Two-scale Homogenization Research Articles

Intra-yarn fibre hybridisation effect on homogenised elastic properties and micro and meso-stress analysis of 2D woven laminae: Two-scale FE model

In this paper, the effect of intra-yarn fibre hybridisation on the homogenised elastic properties and micro- and meso-scale matrix stress fields in 2D woven composite laminae (i.e. plain, 2/2 basket, 2/2 twill and 5-harness satin) is studied with a two-scale homogenisation scheme—employing a representative volume element model at micro-scale and a repeating unit cell model at meso-scale. The study is focused on S-glass/polypropylene/epoxy woven laminae with intra-yarn fibre hybridisation. A modified random sequential expansion algorithm generates microstructure for the micro-mechanical model, and a periodic meso-structure is used to generate the weave architecture for the meso-mechanical model. Both models are verified using analytical models. It is found that intra-yarn fibre hybridisation can significantly alter the homogenised properties as well as the micro- and meso-scale matrix stress fields—depending on the degree of hybridisation (i.e. the combination of S-glass and PP fibre volume fractions). Moreover, the homogenised lamina properties are found to be less sensitive to weave architecture and yarn thickness, but more so to the degree of intra-yarn fibre hybridisation, yarn width and yarn spacing. It is shown that the lamina properties can be tailored, and the micro- and meso-stress fields can be manipulated, by intra-yarn fibre hybridisation and weave architectures.

Homogenization of Smoluchowski-type equations with transmission boundary conditions

Abstract In this work, we prove a two-scale homogenization result for a set of diffusion-coagulation Smoluchowski-type equations with transmission boundary conditions. This system is meant to describe the aggregation and diffusion of pathological tau proteins in the cerebral tissue, a process associated with the onset and evolution of a large variety of tauopathies (such as Alzheimer’s disease). We prove the existence, uniqueness, positivity and boundedness of solutions to the model equations derived at the microscale (that is the scale of single neurons). Then, we study the convergence of the homogenization process to the solution of a macro-model asymptotically consistent with the microscopic one.

Multiscale simulation scheme by using a finite-difference derivative framework for lubrication surface

To exactly describe lubrication problems modelling micro-scale influence in rough lubricated contact is unavoidable. In this study a novel computational scheme for lubricated problem is presented based on two-scale homogenization method, which facilitates solving the two-scale problem. Homogenization model of mass fluxes is implemented at the micro-scale domain, and represents the influence of surface topography. A linearization method is developed to derive finite derivatives of mass fluxes at the micro-scale domain defined by scale separation. The key strength of the proposed procedure is the finite difference derivative coupling which represents the lubricated problem inclusive of the effect of surface topography. The effectiveness of the proposed multiscale scheme is validated by comparing with the deterministic method. The numerical results show that increasing topography amplitude can improve the performance of journal bearing in terms of load carrying capacity and friction coefficient.

Effective magnetoelastic responses of hybrid fiber composites with viscoelastic matrices

This study examines the overall magnetoelastic responses of unidirectional fiber composites constructed by magnetostrictive fibers embedded in viscoelastic polymer matrices that had been reinforced by stiff fillers. In some polymer-based fiber composites, modifying polymer’s property by infiltrating particulate fillers is necessary. To design such hybrid composite systems with magnetoelastic coupling, it requires a detailed understanding of aggregate responses. In this study, we formulate a composite constitutive model based on a two-scale homogenization scheme for effective coupled viscoelastic and magnetoelastic responses for hybrid fiber composites. We first verify the validity of the model by comparing the predictions with experimental results available in literatures. We next investigate the performance of a hybrid composite in light of blocked stress and free strain which are two common factors of characterizing sensors and actuators. We finally elucidate how viscoelastic polymer matrices affect the overall magnetoelastic responses and ΔE-effect phenomenon in terms of some critical parameters such as phase concentrations, loading rates, prestresses and temperatures. The findings of this study serve as a preliminary guidance for the rational design of hybrid magnetostrictive composite materials. The proposed mathematics model can further integrate into a finite element framework for composite structure analysis.

Wave Polarization Control in Anisotropic Locally Resonant Materials

Elastic wave propagation in solids can be controlled and manipulated by properly designed metamaterials. In particular, polarization conversion can be obtained by using anisotropic materials. In this paper, we propose a three-component locally resonant material with non-symmetrically coated inclusions, and we study the effect of the anisotropic equivalent mass on band gap formation and the polarization conversion of elastic waves. The equivalent frequency-dependent mass tensor is obtained through the two-scale homogenization approach. The study of the eigenvalues of the mass tensor enables to predict band gaps and polarization bands, as well as identifying a priori the effect of different geometric and material parameters, thus opening the way to metamaterial optimization.

