Repeating Unit Cell Research Articles

Additively manufactured biodegradable porous iron

Additively manufactured (AM) topologically ordered porous metallic biomaterials with the proper biodegradation profile offer a unique combination of properties ideal for bone regeneration. These include a fully interconnected porous structure, bone-mimicking mechanical properties, and the possibility of fully regenerating bony defects. Most of such biomaterials are, however, based on magnesium and, thus, degrade too fast. Here, we present the first report on topologically ordered porous iron made by Direct Metal Printing (DMP). The topological design was based on a repetitive diamond unit cell. We conducted a comprehensive study on the in vitro biodegradation behavior (up to 28 days), electrochemical performance, time-dependent mechanical properties, and biocompatibility of the scaffolds. The mechanical properties of AM porous iron (E = 1600–1800 MPa) were still within the range of the values reported for trabecular bone after 28 days of biodegradation. Electrochemical tests showed up to ≈12 times higher rates of biodegradation for AM porous iron as compared to that of cold-rolled (CR) iron, while only 3.1% of weight loss was measured after 4 weeks of immersion tests. The biodegradation mechanisms were found to be topology-dependent and different between the periphery and central parts of the scaffolds. While direct contact between MG-63 cells and scaffolds revealed substantial and almost instant cytotoxicity in static cell culture, as compared to Ti-6Al-4V, the cytocompatibility according to ISO 10993 was reasonable in in vitro assays for up to 72 h. This study shows how DMP could be used to increase the surface area and decrease the grain sizes of topologically ordered porous metallic biomaterials made from metals that are usually considered to degrade too slowly (e.g., iron), opening up many new opportunities for the development of biodegradable metallic biomaterials. Statement of SignificanceBiodegradation in general and proper biodegradation profile in particular are perhaps the most important requirements that additively manufactured (AM) topologically ordered porous metallic biomaterials should offer in order to become the ideal biomaterial for bone regeneration. Currently, most biodegradable metallic biomaterials are based on magnesium, which degrade fast with gas generation. Here, we present the first report on topologically ordered porous iron made by Direct Metal Printing (DMP). We also conducted a comprehensive study on the biodegradation behavior, electrochemical performance, biocompatibility, and the time evolution of the mechanical properties of the implants. We show that these implants possess bone-mimicking mechanical properties, accelerated degradation rate, and reasonable cytocompatibility, opening up many new opportunities for the development of iron-based biodegradable materials.

Simulated tissue growth for 3D printed scaffolds.

Experiments have demonstrated biological tissues grow by mechanically sensing their localized curvature, therefore making geometry a key consideration for tissue scaffold design. We developed a simulation approach for modeling tissue growth on beam-based geometries of repeating unit cells, with four lattice topologies considered. In simulations, tissue was seeded on surfaces with new tissue growing in empty voxels with positive curvature. Growth was fastest on topologies with more beams per unit cell when unit cell volume/porosity was fixed, but fastest for topologies with fewer beams per unit cell when beam width/porosity was fixed. Tissue filled proportional to mean positive surface curvature per volume. Faster filling scaffolds had lower permeability, which is important to support nutrient transport, and highlights a need for tuning geometries appropriately for conflicting trade-offs. A balance among trade-offs was found for scaffolds with beam diameters of about [Formula: see text] and 50% porosity, therefore providing the opportunity for further optimization based on criteria such as mechanical factors. Overall, these findings provide insight into how curvature-based tissue growth progresses in complex scaffold geometries, and a foundation for developing optimized scaffolds for clinical applications.

The matrix–twin transition in a perfect Mg crystal: Ab initio study

By means of ab initio calculations, we studied the matrix – twin transition in a perfect Mg crystal where the {101¯2} plane is the twinning plane. The twinning transition is described by the shear of a suitable periodically repeated unit cell and by the atomic shuffling inside the unit cell. We found two hcp – hcp paths that can create the glide and reflection {101¯2} twin boundary, respectively. Assuming a perfect hcp crystal, it turned out that the total energy profiles of both hcp – hcp paths are the same. We further pointed out the equivalence between the description of the matrix – twin transition in the frame of an orthorhombic and of a monoclinic unit cell, both containing four atoms. We have tracked the trajectories of the net atomic motion that are energetically very favorable. In addition, the effect of volume relaxation on the total energy profiles was investigated and its influence on the plastic deformation of Mg crystal was discussed.

