Periodic Representative Volume Element Research Articles

Setting bounds for in-plane shear induced fiber angle deviations in bi-axial non-crimp fabrics

For pick-and-place processes to become widely implemented in industry a consistent and acceptable product quality needs to be achieved. In the state of the art it is assumed that reinforcements will be in perfect condition at the start of forming or draping. In reality the handling process can already result in undesired deformations. The current work will look at fiber angle deviations that occur during this process due to in-plane shear. It is shown that bounds can be set for these fiber angle deviations based on experimental work. Periodic representative volume element homogenization is used to obtain homogenized material properties for a bi-axial non-crimp fabric with a specific construction. With these material properties the in-plane shear strain, and thus the fiber angle deviations, can be predicted. The presented methodology and results obtained using it can be a basis in the design process for automated handling of reinforcements and for in-situ quality control of the pick-and-place process.

A computationally efficient approach for generating RVEs of various inclusion/fibre shapes

A computationally efficient method for generating virtual periodic representative volume element (RVE), capable of handling arbitrary inclusion shapes, is developed. A universal collision/overlap detection and repair method is proposed, where each inclusion shape is represented as a union of n-Spheres (UnS). A constrained optimization problem is formulated and solved to remove inclusion overlaps; a closed-form solution is derived for calculating the degree of inclusions overlap and its gradient vector with respect to inclusion position. RVE generation is illustrated with circular, spherical, four non-circular and four non-spherical inclusion shapes. Computational efficiency is demonstrated using an elaborate RVE generation time study. The generated RVEs are evaluated using various statistical metrics; results confirm the random distribution of inclusions. Effective properties of RVEs, representing unidirectional composites, are determined using homogenization with various fibre cross-section shapes; obtained mechanical properties have shown transverse isotropy.

Influence of the formation of clusters on the effective elastic properties of platelet reinforced polymers

The effect of the clusterization on the effective properties of a composite material reinforced by MXene or graphene platelets is studied using the finite element method with periodic representative volume element (RVE). A hybrid 2D/3D finite element mesh is used to reduce the computational complexity of the numerical model. Several realizations of an RVE were generated with increasing volume fractions of inclusions, resulting in progressive clusterization of the platelets. Numerically obtained effective properties of the composite are compared with analytical predictions by the Mori-Tanaka method and Halpin-Tsai equations, and the limits of the applicability of the analytical models are established. A two-step homogenization scheme is proposed to increase the accuracy and stability of the effective properties of an RVE with a relatively small number of inclusions. Simple scaling relations are proposed to generalize numerical results to platelets with other aspect ratios.

Anisotropic yield surfaces of additively manufactured metals simulated with crystal plasticity

The mechanical anisotropy created by additive manufacturing (AM) is not yet fully understood and can depend on many factors, such as powder material, manufacturing technology and printing parameters. In this work, the anisotropic mechanical properties of as-built, laser powder bed fusion (LPBF) austenitic stainless steel 316L and titanium alloy Ti-6Al-4V are investigated through crystal plasticity simulations. Periodic representative volume elements (RVEs) are used that are specific to each material. The RVE for austenitic stainless steel consists of FCC crystals with a crystallographic texture measured by X-ray diffraction. The α′ martensite microstructure of Ti-6Al-4V is captured with a multi-scale RVE, including internal lamellar structures, using HCP crystals and a synthetically generated texture. For both materials, the crystal plasticity parameters are calibrated against tensile tests carried out on dog-bone specimens printed in different orientations. The RVEs, calibrated to experiments, are applied in virtual material testing and subjected to multiple load cases to generate the Hill-48 and Yld2004-18p yield surfaces of the materials.

Geometric and finite element modeling of biopolymer aerogels to characterize their microstructural and mechanical properties

AbstractBiopolymer aerogels belong to a class of highly open‐porous cellular materials. Their macroscopic mechanical properties (such as elasticity or thermal conductivity) depend on microstructural features (namely pore size distribution (PSD), fiber diameter and solid fraction), which can be tailored by different synthesis and drying routes. The design of modern aerogel materials requires a better perception into the microstructure and its influence on the mechanical properties. To predict the material properties using simulation, it is significant to construct a geometric model which is sufficiently precise to represent the microstructure of real materials. A tessellation approach based on Voronoi diagrams is a powerful tool to model such cellular‐like materials. In this contribution, the diversified cellular morphology of aerogels is described computationally using a Voronoi tessellation‐based approach [1]. Accordingly, Voronoi tessellations are generated to create periodic representative volume elements (RVEs) resembling the microstructural properties of the cellular network. Stress‐strain curves resulting from finite element simulations of these RVEs and experiments of the aerogels under compression are compared. This work is an extension of our previous Voronoi tessellation‐based on the 2‐d description of biopolymer aerogels [2].

