Kinematic Uniform Boundary Conditions Research Articles

Enhancing the electro‐elastic properties of a castor‐oil‐derived polyurethane/barium titanate piezoelectric energy‐harvesting composite by integration of multiwalled carbon nanotubes

AbstractThree‐component piezoelectric composites, with castor‐oil‐derived polyurethane (COPU) as the matrix, reinforced with barium titanate (BTO) microparticles (as the primary piezo‐active inclusion) and with multiwalled carbon nanotubes (MWCNTs) were synthesized. BTO content is varied (3, 12, and 25 vol%), while the MWCNTs content is maintained constant at 0.65 wt% (0.37 vol%) w.r.t. the COPU matrix. The addition of MWCNTs significantly enhances the piezoelectric strain coefficient (), relative dielectric constant (k/k0), and piezoelectric response, that is, open circuit voltage (VOC) of the piezo‐composites. The (COPU/MWCNTs)/25 vol% BTO composite gives a value of 1.8 ± 0.1 pC/N, an increase of ~1.38 times compared to COPU/25 vol% BTO composite (with a value of 1.3 ± 0.1 pC/N). The relative dielectric constant (k/k0) value of two‐component COPU/BTO (25.0 vol%) increased from 10.34 ± 0.06 to 11.15 ± 0.05 by addition of 0.65 wt% MWCNTs. A maximum open circuit voltage of ~19.6 V for three‐component (COPU/MWCNTs)/ 25 vol% BTO composite was obtained at 45 N load and at 5 Hz frequency. This value (VOC) is ~1.29 times higher compared to COPU/25 vol% BTO composite (with VOC ~ 15.2 V). The k/k0 and values and elastic moduli obtained via experiments were further compared with those obtained from finite element analyses based on kinematic uniform boundary conditions (KUBC) and from Eshelby–Mori–Tanaka micromechanics extended to the domain of linear piezoelectric composites.Highlights Castor‐oil‐derived (COPU/MWCNTs/BTO) three‐component composites were synthesized Integration of MWCNTs enhances by ~1.38 times for COPU/25 vol% BTO Elastic modulus and electrical conductivity increased with the addition of MWCNTs Computational predicted k/k0 and values closely match to experimental data

Accuracy of osseointegrated screw-bone construct stiffness and peri-implant loading predicted by homogenized FE models relative to micro-FE models

Computational predictions of stiffness and peri-implant loading of screw-bone constructs are highly relevant to investigate and improve bone fracture fixations. Homogenized finite element (hFE) models have been used for this purpose in the past, but their accuracy has been questioned given the numerous simplifications, such as neglecting screw threads and modelling the trabecular bone structure as a continuum. This study aimed to investigate the accuracy of hFE models of an osseointegrated screw-bone construct when compared to micro-FE models considering the simplified screw geometry and different trabecular bone material models.Micro-FE and hFE models were created from 15 cylindrical bone samples with a virtually inserted, osseointegrated screw (fully bonded interface). Micro-FE models were created including the screw with threads (=reference models) and without threads to quantify the error due to screw geometry simplification. In the hFE models, the screws were modelled without threads and four different trabecular bone material models were used, including orthotropic and isotropic material derived from homogenization with kinematic uniform boundary conditions (KUBC), as well as from periodicity-compatible mixed uniform boundary conditions (PMUBC). Three load cases were simulated (pullout, shear in two directions) and errors in the construct stiffness and the volume average strain energy density (SED) in the peri-implant region were evaluated relative to the micro-FE model with a threaded screw.The pooled error caused by only omitting screw threads was low (max: 8.0%) compared to the pooled error additionally including homogenized trabecular bone material (max: 92.2%). Stiffness was predicted most accurately using PMUBC-derived orthotropic material (error: -0.7 ± 8.0%) and least accurately using KUBC-derived isotropic material (error: +23.1 ± 24.4%). Peri-implant SED averages were generally well correlated (R2 ≥ 0.76), but slightly over- or underestimated by the hFE models and SED distributions were qualitatively different between hFE and micro-FE models.This study suggests that osseointegrated screw-bone construct stiffness can be predicted accurately using hFE models when compared to micro-FE models and that volume average peri-implant SEDs are well correlated. However, the hFE models are highly sensitive to the choice of trabecular bone material properties. PMUBC-derived isotropic material properties represented the best trade-off between model accuracy and complexity in this study.

