Cellular Solids Research Articles

Cellular solids and prestressed affine networks as models of the elastic behavior of soft biological structures.

We reviewed two microstructural models, cellular solid models and prestressed affine network models, that have been used previously in studies of elastic behavior of soft biological materials. These models provide simple and mathematically transparent equations that can be used to interpret experimental data and to obtain quantitative predictions of the elastic properties of biological structures. In both models, volumetric density and elastic properties of the microstructure are key determinants of the macroscopic elastic properties. In the prestressed network model, geometrical rearrangement of the microstructure (kinematic stiffness) is also important. As examples of application of these models, we considered the shear behavior of the cytoskeleton of adherent cells, of the collagen network of articular cartilage, and of the lung parenchymal network since their ability to resist shear is important for their normal biological and physiological functions. All three networks carry a pre-existing stress (prestress). We predicted their shear moduli using the microstructural models and compared those predictions with existing experimental data. Prestressed network models of the cytoskeleton and of the lung parenchyma provided a better correspondence to experimental data than cellular solid models. Both cellular solid and prestressed network models of the cartilage collagen network provided reasonable agreements with experimental values. These findings suggested that the kinematic stiffness and material stiffness of microstructural elements were both important determinants of the shear modulus of the cytoskeleton and of the lung parenchyma, whereas elasticity of collagen fibrils had a predominant role in the cartilage shear behavior.

Ultra-soft cellular solids inspired by marine mussel plaques: scaling of the mechanical properties

The porous core of the marine mussel provides mechanical advantages of achieving substantial deformation and high toughness. Inspired by the unique mechanical behaviours of mussel plaque cores, the present study develops ultra-soft, stretchable and tough cellular solids by bio-mimicking their porosity and properties. Although scaling laws to correlate the mechanical properties of cellular solids with bulk materials have been established, they are primarily based on experimental data from stiff cellular solids. In this article, the scaling law for soft cellular solids is studied to relate tensile properties and toughness with volume fractions. Unlike conventional stiff cellular solids, the scaling law shows that soft lattices experience bending-dominated deformation at the elastic state and fail under stretching-dominated deformation. Uniaxial and planar tensile tests demonstrate that the proposed soft lattices achieved Young’s modulus of 0.04 ∼ 0.17 MPa, failure strain of 135.5 ∼ 213.9 % and toughness of 582 ∼ 941 J m −2 by manipulating the volume fraction ( 0.2 ∼ 0.5 ), positioning them as softer, more stretchable and tougher than the majority of engineering foams in the Ashby diagram. In addition, the toughness of soft lattices is found to be sensitive to volume fraction but insensitive to initial crack length, thickness and height ( ≥ 20 m m ). Two distinct failure modes, truss failure and joint failure, are correlated with volume fraction. The soft lattice with a volume fraction (around 0.4) similar to mussel plaque fails at the transition of these two failure modes and achieves the highest toughness. This study contributes new insights to the materials science community and lays the foundation for the development of lightweight, damage-tolerant and high-performance metamaterials.

Crashworthiness Investigations for 3D-Printed Multi-Layer Multi-Topology Engineering Resin Lattice Materials.

In comparison to monolithic materials, cellular solids have superior energy absorption capabilities. Of particular interest within this category are the periodic lattice materials, which offer repeatable and highly customizable behavior, particularly in combination with advances in additive manufacturing technologies. In this paper, the crashworthiness of engineering multi-layer, multi-topology (MLMT) resin lattices is experimentally examined. First, the response of a single- and three-layer single topology cubic and octet lattices, at a relative density of 30%, is investigated. Then, the response of MLMT lattices is characterized and compared to those single-topology lattices. Crashworthiness data were collected for all topology arrangements, finding that while the three-layer cubic and octet lattices were capable of absorbing 9.8 J and 7.8 J, respectively, up to their respective densification points, the unique MLMT lattices were capable of absorbing more: 19.0 J (octet-cube-octet) and 22.4 J (cube-octet-cube). These values are between 94% and 187% greater than the single-topology clusters of the same mass.

