A computational homogenization approach for the yield design of periodic thin plates. Part I: Construction of the macroscopic strength criterion

International audienceThe purpose of this paper is to propose numerical methods to determine the macroscopic bending strength criterion of periodically heterogeneous thin plates in the framework of yield design (or limit analysis) theory. The macroscopic strength criterion of the heterogeneous plate is obtained by solving an auxiliary yield design problem formulated on the unit cell, that is the elementary domain reproducing the plate strength properties by periodicity. In the present work, it is assumed that the plate thickness is small compared to the unit cell characteristic length, so that the unit cell can still be considered as a thin plate itself. Yield design static and kinematic approaches for solving the auxiliary problem are, therefore, formulated with a Love-Kirchhoff plate model. Finite elements consistent with this model are proposed to solve both approaches and it is shown that the corresponding optimization problems belong to the class of second-order cone programming (SOCP), for which very efficient solvers are available. Macroscopic strength criteria are computed for different type of heterogeneous plates (reinforced, perforated plates,...) by comparing the results of the static and the kinematic approaches. Information on the unit cell failure modes can also be obtained by representing the optimal failure mechanisms. In a companion paper, the so-obtained homogenized strength criteria will be used to compute ultimate loads of global plate structures

Charge-Exchange Collision Dynamics and Ion Engine Grid Geometry Optimization

The development of a new three-dimensional model for determining the absolute energy distribution of ions at points corresponding to spacecraft surfaces to the side of an ion engine is presented. The ions resulting from elastic collisions, both charge-exchange (CEX) and direct, between energetic primary ions and thermal neutral xenon atoms are accounted for. Highly resolved energy distributions of CEX ions are found by integration over contributions from all points in space within the main beam formed by the primary ions. The sputtering rate due to impingement of these ions on a surface is calculated. The CEX ions that obtain significant energy (10 eV or more) in the collision are responsible for the majority of the sputtering, though this can depend on the specific material being sputtered. In the case of a molybdenum surface located 60 cm to the side of a 30 cm diameter grid, nearly 90% of the sputtering is due to the 5% of ions with the highest collision exit energies. Previous models that do not model collision energetics cannot predict this. The present results agree with other models and predict that the majority of the ion density is due to collisions where little to no energy is transferred. The sputtering model is combined with a grid-structure model in an optimization procedure where the sputtering rate at specified locations is minimized by adjustment of parameters defining the physical shape of the engine grids. Constraints are imposed that require that the deflection of the grid under a specified load does not exceed a maximum value, in order to ensure survivability of the grids during launch. To faciliate faster execution of the calculations, simplifications based on the predicted behavior of the CEX ions are implemented. For diametrically opposed sputtering locations, a rounded barrel-vault shape reduces the expected sputtering rate by up to 30% in comparison to an NSTAR-shaped grid.</p

Design and Analysis of Honeycomb Structures with Advanced Cell Walls

Honeycomb structures are widely used in engineering applications. This work consists of three parts, in which three modified honeycombs are designed and analyzed. The objectives are to obtain honeycomb structures with improved specific stiffness and specific buckling resistance while considering the manufacturing feasibility. The objective of the first part is to develop analytical models for general case honeycombs with non-linear cell walls. Using spline curve functions, the model can describe a wide range of 2-D periodic structures with nonlinear cell walls. The derived analytical model is verified by comparing model predictions with other existing models, finite element analysis (FEA) and experimental results. Parametric studies are conducted by analytical calculation and finite element modeling to investigate the influences of the spline waviness on the homogenized properties. It is found that, comparing to straight cell walls, spline cell walls have increased out-of-plane buckling resistance per unit weight, and the extent of such improvement depends on the distribution of the spline’s curvature. The second part of this research proposes a honeycomb with laminated composite cell walls, which offer a wide selection of constituent materials and improved specific stiffness. Analytical homogenization is established and verified by FEA comparing the mechanical responses of a full-detailed honeycomb and a solid cuboid assigned with the calculated homogenization properties. The results show that the analytical model is accurate at a small computational cost. Parametric studies reveal nonlinear relationships between the ply thickness and the effective properties, based on which suggestions are made for property optimizations. The third part studies honeycomb structures with perforated cell walls. The homogenized properties of this new honeycomb are analytically modeled and investigated by finite element modeling. It is found that comparing to conventional honeycombs, honeycombs with perforated cell walls demonstrate enhanced in-plane stiffness, out-of-plane bending rigidity, out-of-plane compressive buckling stress, approximately the same out-of-plane shear buckling strength, and reduced out-of-plane stiffness. For the future design, empirical formulas, based on finite element results and expressed as functions of the perforation size, are derived for the mechanical properties and verified by mechanical tests conducted on a series of 3D printed perforated honeycomb specimens