Randomly Dispersed Coated Composites Study by Statistical Approach and Numerical Homogenization Method

The aim of this work is the computation of effective elastic properties of 3D 3-phase random heterogeneous coated materials. For that, a new expression of the integral range for 3-phase random coated heterogeneous materials is used. The computation is achieved using a representative microstructure with non-overlapping inclusions. Numerical simulations is used under periodic boundary conditions (PBC) and kinematic uniform boundary conditions (KUBC) prescribed over Representative Volume Element (RVE). The obtained effective elastic properties are compared with different analytical models as Hashin and Shtrikman bounds and the n+1 phase model. Using the statistical methods, a new extension of the integral range for 3-phase coated materials is proposed.

Evolution of effective mechanical and interphase properties during natural ageing of glass-fibre/epoxy composites using micromechanical approach

This work proposes, by a micromechanical approach, to simulate the effects of humid tropical aging on a unidirectional composite (E-glass fibers/epoxy) by taking into account an interphase region. An interphase area (500 nm width) is identified by AFM measurements around each fibre at the initial state. During the ageing this area increases up to 1 µm width. Numerical micro-models, using ABAQUS™, are developed to predict the mechanical behaviour of the composite under monotonic transverse traction. A Representative Volume Element description with a random distribution of the fibres is generated from the plug-in “RVE for Composites”. Linear elastic behaviour laws are used to describe the behaviour of materials and Kinematic Uniform Boundary Conditions are considered to describe the boundaries conditions. Numerical, analytical and experimental results are compared to validate the micro-models. The obtained results are in good agreement with the analytical and experimental values. These results highlight the importance of taking composite interphase into account in numerical models for mechanical behaviour.

Predicting Effective Electromechanical Properties of Pb-Free Polymer Composite Using Finite Element Method

Polyvinylidene fluoride (PVDF) – Lead Zirconate Titanate (PZT) is a polymer composite that is becoming increasingly popular in micro-scale sensors and actuators because of its unique properties such as high flexibility, low density and high piezoelectric constants. However, lead-based piezoceramics, despite their superior properties, are toxic and are known to damage the environment, and as such a conscientious effort is being made by the scientific community towards replacing lead-containing piezoceramics with environmentally-friendlier and lead-free piezoceramics. Barium Titanate (BaTiO3) is one such piezoceramics that is widely studied today to be a potential replacement of PZT in many applications. As such, in this work, effort has been made to predict the effective mechanical, dielectric and piezoelectric properties of PVDF-BaTiO3 composite system using Finite Element Method (FEM). Kinematic Uniform Boundary Conditions (Displacement and Voltage) are used for this analysis. For evaluation of the effective material constants of the composite, several types of representative volume elements are considered. The effects of volume fraction, effect of the size of the micro-particles i.e. mono-modal versus multi-modal size distribution, effect of periodic versus quasi-random distribution of microparticles in the matrix, the effect of clustering of the particles, effect of orientation of the microparticles i.e. unidirectional or randomly oriented are discussed. Finally, a comparison of properties between PVDF-PZT and PVDF-BaTiO3 is made, so as to see whether PVDF-BaTiO3 can be a potential replacement for PVDF-PZT composite.

Effect of interphase region on the elastic behavior of unidirectional glass-fiber/epoxy composites

Using a FEA homogenization technique, this study presents the influence of interphase region on composites. Numerical models using Abaqus™ are developed in order to predict the mechanical behavior of a unidirectional composite (E-glass fibers/epoxy) under monotonic transverse traction. A Representative Volume Element description with a random distribution of the fibers is used. Linear elastic constitutive materials and Kinematic Uniform Boundary Conditions are considered.A numerical/experimental comparison is made in order to validate the model. The thickness and the modulus of the interphase have been determined by experimental measurements based on Atomic Force Microscopy and introduced in the numerical simulation, the effect of these properties as well as the effect of the interphase Poisson’s ratio, on the effective elastic properties of the composite is discussed. The obtained results are in good agreement with the experimental values previously measured by tensile test on macroscopic samples. The fact of considering the interphase improves the accuracy of the prediction of the composite's mechanical elastic behavior.