Blast impact on the density-based tri-layered polyurethane foam

Cellular solids are interesting materials for blast energy absorption because of their high porosity, cell structure, and unique mechanical properties. Also, it will undergo graded compression over the same foam density. Hence, in this study, an experimental investigation of blast pressure impact on the trilayered sequences made of three different polyurethane (PU) foam densities of equal thickness, such as D1-29.201 kg/m3, D2-59.692 kg/m3, and D3-107.720 kg/m3 is carried out. The force transmitted (FT) on the reaction plate and incident force on the blast impact face are recorded. The maximum force amplification (Famp) of the 63.79% was observed in the S5 and the minimum of 6.19% in S4. Thus the reduction in Famp in S4 compared to S5 is 90.3%. Similarly, the energy absorbed (Eabs) by the trilayer is a maximum of 24.80 J in S2 and a minimum of 3.60 J in S3. The Eabs increased to 85.48% in S2 solely by altering the layer sequences in S3. Hence, the location of the density in the layer sequences plays a key role in effective blast mitigation on different response measures.

Bidirectionally graded honeycombs under quasi-static loading: Experimental and numerical study

Density-graded cellular solids possess tailorable mechanical properties through variable localized densities, made possible with design freedom offered by additive manufacturing. In this paper, a novel design strategy is proposed to generate bidirectionally graded honeycombs where the gradient direction is both parallel and perpendicular to the loading direction. A gradient function is developed to design three density gradient honeycombs having three different thickness gradients. The graded honeycombs along with their uniform density regular honeycomb counterparts of similar relative density are manufactured using material extrusion process. In-plane compression tests are carried out to perform a comparative study of the honeycombs. Unlike the regular honeycombs, graded honeycombs show layer-by-layer deformation, in addition to cell-wise collapse in lateral direction. Graded honeycombs show better energy absorption characteristics in low and high energy compressions, and high strain regime. Also, graded honeycombs have similar or higher specific energy absorption, densification strain, mean crushing force, and peak crushing force. The compressive responses of the honeycombs are simulated with finite element analysis and the simulation results agree well with the experimental results. The results substantiate the significance of thickness gradient in controlling the density gradation, and effectively tailoring the load-bearing capacity, deformation behavior, and energy absorption characteristics of cellular structures.

Three-dimensional cellular structures for viscous and thermal energy control in acoustic and thermoacoustic applications

The multi-scale asymptotic method provides a separate description of the viscous and thermal response functions governing acoustic waves propagation through porous materials. However, these response functions are inherently interdependent for simple porous structures such as slits or tubes – their determination being directly influenced by the microstructural features of the geometry. This study aims to identify, characterize, and realize a microgeometry providing the ability to independently modify the viscous and thermal behaviours. A relative autonomy in the control of these phenomena would make it possible to regulate energy conversion processes within the structure, which can be either dissipative (e.g., sound absorption) or generative (e.g., thermoacoustic gain). Among the various microgeometries, cellular solids made by an assembly of cells with solid edges or faces, packed together so that they fill space, emerge as promising candidates for achieving such a goal. Within the interconnected cells (pores), the viscous losses are governed by the aperture sizes of the faces (throat section). On the other hand, the thermal exchanges are closely related to the interface between the fluid and the solid and consequently to the dimensions of the cells. The identified microstructure is a typical Kelvin cell-based geometry, modified to account for manufacturing constraints, and characterized by faces with well-defined opening ratio and thickness. This work presents a model, experimentally validated, to predict the transport parameters of such cellular solids. It provides a valuable tool for designing microstructures having the attributes to independently tune thermal and viscous effects for specific application requirements.

