Continuum Mechanics and Thermodynamics

This work discusses some alternate models of a mixed assumed strain finite element which has been developed for laminated plates. After a brief theoretical review about this kind of plates and their possible finite element formulation, specifically devised for predicting the mechanical behavior of such structures, we discuss four possible assumptions for strains generating four kinds of mixed assumed strain finite elements. Several numerical tests performed on the aforementioned finite elements are thoroughly discussed in order to sketch some guidelines which can be useful when dealing with laminated plate problems.

In this paper we aim to introduce a systematic way to derive relaxation terms for the Boltzmann equation based on the minimization problem for the entropy under moments constraints (Levermore in J. Stat. Phys. 83:1021–1065, 1996; Schneider in M2AN 38:541–561, 2004). In particular the moment constraints and corresponding coefficients are linked with the eigenfunctions and eigenvalues of the linearized collision operator through the Chapman–Enskog expansion. Then we deduce from this expansion a single relaxation term of BGK type. Here we stop the moments constraints at order two in the velocity v and recover the ellipsoidal statistical model (Holway in Rarefied Gas Dynamics, vol I, pp 193–215, 1966).

In geomechanics, a relevant role is played by coupling phenomena between compressible fluid seepage flow and deformation of the solid matrix. The behavior of complex porous materials can be greatly influenced by such coupling phenomena. A satisfactorily theoretical framework for their description is not yet completely attained. In this paper, we discuss how the model developed in dell’Isola et al. (Int J Solids Struct 46:3150–3164, 2009) can describe how underground flows or, more generally, confined streams of fluid in deformable porous matrices affect compression wave propagation and their reflection and transmission at a solid-material discontinuity surface. Further work will investigate the effect of stream flow in porous media on shear waves, generalizing what done in Djeran Maigre and Kuznetsov (Comptes Rendus Mécanique 336(1–2):102–107, 2008) for shear waves in one-constituent orthotropic two-layered plates. The presented treatment shows that the presence of fluid streams considerably affect reflection and transmission phenomena in porous media.

Recently, the authors have focused on the shear behavior of interface between granular soil body and very rough surface of moving bounding structure. For this purpose, they have used finite element method and a micro-polar elasto-plastic continuum model. They have shown that the boundary conditions assumed along the interface have strong influences on the soil behavior. While in the previous studies, only very rough bounding interfaces have been taken into account, the present investigation focuses on the rough, medium rough and relatively smooth interfaces. In this regard, plane monotonic shearing of an infinite extended narrow granular soil layer is simulated under constant vertical pressure and free dilatancy. The soil layer is located between two parallel rigid boundaries of different surface roughness values. Particular attention is paid to the effect of surface roughness of top and bottom boundaries on the shear behavior of granular soil layer. It is shown that the interaction between roughness of bounding structure surface and the rotation resistance of bounding grains can be modeled in a reasonable manner through considered Cosserat boundary conditions. The influence of surface roughness is investigated on the soil shear strength mobilized along the interface as well as on the location and evolution of shear localization formed within the layer. The obtained numerical results have been qualitatively compared with experimental observations as well as DEM simulations, and acceptable agreement is shown.

This research originally was aimed at modeling all flows (except free-molecular) by systems of hyperbolic-relaxation equations (moments of the Boltzmann equation), and developing efficient numerical methods for these. Such systems have many potential numerical advantages, mainly because there are no second or higher derivatives to be approximated. This avoids accuracy problems on adaptive unstructured grids, and the source terms, though often stiff, are only local; the compact stencils facilitate code parallelization. A single code could simulate flows up to intermediate Knudsen numbers, and be hybridized with DSMC where needed. In this project, one major problem arose that we have not yet solved: the accurate representation of shock structures. This makes the methodology currently unsuited for, e.g., re-entry flows. We have validated it for subsonic and transonic flows and are concentrating on applications to MEMS-related flows. We discuss the challenges of our approach, present numerical algorithms and results based on the 10-moment model, and report progress in our latest research topic: formulating accurate solid-boundary conditions.

In this paper, we consider compliance minimization problems within the variable-thickness approach for the rib-stiffened plates subjected to a transverse loading. It is known, that such optimization problems are usually not well posed and their solutions become strongly mesh-dependent. To overcome this issue, we introduce additional regularization constraint on the thickness gradient and evaluate the convergence and efficiency of considered method. Variable thickness is defined based on topology optimization approach introducing additional design variables in the nodes of the shell-type elements. Numerical solutions are provided by using finite element simulations within Mindlin–Reissner theory and method of moving asymptotes. Possibility for the well-converged optimal solutions for the benchmark problems with rib-stiffened panels loaded by the systems of concentrated forces is shown. Parametric studies are provided to analyse the effects of the shape functions order, values of penalty factors and initial conditions for the plate thickness. Recommendations for the optimal settings of the considered method are established. Theoretical and experimental assessments on the advantages and accuracy of the variable-thickness approach are given based on comparison of the obtained solutions to the standard design for the plates with regular stiffening.

3D printing technology has opened application perspectives which were difficult to imagine only few years ago. In this paper, we show how it is possible to design and print some microstructures in which relative displacements are allowed at micro-level. Some structural elements have been built in the aforementioned structures which can be confidently modeled as perfect pivots or as soft elastic connections. The obtained specimens can be regarded as constituted by very exotic materials, as forecast theoretically. We numerically study the behavior of pantographic structures including soft or nearly perfect pivots in large deformations, and we experimentally observe an enlarged elastic range and peculiar buckling mechanisms. The presented results are extremely promising: We consider now as proven that higher gradient metamaterials can be realized by using microstructures having micro-characteristic lengths of the order on tenth of millimeters.

A system of equations from extended thermodynamics is proposed for the calculation of the temperature field in a rarefied gas at rest. Numerical simulations suggest that the temperature field in this case should exhibit boundary layers superposed on the classical Fourier solution. In order to derive such a field from the extended thermodynamics of moments, one needs boundary values for higher moments, which in practice cannot be assigned and controlled. We suggest that such boundary data emerge as mean values of thermal fluctuations and thus calculate them. The result agrees qualitatively with numerical simulations and it is quantitatively of the same order of magnitude.

The analytical formulas for spectrum of turbulence on the basis of the new theory of stochastic hydrodynamics are presented. This theory is based on the theory of stochastic equations of continuum laws and equivalence of measures between random and deterministic movements. The purpose of the article is to present a solutions based on these stochastic equations for the formation of the turbulence spectrum in the form of the spectral function $$ E(k)_j$$ depending on wave numbers k in form $$E(k)_{{j}}\sim k^{n}$$ . At the beginning of the article two formulas for the viscous interval were obtained. The first analytical formula gives the law $$E(k)_{j}\sim k^{-3}$$ and agrees with the experimental data for initial period of the dissipation of turbulence. The second analytical formula gives the law which is in a satisfactory agreement with the classical Heisenberg’s dependence in the form of $$E(k)_{j}\sim k^{-7}$$ . The final part of the paper presents four analytical solutions for a spectral function on the form $$E(k)_{j}\sim k^{n}$$ , $$\hbox {n}=(-1,4;-5/3;-3;-7)$$ which are derived on the basis of stochastic equations and equivalence of measures. The statistical deviation of the calculated dependences for the spectral function from the experimental data is above 20%. It should be emphasized that statistical theory allowed to determine only two theoretical formulas that were determined by Kolmogorov $$E(k)_{j}\sim k^{{-5/3}}$$ and Heisenberg $$E(k)_{j}\sim k^{{-7}}$$ .