Estimation of the effective yield properties of human trabecular bone using nonlinear micro-finite element analyses.

Micro-finite element ([Formula: see text]FE) analyses are often used to determine the apparent mechanical properties of trabecular bone volumes. Yet, these apparent properties depend strongly on the applied boundary conditions (BCs) for the limited size of volumes that can be obtained from human bones. To attenuate the influence of the BCs, we computed the yield properties of samples loaded via a surrounding layer of trabecular bone ("embedded configuration"). Thirteen cubic volumes (10.6mm side length) were collected from [Formula: see text]CT reconstructions of human vertebrae and femora and converted into [Formula: see text]FE models. An isotropic elasto-plastic material model was chosen for bone tissue, and nonlinear [Formula: see text]FE analyses of six uniaxial, shear, and multi-axial load cases were simulated to determine the yield properties of a subregion (5.3mm side length) of each volume. Three BCs were tested. Kinematic uniform BCs (KUBCs: each boundary node is constrained with uniform displacements) and periodicity-compatible mixed uniform BCs (PMUBCs: each boundary node is constrained with a uniform combination of displacements and tractions mimicking the periodic BCs for an orthotropic material) were directly applied to the subregions, while the embedded configuration was achieved by applying PMUBCs on the larger volumes instead. Yield stresses and strains, and element damage at yield were finally compared across BCs. Our findings indicate that yield strains do not depend on the BCs. However, KUBCs significantly overestimate yield stresses obtained in the embedded configuration (+43.1 ± 27.9%). PMUBCs underestimate (-10.0 ± 11.2%), but not significantly, yield stresses in the embedded situation. Similarly, KUBCs lead to higher damage levels than PMUBCs (+51.0 ± 16.9%) and embedded configurations (+48.4 ± 15.0%). PMUBCs are better suited for reproducing the loading conditions in subregions of the trabecular bone and deliver a fair estimation of their effective (asymptotic) yield properties.

Modeling of Cementitious Representative Volume Element with Additives

CEMHYD3D has been employed to simulate the representative volume element (RVE) of cementitious systems (Type I cement) containing fly ash (Class F) through a voxel-based finite element analysis (FEA) approach. Three-dimensional microstructures composed of voxels are generated for a heterogeneous cementitious material consisting of various constituent phases. The primary focus is to simulate a cementitious RVE containing fly ash and to present the homogenized macromechanical properties obtained from its analysis. Simple kinematic uniform boundary conditions as well as periodic boundary conditions were imposed on the RVE to obtain the principal and shear moduli. Our current work considers the effect of fly ash percentage on the elastic properties based on the mass and volume replacements. RVEs with lengths of 50, 100 and 200[Formula: see text][Formula: see text] at different degrees of hydration are generated, and the elastic properties are modeled and simulated. In general, the elastic properties of a cementitious RVE with fly ash replacement for cement based on mass and volume differ from each other. Moreover, the finite element (FE) mesh density effect is studied. Results indicate that mechanical properties decrease with increasing mesh density.

The effective elastic properties of human trabecular bone may be approximated using micro-finite element analyses of embedded volume elements.