On the infiltration of cellular solids by sheet molding compound: process simulation and experimental validation

This study examines a numerical method to simulate the production of novel multi-material metal-composite components, where an additive-manufactured cellular solid is infiltrated by a sheet molding compound (SMC) in a single-step compression molding operation. A single-fiber numerical approach is adopted to predict microstructural changes, such as fiber orientation, fiber-matrix separation, and fiber volume content variations during molding. The accuracy of the numerical predictions is confirmed by physical samples using micro-computed tomography and optical microscopy investigations at both the qualitative and quantitative scales. From optical microscopy observations, there emerged a positive correlation between experimental outcomes and simulation results, accurately capturing fiber swirling, wrinkling, and draping that occurred during molding. At a quantitative scale, a 0.6% mismatch was observed when void volume and unfilled areas were compared, as measured by micro-computed tomography and numerical simulation.

Review Study on Mechanical Properties of Cellular Materials.

Cellular materials are fundamental elements in civil engineering, known for their porous nature and lightweight composition. However, the complexity of its microstructure and the mechanisms that control its behavior presents ongoing challenges. This comprehensive review aims to confront these uncertainties head-on, delving into the multifaceted field of cellular materials. It highlights the key role played by numerical and mathematical analysis in revealing the mysterious elasticity of these structures. Furthermore, the review covers a range of topics, from the simulation of manufacturing processes to the complex relationships between microstructure and mechanical properties. This review provides a panoramic view of the field by traversing various numerical and mathematical analysis methods. Furthermore, it reveals cutting-edge theoretical frameworks that promise to redefine our understanding of cellular solids. By providing these contemporary insights, this study not only points the way for future research but also illuminates pathways to practical applications in civil and materials engineering.

Fast optimisation of honeycombs for impact protection

Cellular solids such as honeycombs are often used in impact protection systems. Those with precisely defined geometries allow responses to be programmed to specific functions, but can be costly to design. This article presents an analytical model for fast, parametric optimisation. The analytical model, and a numerical one, were validated against experimental data for three honeycomb variants, during compression tests to 80% engineering strain, and 1, 3 and 5 J impacts. The numerical model force readings remained within 5% of the experiments. A further verification of the analytical model, varying all parameters within the honeycomb geometry, was undertaken for 5 J and 15 J impacts. The analytical model predicted energy absorption at all displacement increments to be (on average) 3% higher than the numerical one. The limits of agreement (with 95% confidence) were between +15 and −9% of the numerical model. The analytical model provided solutions almost instantly, so was used in a demonstrative parametric optimisation study, for a 10 J impact. The input and output solutions were verified in the numerical model, showing a four-fold reduction in peak force (from ∼2000 to ∼500 N).

Crashworthiness of 3D Lattice Topologies under Dynamic Loading: A Comprehensive Study.

Periodic truss-based lattice materials, a particular subset of cellular solids that generally have superior specific properties as compared to monolithic materials, offer regularity and predictability that irregular foams do not. Significant advancements in alternative technologies-such as additive manufacturing-have allowed for the fabrication of these uniquely complex materials, thus boosting their research and development within industries and scientific communities. However, there have been limitations in the comparison of results for these materials between different studies reported in the literature due to differences in analysis approaches, parent materials, and boundary and initial conditions considered. Further hindering the comparison ability was that the literature generally only focused on one or a select few topologies. With a particular focus on the crashworthiness of lattice topologies, this paper presents a comprehensive study of the impact performance of 24 topologies under dynamic impact loading. Using steel alloy parent material (manufactured using Selective Laser Melting), a numerical study of the impact performance was conducted with 16 different impact energy-speed pairs. It was possible to observe the overarching trends in crashworthiness parameters, including plateau stress, densification strain, impact efficiency, and absorbed energy for a wide range of 3D lattice topologies at three relative densities. While there was no observed distinct division between the results of bending and stretching topologies, the presence of struts aligned in the impact direction did have a significant effect on the energy absorption efficiency of the lattice; topologies with struts aligned in that direction had lower efficiencies as compared to topologies without.