WAVE PROPAGATION IN ELASTIC MEDIA WITH INTERNAL STRUCTURE. PERIODIC TRANSFORMATIONS AND CURVED BEAMS

In this thesis we will focus on the linear elastodynamic properties of complex materials. In Chapter 2, we review the linear theory of elasticity. This chapter provides a brief overview of the basic laws of elasticity theory for di_erent coordinate systems, that will facilitate the further development of our research. Dispersion properties of 2D periodic systems for di_erent lattices are reported in Chapter 3. The governing equations of out of plane wave and examples of coordinate transformations in Cartesian and cylindrical are considered in Chapter 4. Also the scattering problems by square and cylindrical uncloaked and cloaked holes are investigated analytically and numerically. In Chapter 5 a periodic transformation approach has been applied to the problem of out of plane shear wave propagation in an isotropic linear elastic material. The Chapter is organized as follows. In Section 5.1 we present initial and transformed equations of motion and corresponding boundary conditions, describing the periodic locally radial geometric transformation. In Section 5.2 we report the comparative analysis of dispersion properties and briey describe the applied multipole expansion method. In particular, we focus our attention on classical, overlapping and unfolding transformations by also performing a low-frequency, long wavelength homogenisation. In Section 5.3 we show several application including a transmission problems in a continuum and in a waveguide, the detection of defect modes and the design of the transformation for the existence of Dirac points. In Chapter 6, the mathematical model of a curved beam that is connected to two semi-in_nite straight beams is developed. Dispersion properties of curved beams are derived, characterized by three di_erent propagating regimes. By implementing the Transfer matrix approach, the reection and transmission coe_cients that depend on the curvature, frequency and total angle of the curved beam are determined. By analysing the e_ect of the curvature, frequency and total angle on energy ux, separation between high frequency/low curvature regime, where the incident wave is practically totally transmitted, and low frequency/high curvature regime where, in addition to reection there is a strong coupling between longitudinal and exural waves, are de_ned. Finally, general conclusions are given in the last chapter

Low frequency acoustic stop bands in cubic arrays of thick spherical shells with holes

We analyse the propagation of pressure waves within a fluid filled with a three-dimensional array of rigid coated spheres (shells). We first draw band diagrams for corresponding Floquet-Bloch waves. We then dig a channel terminated by a cavity within each rigid shell and observe the appearance of a low frequency stop band. The underlying mechanism is that each holey shell now acts as a Helmholtz resonator supporting a low frequency localized mode: Upon resonance, pressure waves propagate with fast oscillations in the thin water channel drilled in each shell and are localized in each fluid filled inner cavity. The array of fluid filled shells is approximated by a simple mechanical model of springs and masses allowing for asymptotic estimates of the low frequency stop band. We finally propose a realistic design of periodic macrocell with a large defect surrounded by 26 resonators connected by thin straight rigid wires, which supports a localized mode in the low frequency stop band

Numerical investigation of necking in perforated sheets using the periodic homogenization approach

Due to their attractive properties, perforated sheets are increasingly used in a number of industrial applications, such as automotive, architecture, pollution control, etc. Consequently, the accurate modeling of the mechanical behavior of this kind of sheets still remains a valuable goal to reach. This paper aims to contribute to this effort by developing reliable numerical tools capable of predicting the occurrence of necking in perforated sheets. These tools are based on the coupling between the periodic homogenization technique and three plastic instability criteria. The periodic homogenization technique is used to derive equivalent macroscopic mechanical behavior for a representative volume element of these sheets. On the other hand, the prediction of plastic instability is based on three necking criteria: the maximum force criterion (diffuse necking), the general bifurcation criterion (diffuse necking), and the loss of ellipticity criterion (localized necking). The predictions obtained by applying the three instability criteria are thoroughly analyzed and compared. A sensitivity study is also conducted to numerically investigate the influence on the prediction of necking of the design parameters (dimension, aspect-ratio, orientation, and shape of the holes), the macroscopic boundary conditions and the metal matrix material parameters (plastic anisotropy, hardening)

14th Conference on Dynamical Systems Theory and Applications DSTA 2017 ABSTRACTS

From Preface: This is the fourteen time when the conference “Dynamical Systems – Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and the Ministry of Science and Higher Education. It is a great pleasure that our invitation has been accepted by so many people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcome nearly 250 persons from 38 countries all over the world. They decided to share the results of their research and many years experiences in the discipline of dynamical systems by submitting many very interesting papers. This booklet contains a collection of 375 abstracts, which have gained the acceptance of referees and have been qualified for publication in the conference proceedings [...]