Asymptotic Homogenization Research Articles

The effect of process parameters on the mechanical properties of additively manufactured parts using a hierarchical multiscale model

Purpose This paper aims to present a hierarchical multiscale model to evaluate the effect of fused deposition modeling (FDM) process parameters on mechanical properties. Asymptotic homogenization mathematical theory is developed into two scales (micro and macro scales) to compute the effective elastic and shear modulus of the printed parts. Four parameters, namely, raster orientation, layer height, build orientation and porosity are studied. Design/methodology/approach The representative volume elements (RVEs) are generated by mimicking the microstructure of the printed parts. The RVEs subjected to periodic boundary conditions were solved using finite element. The experimental characterization according to ASTM D638 was conducted to validate the computational modeling results. Findings The computational model reports reduction (E1, ∼>38%) and (G12, ∼>50%) when porosity increased. The elastic modulus increases (1.31%–47.68%) increasing the orthotropic behavior in parts. Quasi-solids parts (100% infill) possess 10.71% voids. A reduction of 11.5% and 16.5% in elastic modulus with layer height is reported. In total, 45–450 oriented parts were highly orthotropic, and 0–00 parts were strongest. The order of parameters affecting the mechanical properties is porosity > layer height > raster orientation > build orientation. Originality/value This study adds value to the state-of-the-art terms of construction of RVEs using slicing software, discarding the necessity of image processing and study of porosity in FDM parts, reporting that the infill density is not the only measure of porosity in these parts.

Optimal heat transport induced by magnetic nanoparticle delivery in vascularised tumours

We describe a novel mathematical model for blood flow, delivery of nanoparticles, and heat transport in vascularised tumour tissue. The model, which is derived via the asymptotic homogenisation technique, provides a link between the macroscale behaviour of the system and its underlying, tortuous micro-structure, as parametrised in Penta and Ambrosi (2015). It consists of a double Darcy’s law, coupled with a double advection–diffusion–reaction system describing heat transport, and an advection–diffusion–reaction equation for transport and adhesion of particles. Particles are assumed sufficiently large and do not extravasate to the tumour interstitial space but blood and heat can be exchanged between the two compartments. Numerical simulations of the model are performed using a finite element method to investigate cancer hyperthermia induced by the application of magnetic field applied to injected iron oxide nanoparticles. Since tumour microvasculature is more tortuous than that of healthy tissue and thus suboptimal in terms of fluid and drug transport, we study the influence of the vessels’ geometry on tumour temperature. Effective and safe hyperthermia treatment requires tumour temperature within certain target range, generally estimated between 42 °C and 46 °C, for a certain target duration, typically 0.5h to 2h. As temperature is difficult to measure in situ, we use our model to determine the ranges of tortuosity of the microvessels, magnetic intensity, injection time, wall shear stress rate, and concentration of nanoparticles required to achieve given target conditions.

Multiscale structural analysis for plain weave composite plates using asymptotic homogenization method

Asymptotic homogenization method (AHM) is used to perform multiscale structural analysis for plain weave composite plates (PWCPs). First, AHM is used to obtain the effective elastic properties of yarns. Then, the effective plate properties of PWCPs are predicted using AHM. Finally, an equivalent plate model is established to perform the macroscale structure analysis, and recover the mesoscale fields. The accuracy and computational efficiency of the multiscale structural analysis are verified by comparing the results with those of direct numerical simulation. Furthermore, the influences of the geometrical parameters and laying angle of yarns on the equivalent plate stiffness are investigated.