On the mechanical properties of dual-scale microlattice of starfish ossicles: A computational study

While engineering cellular solids, either stochastic foams or architected lattices, are usually made of polycrystalline or amorphous materials, echinoderms (e.g., sea urchins, starfish, and brittle stars) build their microporous skeletal elements with single-crystalline calcite. This class of biomineralized cellular solid, known as stereom, also exhibits a vast diversity in pore morphology, ranging from random open-cell foams to fully periodic lattices. In particular, the skeletal elements (also known as ossicles) of some starfish species are composed of a diamond-triply periodic minimal surface (diamond-TPMS) microlattice. In addition, the crystallographic symmetries at both atomic (calcite) and lattice (diamond-TPMS) levels are precisely aligned. Here we investigate the mechanical performance of this unique dual-scale microlattice and discuss the synergistic effects of atomic- and lattice-scale crystallographic coalignment. Our computational and theoretical analysis suggests that the mechanical isotropy of ossicles is enhanced due to the property compensation between the atomic-level and lattice-level architectures. Moreover, the observed 50 vol% relative density in the diamond-TPMS microlattice of ossicles may be a result of achieving overall mechanical isotropy, minimal surface curvature, and maximized surface area. We believe the methodology introduced here will be useful for understanding the mechanical behavior of natural crystalline cellular solids such as echinoderms’ stereom and designing dual-scale lattice materials.

Collagen Networks under Indentation and Compression Behave Like Cellular Solids.

Simple synthetic and natural hydrogels can be formulated to have elastic moduli that match biological tissues, leading to their widespread application as model systems for tissue engineering, medical device development, and drug delivery vehicles. However, two different hydrogels having the same elastic modulus but differing in microstructure or nanostructure can exhibit drastically different mechanical responses, including their poroelasticity, lubricity, and load bearing capabilities. Here, we investigate the mechanical response of collagen-1 networks to local and bulk compressive loads. We compare these results to the behavior of polyacrylamide, a fundamentally different class of hydrogel network consisting of flexible polymer chains. We find that the high bending rigidity of collagen fibers, which suppresses entropic bending fluctuations and osmotic pressure, facilitates the bulk compression of collagen networks under infinitesimal applied stress. These results are fundamentally different from the behavior of flexible polymer networks in which the entropic thermal fluctuations of the polymer chains result in an osmotic pressure that must first be overcome before bulk compression can occur. Furthermore, we observe minimal transverse strain during the axial loading of collagen networks, a behavior reminiscent of open-celled cellular solids. Inspired by these results, we applied mechanical models of cellular solids to predict the elastic moduli of the collagen networks and found agreement with the moduli values measured through contact indentation. Collectively, these results suggest that unlike flexible polymer networks that are often considered incompressible, collagen hydrogels behave like rigid porous solids that volumetrically compress and expel water rather than spreading laterally under applied normal loads.

Crashworthiness investigations for 3D printed multi-layer multi-topology carbon fiber nylon lattice materials

Cellular solids have superior energy absorption capabilities as compared to monolithic materials. Within this category of materials, lattice materials are of particular interest since their periodicity offers repeatable – and thus predictable – behavior. In combination with the advancements in additive manufacturing technologies, these lattice materials can be highly customized for a desired response. In this paper, the crashworthiness of unique multi-layer, multi-topology (MLMT) lattices is investigated. First, the nylon-carbon fiber composite material properties within a developed numerical model were tuned based on strut orientation. Then, the response of single-layer and three-layer cubic and octet lattices was investigated, where all lattices were designed with a relative density of 30%. Following the characterization of single-topology lattices, the response of MLMT lattices were investigated. Stress-strain, efficiency-strain, and multiple crashworthiness parameter data was collected for all lattices to facilitate in the comparison of those lattices. It was found that, experimentally, the unique MLMT lattices did not absorb more energy than their constituent layers combined, though modifications to the interface between layers could increase the energy absorption capability; the prediction of energy absorption of the MLMT lattices based on constituent layers was similar to actual numerical results. As all lattices were designed at the same relative density, the mass-specific energy absorption of the cubic-octet-cubic MLMT lattice (1.56 x103 J/kg) outperforms the single-topology octet lattice by 19% to 36% (1.15–1.31 x103 J/kg). While the octet-cubic-octet MLMT lattice (0.71 x103 J/kg) is outperformed by the single-topology cubic lattices (1.69–3.76 x103 J/kg), they see an increase of 59% to 77% in plateau stress (5.1–9.2 MPa) as compared to the MLMT lattice (2.1 MPa).