Thermo-mechanical behavior of 3D multi-directional braided composites based on a two-scale method

Two-scale modeling is adopted to investigate the thermo-mechanical behavior of 3D four-directional (3D4D), 3D five-directional (3D5D), and 3D full five-directional (3DF5D) braided composites. Based on the stress-strain relationship considering thermal expansion and the periodic boundary conditions, the elastic constants and the coefficients of thermal expansion (CTE) of the three types of braided composites are predicted by a two-scale homogenization method. The micro stress under free expansion and thermo-mechanical coupling is also simulated. The calculated results are in good agreement with experimental results from relevant references. The numerical results show that the longitudinal elastic and thermal expansion properties are gradually improved with the increase of axial yarn content from 3D4D to 3D5D and then to 3DF5D braided composites. The braiding angle corresponding to the zero longitudinal CTE of each braided structure is basically about 40°. Furthermore, with the increase of temperature, the longitudinal micro-stress in yarns increases gradually, but that in matrix drops. These conclusions will provide a reliable basis for the structural optimization design and safety evaluation of 3D multi-directional braided composites in a thermal environment.

An effective Fourier spectral phase-field approach for ferroelectric materials

In this article, we propose an efficient iterative numerical algorithm applying a Fourier spectral method to accurately compute the coupled elasto–electrostatic equilibrium equations in ferroelectric microstructures. The method is implemented for a phase-field model for domain pattern formation and domain kinetics evolution in tetragonal perovskite ferroelectrics, that uses spontaneous polarization as order parameter and includes the piezoelectric effect in the constitutive equation. The scheme is based on a two-scale homogenization approach, introducing local perturbations for stress and dielectric displacement. Green’s functions are derived in Fourier space for the coupled constitutive equations. Furthermore, using model parameters for BaTiO3 we investigate domain formation in 2D and 3D, and the polarization switching process starting from a critical nucleus subjected to an applied electric field and compressive stress. The simulation results based on the proposed method reveal that for polarization switching under external fields, intermediately appearing domains with 90° rotated polarization are formed, that reduce electrostatic or elastic strain energy.

Determination of the Effective Permeability of Doubly Porous Materials by a Two-Scale Homogenization Approach

Determination of the Effective Permeability of Doubly Porous Materials by a Two-Scale Homogenization Approach

Two-Scale Homogenization of Piezoelectric Perforated Structures

We are interested in the homogenization of the elastic-electric coupling equation with rapidly oscillating coefficients, in a periodically perforated piezoelectric body. The holes, whose size are supposed to tend to zero, are periodically distributed. We give a new approach, based on the two-scale convergence, and we justify the two first terms in the usual asymptotic development of the solution. A two-scale homogenized system is obtained as the limit of the periodic problem, and explicit formulae of elastic, piezoelectric and dielectric homogenized coefficients are reported. In the static limit, the method provides homogenized electroelastic coefficients coinciding with those deducted from alternative approaches.

Machine learning-based multiscale constitutive modelling: Development and application to dual-porosity mass transfer

In multiscale modelling, multiple models are used simultaneously to describe scale-dependent phenomena in a system of interest. Here we introduce a machine learning (ML)-based multiscale modelling framework for modelling hierarchical multiscale problems. In these problems, closure relations are required for the macroscopic problem in the form of constitutive relations. However, forming explicit closures for nonlinear and hysteretic processes remains challenging. Instead, we provide a framework for learning constitutive mappings given microscale data generated according to micro and macro transitions governed by two-scale homogenisation rules. The resulting data-driven model is then coupled to a macroscale simulator leading to a hybrid ML-physics-based modelling approach. Accordingly, we apply the multiscale framework within the context of transient phenomena in dual-porosity geomaterials. In these materials, the inter-porosity flow is a complex time-dependent function making its adoption within flow simulators challenging. We explore nonlinear feedforward autoregressive ML strategies for the constitutive modelling of this sequential problem. We demonstrate how to inject the resulting surrogate constitutive model into a simulator. We then compare the resulting hybrid approach to traditional dual-porosity and microscale models on a variety of tests. We show the hybrid approach to give high-quality results with respect to explicit microscale simulations without the computational burden of the latter. Lastly, the steps provided by the multiscale framework herein are sufficiently general to be applied to a variety of multiscale settings, using different data generation and learning techniques accordingly.

Traveling Fronts in a Reaction–Diffusion Equation with a Memory Term

Based on a recent work on traveling waves in spatially nonlocal reaction–diffusion equations, we investigate the existence of traveling fronts in reaction–diffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reaction–diffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that two-scale homogenization of spatially periodic systems can induce spatially homogeneous systems with temporal memory. The existence of fronts is proved using comparison principles as well as a reformulation trick involving an auxiliary speed that allows us to transform memory terms into spatially nonlocal terms. Deriving explicit bounds and monotonicity properties of the wave speed of the arising traveling front, we are able to establish the existence of true traveling fronts for the original problem with memory. Our results are supplemented by numerical simulations.

