The symmetric buckling mode in laminated elastoplastic micro-structures under plane strain

The present work considers lamellar (micro) structures of thin, elastic lamellae embedded in a yielding matrix as a stability problem in the context of the theory of stability and uniqueness of path-dependent systems. The volume ratio of the stiff lamellae to the relatively soft matrix is assumed low enough to initiate a symmetric buckling mode, which is investigated by analytical and numerical means. Using a highly abstracted, incompatible model, a first approach is made, and the principal features of the problem are highlighted. Assuming plane strain deformation, an analytic expression for the bifurcation load of a refined, compatible model is derived for the special case of ideal plasticity and verified by numerical results. The effect of lamella spacing and matrix hardening on the bifurcation load is studied by a finite element unit cell model. Some of the findings for the ideal plastic matrix are shown to also apply for a mildly hardening matrix material. Furthermore, the postbuckling behaviour and the limit load are investigated by simulating a bulk lamella array

Modelling of air chamber supported floating platforms – coupling free surface flow, compressible air, and flexible structures

Air chamber supported floating platforms can significantly decrease wave induced structural responses. Novel applications, like floating arrays of solar collectors, with low payload requirements allow the design of floating platforms supported by large, cylindrical air chambers made of highly flexible membranes. In order to predict the dynamics of such systems a modelling strategy capturing all important phenomena: incompressible free surface flow, compressible air and flexible structures is presented. The governing partial differential equations and boundary conditions are given in their linearised form, and subsequently solved by the finite element method. A frequency domain formulation is chosen to compute the steady state response to harmonic excitation. In order to handle problems in unbounded domains a perfectly matched layer formulation is used. Thereby, radiating waves are efficiently damped at the edge of the computational domain. For the sake of simplicity we present two-dimensional, test problems used for the validation of the developed modelling strategy. Finally, we present a fully coupled simulation of wave interactions with a flexible, air chamber supported floating platform

Stability of Rod-Shaped Nanoparticles Embedded in an Elastic Matrix

International audienceIn many biological tissues as well as in technical materials we find nano-sized rod-shaped particles embedded in a relatively soft matrix. Loss of stability of equilibrium, i.e. buckling, is one of the possible failure modes of such materials. In the present paper different kinds of load transfer between matrix and reinforcing particles, which are typical for ro-shaped nanostructures in biological tissues, are considered with respect to stability of equilibrium. Two regimes of matrix stiffnesses leading to different modes of buckling, and a transition regime in between, have been found: soft matrix materials leading to the so-called "flip mode" (also called "tilt mode") and hard matrix materials resulting in "bending mode" buckling. The transition regime is of particular relevance for biological tissues. Numerical and semi-analytical as well as asymptotic concepts are employed leading to results for estimating the critical load intensities both in the form of closed form solutions and diagrams. The analytical solutions are compared with results of finite element analyses. From these comparisons indiations are gained for deciding, which of the different analytical approaches should be chosen for a particular nanostructure configuration in terms of associated buckling modes

Increase in buckling loads of plates by introduction of cutouts

It might seem amazing: Cutting a hole in a plate can increase the buckling strength! It is the objective of this paper to present and clarify this astounding phenomenon. In lightweight design, typically thin-walled structures are used. Therefore, buckling must be considered as a possible failure mode. One might assume that removing material, and thus, reducing stiffness must result in a reduction of the buckling strength. However, perhaps surprisingly, it can be shown that introduction of cutouts, placed appropriately, can under certain conjunctures increase buckling loads. At the same time, the structural mass is reduced. Thus, the paper presents a measure, which can be used for fulfillment of a requirement in lightweight design in a twofold manner: increase in buckling strength by reduction of mass! In addition to describing a nice theoretical peculiarity, it might be of more importance from the engineering point of view that the presented methodology may help designers of lightweight structures, e.g., for aerospace applications, to place openings, which are required for some reasons in a plate being part of the construction, at such positions, at which the plate’s buckling resistance is just slightly or not at all reduced or even increased, and to avoid placing holes in unfavorable areas. Based on the Rayleigh–Ritz method in terms of the Rayleigh quotient, criteria and procedures are derived which can be used to find beneficial positions for cutouts and such ones, at which cutouts should not be placed

Modelling of air chamber supported floating platforms – coupling free surface flow, compressible air, and flexible structures

Air chamber supported floating platforms can significantly decrease wave induced structural responses. Novel applications, like floating arrays of solar collectors, with low payload requirements allow the design of floating platforms supported by large, cylindrical air chambers made of highly flexible membranes. In order to predict the dynamics of such systems a modelling strategy capturing all important phenomena: incompressible free surface flow, compressible air and flexible structures is presented. The governing partial differential equations and boundary conditions are given in their linearised form, and subsequently solved by the finite element method. A frequency domain formulation is chosen to compute the steady state response to harmonic excitation. In order to handle problems in unbounded domains a perfectly matched layer formulation is used. Thereby, radiating waves are efficiently damped at the edge of the computational domain. For the sake of simplicity we present two-dimensional, test problems used for the validation of the developed modelling strategy. Finally, we present a fully coupled simulation of wave interactions with a flexible, air chamber supported floating platform

Evolution of chemically induced cracks in alkali feldspar: thermodynamic analysis

A system of edge cracks was applied to polished (010) surfaces of K-rich gem-quality alkali feldspar by diffusion-mediated cation exchange between oriented feldspar plates and a Na-rich NaCl–KCl salt melt. The cation exchange produced a Na-rich layer at and beneath the specimen surface, and the associated strongly anisotropic lattice contraction lead to a tensile stress state at the specimen surface, which induced fracturing. Cation exchange along the newly formed crack flanks produced Na-enriched diffusion halos around the cracks, and the associated lattice contraction and tensile stress state caused continuous crack growth. The cracks nucleated with non-uniform spacing on the sample surface and quickly attained nearly uniform spacing below the surface by systematic turning along their early propagation paths. In places, conspicuous wavy cracks oscillating several times before attaining their final position between the neighboring cracks were produced. It is shown that the evolution of irregularly spaced towards regularly spaced cracks including the systematic turning and wavyness along the early propagation paths maximizes the rate of free energy dissipation in every evolutionary stage of the system. Maximization of the dissipation rate is suggested as a criterion for selection of the most probable evolution path for a system undergoing chemically induced diffusion mediated fracturing in an anisotropic homogeneous brittle material