Impact resistance of a double re-entrant negative poisson’s ratio honeycomb structure

Auxetic metamaterials, usually consisting of cellular solids or honeycombs, exhibit the advantages of high designability and tunability. In particular, the negative Poisson’s ratio (NPR) property endows them with innovative mechanical properties and makes them promising for a wide range of applications. This paper proposes a modified double re-entrant honeycomb (MDRH) structure and explores its Young’s modulus and Poisson’s ratio through theoretical derivation and finite element analysis. Additionally, it discusses the relationship between these parameters and the concave angle. Furthermore, the deformation mode, nominal stress–strain curve, and specific energy absorption of this MDRH are investigated for different impact velocities and compared with traditional re-entrant honeycomb (TRH) materials. The results show that the MDRH honeycomb structure greatly widens the range of effective modulus and NPR values. At different impact velocities, the MDRH exhibits high plateau stress and specific energy absorption, indicating good impact resistance. These results provide a theoretical foundation for the design and implementation of new energy-absorbing structures.

Uninterrupted in-situ compression of automotive expanded polypropylene, imaged via synchrotron X-ray computed tomography

Synchrotron radiation at the ESRF was used to scan an uninterrupted uniaxial compression test of Expanded Polypropylene samples sourced from two automotive component suppliers. The samples, nominally 80 kg.m-3 were compressed at a constant quasi-static rate of 10 μm.s-1, avoiding an interrupted, stop-start approach that has previously been adopted for scanning cellular solids.

Predicting buckling resistance of two three-dimensional lattice architectures

Lattice structures are an important class of architected cellular solids and structures with high potential for multifunctional and lightweight applications. Novel technologies such as additive manufacturing have vastly extended the design freedom to develop such architectures. In this work, a reliable theoretical model for optimizing unit cell design against buckling is developed for two different cell architectures: pyramidal and tetrahedral. The model’s accuracy was evaluated through extensive finite element analysis and compared to existing methods available in the literature.

Effective Mechanical Properties of Periodic Cellular Solids with Generic Bravais Lattice Symmetry via Asymptotic Homogenization.

In this paper, the scope of discrete asymptotic homogenization employing voxel (cartesian) mesh discretization is expanded to estimate high fidelity effective properties of any periodic heterogeneous media with arbitrary Bravais's lattice symmetry, including those with non-orthogonal periodic bases. A framework was developed in Python with a proposed fast-nearest neighbour algorithm to accurately estimate the periodic boundary conditions of the discretized representative volume element of the lattice unit cell. Convergence studies are performed, and numerical errors caused by both voxel meshing and periodic boundary condition approximation processes are discussed in detail. It is found that the numerical error in periodicity approximation is cyclically dependent on the number of divisions performed during the meshing process and, thus, is minimized with a refined voxel mesh. Validation studies are performed by comparing the elastic properties of 2D hexagon lattices with orthogonal and non-orthogonal bases. The developed methodology was also applied to derive the effective properties of several lattice topologies, and variation of their anisotropic macroscopic properties with relative densities is presented as material selection charts.

