In-plane Elastic Moduli Research Articles

Fracture behavior of 3D-printed thermoplastics—A dilemma of fracture toughness versus fracture energy

The Material Extrusion (ME) technique in 3D printing facilitates the production of thermoplastic materials with diverse cellular or lattice-like infill geometries, enabling the creation of lightweight, high-performance materials. This study investigates the influence of infill geometry and relative density on the fracture behavior of 3D-printed thermoplastics produced through ME. Polylactic acid (PLA) is used as the model material, with unit cell geometries-square, triangle, and Kagome-and relative densities ranging from 0.4 to 0.8 explored. Tensile tests are first conducted to determine in-plane elastic moduli in two orthogonal directions, accounting for unit cell anisotropy. These results are employed in effective fracture energy calculations. Compact tension fracture tests follow, quantitatively characterize crack growth characteristics in both directions. In both tests, digital image correlation method is used to measure full field strain distributions. Three significant findings emerge. First, there is a power scaling relationship between Elastic Modulus and relative density across all unit cells, with Kagome exhibiting the highest correlation constant. Triangle and Kagome unit cells show negligible orthotropic responses in perpendicular directions for varying relative densities. Second, the relationship between nondimensional fracture toughness and relative density follows a similar scaling law, with minor variations depending on unit cell type. The correlation factor slightly dependens on unit cell geometry but remains around 3 across the the examined relative density range. Third, considering the dissipation of inelastic fracture energy, both triangle and Kagome unit cell geometries demonstrate an order of magnitude higher effective fracture energy (i.e., crack growth resistance) compared to the square unit cell. This is associated with toughening mechanisms such as crack deflection and bifurcation. These findings highlight the disparity between fracture toughness and effective fracture energy in assessing the fracture resistance of 3D-printed thermoplastics, clearly demonstrating that fracture toughness alone is insufficient for a comprehensive assessment. The proposed scaling laws provide predictive insights into the fracture toughness of polymeric materials produced via the ME method.

Stability and Elasticity of Ultrathin Sphere-Patterned Block Copolymer Films.

Sphere-patterned ultrathin block copolymer films are potentially interesting for a variety of applications in nanotechnology. We use self-consistent field theory to investigate the elastic response of sphere monolayer films with respect to in-plane shear, in-plane extension, compression deformations, and bending. The relations between the in-plane elastic moduli are roughly compatible with the expectations for two-dimensional elastic systems with hexagonal symmetry, with one notable exception: The pure shear and the simple shear moduli differ from each other by roughly 20%. Even more importantly, the bending constants are found to be negative, indicating that free-standing block copolymer membranes made of only a sphere monolayer are inherently unstable above the glass transition. Our results are discussed in view of the experimental findings.

Equivalent in-plane elastic moduli of honeycomb materials under hypergravity conditions

Honeycomb materials frequently encounter hypergravity conditions in both aerospace and biological contexts during phases such as launch, reentry or under centrifugal motion. The significant body force engendered by hypergravity induces alterations in the microstructure of honeycomb materials, which in turn, influences their macroscopic mechanical behaviour. Leveraging the stiffness of the beam element as a pivotal variable, we successfully derived the equivalent moduli of the honeycomb material under hypergravity conditions. We further proposed the concept of a ‘hypergravity factor’, elucidating that the density of the base material, the dimensions of honeycomb cells and the magnitude of the hypergravity contribute to amplifying hypergravity effects. The results, numerically validated through finite-element simulations, could be reduced to the case that neglects body force. The critical buckling load of the honeycomb material under hypergravity can be assessed by setting the derived moduli to zero. In the presence of hypergravity, a honeycomb material undergoes a transition into a gradient material along the hypergravity direction, thereby exacerbating anisotropy. This phenomenon is theoretically expected to occur in virtually all porous materials. The analytical framework adopted, which employs beam stiffness as an intermediary variable, facilitates the extension of these results to honeycomb materials which encompass beam elements with functional gradients or varying cross-sectional morphologies.

