Repeating Unit Cell Research Articles

Compressive behaviour of cellular structures with aperiodic order

Cellular structures are commonplace in engineering applications, such as aerospace and medical engineering, because material-air composites offer significant mechanical benefits, for example due to improved weight-to-strength ratio. Typically, cellular structures are based on patterns of periodically repeating unit cells, such as squares or hexagons, but the periodic nature and the available symmetries of the patterns give rise to anisotropic performance. This is where patterns with aperiodic order are a viable alternative. Patterns created with rotational symmetry, yet no translational repetition do not possess the orders of symmetry from which mechanical anisotropy originates and therefore have the potential to mitigate this issue. In this study, additive manufacturing was used to create 2.5D, 45% dense, honeycomb cuboids based on the Penrose P3 aperiodic tiling. These were then tested under compression loading. Honeycomb cuboids based on periodic patterns were also manufactured using identical processes for the purpose of comparison. The outcome shows a significant improvement in isotropy and notably different progression of strain localisation for the honeycombs based on Penrose P3 patterns compared to the periodic comparisons during both elastic and plastic deformation.

Dynamics of quasiperiodic and quasicrystalline elastic metamaterials

This talk describes recent progress on quasiperiodic and quasicrystalline metamaterial research done by the group. While the vast majority of metamaterials are based on periodic designs based on a repeating unit cell, quasiperiodic and quasicrystalline metamaterials break the periodicity paradigm and open new interesting possibilities. The first category is based on modulations of the properties of inclusions of otherwise periodic lattices, such as point masses, springs, resonators, or stiffeners, that are given by a deterministic non-periodic pattern. In the second case, quasicrystalline materials are based on rotational symmetries which are forbidden in periodic materials, such as 5,7,8, and 10-fold rotational symmetries. The investigation of the dynamics of such materials reveals intriguing properties such as topological bandgaps and localized vibration modes that can be manipulated by a few key parameters, and wave directionalities enabled by higher order rotational symmetries that expand the behavior known to be possible in periodic metamaterials. The talk will summarize the progress done in the area, including a large number of numerical and experimental investigations.

Meso-scale Finite Element (FE) modelling of biaxial carbon fibre non-crimp-fabric (NCF) based composites under uniaxial tension and in-plane shear

Non-crimp-fabrics (NCF) are promising materials in aerospace applications. The complex internal structure of NCF composites could influence the in-plane performances, which needs to be comprehensively studied. The novel three-dimensional (3D) meso-scale repeated unit cell (RUC) models were proposed for biaxial NCF composites based on the Finite Element (FE) method to conduct a systematic parameter study, including layup sequence, out-of-plane tow waviness, resin-rich areas, transverse tow placements and delamination. The meso RUC model could effectively predict the homogenised uniaxial tensile and in-plane shear properties of biaxial NCF composites based on their meso-scale constituent and material properties. A multiscale framework was also developed for biaxial NCF composites. A micromechanical representative volume element (RVE) model provided homogenised mechanical properties for tows, and a macroscopical FE model validated the test results using the homogenised results obtained from meso RUC models. The numerical results were in good agreement with the experiment results. Therefore, the multiscale framework provides an insight into the critical parameters influencing the in-plane properties of NCF composites and an analysis tool for NCF material design.

Lattice structures in stainless steel 17-4PH manufactured via selective laser melting (SLM) process: dimensional accuracy, satellites formation, compressive response and printing parameters optimization

By the term, lattice structures are intended topologically ordered open-celled structures consisting of one or more repeating unit cells. Technological development and especially the growth of the additive manufacturing (AM) industry allows innovative structural design, including complex lattice structure. Selective laser melting (SLM) is an AM process that enables the manufacture of space filling structures. This work investigated the influence of the most important process parameter settings on lattices printability, focusing on the geometrical accuracy, the quantity of powders adhered to the main frame (satellites) and their compression behaviour. The process parameters such as the laser power, scan speed and layer height affect vigorously the design, quality and mechanical properties of the part. The aim of the paper is to evaluate how different parameter combinations affect the cellular structures’ printing. Twenty-four lattice structures with cubic and rhombic dodecahedron unit cells made of stainless steel 17-4PH (AISI-630) were printed using different combinations of SLM process parameters. Each structure was analysed considering its geometrical, topological and mechanical properties. Finally, the best parameter combination was evaluated comparing results achieved. Although this work investigated the 17-4PH stainless steel, physical principles related to the printing process described are generally true for the SLM process. Therefore, the adopted approach could still be suitable also for all the other materials commonly used with this AM technology.