Boundary conditions (BCs) and sample size affect the measured elastic properties of cancellous bone. Samples too small to be representative appear stiffer under kinematic uniform BCs (KUBCs) than under periodicity-compatible mixed uniform BCs (PMUBCs). To avoid those effects, we propose to determine the effective properties of trabecular bone using an embedded configuration. Cubic samples of various sizes (2.63, 5.29, 7.96, 10.58 and 15.87mm) were cropped from [Formula: see text] scans of femoral heads and vertebral bodies. They were converted into [Formula: see text] models and their stiffness tensor was established via six uniaxial and shear load cases. PMUBCs- and KUBCs-based tensors were determined for each sample. "In situ" stiffness tensors were also evaluated for the embedded configuration, i.e. when the loads were transmitted to the samples via a layer of trabecular bone. The Zysset-Curnier model accounting for bone volume fraction and fabric anisotropy was fitted to those stiffness tensors, and model parameters [Formula: see text] (Poisson's ratio) [Formula: see text] and [Formula: see text] (elastic and shear moduli) were compared between sizes. BCs and sample size had little impact on [Formula: see text]. However, KUBCs- and PMUBCs-based [Formula: see text] and [Formula: see text], respectively, decreased and increased with growing size, though convergence was not reached even for our largest samples. Both BCs produced upper and lower bounds for the in situ values that were almost constant across samples dimensions, thus appearing as an approximation of the effective properties. PMUBCs seem also appropriate for mimicking the trabecular core, but they still underestimate its elastic properties (especially in shear) even for nearly orthotropic samples.

Multiscale modeling of macroscopic and microscopic residual stresses in metal matrix composites using 3D realistic digital microstructure models

This work presents a hierarchical multiscale method for predicting accurately and efficiently the macroscopic and microscopic residual stresses (RSes) in MMCs based on large-size realistic digital microstructure models. Effects of various conditions on the multiscale modeling are systematically studied. Results indicate that the hierarchical multiscale model shows both a good self-consistency and a good accuracy. Compared with the kinematic uniform boundary conditions, the static uniform boundary conditions lead to more accurate prediction of the thermal misfit RS. The size of the volume element has significant effects on the predicted values of the thermal misfit RS. The hierarchical multiscale model gains significant advantages over the hybrid-semiconcurrent one with respect to both computational efficiency and computer memory cost. The local fluctuation profiles and total variations of the total RS are dominated by those of the thermal misfit RS at the microscale and by those of the macroscopic RS at the macroscale, respectively.

Effect of boundary conditions on yield properties of human femoral trabecular bone.

Trabecular bone plays an important mechanical role in bone fractures and implant stability. Homogenized nonlinear finite element (FE) analysis of whole bones can deliver improved fracture risk and implant loosening assessment. Such simulations require the knowledge of mechanical properties such as an appropriate yield behavior and criterion for trabecular bone. Identification of a complete yield surface is extremely difficult experimentally but can be achieved in silico by using micro-FE analysis on cubical trabecular volume elements. Nevertheless, the influence of the boundary conditions (BCs), which are applied to such volume elements, on the obtained yield properties remains unknown. Therefore, this study compared homogenized yield properties along 17 load cases of 126 human femoral trabecular cubic specimens computed with classical kinematic uniform BCs (KUBCs) and a new set of mixed uniform BCs, namely periodicity-compatible mixed uniform BCs (PMUBCs). In stress space, PMUBCs lead to 7-72% lower yield stresses compared to KUBCs. The yield surfaces obtained with both KUBCs and PMUBCs demonstrate a pressure-sensitive ellipsoidal shape. A volume fraction and fabric-based quadric yield function successfully fitted the yield surfaces of both BCs with a correlation coefficient [Formula: see text]. As expected, yield strains show only a weak dependency on bone volume fraction and fabric. The role of the two BCs in homogenized FE analysis of whole bones will need to be investigated and validated with experimental results at the whole bone level in future studies.

Modeling of the effect of particles size, particles distribution and particles number on mechanical properties of polymer-clay nano-composites: Numerical homogenization versus experimental results

The main goal of this paper is to predict the elastic modulus of partially intercalated and exfoliated polymer-clay nano-composites using numerical homogenization techniques based on the finite element method. The representative volume element was employed here to capture nano-composites microstructure, where both intercalated exfoliated and clay platelets coexisted together. The effective macroscopic properties of the studied microstructure are obtained with two boundary conditions: periodic boundary conditions and kinematic uniform boundary conditions. The effect of particle volume fractions, aspect ratio, number and distribution of particles and the type of boundary conditions are numerically studied for different configurations. This paper investigate also the performance of several classical analytical models as Mori and Tanaka model, Halpin and Tsai model, generalized self consistent model through their ability to estimate the mechanical properties of nano-composites. A comparison between simulation results of polypropylene clay nano-composites, analytical methods and experimental data has confirmed the validity of the set results.

