By Bettina Albers, Mieczyslaw Kuczma
This publication offers study advances within the box of continuing Media with Microstructure and considers the 3 complementary pillars of mechanical sciences: concept, learn and computational simulation. It specializes in the subsequent difficulties: thermodynamic and mathematical modeling of fabrics with extensions of classical constitutive legislation, unmarried and multicomponent media together with sleek multifunctional fabrics, wave propagation, multiscale and multiphysics techniques, section modifications, and porous, granular and composite fabrics. The e-book offers the court cases of the second convention on non-stop Media with Microstructure, which used to be held in 2015 in Łagów, Poland, in reminiscence of Prof. Krzysztof Wilmański.
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Additional info for Continuous Media with Microstructure 2
Here, we are trying to unify the geometric approach and use a truly modern language of fibered bundles which is systematically developed in the monograph Natural Operations in Differential Geometry  of Koláˇr et al. Besides this unifying language there are also some of our calculations (principal morphisms between frame bundles with different structure groups, the form of elements of Toupin subgroups) completely new in terms of the explicit expression. We are convinced that local coordinate expressions are necessary to be truly effective in applications.
As Y is supposed to be invertible, B(τ, x) becomes also available. Then one may split w∗ into three addenda by the introduction of a new peculiar velocity c: c(τ, x; y) := w∗ (τ, x; y) − v(τ, x) − [B(τ, x)]y, (15) which, contrary to c∗ , is observer independent. At first, we do not pursue a detailed knowledge of the fields c; but, rather, as announced, we introduce a new tensor H based on c: H(τ, x) := ω−1 c(x) θ c ⊗ c dvol, (16) which, multiplied by ρ, leads to Reynolds’ tensor measuring, as declared in , ‘the apparent stress due to change of moment of momentum’; in fact ρH has the physical dimension of a stress (also of an energy density, of course), but, in our opinion and in accordance with the declaration just mentioned, it is a kinetic quantity gauging a contribution to the time-rate of change of K.
Hence we are led later to suggest for it an evolution equation to replace, in a sense, one proposed in the theory of hypoelasticity for the tensor of Cauchy’s stress . Because c may be interpreted as the average relative velocity of the molecules belonging to the bunch in the immediate neighbourhood of y within c(x), it could be imagined also to be the relative velocity of a fictitious sub-element presently at y. Consequently, one must account for a sub-local relative momentum and an associated kinetic energy not yet accounted for at the mesolevel.
Continuous Media with Microstructure 2 by Bettina Albers, Mieczyslaw Kuczma