The main focus of this paper lies in the drastic model reduction for a complex multi‐scale problem of linear elasticity. The core of the work lies in the structural optimization of periodic perforated cylindrical shells under a given load on a small portion of the surface. Periodic structure of the shell is a frame of beams. Algorithm presented in the paper utilizes two our recent analysis works on homogenization and dimension reduction of the problem in the perforated 3D shell to two‐dimensional homogenized shell, and then dimension reduction w.r.t. the beam thickness in auxiliary cell‐problems to problem on a frame of beams, by asymptotic method. In this paper, we found the analytical solution for the orthotropic cylindrical shell, obtained in the limit, by variable separation and Fourier analysis. The solution depends explicitly on the effective properties, which are computed symbolically, and on the design variables of the structure. Cell‐problems are solved by symbolic means and the structural design for this type of the loading has been optimized. Moreover, this yields a semi‐analytic optimization problem and the practical usage of the underlying theoretical derivation. We stress that the homogenization and dimension reduction of a shell with holes and the analytic solution to the corresponding macroscopic problem are new.
A simple formula for buckling load was derived from the asymptotic analysis of nonlinear behavior of a thin spherical shell. Firstly, two asymptotic cases were studied: the initial post-buckling regime of a perfect structure with small (compared to shell thickness) deflections and equilibrium states with large deflections. Two asymptotic formulae were jointed to obtain the solution for the entire range of deflection amplitude. Then the solution was modified for an imperfect shell. Initial deflections were introduced by only one parameter: the slope of the load–deflection diagram at small pressure. This minimal information was enough to predict the buckling load of the structure with localized imperfections. The suggested asymptotic result was validated by the finite element method and by comparison with experimental data.
Simulation-based prediction of mechanical properties is highly desirable for optimal choice and treatment of leather. Nowadays, this is state-of-the-art for many man-made materials. For the natural material leather, this task is however much more demanding due to the leather’s high variability and its extremely intricate structure. Here, essential geometric features of the leather’s meso-scale are derived from 3D images obtained by micro-computed tomography and subsumed in a parameterizable structural model. That is, the fiber-bundle structure is modeled. The structure model is combined with bundle properties derived from tensile tests. Then the effective leather visco-elastic properties are simulated numerically in the finite element representation of the bundle structure model with sliding contacts between bundles. The simulation results are validated experimentally for two animal types, several tanning procedures, and varying sample positions within the hide. Finally, a complete workflow for assessing leather quality by multi-scale simulation of elastic and visco-elastic properties is established and validated.
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