Crystal plasticity modeling and experimental characterization of strain localization and forming limits in ferrite-pearlite steels

The main objective of this study is to predict the deformation behavior, strain localization, and forming limits of ferrite-pearlite steels by incorporating the contributions of the microstructural characteristics and mechanical properties of the underlying microstructure. A realistic microstructure-based micromechanical approach in the framework of the crystal plasticity (CP) model was carried out using the periodic representative volume element (RVE) generated from the scanning electron microscopy (SEM) image. The homogenized stress–strain curve of the realistic RVE was validated with the experimental data with an error of less than 6.71% at large strains. Afterward, the initial microstructural inhomogeneity criterion was applied to the realistic RVE under various loading paths to predict the forming limit diagram (FLD), which compared with the experimental results of the Nakazima-stretch forming test. Consequently, the contributions of the pearlite volume fraction and the microstructural morphology were studied by extending this approach to the synthetic microstructures with various pearlite volume fractions. In conclusion, the microstructure-level inhomogeneity at the microscopic level results in intense stress/strain partitioning and strain localization, emerging in the form of micro-localized deformation bands. Moreover, it has been found that the local deformation pattern at the micron-scale significantly depends on the microstructural morphology and loading direction.

Micro-mechanical modeling of irreversible hygroscopic strain in paper sheets exposed to moisture cycles

Paper is a complex material consisting of a network of cellulose fibres at the micro-level. During manufacturing, the network is dried under restraint conditions due to tension in the paper web in machine direction. This gives rise to internal strains that are stored in the produced sheet. Upon exposure to a moisture cycle, these strains may be released. This results in permanent shrinkage that may cause instabilities such as curl or waviness of the sheet. The prime objective of this paper is to model this irreversible shrinkage and to link its magnitude to the properties of the fibres and of the network. For this purpose, randomly generated fibrous networks of different coverages (i.e.ratio of the area occupied by fibres and that of the sheet) are modeled by means of a periodic representative volume element (RVE). Within such RVEs, a finite element method combined with a kinematic hardening plasticity model at the scale of the fibres is used to capture the irreversible response. The computational results obtained demonstrate that the magnitude of the irreversible strains increases with coverage until a certain coverage and beyond that coverage decreases in magnitude. This phenomenon is explained by considering the area fraction of free-standing fibre segments relative to bonded fibre segments in the network. A structure–property dependency of irreversible strains at the sheet-level on the micro-structural parameters of the network is thereby established.

A refinement indicator for adaptive quasicontinuum approaches for structural lattices

AbstractThe quasicontinuum (QC) method is a concurrent multiscale approach in which lattice models are fully resolved in small regions of interest and coarse‐grained elsewhere. Since the method was originally proposed to accelerate atomistic lattice simulations, its refinement criteria—that drive refining coarse‐grained regions and/or increasing fully‐resolved regions—are generally associated with quantities relevant to the atomistic scale. In this contribution, a new refinement indicator is presented, based on the energies of dedicated cells at coarse‐grained domain surfaces. This indicator is incorporated in an adaptive scheme of a generalization of the QC method able to consider periodic representative volume elements, like the ones employed in most computational homogenization approaches. However, this indicator can also be used for conventional QC frameworks. Illustrative numerical examples of elastic indentation and scratch of different lattices demonstrate the capabilities of the refinement indicator and its impact on adaptive QC simulations.

Modeling the microstructure of biopolymer aerogels using Voronoi tessellation method

AbstractThe bulk properties of biopolymer aerogels depend on their microstructure, which can be tailored by different synthesis and drying methods. Biopolymer aerogels are characterized by a fibrillar morphology having a cellular‐like network. The recently proposed constitutive modeling approach by Rege et al. [1] has shown good predictive capabilities in describing the mechanical behavior of such aerogels. Although the model describes the cellular nature and adheres to the cell‐size distributions of aerogels, it is based on the assumption that the network is made up of idealized square‐shaped cells. In this contribution, the diversified cellular morphology of aerogels is described computationally using a Laguerre‐Voronoi tessellation based approach [2]. The pore‐size distribution (PSD) data obtained from experiments accounts for the random cell sizes within the network. Accordingly, Voronoi tessellations are generated to create periodic representative volume elements (RVEs) resembling the microstructural properties of the cellular network. This work is an extension of our previous Voronoi tessellation‐based 2‐d description of biopolymer aerogels [3].

