Proposed Homogenization Method Research Articles

Stress-driven nonlocal homogenization method for cellular structures

Strong size-dependent mechanical behaviors can be observed in cellular structures violating the principle of scale separation and cannot be captured by classical homogenization methods. This paper proposes a stress-driven nonlocal homogenization method to capture the size-dependent mechanical behavior due to the nonlocal force of cellular structures. A nonlocal discrete element model is proposed first to describe the nonlocal deformation mechanism of cellular structures. Then, a continuum stress-driven nonlocal homogenization method is calibrated by deriving the continuum limit from the nonlocal discrete element model. The continuum homogenization method releases the assumption of scale separation by introducing an intrinsic length, which can be calibrated by high throughput numerical computation. Also, for efficient prediction of size-dependent mechanical behaviors, an offline dataset of the intrinsic length is constructed for different unit cells. With the help of the offline dataset, the proposed homogenization method improves the accuracy of the classic homogenization method and reduces the computational cost of the high-fidelity finite element method. Finally, a cellular rod under tension is used as an application to illustrate the efficiency and accuracy of the proposed homogenization method. Results indicate that compared with the classic homogenization method, the relative error of the proposed homogenization method is less than 1% which has a good consistency with the high-fidelity method. Moreover, the computational efficiency of the proposed homogenization method is more than five times that of the high-fidelity finite element method.

Two-scale asymptotic homogenization analysis of piezoelectric composite materials in generalized curvilinear coordinates

This paper presents a comprehensive study on applying the two-scale asymptotic homogenization method to derive the effective properties of piezoelectric composite materials characterized by generalized periodicity in curvilinear coordinates. The methodology involves formulating the homogenization problem in a general curvilinear framework suitable for materials with complex geometric configurations. By systematically deriving the homogenized equations and effective coefficients, the paper provides a robust theoretical foundation for understanding the macroscopic behavior of piezoelectric composites under various loading conditions. The model’s versatility is demonstrated through its application to specific curvilinear coordinate systems, including the helix coordinate system and cylindrical coordinates with wavy geometry. These examples highlight the potential of the proposed approach in analyzing and designing piezoelectric materials with intricate geometries. The state-of-the-art numerical homogenization methods have also validated the proposed homogenization method, ensuring its accuracy and robustness.

A novel homogenization method for periodic piezoelectric composites via diffused material interface

In this article, we propose a novel homogenization method for piezoelectric composites with periodic microstructures. In the proposed homogenization method, one can find the analytical expressions for the material properties of the composites in the representative volume element (RVE) via diffused material interface method. The availability of analytical expressions for the material properties in the entire domain of the RVE enables one to determine the effective homogenized material properties with the standard finite element method. As the proposed homogenization method regularizes the discontinuity in material properties across the interfaces, there is no need to explicitly track the material interfaces while implementing the finite element method for determining the effective material properties. Hence, one can implement the proposed homogenized method for any piezoelectric composites with complicated material interfaces and determine the effective material properties. In this study, we have considered the piezoelectric composites to be comprised of periodic microstructures where the RVE constitutes a matrix with an inclusion of a specific shape. We have carried out a study on the effect of the shape of inclusion viz. square-shaped, I-shaped, T-shaped, and plus-shaped, and the size of inclusion on the effective piezoelectric material properties. We have also studied the influence of shape and size of inclusion on the electro-mechanical response of a homogenized piezoelectric continuum. To implement the proposed homogenization method, we have used Gridap, an open-source finite element toolbox in Julia that provides very compact codes freely available to all the researchers and makes a third-party verification of the proposed method straightforward.

Chiral Cosserat homogenized constitutive models of architected media based on micromorphic homogenization

The present contribution aims to build chiral Cosserat effective models of architectured media prone to such effects, recoursing to dedicated homogenization methods. In order to develop a method in full generality, the effective Cosserat model is obtained from the degeneration of the micromorphic effective continuum, by projecting the micromorphic kinematic degrees of freedom onto the Cosserat variables. The proposed homogenization method relies on variational principles and a decomposition of the microscopic displacement into a so-called homogeneous part and a periodic fluctuation. The homogeneous contribution to the total microscopic displacement accounts for the postulated effective generalized continuum, and it is corrected by a displacement fluctuation. The energy of the fluctuating displacement is shown to define a measure of the accuracy of the homogenized chiral Cosserat model. Applications of the proposed homogenization framework are done for centrosymmetric and non-centrosymmetric chiral architected media.