Nonlocal strain gradient-based isogeometric analysis of graphene platelets-reinforced functionally graded triply periodic minimal surface nanoplates

Recent developments in additive manufacturing (AM) technologies have empowered the design and fabrication of intricate bioinspired engineering structures at the nano/micro scale. However, mathematical modeling and computation of these structures are still challenging. The main target of this study is to address an efficient computational approach for predicting the mechanical behavior of graphene platelets (GPLs)-reinforced functionally graded triply periodic minimal surface (FG-TPMS) nanoplates. The computational model integrates both nonlocal elasticity and strain gradient effects into the NURBS-based isogeometric analysis of these small-scale structures. We establish advanced nanoplate models by combining three sheet-based TPMS architectures with two new porosity distribution patterns and three distribution patterns of GPLs across the thickness direction. Moreover, the present work makes a pioneering attempt to elucidate how the stiffness-hardening and stiffness-softening mechanisms influence FG-TPMS nanoplates reinforced with GPLs. Compared with two common cellular solids, the superiority of TPMS architectures' mechanical performance is demonstrated. Among all, P and IWP TPMS types along with symmetric porosity and GPLs distributions exhibit outstanding behaviors under static bending, free vibration, and dynamic instability. Furthermore, we conducted a performance analysis for the first time, showcasing the superior capabilities of TPMS architectures under dynamic in-plane compressive loads, especially when compared to isotropic plates of equal weight. The findings of this study greatly enhance our understanding of the intricate mechanical responses of GPLs-reinforced TPMS architectures at the small-scale level, contributing to future interdisciplinary applications.

Three novel computational modeling frameworks of 3D-printed graphene platelets reinforced functionally graded triply periodic minimal surface (GPLR-FG-TPMS) plates

This paper presents a comprehensive investigation of novel frameworks for graphene platelets reinforced functionally graded triply periodic minimal surface (GPLR-FG-TPMS) plates. Three functional grading frameworks, which focus on controlling mass density, elastic modulus, and shear modulus, are proposed, each incorporating three porosity distributions in combination with three GPL volume fraction distributions. Three sheet-based TPMS structures including Primitive, Gyroid, and I-graph and Wrapped Package-graph are explored in this study. The static and free vibration analyses of these plates are conducted using a five-variable plate theory model with isogeometric analysis (IGA). The key parameters in each framework, including porosity coefficients and GPL weight fractions, are attentively studied. Their influences on plate responses are indicated corresponding to each porosity and GPL volume distribution. Notably, the mass density framework exhibits significant potential as a means of comparing different porous cores. It can further provide an approximate stiffness-to-weight ratio with a great lower weight compared to others. In this framework, the fundamental frequency of the plate can be achieved at approximately 95% of the frequency obtained with 0.97 and 0.98 weight in the elastic modulus and shear modulus frameworks, respectively, while utilizing only 0.35 of the total weight. To demonstrate the efficiency of the novel FG plate models, two cellular solids are also adopted in computations. The comparison of these structures in all three frameworks together with different porosity distributions is presented using polar charts. Remarkably, we find that in the context of thick plates, Primitive plates exhibit superior performance among all the plates. In general, this research tries to shed light on the development of advanced GPLR-FG-TPMS plates, elucidating the effects of different functional grading frameworks and guiding the design of lightweight and mechanically efficient structures.

Properties, Applications and Recent Developments of Cellular Solid Materials: A Review.

Cellular solids are materials made up of cells with solid edges or faces that are piled together to fit a certain space. These materials are already present in nature and have already been utilized in the past. Some examples are wood, cork, sponge and coral. New cellular solids replicating natural ones have been manufactured, such as honeycomb materials and foams, which have a variety of applications because of their special characteristics such as being lightweight, insulation, cushioning and energy absorption derived from the cellular structure. Cellular solids have interesting thermal, physical and mechanical properties in comparison with bulk solids: density, thermal conductivity, Young's modulus and compressive strength. This huge extension of properties allows for applications that cannot easily be extended to fully dense solids and offers enormous potential for engineering creativity. Their Low densities allow lightweight and rigid components to be designed, such as sandwich panels and large portable and floating structures of all types. Their low thermal conductivity enables cheap and reliable thermal insulation, which can only be improved by expensive vacuum-based methods. Their low stiffness makes the foams ideal for a wide range of applications, such as shock absorbers. Low strengths and large compressive strains make the foams attractive for energy-absorbing applications. In this work, their main properties, applications (real and potential) and recent developments are presented, summarized and discussed.