IUTAM Symposium on Theoretical, Computational and Modelling by B. Daya Reddy

By B. Daya Reddy

This paintings includes papers in accordance with many of the talks offered on the IUTAM Symposium of an analogous identify, held in Cape city, January 14-18, 2008. This volume treats state of the art matters in modelling, the behaviour of assorted periods of inelastic media, and linked algorithms for conducting computational simulations. A key function of the contributions are works directed at modelling behaviour on the meso and micro-scales, and at bridging the micro-macro scales.

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Additional info for IUTAM Symposium on Theoretical, Computational and Modelling Aspects of Inelastic Media: Proceedings of the IUTAM Symposium Held at Cape Town, South Africa, January 14-18, 2008

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For linear FE functions the infimum is the integral of the quasiconvex envelope of W , E (v ) = W qc (x, v (x), Dv (x))dx − L(v ). T ∈T T The numerical outcome for the computational microstructures benchmark is shown in Figure 2 [9]. This model is a typical example which is in its original formulation not quasiconvex and therefore oscillations and meshdependance of numerical solutions are observed. The relaxation in closed form for this example yields meshindependent solutions. In this example we are looking for ϕ : R2 → R and define the energy density W (F ) =|F − F1 |2 |F − F2 |2 √ for two different wells F1 = −F2 = −(3, 2)/ 13 ∈ R2 .

One calls function W : Rm×n → R quasiconvex if for some open subset ω ⊂ Rn and all E ∈ Rm×n ; W (F ) = inf ϕ∈Cc∞ (ω,Rm ) 1 |ω| W (F + Dϕ)dx ω 44 C. Carstensen and R. Huth m1 m F s1 Fp m Fe s s Fig. 1 Split of plastic and elastic deformation. holds. The quasiconvex envelope W qc is given W qc (F ) = sup W ∗ (F ) : W ∗ ≤ W and W ∗ is quasiconvex . It is clear that W qc is also quasiconvex. The quasiconvex energy E qc (v) = W qc (x, v(x), Dv(x)) dx for all v ∈ V (Q) models relevant macroscopic properties like the displacement- or stress-field.

Kochmann 6 Numerical Results The numerical scheme outlined above can be applied to arbitrary deformations, so long as the deformation remains volume-preserving to account for incompressible material. The first example treats the microstructure evolution during a plane-strain simple shear test parametrised by the macroscopic deformation gradient ⎞ ⎛ 1γ 0 (37) F = ⎝0 1 0⎠. 001 up to the extremal loads γmax = 6 and γmin = −3. The slip-system was oriented under an angle of ϕ = 150◦ (see Figure 3(a) for the definition of ϕ).

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