Mechanism exploration and effective analysis method of shear effect of helically wound structures

Helically wound structures are widely used in practical engineering due to its excellent mechanical property, such as the steel rope, the reinforced armor layer of the marine flexible pipe/cables. The shear stiffness plays an important role in exactly predicting the mechanical response of the helically wound structure, especially for the short structure. There has been no general methodology to directly calculate the shear effect of this type of structure because of the geometrical complexity. This paper introduces and modifies a novel implementation of asymptotic homogenization method so-called NIAH to effectively calculate the shear property of the helically wound structure. This modification of NIAH is derived based on the strain energy equivalence of the macroscopic structure and the microscopic structure so-called the microscopic unit cell, which can be used to quickly calculate the effective stiffness and effective stress. In this paper, the effective stiffness is only discussed. The mechanical mechanism of the shear effect of helically wound structures is firstly explained, and then taking into account the shear effect, a quickly effective analysis method of the mechanical response for the helically wound structure is proposed. The efficient and accurate finite-element model of the unit cell which is used in the numeric implementation of the effective analysis method, is determined through the sensitivity analysis of meshing methods and periodic boundary conditions. Considering the practical application, the implementation of this method is validated for helically wound structures with equal-scale subcomponents and non-equal-scale subcomponents. The influence of the slenderness ratio on the shear effect is also explored in this work. This study provides a meaningful reference for the loading analysis and structural design of helically wound structures. • A method for calculating shear property of helically wound structure is proposed. • Mechanical mechanism of shear effect of helically wound structure is revealed. • The influence of the slenderness ratio of structure on shear effect is studied. • An accurate and efficient model of unit cell is determined by sensitivity analysis. • Shear property of equal-scale and non-equal-scale structures is discussed.

Micromechanical analysis of the effective stiffness of poroelastic composites

Within this work we investigate the role that the microstructure of a poroelastic material has on the resulting elastic parameters. We are considering the effect that multiple elastic and fluid phases at the same scale (LMRP model (L. Miller and R. Penta, 2020)) have on the estimation of the materials elastic parameters when compared with a standard poroelastic approach. We present a summary of both the LMRP model and the comparable standard poroelastic approach both derived via the asymptotic homogenization approach. We provide the 3D periodic cell problems with associated boundary loads that are required to be solved to obtain the effective elasticity tensor for both model setups. We then perform a 2D reduction of the cell problems, again presenting the 2D boundary loads that are required to solve the problems numerically. The results of our numerical simulations show that whenever investigating a poroelastic composite material with porosity exceeding 5% then the LMRP model should be considered more appropriate in incorporating the structural details in the Young’s moduli E1 and E3 and the shear C44. Whenever the porosity exceeds 20% it should also be used to investigate the shear C66. We find that for materials with less than 5% porosity that the voids are so small that a standard poroelastic approach or the LMRP model produce the same results.

Generalized thermo-mechanical framework for heterogeneous materials through asymptotic homogenization

A fundamental understanding of the interaction between microstructure and underlying physical mechanisms is essential, especially for developing more accurate multi-physics models for heterogeneous materials. Effects of microstructure on the material response at the macroscale are modeled by using the generalized thermomechanics. In this study, strain gradient theory is employed as a higher-order theory on the macroscale with thermodynamics modeled as a first-order theory on the microscale. Hence, energy depends only on the temperature such that we circumvent an extension of Fourier’s law and analyze the “simplest” thermo-mechanical model in strain gradient elasticity. Developing multiphysics models for heterogeneous materials is indeed a challenge and even this “simplest” model in generalized thermomechanics creates dozens of parameters to be determined. We develop a thermo-mechanical framework, in which microstructure is modeled as a periodic structure and through asymptotic homogenization approach, higher-order parameters at macroscopic scale are calculated. To illustrate the importance of higher-order parameters in overall thermo-mechanical response of a heterogeneous materials, finite element method (FEM) is employed with the aid of open-source codes (FEniCS). Verification example of a bulk system and several case studies of porous structures demonstrate how such numerical framework can be beneficial in the design of materials with tailored microstructures.

Two-Scale Asymptotic Homogenization Method for Composite Kirchhoff Plates with in-Plane Periodicity

This paper develops a two-scale asymptotic homogenization method for periodic composite Kirchhoff plates. In this method, a three-dimensional (3D) periodic plate problem is simplified as a Kirchhoff plate problem, which is governed by a fourth-order uniformly elliptic partial differential equation (PDE) with periodically oscillating coefficients. Then, a two-scale solution in an asymptotic expansion form is presented for the PDE, and it is found that the first-order perturbed displacement in the asymptotic solution is zero. Additionally, periodic boundary and normalization constraint conditions are proposed to determine the unique solution to unit cell problems. Moreover, standard finite element formulations for calculating the perturbed displacements are derived from the principle of virtual work. Physical interpretations of the influence functions are presented by analyzing the properties of self-balanced quasi-load vectors used for solving the influence functions. Numerical comparisons show that the present method is physically acceptable and highly accurate.