Assessment of structural and hemodynamic performance of vascular stents modelled as periodic lattices

This work considers vascular stents with tubular geometry assumed to follow a periodic arrangement of repeating unit cells. Structural and hemodynamic metrics are presented to assess alternative stent geometries, each defined by the topology of the unit cell. Structural metrics include foreshortening, elastic recoil and radial stiffness, whereas hemodynamic performance is described by a wall shear stress index quantifying the impact of in-stent restenosis. A representative volume element (RVE) modelling approach is used, and results are compared to those obtained from full simulations of entire stents. We demonstrate that the RVE approach can be used to quantify the impact of the topology of the repeating unit on the structural and hemodynamic properties of a stent, and thus support clinicians in making proper choices among alternative stent geometries.

Ultralight cellular composite materials with architected geometrical structure

A novel computational approach is presented to predict the overall hyperelastic properties of ultralight cuboct cellular lattices made of brittle carbon fiber-reinforced polymer composites, such as the ones recently fabricated at the MIT Media Lab-Center for Bits and Atoms. The repetitive unit cell (RUC) approach is employed to model the fabricated cellular micro-lattices. Each member of the cellular structure is modeled using only one finite beam element with 12 degrees of freedom, and the nonlinear coupling of axial, bidirectional-bending, and torsional deformations is studied for each 3D spatial beam element. Since the cellular composite material is fabricated via the assemblage of building blocks by mechanical interlocking connections, we utilize the standardized Ramberg-Osgood function for the moment-rotation relation at the ends of adjacent members to enable tuning the appropriate flexibility for connections between two extreme limits of pin-jointed or rigid-jointed connections. The mixed variational functional in the updated Lagrangian co-rotational reference frame is obtained to derive explicitly the stiffness matrix. Then, we use newly proposed homotopy methods to solve the algebraic equations.

Prediction of mechanical properties of knitted fabrics under tensile and shear loading: Mesoscale analysis using representative unit cells and its validation

Abstract This article presents a numerical framework to predict the mechanical behavior of knitted fabrics from their discrete structure at the fabric yarn level, i.e., the mesostructure, utilizing the hierarchical multiscale method. Due to the regular distribution of yarn loops in a knitted structure, the homogenization theory for periodic materials can be employed. Thus, instead of considering the whole fabric sample under loading, a significantly less computationally demanding analysis can be done on a repeated unit cell (RUC). This RUC is created based on simple structural parameters of knitted yarn loops and its fabric yarns are assumed to behave transversely isotropic. Nonlinear finite element analyses are performed to determine the stress fields in the RUC under tensile and shear loading. During this analysis, contact friction among yarns is considered as well as the periodic boundary conditions are employed. The macroscopic stresses then can be derived from the stress fields in the RUC by means of the numerical homogenization scheme. The physical fidelity of the proposed framework is shown by the good agreement between the predicted mechanical properties of knitted fabrics and corresponding experimental data.

Investigation on the Yarn Squeezing Effect of Three Dimensional Full Five Directional Braided Composites

The influence of yarn squeezing effect on the geometric morphology and mechanical property of the three dimensional full five directional (3DF5D) braided composites is explored. Spatial path and cross-section shape of the yarns in the braided structure are characterized based on the micro computed tomography (micro CT) scanning images. The yarn distortion due to the squeezing effect is discussed and mathematical morphology of the yarn geometry is established. A new repeated unit cell (RUC) model of 3DF5D braided composites considering yarn squeezing effect is developed. Based on this model, mechanical properties of 3DF5D braided composites are analyzed. Good agreement is obtained between the predicted and experiment results. Moreover, the stress distribution of the new RUC model are compared with original RUC model, showing that the squeezing effect significantly increases the stress concentration level of the axial yarns.