Multifield constitutive identification of non-conventional thermo-viscoelastic periodic Cauchy materials

The present work proposes a variational-asymptotic homogenization technique for non-conventional thermo-viscoelastic periodic microstructured materials. According to second sound theories, the heat flux vector linearly depends upon the history of temperature gradient and the heat conduction tensor represents the kernel of the relative hereditary integral, thus overcoming the paradox inferred from the usual Fourier’s law of thermal waves propagating at infinite speed. Down-scaling relations have been provided in the frequency domain, relating the transformed micro displacement and relative temperature fields to the corresponding macro variables and their gradients through frequency-dependent perturbation functions which convey the influence of the underlying microstructural heterogeneity. Average field equations of infinite order have been derived. Transformed field equations of the equivalent first-order medium have been obtained as Euler–Lagrange equations of a suitable functional whose first variation is properly truncated. They are characterized by frequency-dependent overall constitutive tensors, whose closed form has been provided. A benchmark test assesses the capabilities of the proposed homogenization method, where the solutions relative to a periodic, bi-phase, thermo-viscoelastic material and to the equivalent homogenized medium match under the hypothesis of periodic source terms.

An improved homogenization method for the treatment of strong absorbers in pebble-bed HTGR

In pebble-bed high temperature gas-cooled reactors (HTGRs), strong absorbers including control rods and absorber balls are placed in the reflector. The strong absorber regions need to be homogenized for the convenience of downstream whole-core neutronics diffusion calculation. Previously, a homogenization method based on the generalized equivalence theory (GET) was studied. This paper proposes an improved homogenization method by employing a more rigorous discontinuity factor (DF) model, for more accurate treatment of the strong absorbers in the HTGR reflector. The proposed method has been implemented in the PANGU code developed by Institute of Nuclear and New Energy Technology (INET), Tsinghua University. Numerical results suggest that the newly proposed homogenization method achieves good agreement with the heterogeneous Monte Carlo transport solution and experimental result.

Homogenized characterization of cylindrical Li‐ion battery cells using elliptical approximation

Homogenization and finding the constitutive model of jellyroll in cylindrical lithium-ion batteries can be challenging because of their form factor. Taking samples out of the original jellyroll wounding or compressing cell assembly in its cylindrical coordinates are two possibilities for measuring the homogenized lateral strength of the cell. However, the former causes loss of accuracy due to changing constraints and electrolyte environment, and the latter requires complex fixtures that are not readily available or even practical to manufacture. Various approaches have been suggested by researchers to circumvent the above difficulties and allow the extraction of hardening curves. However, the precision of those approaches diminishes when the cells are under global compression vs local punch deformations. In this study, an updated homogenization method is established, using a lateral compression test on the jellyroll. The homogenization method is based on the assumption that the circular cross-section of the jellyroll under compression is deformed in an elliptical shape. Then the principle of virtual work is used to extract the hardening curve. To validate the above characterization model, isotropic and anisotropic finite element models were developed using crushable foam and modified honeycomb material models from the LS-DYNA library. Four sets of cell-level experiments were performed on cylindrical batteries using custom-designed fixtures, including flat lateral compression, rod indentation, hemispherical punch, and three-point bending. The voltage and surface temperature of the batteries were measured to capture the onset of short circuit during the tests. Comparison of the simulation results confirmed that the proposed homogenization method and the FE models can predict the behavior of cylindrical lithium-ion batteries with much higher accuracy compared to the currently available methods presented in the literature.