Calculation of the Single-Walled Carbon Nanotubes' Elastic Modulus by Using the Asymptotic Homogenization Method

This article uses the asymptotic homogenization method to investigate the mechanical characteristics of single-walled carbon nanotubes. The asymptotic homogenization method has been used to derive the equations for the homogenized elastic properties matrix. Then, MATLAB is used to model nanotubes and implement the finite element simulation. Two types of chiralities, including armchair and zigzag, are taken into account in this regard. Young's modulus and shear modulus have been estimated for both armchair and zigzag nanotubes in accordance with the relationships between the derived coefficients and molecular mechanical characteristics. Investigations have been done into how diameter and orientation affect the mechanical characteristics of carbon nanotubes. The research's findings are corroborated and generally consistent with those found in other articles.

On gas transport modes in matrix-fracture systems with arbitrary multiscale configurations

Tight shale reservoirs exhibit high heterogeneity and strong anisotropy in multiscale pore/fracture networks, with highly variable properties. The local equilibrium or non-equilibrium states vary spatially and are strongly controlled by the gas transport modes at each scale. A fundamental understanding of the coupling effects of gas flow in heterogeneous porous media with arbitrary scale ratios is crucial but not yet available. Here, we systematically and theoretically study the gas transport modes and gas flow velocity in multiscale matrix-fracture systems using the asymptotic homogenization method. A series of exact scaling laws for the gas velocity in heterogeneous porous media with arbitrary multiscale configurations are established, and the local equilibrium/non-equilibrium effects at each scale are analyzed in detail. It is shown that the gas transport modes between two adjacent porous media can be classified into four distinct types governed by two characteristic time scales (rather than two types as commonly reported). We demonstrate an ultrahigh pressure gradient in a thin depressurized zone in the matrix that can reach 103∼105 times the macroscopic pressure gradient, greatly increasing gas flow rates by three to five orders of magnitude. The hydraulically-created fractures not only provide preferential flow pathways, but more importantly, they increase the gas velocity in the matrix (which does not contain any fractures) by several orders of magnitude. The work also sheds light on the discrepancy between the observed high gas production and the experimentally measured low permeability in drilled cores.

Negative refraction in a single-phase flexural metamaterial with hyperbolic dispersion

We analyze the band structure of a single-phase metamaterial involving low-frequency flexural resonances by combining asymptotic homogenization and Bloch–Floquet analysis. We provide the closed-form expression of the dispersion relation in the whole Brillouin zone. The dispersion relation involves two effective, frequency-dependent, mass densities associated with symmetric and antisymmetric flexural resonances of the beams at the microscopic scale. We demonstrate that our simple locally-resonant structure produces at low-frequency band-gaps and, in the hyperbolic regions of the dispersion diagram, negative refraction. Our findings are validated by direct numerical calculations.

Cosserat continuum modelling of chiral scale-size effects and their influence on effective constitutive laws

We derive effective models from heterogeneous Cosserat media using asymptotic homogenization and taking into account size effects observed experimentally. The classification of the effective equations depends on the hierarchy of four characteristic lengths: the size of the heterogeneities ℓ, the Cosserat intrinsic lengths of the constituents ℓcoss and ℓchiral, and the overall characteristic length of the domain L. Depending on the different scale interactions between ℓcoss, ℓchiral, ℓ, and L we obtain either an effective Cauchy continuum or an effective Cosserat continuum with or without centrosymmetry. Herewith, we extend the work of Forest, Pradel, and Sab (Int. J. Solids Structures 38 (26–27): 4585–4608 ’01) by accounting for chiral effects.

