Comparison of microscopic and macroscopic instabilities in a class of two-dimensional periodic composites
The quantitative comparison between the microscopic (local) and the corresponding macroscopic (global) failure is carried out in a two- dimensional fiber reinforced composite. The latter is modeled as an infinite periodic grillage (with a rectangular unit cell) of axially compressed in both directions beams with an average shear stiffness. At first, an analytical solution for the exact buckling problem of the model is represented for the onset of the first bifurcation in the infinite periodic structure using Bloch wave theory. The solution provides the lowest buckling load corresponding to a mode with given wavelengths in each direction. The critical load for a given geometry of the model is defined as the lowest point of the characteristic surface. Then, the averaged (macroscopic) behavior of the model is considered, and a study of its stability is presented. The macroscopic behavior of the periodic structure is characterized by the homogenized incremental moduli of the structure, where the average is taken over a unit cell which is the characteristic volume for the periodic material. The stability of the homogenized material results in the ellipticity of the derived spatially constant homogenized incremental moduli. The loss of ellipticity in a homogeneous material with spatially constant properties means that the homogeneous stress state in a Dirichlet problem is no longer an energy minimizer and is unstable. Further, it is shown that the loss of elliptic instability for the homogenized (macroscopic) model corresponds to a bifurcation instability of the exact (microscopic) model when the wavelength of its buckling mode is much larger than the unit cell. The numerical results are obtained in the form of the critical load dependence on various microstructure geometries. At first, attention is focussed on the stability of the grillage in the absence of interstitial stiffness \((g=0)\). The common feature for all the critical load calculations in this case is that the critical buckling mode always corresponds to the long wavelength mode. Next, there is a domain of ellipticity for the homogenized moduli which covers the stable stress states for the grillage. Moreover, the effects of the various geometric parameters of the grillage on the stability region of the structure are investigated, namely: (i) the influence of the grillage shape (the structure made up of square unit cells possesses the largest region of stable stress states); (ii) the influence of the stubiness of the beams (with the largest region of stable stress states corresponding to the slenderness angle \(\chi_{\mathfrak r}= \pi/4\)); (iii) the effect of the beam shape constants (their influence is negligible compared to above ones). From numerical results one concludes that the beams stubbiness parameters determine the type of failure mode for the structure, and hence the shape of the stable stress states region. Then it is noted that for \(g\neq 0\) (at the adequately large values of \(g\)) a local buckling mode can precede the global one. Finally, it is shown that the determining factor, whether or not the critical mode is global or local for a given microstructure, is the critical value of the interstitial stiffness parameter \(g_ c\). For \(0\leq g\leq g_ c\), the critical mode is a global one (determined by the macroscopic homogenized moduli), while for \(g>g_ c\), the critical mode is a local one which is not obtainable from macroscopic information. Consequently, the dependence of \(g_ c\) on the load ratio parameter for a particular grillage is of fundamental importance in characterizing the domains of stable stress states for that grillage.
scientific article; zbMATH DE number 480388