A homogenization method for nonlinear inhomogeneous elastic materials

BackgroundFast simulation techniques are strongly favored in computer graphics, especially for the nonlinear inhomogeneous elastic materials. The homogenization theory is a perfect match to simulate inhomogeneous deformable objects with its coarse discretization, as it reveals how to extract information at a fine scale and to perform efficient computation with much less DOF. The existing homogenization method is not applicable for ubiquitous nonlinear materials with the limited input deformation displacements. MethodsIn this paper, we have proposed a homogenization method for the efficient simulation of nonlinear inhomogeneous elastic materials. Our approach allows for a faithful approximation of fine, heterogeneous nonlinear materials with very coarse discretization. Modal analysis provides the basis of a linear deformation space and modal derivatives extend the space to a nonlinear regime; based on this, we exploited modal derivatives as the input characteristic deformations for homogenization. We also present a simple elastic material model that is nonlinear and anisotropic to represent the homogenized materials. The nonlinearity of material deformations can be represented properly with this model. The material properties for the coarsened model were solved via a constrained optimization that minimizes the weighted sum of the strain energy deviations for all input deformation modes. An arbitrary number of bases can be used as inputs for homogenization, and greater weights are placed on the more important low-frequency modes. ResultsBased on the experimental results, this study illustrates that the homogenized material properties obtained from our method approximate the original nonlinear material behavior much better than the existing homogenization method with linear displacements, and saves orders of magnitude of computational time. ConclusionsThe proposed homogenization method for nonlinear inhomogeneous elastic materials is capable of capturing the nonlinear dynamics of the original dynamical system well.

Homogenized model of environmental barrier coatings for evaluation of energy release rate

AbstractFor efficient calculation of energy release rate (ERR) for interface cracks in environmental barrier coatings (EBCs) with a columnar layer, we propose a simple and versatile homogenization method of the columnar layer. The columnar layer consisting of a number of fine columns is modeled as an anisotropic homogenized layer whose elastic properties are isotropic in the plane perpendicular to the layer thickness direction. The homogenization was achieved by a finite element method (FEM) analysis for a single columnar model to obtain elastic parameters that characterize the homogenized layer. For validation of the homogenization method, we conduct an FEM analysis for a typical EBC structure and obtain the ERRs of an interface crack with various crack length and columnar size. We demonstrate that the homogenization method provides estimation of ERR with a good accuracy on the safer side, which is advantageous for EBC structure design. In addition, the method can reduce the calculation time by more than 50%. From these results, we conclude that the proposed homogenization method can be applied generally for estimation of ERR in EBCs with a columnar layer.

Influence of Micro‐Inertia on the Macroscale ‐ A Fully‐Coupled Direct Homogenization Framework for Dynamics

AbstractThis contribution presents a numerical example of a locally‐resonant microstructure, analyzed using a fully‐coupled large strain homogenization framework including dynamics at both scales. The fundamental assumptions the framework is based on are summarized and results showcasing the ability of the proposed homogenization method are given.

Model reduction in computational homogenization for transient heat conduction

This paper presents a computationally efficient homogenization method for transient heat conduction problems. The notion of relaxed separation of scales is introduced and the homogenization framework is derived. Under the assumptions of linearity and relaxed separation of scales, the microscopic solution is decomposed into a steady-state and a transient part. Static condensation is performed to obtain the global basis for the steady-state response and an eigenvalue problem is solved to obtain a global basis for the transient response. The macroscopic quantities are then extracted by averaging and expressed in terms of the coefficients of the reduced basis. Proof-of-principle simulations are conducted with materials exhibiting high contrast material properties. The proposed homogenization method is compared with the conventional steady-state homogenization and transient computational homogenization methods. Within its applicability limits, the proposed homogenization method is able to accurately capture the microscopic thermal inertial effects with significant computational efficiency.

Determination of Fractured Rock’s Representative Elementary Volume by a Numerical Simulation Method

The present study aims at developing a numerical program called DISSIM which can analyze the homogenization of rock massifs using a new subroutine which calculates Representative Elementary Volume (REV). The DISSIM methodology consists of two steps. The first step involves the modeling of the fractured network in order to provide a surface simulation that represents the real fracture of the examined front. The second step is to numerically model the wave propagation through the simulated fracture network while characterizing the attenuation of vibrations due to the effect of discontinuities. This part allows us to determine in particular the wave propagation velocity through the fractured mass, from which we can determine the homogenized Young's modulus. However, after extensive bibliographic research, it was realized that a third step appeared to be necessary. In fact, it is necessary to look for a representative elementary volume on which we apply the proposed homogenization method. Two types of the representative elementary volume are proposed in this article, the geometric REV and the mechanical REV. The presentation of these two types of REV and the DISSIM methodology are detailed in this paper. Then, this methodology was applied to the study of a real case. The present research provides a method allowing the calculation of both types of REV for fissured rocks. The case study yielded comparable results between the mechanical REV and the geometric REV, which is compatible with previous research studies.