New boundary conditions for the computation of the apparent stiffness of statistical volume elements

We present a new auxiliary problem for the determination of the apparent stiffness of a Statistical Volume Element (SVE). The SVE is embedded in an infinite, homogeneous reference medium, subjected to a uniform strain at infinity, while tractions are applied to the boundary of the SVE to ensure that the imposed strain at infinity coincides with the average strain over the SVE. The main asset of this new auxiliary problem resides in the fact that the associated Lippmann–Schwinger equation involves without approximation the Green operator for strains of the infinite body, which is translation-invariant and has very simple, closed-form expressions. Besides, an energy principle of the Hashin and Shtrikman type can be derived from this modified Lippmann–Schwinger equation, allowing for the computation of rigorous bounds on the apparent stiffness. The new auxiliary problem requires a cautious mathematical analysis, because it is formulated in an unbounded domain. Observing that the displacement is irrelevant for homogenization purposes, we show that selecting the strain as main unknown greatly eases this analysis. Finally, it is shown that the apparent stiffness defined through these new boundary conditions “interpolates” between the apparent stiffnesses defined through static and kinematic uniform boundary conditions, which casts a new light on these two types of boundary conditions.

Finite element analysis of an accordion-like honeycomb scaffold for cardiac tissue engineering

Optimizing the function of tissue engineered cardiac muscle is becoming more feasible with the development of microfabricated scaffolds amenable to mathematical modeling. In the current study, the elastic behavior of a recently developed poly(glycerol sebacate) (PGS) accordion-like honeycomb (ALH) scaffold [Engelmayr et al., 2008. Nature Materials 7 (12), 1003–1010] was analyzed. Specifically, 2D finite element (FE) models of the ALH unit cell (periodic boundary conditions) and tessellations (kinematic uniform boundary conditions) were utilized to determine a representative volume element (RVE) and to retrospectively predict the elastic effective stiffnesses. An RVE of 90 ALH unit cells ( ≃ 3.18 × 4.03 mm ) was found, indicating that previous experimental uni-axial test samples were mechanically representative. For ALH scaffolds microfabricated from PGS cured 7.5 h at 160 °C, FE predicted effective stiffnesses in the two orthogonal material directions (0.081±0.012 and 0.033±0.005 MPa) matched published experimental data (0.083±0.004 and 0.031±0.002 MPa) within 2.4% and 6.4%. Of potential use as a design criterion, model predicted global strain amplifications were lower in ALH (0.54 and 0.34) versus rectangular honeycomb (1.19 and 0.74) scaffolds, appearing to be inversely correlated with previously measured strains-to-failure. Important in matching the anisotropic mechanical properties of native cardiac muscle, FE predicted ALH scaffolds with 50 μ m wide PGS struts to be maximally anisotropic. The FE model will thus be useful in designing future variants of the ALH pore geometry that simultaneously provide proper cardiac anisotropy and reduced stiffness to enhance heart cell-mediated contractility.

Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume

Previous results by Huet [ J. Mech. Phys. Solids 38, 813–841 (1990)] for heterogeneous elastic bodies smaller than the representative volume and submitted to static or kinematic uniform boundary conditions are partly extended to the case of mixed boundary conditions. Apparent elastic modulus and compliance tensors denned through the energetic procedure on a single specimen are considered. Through a variational approach, it is shown that each of them is bounded on one side. For cases fulfilling the Hill condition, these apparent modulus and compliance tensors are reciprocals. This provides two-sided bounds on each one. In this case, for a body smaller than the representative volume, the apparent elasticity tensor for mixed boundary conditions falls between the tensors associated with the static and kinematic uniform boundary conditions. Illustrating examples are studied. Various possible fields of application and other extensions are considered.