Tunable bandgaps in a deployable metamaterial

In this manuscript, we investigate deployable structures (such as solar arrays) and origami-inspired foldable structures as metamaterials capable of tunable wave manipulation. Specifically, we present a metamaterial whose bandgaps can be modulated by changing the fold angle of adjacent panels. The repeating unit cell of the structure consists of a beam (representing a panel) and a torsional spring (representing the folding mechanism). Two important cases are considered. Firstly, the fold angle (angle between adjacent beams), Ψ, is zero and only flexural waves propagate. In the second case, the fold angle is greater than zero (Ψ>0). This causes longitudinal and transverse vibration to be coupled. FEM models are used to validate both these analyses.Increasing the fold angle was found to inflict notable changes to the wave transmission characteristics of the structure. In general, increasing the fold angles caused the bandwidth of bandgaps to increase. For the lowest four bandgaps we found bandwidth increases of 252%, 177%, 230% and 163% respectively at Ψ=90deg (relative to the bandwidths at Ψ=0). In addition, non-trivial increases in bandwidth of the odd-numbered bandgaps occurs even at small fold angles-the bandwidth for the first and third bandgaps effectively double in size (increase by 100%) at Ψ=20deg relative to those at Ψ=0. This could have ramifications in the context of tunable wave manipulation and adaptive filtering.In addition, by expanding out the characteristic equation of transfer matrix for the straight structure, we prove that the upper band edge of the nth bandgap will always equal the nth simply supported natural frequency of the constituent beam. Further, we found that the ratio (EIkt) is a pertinent parameter affecting the bandwidth of bandgaps. For low values of the ratio, effectively, no bandgap exists. For higher values of the ratio (EIkt), we obtain a relatively large bandgap over which no waves propagate. This can have ramifications for the design of foldable structures. As an alternative to impedance-based structural health monitoring, these insights can aid in health monitoring of deployable structures by tracking the bandwidth of bandgaps which can provide clues about the mechanical parameters of the structure.

Characterization of intralaminar strengths of virtually cured polymer matrix composites

Abstract The effect of the curing process on the mechanical response of fiber reinforced polymer matrix composites, IM7/Epoxy 862, is studied using a novel computational model. Virtual testing is performed using the Finite Element Method (FEM) at the microscale where Repeating Unit Cells (RUCs) are analyzed to determine strengths and elastic properties of a thin lamina. Two sets of Boundary Conditions (BCs) during curing are studied; Periodic (PBCs) and Flat (FBCs). The commercial codes, ABAQUS/Standard and ABAQUS/Explicit have been used as the FEM solver, supplemented by user written subroutines, that model the curing process and the mechanical response including the onset of damage and subsequent failure. The transition from a continuum to damage/failure state is effected by using the Bazant-Oh Crack Band model, which preserves mesh objectivity. Results are presented for a randomly packed RUC which is first subjected to curing and subsequently to mechanical loading. It is assumed that damage can initiate during the cure cycle if the internal stresses exceed prescribed values and that this damage can grow in the form of cohesive cracks. After the curing of the RUCs, virtual mechanical loading programs are studied. In this latter stage, the cured RUCs, which may contain internal damage and residual stresses are further subjected to additional stresses induced by the mechanical loading. The results clearly show the influence of the curing cycle (the effects of manufacturing) on the mechanical response.

Acoustic properties of additive manufactured narrow tube periodic structures

Quarter tube periodic resonators are used to attenuate the acoustic energy of discrete frequencies wherein their sizes can vary from narrow to large. This paper reports acoustical properties of additive manufactured, narrow tube, single and multi-periodic structures, both, theoretically and experimentally. Herein, multi-periodic structures are defined as periodically arranged unit sections of tubes, where each section is composed of periodically repeated unit cells of different sizes and shapes. Structures with hexagonal narrow tubes and octagonal narrow tubes with interfacial gaps are considered for the study, and normal absorption coefficients of these samples are measured by using impedance tube. the theoretical absorption coefficient of these structures is predicted using unit section analysis method and narrow tube theory, where shape dependent viscous and thermal losses are incorporated. Estimated theoretical absorption coefficients are in good agreement with measured results. The result shows that the frequency and amplitude of maximum absorption can be varied by altering the aperture ratio and/or the length of periodic structure. The proposed theoretical method gives an alternative approach for designing and manufacturing periodic narrow tubes for different applications such as absorbing panels, acoustic transducers, and engine filter elements.

General shell section properties and failure model for cross-laminated timber obtained by numerical homogenization

In this study homogenized mechanical properties are derived for structural finite element analysis of cross-laminated timber (CLT) discretized by means of shell elements. In the first step, based on the results of experimental three-point bending tests on a CLT slab component, a numerical optimization procedure is applied to identify the elastic and plastic properties of the individual spruce wood laminates, considering orthotropic elasticity and Hill’s yield criterion. Taking into account its periodic structure, a repeating unit cell (RUC) of the CLT is defined in the subsequent step. To this RUC specific loading conditions are imposed successively, yielding the entries of the homogenized elastic stiffness matrix of a general shell section. Extending the RUC simulations into the inelastic domain of deformation results in a homogenized failure surface on the structural scale, which is implemented as a postprocessing variable in a commercial finite element program. Throughout all analyses, in sensitivity studies the most decisive material properties are identified. Results of comparative simulations on a homogenized structural model and an elaborate full three-dimensional finite element model of a point-supported CLT slab show the accuracy and efficiency of the proposed approach based on homogenized mechanical properties.