Analytical homogenization for equivalent in-plane elastic moduli of multi-material honeycombs

3D printable multi-material honeycombs have attracted increasing interest recently due to the improved elastic moduli, buckling and energy absorption properties. This paper proposes an analytical homogenization for the equivalent in-plane elastic moduli (EIEM) of multi-material honeycombs. First, the axial and bending stiffness of a cantilever beam consist of three sections made of different materials is formulated. Then based on unit cell method and cantilever beam model, the closed-form expressions of EIEM are proposed by fully considering the deformations of both joints and cell walls with arbitrary stiffness, which are sufficiently general to be applied to hexagonal, auxetic and rectangular multi-material honeycombs and validated very well by numerical simulations and experiments. Furthermore, the effects of geometric and material distribution ratios on EIEM are discussed. The results show that the geometric and material distribution ratios of inclined cell wall have a significant effect on all five EIEM while that of vertical cell wall shows a significant effect on shear modulus but only a slight effect on y- direction elastic moduli. Compared with single-material honeycombs, changing joint stiffness has a significant effect on the equivalent Young’s moduli and shear modulus but a slight effect on Poisson’s ratios.

Effective elastic moduli of space-filled multi-material composite lattices

Traditionally lattice materials are made of a network of beams in two and three dimensions with majority of the lattice volume being void space. Recently researchers have started exploring ways to exploit this void space for multi-physical property modulation of lattices such as global mechanical behaviour including different elastic moduli, wave propagation, vibration, impact and acoustic features. The elastic moduli are of crucial importance to ensure the structural viability of various multi-functional devices and systems where a space-filled lattice material could potentially be used. Here we develop closed-form analytical expressions for the effective elastic moduli of space-filled lattices based on an exact stiffness matrix approach coupled with the unit cell method, wherein transcendental shape functions are used to obtain exact solutions of the underlying differential equation. This can be viewed as an accurate multi-material based generalisation of the classical formulae for elastic moduli of honeycombs. Numerical results show that the effective in-plane elastic moduli can increase by orders of magnitude with a relatively lower infill stiffness (∼10%). This gives an exceptional opportunity to engineer multi-material lattices with optimal specific stiffness along with characterising the mechanical properties of a multitude of lattice-like artificial and naturally occurring structural forms with space filling.

Analytical homogenization for equivalent in-plane elastic moduli of honeycomb structures with stiffened joints

The joint-stiffened honeycombs have attracted increasing interest recently due to the improved stiffness, strength and energy absorption properties. Undoubtedly, stiffened joints play a significant role in the equivalent in-plane elastic moduli (EIEM). This paper proposes an analytical homogenization for EIEM of honeycomb with stiffened joints (HSJ). First, the stiffness matrix of a honeycomb cell wall (based on rod and Timoshenko theories) connected to stiffened joints (modelled as rigid bodies) is formulated. Based on the unit cell method, closed-form expressions of EIEM are proposed, which are sufficiently general to be applied to three types of honeycombs. The closed-form expressions are validated very well by numerical simulation and experiment. Furthermore, the proposed expressions are applied to hexagonal, auxetic and square HSJ and the dependency of EIEM on inclined and vertical sizes of stiffened joints is discussed. The results show that Young’s moduli and Poisson’s ratio are strongly related to the size of the inclined section of stiffened joints; while shear modulus is strongly related to the size of the vertical section. It is also found that the improvement of Young’s modulus for aluminium joint-stiffened nylon honeycomb can reach up to 41% compared to nylon honeycomb under the same weight.