Homogenization of large deforming fluid-saturated porous structures

The two-scale computational homogenization method is proposed for modeling of locally periodic fluid-saturated media subjected to a large deformation induced by quasistatic loading. The periodic heterogeneities are relevant to the mesoscopic scale at which a double porous medium constituted by a hyperelastic skeleton and an incompressible viscous fluid is featured by large contrasts in the permeability. Within the Eulerian framework related to the current deformed configuration, the two-scale homogenization approach is applied to a linearized model discretized in time, being associated with an incremental formulation. For this, the equilibrium equation and the mass conservation expressed in the spatial configuration are differentiated using the material derivative with respect to a convection velocity field. The homogenization procedure of the linearized equations provides effective (homogenized) material properties which are computed to constitute the incremental macroscopic problem. The coupled algorithm for the multiscale problem is implemented using the finite element method. Illustrative 2D numerical simulations of a poroelastic medium are presented including a simple validation test.

Theoretical formulation and computational aspects of a two-scale homogenization scheme combining the TPM and FE[formula omitted] method for poro-elastic fluid-saturated porous media

The focus of this investigation lies on the development of a two-scale homogenization scheme for poro-elastic fluid-saturated porous media. For this purpose, the general concepts of the Theory of Porous Media (TPM) are combined with the FE2 method. After an introduction of the basics of TPM, the weak forms for the macroscopic and the microscopic scale will be formulated and the averaged macroscopic tangent moduli considering the microscale will be derived. Additionally, the formulation of lower level boundary conditions, which refer to the quantities that will be transmitted from the macro- to the microscale, in strict compliance with the Hill–Mandel homogeneity condition, is derived. Finally, a numerical example will be presented, pointing out the gained features of the methodology.

A simplified homogenization model applied to viscoelastic behavior of cortical bone at ultrasonic frequencies

Cortical bone is a complex multiscale medium and its study is of importance for clinical fracture prevention. In particular, cortical attenuation is known to be linked with shock energy absorption and ability to resist fracture. However, the links between cortical bone absorption and its multiscale structure are still not well understood. This work is about the use of homogenized tensors in order to characterize the viscoelastic behavior of cortical bone at ultrasonic frequencies, i.e., about 0.1 to 10 MHz. Such tensors are derived from the cell problem via two-scale homogenization theory for linear elastic and Kelvin–Voigt viscoelastic descriptions. The elliptic formulations obtained from the cell problems are implemented within the range of medically-observed porosities. Microstructure is assessed considering cubic cells with cylindrical inclusion and transverse isotropic assumption. A simplified model, adding one temporal parameter τ per phase, allows a good agreement with experimental data. The corresponding attenuation is proportional to the square of the frequency, in agreement with Kramer–Kronig relations. This development is proposed in the context of robust clinical inverse problem approaches using a restricted number of parameter. Two main properties for the material filling the pores are adjusted and discussed: absorption and shear contribution. Best agreement with experimental data is observed for material inside the pores being solid and highly attenuating.

Viscoelastic modeling and characterization of thin-ply composite laminates

Thin-ply composite laminates are promising materials for deployable spacecraft structures owing to their low mass and high folding curvature. However, the deployment force and the shape accuracy of the deployed configuration after a long duration of stowage is adversely affected by the inherent viscoelastic behavior of the polymer matrix. This paper presents a computational and experimental investigation into the effective viscoelastic response of thin-ply composite laminates made of woven carbon fabrics and an epoxy matrix intended for deployable spacecraft applications. A two-scale homogenization approach is utilized to model thin-ply composites as viscoelastic plates represented with relaxation ABD matrices. Micrographic analysis and relaxation tests are conducted to obtain the geometric and material characteristics that are incorporated into the model. The bending relaxation behavior is experimentally characterized using the recently developed column bending test methodology to compare against model predictions. The relaxation response is studied in two different orientations. The effects of ply phase arrangement, laminate thickness, fiber volume fraction, specimen width, and coordinate rotation are investigated.

Level set-based topology optimization for graded acoustic metasurfaces using two-scale homogenization

This paper proposes a level set-based topology optimization method for acoustic metasurfaces consisting of multiple types of unit cells. As a type of acoustic metasurface, we focus on a two-layered metasurface, where each layer contains various types of unit cell designs. To handle this metasurface, we first propose a two-scale homogenization method based on previous studies targeting metasurfaces composed of a single type of unit cell structure. As a result of homogenization, each layer of the array of unit cells is replaced with an interface with a nonlocal transmission condition. Next, an optimization problem is formulated. We set an objective function to achieve negative refraction, and macroscopic responses obtained by the homogenization method are used to define the objective function. The material distributions in each unit cell of the metasurface are set as the design variables. Based on these settings, we perform a sensitivity analysis based on the concept of the topological derivative. A level set-based topology optimization method is introduced to solve the optimization problem, and two-dimensional numerical examples are provided to demonstrate the validity of the proposed method. An optimal design of a two-layered acoustic metasurface that induces negative refraction is proposed, and a discussion of its mechanism is provided.

