogden The second term can be integrated explicitly to give, with the help of (102), πa2 P , which is interpreted as the load contribution on the ends of a tube with closed ends due to the internal pressure.

Ogden The invariants of τ are not affected by the deformation, while in the configuration Br the other invariants reduce to I1 = I2 = 3, I3 = 1, I5 = I6 = tr τ , I7 = I8 = tr (τ 2 ). (140) We emphasize that the above set of 10 invariants, or an equivalent set of alternative invariants, forms a complete set of invariants of C and τ in three dimensions. For an incompressible material, since I3 = 1, there are nine independent invariants in general. 4 for transverse isotropy. Stress tensors. Expanded expressions for the stress tensors given in (130) and (132) require the calculation of ∂W = ∂F Wi i∈I ∂Ii , ∂F (141) where I is the index set {1, 2, 3, 5, 6, 7, 8} for an unconstrained material and {1, 2, 5, 6, 7, 8} for incompressible material, and the notation Wi = ∂W/∂Ii , i ∈ I has been adopted.

From (65) and (71) we see that for the considered pure homogeneous strain I1 , I2 , I4 , I5 depend on only two independent deformation variables, namely λ1 and λ2 . They also depend on the angle ϕ, but this is not a deformation variable and can be regarded as a material parameter. ˆ (λ1 , λ2 ) for the present situation, but We can then adapt the notation W to emphasize the presence of fibre reinforcement here we include ϕ in the ˆ and write argument of W ˆ (λ1 , λ2 , ϕ) = W (I1 , I2 , I4 , I5 ), W (78) where I1 and I2 are given by (65) and I4 and I5 by (71).

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