Failure zone homogenization for modeling damage- and debonding-induced softening in composites including gradient-extended damage at finite strains

Within multiscale modeling strategies for composite materials, standard Hill averaging is widely used for homogenization of repeating unit cells (RUCs). However, Hill’s homogenization approach reaches its limitations at the presence of strain softening, because in the standard averaging sense the representativeness of the considered micro-scale volume is lost. Hence, in order to overcome these limitations, we propose a new failure zone averaging scheme at finite strains, which is also applicable in the softening regime. First, starting from Hill’s assumption of equal virtual work densities on different scales, a novel homogenization approach is derived analytically. Then, the analytical results are verified by numerical simulations evaluating different RUC sizes and accounting for varying geometric realizations in a statistical manner. It is shown that the proposed homogenization scheme yields size-independent stress–strain relations even after localized damage has occurred leading to strain softening within the RUCs. In addition, the material model for the epoxy bulk is valid for the description of damage progression at large deformations using a gradient-enhanced damage model. It turns out that the micromorphic power density – related to the gradient enhancement – of the failure zone gives significant values during the formation of localization while the overall work density is small. Thus, it can be concluded that for accurate failure zone homogenization, the contribution of gradient terms has to be considered. Further, cohesive zone elements are incorporated to consider the decohesion between fibers and matrix. As a result, the stress–strain curves show a lower peak stress and a higher scatter in the stress response compared to samples with perfect interfaces. In contrast, the work density plots show a higher dissipation for the RUC with damaging interfaces.

Acoustic Metamaterials and Acoustic Foams: Recent Advances

Acoustic metamaterials are synthetic materials, made of repeating unit cells that are designed to address an acoustic problem, through the rational design of their micro-features [...]

Synergistic Energy Absorption Mechanisms of Architected Liquid Crystal Elastomers.

A unique rate-dependent energy absorption behavior of liquid crystal elastomer (LCE)-based architected materials is reported. The architected materials consist of repeating unit cells of bistable tilted LCE beams sandwiched between stiff supports. The viscoelastic behavior of the LCE causes the energy absorption to increase with strain rate according to a power-law relationship, which can be modulated by changing the degree of mesogen alignment and the loading direction relative to the director. For a strain rate of 600 s-1 , the unit cell exhibits up to a 5MJ m-3 energy absorption density, which is two orders of magnitude higher than the same structure fabricated from poly(dimethylsiloxane) elastomer and is comparable to the dissipation from irreversible plastic deformation exhibited by denser metals. For a multilayered structure of unit cells, nonuniform buckling of the different layers produces additional viscoelastic dissipation. This synergistic interaction between viscoelastic dissipation and snap-through buckling causes the energy absorption density to increase with the number of layers. The sequence of cell collapse can be controlled by grading the beam thickness to further promote viscous dissipation and enhance the energy absorption density. It is envisioned that the study can contribute to the development of lightweight extreme energy-absorbing metamaterials.

Multiscale failure analysis of a 3D woven composite containing manufacturing induced voids and disbonds

Multiscale failure simulations of a three-dimensional woven composite repeating unit cell considering six scales, spanning the woven composite mesoscale to the sub-microscale voids, have been performed. The multiscale recursive micromechanics approach, which enables recursive integration of general micromechanics theories over an arbitrary number of length scales, has been employed within the NASA Multiscale Analysis Tool. The multiscale model uses both the generalized method of cells and Mori-Tanaka micromechanics theories and considers failure in the constituent materials using a simple damage model. Baseline results, from a model of a composite containing distributed voids, are compared to uniaxial experimental data for an AS4 carbon fiber/ RTM6 epoxy matrix orthogonal three-dimensional woven composite with good agreement in terms of global stiffness and global failure stress. The multiscale model is then used to examine the nonlinear response of the material to other loading conditions. Case studies, motivated by X-ray computed tomography data, are presented on the effects of manufacturing induced voids and cracks.