A two-scale strategy for the modeling of hook and loop fasteners

Hook-and-loop-like joints have been widely used in many engineering applications and even observed in nature. The demand for computational models to predict the behavior of fasteners is increasing as they will be key for reducing design process time and manufacturing costs. Here, we develop a bottom-up two-scale modeling strategy, which enables us to capture the mechanical performance of a hook-and-loop fastener at both micro- and macroscales. In particular, we employ a two-scale homogenization approach capable of predicting the mechanical behavior of the hook-and-loop fastener. The model starts with a micromechanical model of a hook-and-loop fastener which considers the detailed geometry of the individual building blocks of the hook-and-loop fastener and their statistical geometrical variability through a so-called Representative Hook and Loop Element (RHLE) computational domain. The second scale considers the homogenization of the micromechanical model into a Detachment Process Zone (DPZ) which reveals an emerging length scale. The resulting effective traction-separation law concept is then employed at the macroscale with a Double Cantilever Beam (DCB) test using a macroscale model, which is then validated with experiments. The results suggest that the two-scale model strategy is able to predict the mechanical behavior of hook-and-loop fastener and to capture the main deformation and dissipative mechanisms at the relevant length scales.

Double poroelasticity derived from the microstructure

We derive the balance equations for a double poroelastic material which comprises a matrix with embedded subphases. We assume that the distance between the subphases (the local scale) is much smaller than the size of the domain (the global scale). We assume that at the local scale both the matrix and subphases can be described by Biot’s anisotropic, heterogeneous, compressible poroelasticity (i.e. the porescale is already smoothed out). We then decompose the spatial variations by means of the two-scale homogenization method to upscale the interaction between the poroelastic phases at the local scale. This way, we derive the novel global scale model which is formally of poroelastic-type. The global scale coefficients account for the complexity of the given microstructure and heterogeneities. These effective poroelastic moduli are to be computed by solving appropriate differential periodic cell problems. The model coefficients possess properties that, once proved, allow us to determine that the model is both formally and substantially of poroelastic-type. The properties we prove are a) the existence of a tensor which plays the role of the classical Biot’s tensor of coefficients via a suitable analytical identity and b) the global scale scalar coefficient bar{mathcal {M}} is positive which then qualifies as the global Biot’s modulus for the double poroelastic material.

Two-scale homogenization of hydrodynamic lubrication in a mechanical seal with isotropic roughness based on the Elrod cavitation algorithm

A two-scale homogenization method for modelling the hydrodynamic lubrication of mechanical seals with isotropic roughness was developed and presented the influence of surface topography coupled into the lubricating domain. A linearization approach was derived to link the effects of surface topography across disparate scales. Solutions were calculated in a polar coordinate system derived based on the Elrod cavitation algorithm and were determined using homogenization of periodic simulations describing the lubrication of a series of surface topographical features. Solutions obtained for the hydrodynamic lubrication regime showed that the two-scale homogenization approach agreed well with lubrication theory in the case without topography. Varying topography amplitude demonstrated that the presence of surface topography improved tribological performance for a mechanical seal in terms of increasing load-carrying capacity and reducing friction coefficient in the radial direction. A Stribeck curve analysis was conducted, which indicated that including surface topography led to an increase in load-carrying capacity and a reduction in friction. A study of macro-scale surface waviness showed that the micro-scale variations observed were smaller in magnitude but cannot be obtained without the two-scale method and cause significant changes in the tribological performance.

Topology optimization of acoustic metasurfaces by using a two-scale homogenization method

In this paper, we propose a level set-based topology optimization method for the unit-cell design of acoustic metasurfaces by using a two-scale homogenization method. Based on previous works, we first propose a homogenization method for acoustic metasurfaces that can be combined with topology optimization. In this method, a nonlocal transmission condition depending on the unit cell of the metasurface appears in a macroscale problem. Next, we formulate an optimization problem within the framework of a level set-based topology optimization method, wherein an objective functional is expressed as the macroscopic responses obtained through the homogenization, and material distributions in the unit cell are set as design variables. A sensitivity analysis is conducted based on the concept of the topological derivative. To confirm the validity of the proposed method, two-dimensional numerical examples are provided. First, we provide a numerical example that supports the validity of the homogenization method, and we then perform optimization calculations based on the waveguide settings of the acoustic metasurfaces. In addition, we discuss the mechanism of the obtained optimized structures.