Download e-book for iPad: Advances in Extended and Multifield Theories for Continua by Bob Svendsen (auth.), Bernd Markert (eds.)

By Bob Svendsen (auth.), Bernd Markert (eds.)

ISBN-10: 3642227376

ISBN-13: 9783642227370

Modern computational thoughts, similar to the Finite aspect process, have, due to the fact that their improvement numerous a long time in the past, effectively exploited continuum theories for varied functions in technology and know-how. even supposing usual continuum equipment established upon the Cauchy-Boltzmann continuum are nonetheless of serious significance and are ordinary, it more and more seems that fabric houses stemming from microstructural phenomena need to be thought of. this is often quite actual for inhomogeneous load and deformation states, the place lower-scale dimension results start to have an effect on the macroscopic fabric reaction; anything commonplace continuum theories fail to account for. Following this concept, it really is glaring that normal continuum mechanics needs to be augmented to catch lower-scale structural and compositional phenomena, and to make this knowledge available to macroscopic numerical simulations.

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Extra resources for Advances in Extended and Multifield Theories for Continua

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2⎥ C ⎥⎥⎥ ⎥⎦ 2⎥ ⎥ (20) In Eq. (20) p represents the Cosserat parameters and i is the enumerator for the different sample-heights investigated in the virtual shear experiments. x2,k are the x2 -coordinates for the centroids of the honeycomb unit-cells contained in each shear sample corresponding to Fig. 5. There is a reference value at each x2,k for a unitcell’s average rotation evaluated in the micro-computations available, cf. Sect. 3. By substituting μ = μe f f according to Table 2 the third free material parameter is eliminated.

B ∂B ˙ ς) ˙ ∇nχ, ˙ . rf (. . , χ, (40) 12 B. Svendsen To see this, consider its first variation δR = B + B + B + + ∂ς˙ r · δ ς˙ + ∂∇ ς˙ r · δ∇ ς˙ + ∂ ˙ r · δ ˙ ∂B = + ∂χ˙ r · δ χ˙ + ∂∇ χ˙ r · δ∇ χ˙ + ∂∇∇ χ˙ r · δ∇∇ χ˙ ∂χ˙ rf · δ χ˙ + ∂∇nχ˙ rf · δ∇nχ˙ + ∂ς˙ rf · δ ς˙ δ2χ˙ r · δ χ˙ + δς˙ r · δ ς˙ + ∂ ˙ r · δ ˙ ∂B t (41) {∂χ˙ rf + δ∇nχ˙ r + (divs n) ∂∇n∇nχ˙ r − divs (∂∇n∇ χ˙ r)} · δ χ˙ ∂B s ∂Bφ {∂∇nχ˙ rf + ∂∇n∇nχ˙ r} · δ∇nχ˙ {∂ς˙ rf + ∂∇nς˙ r} · δ ς˙ in the rates, with δ∇nχ˙ r := (δ∇ χ˙ r) n , ∂∇n∇nχ˙ r := (∂∇∇ χ˙ r) n ⊗ n , ∂∇nς˙ r := (∂∇ ς˙ r) n .

E. constraining the rotations at the top and bottom has a different influence on the solution in config. 1 and config. 2. 0 μconfig. 1 μconfig. 0 0 20 40 60 80 h [no. ] Fig. 4 Effective shear modulus for varying sample heights, cf. Table 2. 100 From Lattice Models to Extended Continua 25 Table 2 Effective shear moduli and corresponding relative deviations from μGA . config. 1 no. sample h [no. ] 1 2 3 4 5 6 7 8 1 3 5 9 11 23 47 95 μe f f config. 26 μe f f This verification of the micro-models shows their reliability and allows to extract the required reference data sets for the shear stress as well as the nodal displacements and rotations.

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Advances in Extended and Multifield Theories for Continua by Bob Svendsen (auth.), Bernd Markert (eds.)

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