Modeling and computational homogenization of chloride diffusion in three-phase meso-scale concrete

A computational homogenization technique for modeling diffusion in concrete is introduced with emphasis on the influence of the aggregate content and variability. The highly heterogeneous material is investigated on different scales by combining Variationally Consistent Homogenization on numerical microstructures with analytical techniques accounting for lower, unresolved, length scales. The concrete structure consists of the cement paste, the embedded aggregates, and the Interfacial Transition Zone (ITZ) in between the two. Diffusion takes place in the cement phase, as well as in the ITZ. Since the thickness of the ITZ is, typically, much smaller than the diameter of the aggregates, the effect of the ITZ can be modelled as a surface transport around the aggregates. The occurrence of different aggregate sizes is described via the Particle Size Distribution for given sieve curves, as described in design codes. The Particle Size Distribution curve is split into two parts. The effect of smaller aggregates is homogenized analytically using a mixture rule. This results in an effective matrix material consisting of cement paste and the smaller aggregates. Synthetic structures are then generated numerically to account for the larger aggregates. At first, a dense sphere packing is created based on the Particle Size Distribution. This information is used to generate a weighted Voronoi diagram, which is modified by a shrinking process. This procedure allows us to create periodic Representative Volume Elements for numerical investigations.The overall diffusivity of the concrete mixture is evaluated upon using Variationally Consistent Homogenization, in the context of Finite Element analysis, for the generated RVEs and compared with analytical homogenization results and experimental data. It is found that, depending on the Particle Size Distribution, the ITZ has a large effect on the effective properties.

Development of a finite element based strain periodicity implementation method

The current work proposes a novel approach for the implementation of periodic representative volume element (RVE) based multiscale modeling using the finite element method. The approach is based on a rigorous mechanics foundation that implements appropriate boundary conditions for the RVE analysis and simplifies the homogenization technique. Stress components are relaxed in the chosen periodic directions and associated strains are calculated while maintaining periodicity. Our concept ensures that for a displacement-based mechanics approach, the stresses and strains in the domain are periodic, rather than the displacements. This approach allows for computing homogenized effective mechanical properties of heterogeneous materials and capturing appropriate volume change and shape distortion of the RVE at lower length scales. We use the finite element based Multiphysics Object Oriented Simulation Environment (MOOSE, https://mooseframework.org) for implementing the proposed periodic RVE scheme. Our model has been verified with analytical calculations and effective property estimation from various micromechanical approaches. The effective stiffness of a composite material calculated from this approach is in good agreement with other computational and experimental findings. The proposed model has also been applied to different multiscale and multiphysics engineering applications.

Reconstruction of Microstructural and Morphological Parameters for RVE Simulations with Machine Learning

Advanced High Strength Steels (AHSS) are of great interest for the automotive industry. Since they are applied in a diverse context their properties have to adhere to this diversity. The mechanical properties of modern AHSS are therefore tailored to their specific field of application. Thus they are able to fulfill the desired requirements of the technology. By adapting the steel to a specific area of application, however, these materials obtain a multi-phase microstructure with increasingly complex morphologies. To investigate the influence of these morphological properties on the mechanical properties, Representative Volume Elements (RVE) are commonly used. These RVE are generated based on statistical data and algorithms using randomization methods to get virtual microstructure models that adhere to the real material’s microstructural parameters. This study introduces an RVE generator which is capable of generating statistically representative and periodic RVE. Additionally, the RVE are periodically meshed, so that periodic loading conditions are applicable. Since the input parameters are the most important part of generating the RVE, a machine learning algorithm was trained to reproduce input data equivalent to the real material’s microstructural and morphological parameters.