Mechanical behavior of unidirectional composites with hexagonal and uneven distribution of fibers in the transverse cross-section

Uneven distribution of fibers can adversely affect the mechanical behavior of unidirectional composites. A micromechanical model based on finite element analysis is presented to evaluate elastic and strength properties of such composites under normal loading. Analysis starts with identification of micro unit cells/micro repetitive unit cells and/or representative volume elements. Because of uneven distribution/random distribution of the fibers, fiber volume fraction can be different for different micro unit cells present at different locations in the transverse cross-section of the unidirectional composite. Configuration of the micro unit cell is worked out at the outset considering the fiber distribution having the hexagonal arrangement, but with different volume fraction at different locations. For such micro unit cells, elastic and strength properties are obtained based on finite element analysis starting with the elastic and strength properties of fiber and matrix. With the properties obtained for different micro unit cells, elastic and strength properties of the unidirectional composite with micro unit cells having hexagonal arrangement of fibers with different volume fraction at different locations in the transverse cross-section are determined. Further, elastic and strength properties are evaluated for micro unit cells with uneven distribution of fibers in the transverse cross-section.

A multiscale progressive damage analysis for laminated composite structures using the parametric HFGMC micromechanics

A nonlinear multiscale damage analysis is proposed based on the parametric High Fidelity Generalized Method of Cells (HFGMC) micromodel. Two repeating unit-cells (RUCs), square and hexagonal, for the composite microstructure are discretized into subcells which represents the fiber and matrix phases. Average traction and displacement continuities, periodicity conditions, and volumetric equilibrium are applied. The proposed modeling approach is implemented as a user material subroutine within displacement-based layered-shell FE models, where each layer can be represented with at least one RUC. The RUC through-thickness stress and stiffness contributions are eliminated in case of a failure. Two failure criteria, in conjunction with compressive strength equation, are applied to determine RUC failure. In the first, the Tsai-Wu failure criterion is used at the RUC level. The second employs the strain invariant failure theory (SIFT) within each subcell, leading to its extinction. The in-situ properties and damage parameters of the RUC are determined by calibration using axial, transverse, and shear data of unnoteched laminates. The accuracy of the proposed framework is demonstrated by a comparison with experimental results of open-hole laminates.

Solution of the Nonlinear High-Fidelity Generalized Method of Cells Micromechanics Relations via Order-Reduction Techniques

The High-Fidelity Generalized Method of Cells (HFGMC) is one technique, distinct from traditional finite-element approaches, for accurately simulating nonlinear composite material behavior. In this work, the HFGMC global system of equations for doubly periodic repeating unit cells with nonlinear constituents has been reduced in size through the novel application of a Petrov-Galerkin Proper Orthogonal Decomposition order-reduction scheme in order to improve its computational efficiency. Order-reduced models of an E-glass/Nylon 12 composite led to a 4.8–6.3x speedup in the equation assembly/solution runtime while maintaining model accuracy. This corresponded to a 21–38% reduction in total runtime. The significant difference in assembly/solution and total runtimes was attributed to the evaluation of integration point inelastic field quantities; this step was identical between the unreduced and order-reduced models. Nonetheless, order-reduced techniques offer the potential to significantly improve the computational efficiency of multiscale calculations.

Micromechanical Damage Modeling of Long Fiber Reinforced Composites With the Parametric Method of Cells

AbstractMicromechanical damage modeling is presented with the parametric high‐fidelity generalized method of cells for a long fiber reinforced composite. Two models for a planar single fiber repeating unit cell, including damage, are proposed. The first one, implemented with the spatial continuum damage mechanics, is based on the idea that volumetric defects occur in the material phases. The other one, modeled with the interface damage mechanics, is founded on the view that cracks as surface‐like de‐ fects cause the stress degradation. The potential and ability of both approaches to predict damage in first‐order homogenization is shown by comparing the simulation results with each other as well as with test data under uniaxial and biaxial stress loading. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The effect of free-edges and layer shifting on intralaminar and interlaminar stresses in woven composites