Hybrid scaffolds with a 3D-printed polymer lattice core and a bioactive hydrogel shell for bone regeneration

Recent research in the field of tissue engineering is focusing on the realization of hybrid scaffolds as multi-material systems which allow to successfully combine the advantages of different biomaterials. In addition, additive manufacturing technologies are currently explored for their production, as means to control and personalize the scaffold structure. In this paper, composite scaffolds with a core–shell structure are studied, the core consisting of a rigid poly-l-lactic acid lattice realized by fused deposition modeling, and the shell consisting of a bioactive hydrogel, grafted upon the core and freeze-dried to develop porous microstructure. Different lattice structures are designed and realized as repetition of unit cells having different size and strut arrangement. Compression tests reveal the suitability of the mechanical properties of the scaffolds for bone tissue regeneration, and the possibility to modulate their stiffness and strength upon the lattice parameters. Moreover, the interconnected porous structure of the shell, assessed by morphological analysis at the microscope, may promote cell colonization and proliferation, while its composition may support osteogenic differentiation.

Assessment of mechanical behaviour of functionally graded biomaterials using finite element method

Biomechanics is the application of mechanical principles to the modelling and simulation of biological tissues. The manufacturing of functionally graded biomaterials as orthopaedic bone substitutes and implants is possible due to recent developments in additive manufacturing (AM). Despite the fact that biodegradability is an important consideration for tissue regeneration and the avoidance of implant-associated disorders, no functionally graded structures have been 3D-printed from biodegradable materials. The functionally graded biomaterials (FGBs) are a combination of low density, high yield stress and Young's modulus. The present study uses numerical and analytical techniques to compute the mechanical behaviour (Elastic modulus and yield stress) of biomaterials made by repeating unit cells namely bi-pyramid hexagonal lattice structure. The analytical solution was validated through finite element method (ANSYS). Recent developments in additive manufacturing have established that biomaterial may be created using representative volume elements (RVE). In investigating a single unit cell experiencing the loads and boundary conditions sensed in an infinite lattice structure, analytical relationships based on both Euler-Bernoulli and Timoshenko beam theories were derived. Despite the fact that the analytical and numerical results were mostly in agreement, the mechanical properties obtained by the Timoshenko beam theory were closer to the numerical values. The study proved that the computational methodologies to resolve the uncertainty of FGB mechanical properties, allowing for various strategies for designing bioinspired AM biomaterials.

A hybrid cohesive/volumetric multiscale finite element model for the failure analysis of fiber-reinforced composite structures

Several multiscale approaches have been introduced in the last decades, aimed at investigating the failure behavior of composite materials with reduced computational costs and a high accuracy level at the same time. This is due to their capability to account for the effect of microstructural details on the overall mechanical behavior of such materials. Among the existing multiscale approaches, continuous/discontinuous models have gained increasing interest, by virtue of their ability to overcome the main limitation of purely volumetric approaches, consisting in a spurious mesh sensitivity in the presence of strain localization phenomena. In this work, a novel continuous/discontinuous multiscale model is proposed, used in combination with a hybrid cohesive/volumetric finite element approach for the accurate numerical simulation of pervasive fracture propagation at both the micro- and macro-scales. This approach allows a microscopically informed traction-separation law for the embedded interfaces to be extracted from the homogenized bulk response, derived with reference to a suitably defined Repeating Unit Cell (RUC). The present model is applied to the numerical simulation of microscopic damage phenomena in composite structures. Finally, comparisons with direct numerical simulations on fully meshed specimens are presented for validation purposes, in terms of both loading curves and associated crack patterns.

Extended Locally Exact Homogenization Theory for Effective Coefficients and Localized Responses of Piezoelectric Composites

Herein, the elasticity‐based locally exact homogenization theory (LEHT) is extended to study the effective and localized responses of piezoelectric fiber composites. Based on the Trefftz concept, coupled internal solutions for electric fields and elastic displacements are developed for repeating unit cells (RUCs) that are characterized from the composite domain. The unknown coefficients of the internal fields can be determined by imposing interfacial continuity conditions between fiber and matrix and implementing the newly proposed multiphysics periodic boundary conditions. Finally, the generalized constitutive relations are established to predict the effective coefficients. The accuracy of the proposed technique is tested by validating the present simulations against the analytical Eshelby solution and asymptotic method, classical Mori−Tanaka (M−T) model, finite element (FE)‐ and finite volume (FV)‐based methods, as well as the experimental measurement in the literature with excellent agreement. What is more, several microstructural effects, such as phase volume fraction, fiber arrangement, etc., are varied to test their effects on the piezoelectric composites at both homogenized and localized levels. Based on the presented computational efficiency, the multiphysics LEHT is encapsulated into a blackbox with input/output data constructions for easy implementation by professionals or nonprofessionals.