A primal formulation for imposing periodic boundary conditions on conforming and nonconforming meshes

A variational multiscale Discontinuous Galerkin (VMDG) method is developed for microscale modeling of domains containing conforming and non-conforming meshes. Essentially, the product of the applied volume-average strain (or macro-strain) and the domain diameter acts as an imposed displacement jump within the VMDG terms. Hence, the method is suitable for modeling deformation of both block and truly (self) periodic representative volume elements (RVEs). The primal displacement field and macro-strain are the only unknowns because the method eliminates the Lagrange multiplier (LM) enforcement of the kinematic constraint. Rigorous derivation of the method provides a framework to accommodate either the macro-stress or macro-strain as the driver of the microscale boundary value problem. The method is developed first for finite deformations and then specialized to small deformation kinematics. Algorithmic modifications to the method are also studied for their effects on tangent symmetry and convergence rate. The results from numerical studies for isotropic and anisotropic materials show that the proposed method is robust, accurate, stable and variationally consistent for modeling complicated conforming and nonconforming RVEs.

Thermal conductivity of architected cellular metamaterials

Periodic architected cellular metamaterials, as a novel class of low-density materials, possess unprecedented multifunctional properties mainly due to their underlying microarchitecture. In this paper, we study the thermal conductivity of cellular metamaterials and evaluate their performance for thermal management applications. To understand the relations between the microarchitecture and the thermal response, we analyze the thermal conductivity of a wide range of cellular metamaterials with strategically developed microarchitectures from two-dimensional (2D) cells with Supershape pores to three-dimensional (3D) thin-walled open lattices and shellular materials. We implement standard mechanics homogenization on the periodic representative volume elements (RVEs) of these cellular metamaterials to examine the effect of pore architecture (relative density, pore shape, pore orientation, and pore elongation) on their effective thermal conductivity. The numerical results show how the thermal conductivity of an isotropic material can be modified by pore introduction and how the pore architecture could lead to an anisotropic effective thermal conductivity tensor. To examine the impact of having 2D Supershape cuts on 3D RVEs, thin-walled open lattices are designed as an assembly of thickened 2D RVEs with Supershape pores. A mathematical model is derived based on the effective thermal properties of the constituent 2D RVEs to predict the effective thermal properties of these lightweight cellular materials. Effective thermal conductivity of shellular materials based on triply periodic minimal surfaces is also compared with those of the previously introduced architectures. Unlike the shellular materials, which only cover a narrow region of thermal conductivity versus relative density chart, cellular materials with a wide range of anisotropic effective thermal conductivities can be engineered by using 2D Supershape pores on 2D or 3D thin-walled cells. Finally, we show how the concept of architected functionally graded cellular materials can be used to tune the heat flow within cellular media to guide it in a specific direction to control the temperature inside advanced 3D printed materials. As a case study, the optimum spatial distribution of pore rotation angle is found to maximize or minimize the heat flow passing through different sides of a square-shaped porous slab. This paper opens an avenue for developing thermal metamaterials with programmable anisotropic thermal properties.

Mechanics of biopolymer aerogels based on microstructures generated from 2-d Voronoi tessellations

In this paper, the heterogeneous morphology of biopolymer aerogels is described using a 2-d Voronoi tessellation, which accounts for the randomized cell shapes of aerogels and adheres to the pore-size distributions obtained from experimental data. Accordingly, a two-dimensional periodic representative volume element (RVE) has been generated and simulated under compression. In particular, the sensitivity of the varying microstructural parameters within the RVE is analyzed. It is realized that the fiber diameter, the pore-size distribution, and the density of the cells within the aerogel network show a significant influence on their mechanical response. The trends of the model predictions are qualitatively in good agreement with the macroscopic experimental data of biopolymer aerogels.

Constitutive modeling of plastic deformation behavior of open-cell foam structures using neural networks

This article presents an approach to use neural networks as a constitutive model for the inelastic deformation behavior of open-cell foams or other porous materials. The inelastic deformation behavior of highly porous materials can be very complex and makes the formulation of a closed form constitutive model a very challenging or even impossible task. The presented methods include the creation of a periodic representative volume element (RVE) of the porous structure, the supply of training data by means of finite element simulations using the RVE and the use of trained neural networks within a user material routine (UMAT) for the finite element code ABAQUS. As a proof of concept this approach is applied to relatively simple two dimensional porous structures, where both the homogenized material behavior and a fully resolved porous structure are simulated and compared. The computational effort for simulating a simple foam structure can be reduced by a factor of more than 10,000 using the proposed homogenization approach.