Programmed Out-of-Plane Curvature to Enhance Multimodal Stiffness of Bending-Dominated Composite Lattices

Conventional bending-dominated lattices exhibit less specific stiffness compared to stretching-dominated lattices while showing high specific energy absorption capacity. This paper aims to improve the specific stiffness of bending-dominated lattices by introducing elementary-level programmed curvature through a multilevel hierarchical framework. The influence of curvature in the elementary beams is investigated here on the effective in-plane and out-of-plane elastic properties of lattice materials. The beamlike cell walls with out-of-plane curvature are modeled based on three-dimensional degenerated shell finite elements. Subsequently, the beam deflections are integrated with unit cell level mechanics in an efficient semi-analytical framework to obtain the lattice-level effective elastic moduli. The numerical results reveal that the effective in-plane elastic moduli of lattices with curved isotropic cell walls can be significantly improved without altering the lattice-level relative density, while the effective out-of-plane elastic properties reduce due to the introduction of curvature. To address this issue, we further propose laminated composite cell walls with out-of-plane curvature based on the three-dimensional degenerated shell elements, which can lead to holistic improvements in the in-plane and out-of-plane effective elastic properties. The proposed curved composite lattice materials would enhance the specific stiffness of bending-dominated lattices to a significant extent, while maintaining their conventional multifunctional advantages.

An FFT-based method for uncertainty quantification of Nomex honeycomb’s in-plane elastic properties

The honeycomb manufacturing process involves complex and inter-related procedures spanning multiple physics domains and scales. The geometry heterogeneity in honeycomb cells inevitably exists and propagates to the uncertainty of the honeycomb’s mechanical performance. Quantitatively characterizing the effect of cell geometry variability on the effective mechanical properties of honeycomb hence becomes critically important. In this study, a Fast Fourier Transform (FFT) based method was developed to predict the effective in-plane elastic properties of irregular hexagonal honeycomb. The honeycomb geometry heterogeneity was identified with a digital image technology by examining a large number of cells. Correlation analysis on the resulting statistical data enables determining the interdependence of cell wall lengths and angles. Using a Representative Unit Cell (RUC) with varying cell wall lengths and angles, an analytical function for the effective in-plane elastic parameters was established by incorporating the dependencies. The statistical quantification of cell geometry and the propagation of RUC’s elastic characteristics were further integrated into the FFT-based multiscale framework for predicting the in-plane elastic moduli of irregular honeycomb. The tensile and shear tests on the irregular honeycombs, together with the corresponding finite element analysis, were used to verify the proposed method. Both experimental and numerical results confirmed that the uncertainty of elastic moduli achieved by the FFT-based method provides a reasonable estimation.

Determining the interlayer shearing in twisted bilayer MoS2 by nanoindentation

The rise of twistronics has increased the attention of the community to the twist-angle-dependent properties of two-dimensional van der Waals integrated architectures. Clarification of the relationship between twist angles and interlayer mechanical interactions is important in benefiting the design of two-dimensional twisted structures. However, current mechanical methods have critical limitations in quantitatively probing the twist-angle dependence of two-dimensional interlayer interactions in monolayer limits. Here we report a nanoindentation-based technique and a shearing-boundary model to determine the interlayer mechanical interactions of twisted bilayer MoS2. Both in-plane elastic moduli and interlayer shear stress are found to be independent of the twist angle, which is attributed to the long-range interaction of intermolecular van der Waals forces that homogenously spread over the interfaces of MoS2. Our work provides a universal approach to determining the interlayer shear stress and deepens the understanding of twist-angle-dependent behaviours of two-dimensional layered materials.

An evaluation of the loading direction sensitivity of 3D woven composite with novel web-shaped Z-binder architecture

