Micro‐inertia effects in nonlinear heterogeneous media

Abstract

SUMMARY The manuscript presents a dispersive nonlinear continuum theory for the case where the shortest wavelength is several times larger than the characteristic size of the microstructure and the observation window is large. We develop a general purpose computational framework, which is valid for nonlinear problems and requires standard C 0 continuous formalism. The fine‐scale inertia effect is accounted for by formulating a quasi‐dynamic unit cell problem where the fine‐scale inertia effect is represented by so‐called inertia induced eigenstrain. The solution of the nonlinear quasi‐dynamic unit cell problem gives rise to the modification of either coarse‐scale mass matrix in the implicit solvers or internal force in the explicit solvers. Similarly to the classical homogenization theory, scale‐separation is assumed, but higher order homogenization is not pursued to avoid higher order coarse‐scale gradients, higher order continuity, and higher order boundary conditions. Numerical examples for both the one‐dimensional model problem and three‐dimensional heterogeneous medium with layered and fibrous composite microstructure are used to validate the computational framework proposed. Copyright © 2012 John Wiley & Sons, Ltd.

Liquid marble: A novel liquid nanofoam structure for energy absorption

“…However, at higher strain rates, the stress equilibrium condition which is always satisfied under quasi-static tests, is no longer valid and the local pressure at the impact end builds up first due to the inertia effect caused by the heterogeneous configuration. [28][29][30] As the nanoporous particles at the top of the silica gel layer have immediate contact with the bulk liquid phase, liquid infiltration process is activated and finished locally before the liquid fills all the air gaps. As a result, there is 78% of the nanoporous silica gel involved in the energy absorption (Fig.…”

Dispersive computational continua

“…Thus, the condition of having a set of nontrivial reference solutions (i.e., a zero determinant) provides an additional equation from which the dispersion relation (24) follows. Unlike the reference solution, the algebraic system of equations (18) arising from the dispersive C 2 formulation is non -homogeneous, with the right-hand side depending on the macroscopic solution.…”

“…For comparison, we also consider the dispersion relation of the O(1) micro-inertia formulation [24], in the normalized form U   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Fig. 4a depicts the dispersion diagrams obtained from the dispersion relations of the nondispersive formulation (36), the O(1) micro-inertia (35), the dispersive C 2 (32), and the reference solution (24) for the two cases of 0.5,1   .…”

“…4a depicts the dispersion diagrams obtained from the dispersion relations of the nondispersive formulation (36), the O(1) micro-inertia (35), the dispersive C 2 (32), and the reference solution (24) for the two cases of 0.5,1   . It can be seen that the dispersive C 2 curve coincides with the reference solution (24). Note that for 0.5,1   the dispersive C 2 solution of the Floquet-Bloch type is equivalent to the reference solution (see Remark 8 and Appendix A.3) for a wide range of frequencies.…”

“…Noteworthy are the averaging techniques based on various self-consistent schemes that provide dynamic effective constitutive relations of elastic composites [7,19,20,21,22], methods based on equivalent quasi-dynamic unit cell problems [23,24,25], and approaches aimed at extending the Hill-Mandel macro-homogeneity condition to high frequency dynamics [26] along the lines of the Virial formula [27] used in molecular dynamics.…”