Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices

In the present work, we show that the linearized homogenized model for a pantographic lattice must necessarily be a second gradient continuum, as defined in Germain (1973). Indeed, we compute the effective mechanical properties of pantographic lattices following two routes both based in the heuristic homogenization procedure already used by Piola (see Mindlin, 1965; dell'Isola et al., 2015a): (i) an analytical method based on an evaluation at micro-level of the strain energy density and (ii) the extension of the asymptotic expansion method up to the second order. Both identification procedures lead to the construction of the same second gradient linear continuum. Indeed, its effective mechanical properties can be obtained by means of either (i) the identification of the homogenized macro strain energy density in terms of the corresponding micro-discrete energy or (ii) the homogenization of the equilibrium conditions expressed by means of the principle of virtual power: actually the two methods produce the same results. Some numerical simulations are finally shown, to illustrate some peculiarities of the obtained continuum models especially the occurrence of bounday layers and transition zones. One has to remark that available well-posedness results do not apply immediately to second gradient continua considered here

SPECTROPHOTOMETRIC DETERMINATION OF PHENOL BY CHARGE-TRANSFER COMPLEXATION

The phenol is used in pharmaceutical domain as agent of preservation, a rapid and reliable spectrophotometric method was validated for its determination in routine control. This method is based on the formation of a charge transfer complex between phenol and 2,6-dichloroquinone-4-chloroimide (DCQ) in basic medium. This produced a blue product with maximum absorption at 610nm. Beer's law is obeyed and the calibration curve was linear (r = 0.999) over the range 7.5 10 -6 M -7.5 10 -

Derivation of dual horizon state-based peridynamics formulation based on euler-lagrange equation

The numerical solution of peridynamics equations is usually done by using uniform spatial discretisation. Although implementation of uniform discretisation is straightforward, it can increase computational time significantly for certain problems. Instead, non-uniform discretisation can be utilised and different discretisation sizes can be used at different parts of the solution domain. Moreover, the peridynamic length scale parameter, horizon, can also vary throughout the solution domain. Such a scenario requires extra attention since conservation laws must be satisfied. To deal with these issues, dual-horizon peridynamics was introduced so that both non-uniform discretisation and variable horizon sizes can be utilised. In this study, dual-horizon peridynamics formulation is derived by using Euler–Lagrange equation for state-based peridynamics. Moreover, application of boundary conditions and determination of surface correction factors are also explained. Finally, the current formulation is verified by considering two benchmark problems including plate under tension and vibration of a plate

Computational Homogenization of Architectured Materials

Architectured materials involve geometrically engineered distributions of microstructural phases at a scale comparable to the scale of the component, thus calling for new models in order to determine the effective properties of materials. The present chapter aims at providing such models, in the case of mechanical properties. As a matter of fact, one engineering challenge is to predict the effective properties of such materials; computational homogenization using finite element analysis is a powerful tool to do so. Homogenized behavior of architectured materials can thus be used in large structural computations, hence enabling the dissemination of architectured materials in the industry. Furthermore, computational homogenization is the basis for computational topology optimization which will give rise to the next generation of architectured materials. This chapter covers the computational homogenization of periodic architectured materials in elasticity and plasticity, as well as the homogenization and representativity of random architectured materials

A micromechanical model of woven structures accounting for yarn–yarn contact based on Hertz theory and energy minimization

International audienceThe equilibrium shape of plain weave fabric, accounting for yarn to yarn contact and incorporating the transverse compressibility, is presently investigated, relying on an energetic analysis of the textile. Each yarn is modeled as an undulated beam at a mesoscopic level, with a shape modeled by a Fourier series. In order to get an accurate description of the contact between yarns, we have considered the contact occurring on a surface (distributed contact) of finite extent according to Hertz contact theory. This method allows evaluating the reaction forces between weft and warp from the pressure distribution in the contact zone between yarns, and the evolution of the contact area versus the applied tension. The overall response of fabric under uniaxial and biaxial tension is computed as the minimum of the total potential energy of the set of interwoven yarns, including the work of reaction forces on the contact zones. The calculated nonlinear response for two important materials (glass, carbon) is due to the change of the geometry, namely the crimp variation and the change of contact area during increased loading. The contact area increases with the applied tensile force and is comparable to the yarn diameter. Only the contribution of the displacement due to flexion is shown to be sensitive to the distribution of the forces on the contact surface. A good agreement between the model predictions and tensile measurements of the response of plain weave fabric has been obtained

Surface effects of network materials based on strain gradient homogenized media

The asymptotic homogenization of periodic network materials modeled as beam networks is pursued in this contribution, accounting for surface effects arising from the presence of a thin coating on the surface of the structural beam elements of the network. Cauchy and second gradient effective continua are considered and enhanced by the consideration of surface effects. The asymptotic homogenization technique is here extended to account for the additional surface properties, which emerge in the asymptotic expansion of the effective stress and hyperstress tensors versus the small scale parameters and the additional small parameters related to surface effects. Based on the elaboration of small dimensionless parameters of geometrical or mechanical nature reflecting the different length scales, we construct different models in which the importance of surface effects is dictated by specific choice of the scaling relations between the introduced small parameters. The effective moduli reflect the introduced surface properties. We show in particular that surface effects may become dominant for specific choices of the scaling laws of the introduced small parameters. Examples of networks are given for each class of the considered effective constitutive models to illustrate the proposed general framework