The free-edge effects and relative layer shifting in the interlaminar and intralaminar stresses of plain woven composite laminates under uniaxial extension is investigated numerically using a finite element approach. A computational framework of the free-edge problem for periodic structures with finite width is applied to woven laminates. First, two-layered laminates with three different shifting configurations are studied considering repeating unit cells simulating finite and infinite width. For each configuration, two different widths are considered by trimming the model at different locations in order to investigate different free-edge effects. Then, two four-layered laminates with no shifting and a maximum shifting configuration are analyzed to illustrate the effect of neighboring layers in the stresses. For each shifting configuration, different delamination mechanisms are expected. When considering more layers, it is found that the stacking configuration affects the state of stress and the free-edge effects depending on the shifting. In general, a different behavior than that of unidirectional tape laminates is found, since the interlaminar and intralaminar stresses can be higher than those generated at the free-edges. Particularly, for the maximum shifting configuration results are in agreement with experimental results in the literature where no debonding between yarns was observed at the free-edges.

Image-based modeling of moisture-induced swelling and stress in 2D textile composite materials using a global-local approach

The present paper focuses on the employment of a global-local approach for the simulation of moisture-induced swelling and stress in a 2D textile carbon fiber/epoxy matrix composite material. The approach is based on the development of a robust image-based finite element model built starting from X-ray microtomography scans of a repetitive unit cell of the textile material: simulations aim at investigating both the global (sample scale) level and the local micromechanical (interfaces between matrix and fibrous tows, matrix inside a tow) level behaviour. Three models at the different cited scales are first built (ply level model, tow level and fiber level model): then a non-concurrent (node-based) approach is used for coupling the models at different scales. The results of numerical simulation of moisture-induced swelling and stress are discussed, with emphasis on the identification of the most critical zones at which maximal stress values are calculated, at different scales.

Buckling and vibration of composite lattice elliptical cylindrical shells

An approach to the finite element study of the buckling and dynamic behaviour of composite lattice cylindrical shells with elliptical cross sections is presented in this paper. The lattice shells are modelled as three-dimensional frame structures composed of curvilinear ribs using beam finite elements. A specialised algorithm is developed to generate the finite element model of the lattice shells based on multiple use of the repeating unit cell of the composite lattice structure. Using this model, the buckling behaviour of the shells subjected to axial loading and transverse bending are investigated. Fundamental frequencies of axial and transverse vibrations of the shells with a massive rigid disk attached to their ends are determined based on the modelling approach proposed in this work. The effects of parameters of the lattice structure on the values of critical buckling loads, buckling and vibration mode shapes, and the fundamental frequencies are examined using parametric analyses. Based on the computations, the angles of orientation of helical ribs delivering maximum critical loads and fundamental frequencies are identified. The results of this study can be applied to the design of the composite tubular bodies of spacecraft made in the form of cylindrical lattice shells with elliptical cross sections.

Numerical study of enhanced mixing in pressure-driven flows in microchannels using a spatially periodic electric field.

We propose an innovative mechanism for enhancing mixing in steady pressure driven flow of an electrolytic solution in a straight rectangular microchannel. A transverse electric field is used to generate an electroosmotic flow across the cross-section. The resulting flow field consists of a pair of helical vortices that transport fluid elements along the channel. We show, through numerical simulations, that chaotic advection may be induced by periodically varying the direction of the applied electric field along the channel length. This periodic electric field generates a longitudinally varying, three-dimensional steady flow, such that the streamlines in the first half of the repeating unit cell intersect those in the second half, when projected onto the cross-section. Mixing is qualitatively characterized by tracking passive particles and obtaining Poincaré maps. For quantification of the extent of mixing, Shannon entropy is calculated using particle advection of a binary mixture. The convection diffusion equation is also used to track the evolution of a scalar species and quantify the mixing efficiency as a function of the Péclet number.

Multi-scale modeling of triaxial braided composites for FE-based modal analysis of hybrid metal-composite gears

The paper presents a description of a methodology to accurately estimate the natural frequencies of a hybrid metal-composite gear, where the web is made of a polymer matrix reinforced with two-dimensional (2D) triaxial braided fabrics. The proposed approach is based on a multi-scale composite modeling starting from evaluation of homogenized material properties of the gear web at the meso-scale for a subsequent FE-based modal analysis. For this purpose, the mechanical behavior of a single Repetitive Unit Cell (RUC) at the meso-scale is analyzed with a FE procedure which takes into account the interweaving and cross-section geometry of the yarns, volume fraction and local fiber orientation. It is shown that the described modeling strategy allows to predict gear’s natural frequencies that are close to the experimental results. Significant accuracy enhancement is achieved with respect to the model in which the web material is considered as perfectly isotropic.