A beam model for duoskelion structures derived by asymptotic homogenization and its application to axial loading problems

Duoskelion structures have been recently introduced by Barchiesi et al. (2021) as a proof-of-concept motif for a new class of metamaterials. The properties of these periodic beam-like chiral structural elements have been investigated, up to now, by means of a discrete model formulation whose predictions are obtained by numerical methods. In this paper we select a specific scaling law for micro stiffnesses aimed at deriving, via asymptotic homogenization, an internally-constrained Cosserat one-dimensional planar continuum model as the limit of a duoskelion structure. We analyze qualitatively and quantitatively the family of equilibrium configurations of the homogenized continuum when subjected to axial loading and compare the results of the analysis with those obtained by means of the discrete model formulation.

Multiscale Analysis of Bi-Layer Lattice-Filled Sandwich Structure Based on NIAH Method.

Bi-layer lattice-filled sandwich structures have good application prospects for multi-physics problems; however, high-precision numerical analysis methods are lacking. Recently, the newly proposed asymptotic homogenization method called the novel numerical implementation of asymptotic homogenization (NIAH) was further developed based on the Mindlin plate theory, which is a potential method for overcoming the above limitation. This study investigates the feasibility of this method for Bi-layer lattice-filled sandwich structures. The obtained results are compared to those from homogenization methods developed based on the Kirchhoff theory, and accordingly, the influence of the shear effect on the accuracy of the structural responses of the considered structures is studied. Subsequently, the impacts of the size effect, macrostructure type, and lattice type are also considered. The analysis results showed that, for most cases, the NIAH method can yield high-precision results for Bi-layer lattice-filled sandwich structures. When the number of lattice cells is insufficient or different layers of the lattice have excessive differences in their stiffness, the accuracy of the results obtained using the NIAH method is degraded.

Isogeometric analysis-based buckling optimization framework for grid-stiffened shells using asymptotic homogenization method and Rayleigh–Ritz method

Isogeometric analysis-based buckling optimization framework for grid-stiffened shells using asymptotic homogenization method and Rayleigh–Ritz method

Morphology influence on the acoustic properties of permeo-elastic media

This paper deals with the acoustics of permeo-elastic media. These media comprise a rigid porous skeleton with attached elastic films that interact with the fluid that saturates the pores. The pore-scale fluid/film interaction introduces elastic energy that interacts with viscous dissipation and inertial effects. The corresponding non-conventional flow regimes are studied by the two-scale asymptotic homogenisation method by successively considering permeo-elastic media with (i) fully connected pores, (ii) strictly unconnected pores (i.e. pores closed by films), (iii) partially connected pores, and (iv) mixed morphologies. This analysis allows to precisely describe how the morphology both influences, quantitatively and qualitatively, the interaction of the elastic and kinetic energies of the films with the viscous and kinetic energies of the fluid, and modifies the effective properties. It is shown that, in specific frequency ranges, the film–fluid system behaves as an equivalent visco-elastic fluid and that local film–fluid resonances can be observed. Numerical examples illustrate the main features in the different cases. The presented synthetic approach enables the design of materials that exhibit unconventional conductivity, unconventional compressibility, or both, in different or the same frequency range, by means of tuning their morphological and/or mechanical parameters.

Integrated wavelet-learning method for macroscopic mechanical properties prediction of concrete composites with hierarchical random configurations

In this research, a novel integrated wavelet-learning approach with high-accuracy and efficient prediction capability is developed to predict the effective elastic properties of concrete composites with hierarchical random configurations and heterogeneous elasticity in multiple spatial scales. The significant contributions of this work include: (1) The random structural heterogeneities and material uncertainties of concrete composites in meso-scale and micro-scale are modeled on the basis of modified authors’ previous researches and Weibull probabilistic model separately. (2) Innovative stochastic three-scale asymptotic homogenization method (STAHM), background grid technique and padding strategy are presented to thoroughly acquire the doubly random geometry and material features of concrete composites with hierarchical random configurations for constructing concrete material databases. (3) The wavelet transform is utilized to preprocess the high-dimensional data features of the material databases, and the wavelet coefficients are employed as the novel input neurons for efficiently establishing more plain wavelet-neural networks that possess increasing capacity and resisting noise for approximating mappings of high complexity and nonlinearity. The numerical experiments and achieved results demonstrate that the integrated wavelet-learning approach is robust for the high-accuracy and efficient prediction of effective elastic properties of concrete materials with random heterogeneity patterns in multiple spatial scales.

