Generalized Continua as Models for Materials: with by Holm Altenbach, Samuel Forest, Anton Krivtsov

By Holm Altenbach, Samuel Forest, Anton Krivtsov

This quantity provides contributions describing the micro- and macro-behaviours, new life and forte theorems, the formula of multi-scale difficulties, and so forth. and now it's time to consider back the kingdom of subject and to debate new traits and purposes.
The major concentration is directed at the following goods
- Modelling and simulation of fabrics with major microstructure,
- Generalized continua because of multi-scale types,
- Multi-field activities on fabrics leading to generalized fabric versions, and
- comparability with discrete modelling techniques

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Additional resources for Generalized Continua as Models for Materials: with Multi-scale Effects or Under Multi-field Actions (Advanced Structured Materials)

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This worths being emphasized because in the micromorphic approach tensors do have such a component. Therefore, in order to avoid any misunderstanding, it is important to use the vocabulary in an appropriate way. Furthermore the use of a correct generalization of the harmonic decomposition to higher-order tensors provides useful information on the associated constitutive law. For example: • the number of isotropic moduli associated to the isotropic related constitutive tensor (with great symmetry); • the number and the dimension of eigenspaces of the related isotropic related constitutive tensor; • the structure of anisotropy classes of the associated constitutive law [23]; • etc.

A ” denotes the average of a bulk field A across the 1 AII +AI . The Einstein summation convention is also interface defined by A = 2 used throughout the paper. 1 Kinematics Following [16–18], the particle velocity vector jump at the internal discontinuity surface (GB) denoted hereafter S can be decomposed as follows (Fig. 1) [vi ] = [vi ](1) + [vi ](2) , (1) 44 S. Berbenni et al. Fig. 1 Schematic representation of grain boundary (GB) migration and sliding (kinematics). [vi ] = vi II − vi I denotes the jump of vi at the interface (GB plane) oriented by unit normal vector ni from crystal I to crystal II where [vi ](1) = − [ui,j ] nj ωN (2) is the particle velocity jump due to normal GB propagation assuming linearized kinematics [19, 16] and [vi ](2) is the part of particle velocity jump at the discontinuity surface due to tangential GB sliding.

Invariant under any index permutation, only contains evenorder (resp. odd-order) harmonic spaces. 2. In the decomposition of an even-order (resp. odd-order) even-order (resp. odd-order) components are polar and odd-order axial (resp. even order). 4 Structure of Third-Order Strain Measures of Generalized Continua These techniques can now be applied to the third-order tensors involved in the micromorphic and the strain-gradient elasticity model. Micromorphic Elasticity Let us begin with the space Tijk used in the micromorphic theory to model the ∼ ⊗3 R3 , we have Tijk = ∼ H1 ⊗ H1 ⊗ H1 .

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