Plasmon-induced nonlinearity enhancement and homogenization of graphene metasurfaces.

We demonstrate that the effective 3rd-order nonlinear susceptibility of a graphene sheet can be enhanced by more than 2 orders of magnitude by patterning it into a graphene metasurface. In addition, to gain deeper physical insights into this phenomenon, we introduce a versatile homogenization method, which is subsequently used to characterize quantitatively this nonlinearity enhancement effect by calculating the effective linear and nonlinear susceptibility of graphene metasurfaces. The accuracy of the proposed homogenization method is demonstrated by comparing its predictions to those obtained from the Kramers-Kronig relations. This work may open new opportunities to explore novel physics pertaining to nonlinear optical interactions in graphene metasurfaces.

An isotropic self-consistent homogenization scheme for chemo-mechanical healing driven by pressure solution in halite

Mechanical healing is the process by which a damaged material recovers mechanical stiffness and strength. Pressure solution is a very effective healing mechanism, common in crystalline media. Chemical reactions initiate at the location of microstructure defects, which would be very difficult to account for in a homogenization scheme that separates the solid and the pore phases, as is classically the case. Here, we propose a novel chemo-mechanical homogenization model in which the inclusion is not a grain, but rather, a space that contains a pore and discontinuities, where chemical processes take place. Mass and energy balance equations are rigorously established to predict the chemical eigenstrain of each inclusion, which, added to the elastic deformation, provides the microstrain of each inclusion. From there, Hill’s inclusion-matrix interaction law is used to upscale strains and stresses at the scale of a Representative Elementary Volume (REV). The model was calibrated against experimental results published in the literature for salt rock. Subsequent sensitivity analyses show that in samples with same porosity but with inclusions that have different initial void sizes, inclusions with larger voids have a negligible healing rate and they are slowing down the overall healing rate of the REV. The highest healing rate is reached in samples with uniformly distributed void sizes. In addition, the healing rate increases with the initial porosity, but the final porosity change does not depend on the initial porosity of the sample. Principal stresses of higher magnitude are noted in the inclusions that are part of REVs of high initial porosity. In specimens with smaller inclusions (i.e., smaller grains), principal stresses are more widely distributed in magnitude and the healing rate is higher. The proposed homogenization method paves the way to many future developments for upscaling chemo-mechanical processes in heterogeneous media, and can be used to design self-healing materials.

A general micromechanical model to predict elastic and strength properties of balanced plain weave fabric composites

In the present research, a micromechanical-analytical model was developed to predict the elastic properties and strength of balanced plain weave fabric composites. In this way, a new homogenization method has been developed by using a laminate analogy method for the balanced plain weave fabric composites. The proposed homogenization method is a multi-scale homogenization procedure. This model divides the representative volume element to several sub-elements, in a way that the combination of the sub-elements can be considered as a laminated composite. To determine the mechanical properties of laminates, instead of using an iso-strain assumption, the assumptions of constant in-plane strains and constant out-of-plane stress have been considered. The applied assumptions improve the accuracy of prediction of mechanical properties of balanced plain weave fabrics composites, especially the out-of-plane elastic properties. Also, the stress analysis for prediction of strain–stress behavior and strength has been implemented in a similar manner. In addition, the nonlinear mechanical behavior of balanced plain weave composite is studied by considering the inelastic mechanical behavior of its polymeric matrix. To assess the accuracy of the present model, the results were compared with available results in the literature. The results, including of engineering constants (elastic modulus and Poisson’s ratio) and stress–strain behavior show the accuracy of the present model.

Hysteresis Loss Analysis of Laminated Iron Core by Using Homogenization Method Taking Account of Hysteretic Property

This paper proposes a homogenization method for a laminated iron core taking into account hysteretic property. The proposed method is applicable to a static or low-frequency problem neglecting eddy currents induced by the magnetic flux parallel to the lamination. Furthermore, we apply the proposed homogenization method to finite-element magnetic field analyses to demonstrate its effectiveness. As a consequence, it is verified that the proposed homogenization method can accurately estimate a hysteresis loss of a laminated iron core and decrease computational costs.