Machine learning based asymptotic homogenization and localization: Predictions of key local behaviors of multiscale configurations bearing microstructural varieties

AbstractIn this article, a general framework is devised for reliable predictions over the local behaviors, such as the failure strength and stress intensity factor, of multiscale configurations bearing microstructural varieties. Methodologically, a mutually complementing use of asymptotic analysis and machine learning is addressed. On using asymptotic analysis, we manage to identify several key but implicit interrelationships, which link the local quantities of our interest with other onsite mean‐field quantities. Then these identified interrelationships are represented by neural networks in an offline stage, where the training data are also generated based on the formulation devised through asymptotic analysis. Then for the upcoming online simulation, those desired local behaviors can be quickly captured without resolving the microstructural configurations. Numerical examples are presented showing that for relatively complex multiscale configurations, the predicting error here can be theoretically controlled at a satisfactory level.

Homogenization of discrete mesoscale model of concrete for coupled mass transport and mechanics by asymptotic expansion

Mass transport phenomenon in concrete structures is strongly coupled with their mechanical behavior. The first coupling fabric is the Biot's theory according to which fluid pressure interacts with solid stress state and volumetric deformation rate of the solid induces changes in fluid pressure. Another coupling mechanism emerges with cracks which serve as channels for the fluid to flow through them and provide volume for fluid storage. Especially the second coupling mechanism presents a challenge for numerical modeling as it requires detailed knowledge about cracking process. Discrete mesoscale mechanical models coupled with mass transport offer simple and robust way to solve the problem. On the other hand, however, they are computationally demanding. In order to reduce this computational burden, the present paper applies the asymptotic expansion homogenization technique to the coupled problem to deliver (i) continuous and homogeneous description of the macroscopic problem which can be easily solved by the finite element method, (ii) discrete and heterogeneous mesoscale problem in the periodic setup attached to each integration point of the macroscale along with (iii) equations providing communication between these two scales. The transient terms appear at the macroscale only, as well as the Biot's coupling terms. The coupling through cracking is treated at the mesoscale by changing conductivity of the conduit elements according to the mechanical solution, otherwise the two mesoscale steady state problems are decoupled and can be therefore solved in a sequence. This paper presents verification studies showing performance of the homogenized solution.

Identification of MEMS Geometric Uncertainties through Homogenization

Fabrication imperfections strongly influence the functioning of Micro-Electro-Mechanical Systems (MEMS) if not taken into account during the design process. They must be indeed identified or precisely predicted to guarantee a proper compensation during the calibration phase or directly in operation. In this work, we propose an efficient approach for the identification of geometric uncertainties of MEMS, exploiting the asymptotic homogenization technique. In particular, the proposed strategy is experimentally validated on a MEMS filter, a device constituted by a complex periodic geometry, which would require high computational costs if simulated through full-order models. The complex periodic structure is replaced by an equivalent homogeneous medium, allowing a fast optimization procedure to identify imperfections by comparing a simplified analytical model with the experimental data available for the MEMS filter. The actual over-etch, obtained after the release phase, and the electrode offset of a fabricated MEMS filter are effectively identified through the proposed strategy.

Water vapor permeability of flax fibers reinforced raw earth: Experimental and micro-macro modeling

The expected impact of this study is to propose a bio-sourced composite material coupling mechanical and hygrothermal performances. This paper deals with the prediction of hygric properties of compacted raw-earth material reinforced with the flax fibers. New experiments have been performed on fibers and earth composite specimens. Then two modeling approaches have been applied using mixing theory and asymptotic homogenization. It was showed, that besides adding flax fibers improves the mechanical behavior of raw earth as strength and shrinkage reduction, their impact on the hygric property as water vapor permeability was also proved. For longitudinal arrangement of flax fibers with different fibers contents, experimental and modeling results are in good agreement.