Processing of biomaterials for bone tissue engineering: State of the art

Bones are the functional tissue, to provide structural support, protect vital organs, store minerals, helps in multi-directional motion and bear load of our body. Fracture of bones, in which continuity of bone tissue breaks, which occurred due to trauma (high intensity impact) and over stressing are treated with help of bone implants. Bone implants are fabricated through biomaterials which are categorized as as: (a) first generation, (b) second generation and (c) third generation biomaterials. First generation implants are bioinert, second generation implants are biodegradable, bioinert and bioactive. Third generation implants focuses on stimulating specific cellular response at molecular level along with the properties of second generation. Bone tissue engineering (BTE) is responsive for the evolution of third generation implants, where the main focus is to create the environment which helps to stimulate the cell response in direction of bone regeneration. This aim is achieved with engineered tissue implant which includes: (a) scaffold for architectural support & cell homing, (b) osteogenic cells and (c) signaling growth factors. Tissues in human body are the 3D repeating unit cells also called as extra cellular matrix (ECM). Similarly scaffold is the artificial 3D porous structure of biomaterials (ceramics, metals and polymers) which mimic those 3D repeating unit cells or ECM. Scaffold is a backbone or fundamental unit of BTE. Success of scaffold decides the success of engineered bone tissue implant. This review summarizes the factor influencing BTE, factor involved in selection of material for scaffold fabrication and trending fabrication methods for BTE scaffolds.

Free vibration analysis of PLA based auxetic metamaterial structural composite using finite element analysis

The unique engineered material termed Metamaterials is not seen naturally and were made artificially by repeating unit cell geometry in multiple directions to form a structural composite metamaterial. The present research is to investigate the three types of uniform architecture metamaterials i.e square, auxetic and honeycomb along with three of its hybrids i.e square-auxetic, auxetic-honeycomb and square-honeycomb. All the modelled metamaterials samples were imported to Ansys modal analysis section and free vibration analysis is performed under the workbench graphic user interface via the finite element method. The materials for the simulation test is PLA (polylactic acid) adopted. The results like natural frequency and mode shapes are extracted from the simulation. It was observed that among hybrids the highest natural frequency was seen for auxetic-square hybrid geometry i.e 858 Hz at 3rd natural frequency, which decreased to 35 and 44% when compared to square-honeycomb, auxetic-honeycomb respectively. The emerging area potential application of hybrid PLA auxetic metamaterial includes robotics, aerospace, nautical, sports and bio-medical industries.

Machine-learned prediction of the electronic fields in a crystal

We propose an approach for exploiting machine learning to approximate electronic fields in crystalline solids subjected to deformation. Strain engineering is emerging as a widely used method for tuning the properties of materials, and this requires repeated density functional theory calculations of the unit cell subjected to strain. Repeated unit cell calculations are also required for multi-resolution studies of defects in crystalline solids. We propose an approach that uses data from such calculations to train a carefully architected machine learning approximation. We demonstrate the approach on magnesium, a promising light-weight structural material: we show that we can predict the energy and electronic fields to the level of chemical accuracy, and even capture lattice instabilities.

A frequency response function-based optimization for metamaterial beams considering both location and mass distributions of local resonators

To date, phononic crystals/metamaterials normally adopt classical periodic configurations, and optimization strategies for them are based primarily on dispersion relations from a repeated unit cell. In this study, a frequency response function (FRF) based optimization scheme is presented for a locally resonant metamaterial beam that considers both the resonant frequencies and distribution locations of the resonators. Three optimization objectives involving (1) broadband, (2) multi-band, and (3) high-attenuation characteristics are exploited as study cases, and a single-objective genetic algorithm is used to determine the optimal solutions for the prescribed bandgap targets. The spectral element method is used as an analytic formulation to determine the metamaterial FRFs, and the finite element method is used to validate the effectiveness of the optimization strategy. The results reveal that these objective bandgap characteristics can be enabled without increasing the resonator mass following the proposed optimization procedure. This shows the potential of adjusting the locations and resonance frequencies of resonators in metamaterial beams beyond the widely accepted periodic structures. The most important finding of this study is that promising bandgap properties can be easily realized with only single-degree-of-freedom resonators instead of designing more complex ones or multi-resonators. This FRF-based optimization method can be considered as a simple but instructive strategy for optimal or inverse designs in metamaterials.