Homogenization of elastic–viscoplastic composites by the Mixed TFA

A new homogenization technique, based on the Transformation Field Analysis, able to determine the overall behavior of viscoplastic heterogeneous materials, is proposed. This method is derived developing a variational formulation of the compatibility and evolution equations governing two classical elastic–viscoplastic models, i.e. Perzyna and Perić models. A representation form for the stress field and the plastic multiplier is defined on the representative volume element (RVE) of the composite material, resulting in a significant reduction of the history variables ruling the evolution problem. A numerical procedure is developed integrating the evolution equation by a backward-Euler algorithm and solving the single time step developing a specific form of the predictor corrector technique. The accuracy of the proposed homogenization method is assessed through some numerical examples for elastic–viscoplastic composites. The applications aim to verify the effectiveness of the presented technique in modeling the elastic–viscoplastic composite behavior adopting a reduced number of history variables. The homogenization results are compared with the ones obtained by nonlinear finite element micromechanical analyses.

Symplectic analysis of dynamic properties of hexagonal honeycomb sandwich tubes with plateau borders

A new type of hexagonal honeycomb sandwich tube with plateau borders are introduced in this work and the Symplectic analysis with its high computational efficiency and high accuracy is applied to obtain the structural dynamic properties. The effects of material distribution (β) and relative density (ρ¯) on the dynamic properties of the structure are also studied. Based on the definition of the elastic constants and the homogenization method, the independent elastic constants are obtained. By introducing dual variables and applying the variational principle, the canonical equations of Hamiltonian system are constructed. The precise integration method and extended Wittrick–Williams algorithm are adopted to solve the canonical equations. The dispersion relations of sandwich tubes are obtained, and the effects of material distribution and relative density on the normalized frequencies of the sandwich tubes are investigated. The proposed homogenization method is verified by comparing with other researchers׳ works. Dispersion relations of the sandwich tubes are obtained. The material distribution parameter and the relative density have significant effects on the dynamic properties of the structures. This work expects to offer new opportunities for the optimal design of metallic honeycomb sandwich tubes and future applications in the engineering sector.

Numerical modeling of elastic modulus for cement paste using homogenization method

This paper focused on the evolution over time of elasticity of the cement paste during the hydration, e g, Young’s modulus and Poisson’s ration, by the proposed homogenization method combined the percolation algorithm with individual phase intrinsic elasticity. A cement paste development model, named CEMHYD3D, was used to establish an accurate microstructure. The modelling results are in good agreement with the experimental data and other numerical results available in the open literature. The suitable homogeneous scheme, applied to each level, should be carefully chosen to result in a realistic prediction. The percolation concept should aims to correctly predict the elasticity for cement paste at very early age, especially under low w/c ratios.

Multiscale homogenization modeling for thermal transport properties of polymer nanocomposites with Kapitza thermal resistance

In this study, multiscale homogenization modeling to characterize the thermal conductivity of polymer nanocomposites is proposed to account for the Kapitza thermal resistance at the interface and the polymer immobilized interphase. Molecular dynamics simulations revealed that the thermal conductivity dependent on the embedded particle size originated from the structurally altered interphase zone of surrounding matrix polymer in the vicinity of nanoparticles, and clearly indicate strong dominance of interfacial phonon scattering and dispersion. To account for both the thermal resistance and the immobilized interphase, a four-phase equivalent continuum model composed of spherical nanoparticles, Kapitza thermal interface, effective interphase, and bulk matrix phase is introduced in a finite element-based homogenization method. From the given thermal conductivity of the nanocomposites obtained from MD simulations, the thermal conductivity of the interphase is inversely and numerically obtained. Compared with the micromechanics-based multiscale model, the thermal conductivity of the interphase can be obtained more accurately from the proposed homogenization method. Using the thermal conductivity of the interphase, the random distribution and radius of nanoparticles to describe real nanocomposite microstructure are considered, and the results confirm the applicability of the proposed multiscale homogenization model for further stochastic approaches to account for geometric uncertainties in nanocomposites.