Three-dimensional micromechanical assessment of bio-inspired composites with non-uniformly dispersed inclusions

Bio-inspired composites with hexagonal platelet and cylindrical inclusions were studied. A novel algorithm termed staggered hardcore algorithm (SHCA) was used to rapidly generate 3D periodic representative volume elements (RVE) for bio-inspired composites with staggered non-uniformly dispersed inclusions. The spatial dispersions of inclusions in these generated RVEs were assessed using autocorrelation analysis, demonstrating the effectiveness of the SHCA algorithm. Orthotropic elastic properties of two different bio-inspired composites were computed and compared with analytical models, namely modified shear-lag, Mori-Tanaka and Halpin-Tsai, as well as available experimental data from the literature. For lower inclusion volume fractions, the computed results correlated well with experimental data and the analytical results. However, for higher inclusion volume fractions and aspect ratios the analytical results diverged, particularly Mori-Tanaka and modified shear-lag models which was similarly reported in previous studies. The capabilities of the computational model were further demonstrated through a comparative study of orthotropic elastic constants for the cylindrical and hexagonal inclusion composites. The study revealed the necessity to use 3D micromechanical models with realistic inclusion dispersions for accurately assessing the response of high inclusion volume fraction bio-inspired composites.

Special-purpose elements to impose Periodic Boundary Conditions for multiscale computational homogenization of composite materials with the explicit Finite Element Method

A novel methodology is presented to introduce Periodic Boundary Conditions (PBC) on periodic Representative Volume Elements (RVE) in Finite Element (FE) solvers based on dynamic explicit time integration. This implementation aims at overcoming the difficulties of the explicit FE method in dealing with standard PBC. The proposed approach is based on the implementation of a user-defined element, named a Periodic Boundary Condition Element (PBCE), that enforces the periodicity between periodic nodes through a spring-mass-dashpot system. The methodology is demonstrated in the multiscale simulation of composite materials. Two showcases are presented: one at the scale of computational micromechanics, and another one at the level of computational mesomechanics. The first case demonstrates that the proposed PBCE allows the homogenization of composite ply properties through the explicit FE method with increased efficiency and similar reliability with respect to the equivalent implicit simulations with traditional PBC. The second case demonstrates that the PBCE coupled with Periodic Laminate Elements (PLE) can effectively be applied to the computational homogenization of elastic and strength properties of entire laminates taking into account highly nonlinear effects. Both cases motivate the application of the methodology in multiscale virtual testing in support of the building-block certification of composite materials.

Overall elastic properties of a material containing inhomogeneities of concave shape

In this paper, we consider materials containing superspherical inhomogeneities and compare effects produced by pores and rigid particles of this shape. The overall properties are expressed in terms of compliance contribution tensors calculated for inhomogeneities using Finite Element Method (FEM) and Gaussian Approximation (GA) method. Consistency of the results obtained by two numerical techniques is observed for most of the shapes (except of strongly concave inhomogeneities). Based on the numerical solution for octahedron and analytical results for sphere, contributions of a single inhomogeneity to the effective properties are approximated analytically with high accuracy. Effective properties of materials containing multiple superspherical pores or multiple superspherical rigid inhomogeneities are calculated using Non-Interaction Approximation (NIA) and Mori–Tanaka–Benveniste scheme (MTB) using results for compliance contribution tensors. Results of micromechanical predictions are compared against direct FEM simulations performed on artificial periodic representative volume elements containing parallel oriented pores and rigid particles with volume fraction up to 25%. It is shown that MTB scheme provides reasonably accurate approximations.

Finite element multiscale modelling of elastic behavior of cellulose acetate—Graphene oxide nanocomposites, produced using a SC-CO2 assisted technique

This work focuses on the modelling of the mechanical performance of cellulose acetate-graphene oxide nanocomposite membranes, produced by supercritical CO2 assisted phase inversion, using a set of algorithms able to build and integrate two different parametric variational 3D finite element (FE) models. These models simulate micro- and nano-level morphology of the nanocomposite using a multiscale approach. Microlevel is characterized by interconnected spherical pores; whereas, nanolevel is a composites whose filler is formed by graphene oxide sheets. Information about nanocomposite morphology, derived from electron scanning microscopy analyses, was used to build periodic representative volume elements. A numerical-experimental correlation was performed comparing FE model results with the experimental ones obtained by compression tests, at different weight percentages of graphene oxide with respect to the polymer loaded in the membrane. A good approximation of the experimental trend of Young modulus was obtained by FE simulations.