Model of the low temperature magnetic phases of gadolinium gallium garnet

The magnetic behaviour of gadolinium gallium garnet in an external magnetic field at zero temperature is considered. For high fields a classical spin model of the gadolinium ions predicts a spin configuration that is periodic at the level of the smallest repeating unit cell. The quantum version of the model is treated via a truncated Holstein–Primakoff transformation with axes defined by the classical spin configuration, and the magnon excitation bands are calculated. The model predicts a transition in the field range of 1.9–2.1 T, sensitive to the direction of the applied field, which is caused by one or more magnon modes becoming soft as the field is decreased. In general the soft modes occur at incommensurate wavevectors and therefore break the periodicity of the spin configuration below the transition. One exception occurs when the field aligns with one of the principle crystal axes, in which case periodicity of the spin configuration is found to be maintained on a larger crystallographic cubic cell even below the transition. This simple case is studied in more detail. Comparisons are drawn with existing experimental data, and further experimental tests of the model are suggested.

Neuromorphic metamaterial structures

Computerized structural optimization methods are often used to design the shape of structures to achieve a desired function, such as a specific compliance. In the simplest case of a structure built from a material with small deformations obeying Cauchy elasticity, the compliance is constant and the relationship between the forces applied on the structure and its deformation is linear. This linearity severely limits the types of functions which can be achieved by such materials. Here we study mechanical metamaterials made of repeating unit cells, each with specific dimensional parameters and with one-sided contact non-linearities. We show that the force and displacement equilibrium configurations of these metamaterials are mathematically equivalent to the fixed points of a recurrent artificial neural network. By exploiting this equivalence, we demonstrate mechanical metamaterials that can be designed (trained) to implement complex non-linear functions, using a gradient descent algorithm as in artificial neural networks. One of our metamaterial structures has a higher compliance when it is pressed against a pattern of raised bumps corresponding to the vowels in the Braille alphabet, than when it is pressed against patterns for six consonants. As artificial neural networks are known to be efficient models for numerous problems in machine learning, our results reveal that beneficial features of neural networks can be transferred to physical objects (mechanical structures). These features include the design of systems with complex input–output relationships by using generic methods that only rely on the repetitive processing of pairs of inputs and desired outputs, as well as remarkable generalization capabilities. We anticipate our methodology to be a starting point for the transfer of some of the breakthroughs in the rapidly advancing field of machine learning to highly functional physical devices in applications that are constrained by energy, volume, data processing or response time.

ADDITIVELY MANUFACTURED Ti6Al4V LATTICE STRUCTURES FOR BIOMEDICAL APPLICATIONS

Additive Manufacturing (AM) is a rapidly developing technology which provides opportunity to build up complex geometries due to the freedom of manufacturing. Lattice structures, three-dimensional open-celled structures composed of one or more repeating unit cells, can be produced with unique mechanical, thermal, acoustic, biomedical and electrical properties by optimization of type and dimension of unit cell and additive manufacturing parameters. Lattice structures provide lightweight and porous parts which are widely preferable in biomedical applications. Different type of lattice structures have been used for obtaining bone like implant surface to accelerate osseointegration. There are many studies in this field, but the ideal designs and dimensional accuracy of the various lattice structures for biomedical field have not been completely reached. In this study, octahedral, star and dodecahedron lattice structures with thin strut diameter were manufactured by laser powder bed fusion technology (LPBF) by Ti6Al4V powder. Cubic and plate samples were built on z-direction and their top and side surfaces were inspected in terms of topographical characteristics and dimensional accuracy by scanning electron microscope.

Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries

We prove two Lieb-Schultz-Mattis type theorems that apply to any translationally invariant and local fermionic $d$-dimensional lattice Hamiltonian for which fermion-number conservation is broken down to the conservation of fermion parity. We show that when the internal symmetry group $G^{\,}_{f}$ is realized locally (in a repeat unit cell of the lattice) by a nontrivial projective representation, then the ground state cannot be simultaneously nondegenerate, symmetric (with respect to lattice translations and $G^{\,}_{f}$), and gapped. We also show that when the repeat unit cell hosts an odd number of Majorana degrees of freedom and the cardinality of the lattice is even, then the ground state cannot be simultaneously nondegenerate, gapped, and translation symmetric.