3D orthogonal (ORT) through-thickness woven composites have excellent out-of-plane properties but exhibit inferior in-plane properties and are loading sensitive. This study proposes a novel web-shaped orientation of z-binder architecture in 3D orthogonal through-thickness model with in-plane uniform elastic moduli to address this drawback. On-axis tensile, Off-axis tensile and In-plane Shear tests were performed on the four macroscale models (ORTon, WEBon, ORToff, WEBoff). The ORT model, validated by data from the literature, was used as a reference for the proposed WEB model. The overall fibre volume fraction of the WEBon, and WEBoff models were 7.07% and 8.67% lower than the ORTon, and ORToff models. In the on-axis tensile test, the Ex and Ey were lower by 16.89% and 2.66% for WEBon model, but in the off-axis tensile test, they were higher by 13.75% and 13.77% for WEBoff models. The Gxy was 10.96% lower for WEBon model. The difference between the Ex and Ey under on-axis loading conditions was 14.81% for ORT model, whereas 0.22% for WEB model. Thus, the proposed WEB shaped z-binder architecture model outperforms the ORT model under the off-axis and is comparable in transverse loading conditions. It also possesses loading direction insensitivity, unlike the ORT model.

In-plane elasticity of the re-entrant auxetic hexagonal honeycomb with hollow-circle joint

This work proposed an improved modification on the common re-entrant hexagonal honeycomb (RHH) by replacing the solid junction of the cell walls with a small hollow circle, and thus achieving a novel kind of honeycomb. A simple energy-based approach was adopted to develop theoretical models for predicting the in-plane equivalent elastic moduli of the new honeycomb. Finite element analysis was employed to verify the theoretical models and then the effects of the micro geometrical parameters on the in-plane elastic behaviors were further discussed. The results showed that the new honeycomb possesses an obvious negative Poisson's ratio effect, meanwhile, can achieve higher specific Young's modulus and shear modulus than the RHH. Moreover, the new honeycomb was found to exhibit a more tailored anisotropy than the RHH from the off-axial properties analysis. As a result, the new honeycomb may be an effective substitute for the RHH in application and the new layout can be a good guide for the improved design of auxetic honeycombs.

Curved inserts in auxetic honeycomb for property enhancement and design flexibility

Imperfect situation, like the non-uniform cross-section or non-straight profile of cell walls, can in certain case be a positive contributor for properties of cellular solids. However, auxetic cellular structures with imperfect profiles are rarely reported in the literature. Based on the re-entrant hexagonal honeycomb (RHH), this paper developed two new types of honeycombs with curved inserts and focused on the issue that how microscopic curved inserts influence the auxetic structure’s macroscopic mechanical properties? Theoretical models for the in-plane equivalent elastic moduli of the two types of new honeycombs, including the Young’s modulus, Poisson’s ratio and shear modulus, were first established, and then verified by numerical simulations. The results showed that the two types of new honeycombs possess the same ability to achieve auxeticity, in the meantime, exhibit improved Young’s modulus and shear modulus compared to the RHH. Furthermore, the new honeycombs demonstrate more flexibility in design as the curved inserts can offer more geometric parameters for tailoring their macroscopic mechanical properties. The study also showed that the curved inserts played a key role in determining the dominated deformation mode of the new honeycombs, and a special dominated deformation mode intermediate between pure bending and stretching was found in these honeycombs. This work would provide a useful guide for the development and optimization of auxetic materials.

Estimation of improvement in elastic moduli for functionalised defective graphene-based thermoplastic polyurethane nanocomposites: a molecular dynamics approach

ABSTRACT This molecular dynamics (MD) simulation-based study provides a comprehensive investigation about improvement in elastic moduli and strength properties of Stone–Wales defective hydrogen functionalised graphene-reinforced thermoplastic polyurethane (SW-fGr/TPU) nanocomposite. Three models, namely concentration model (CM), stacked model (SM) and interfacial model (IM), have been used here to estimate improvements under the effect of weight concentrations, aspect ratios, dispersion and stacking conditions using the constant strain method with COMPASS II force field. High enhancement in moduli values is observed for CM models due to the high aspect ratio and 4 wt-% of SW-fGr addition in pure TPU. The values of in-plane elastic moduli of stacked dispersed models are found to be higher than the values of stacked agglomerated models. Out of all SM models, maximum in-plane and out-of-plane shear moduli are obtained in the agglomerated (stacked) Gr conditions. The interfacial normal strength is found to be substantially higher than interfacial shear strength within the IM models. The enhancement of elastic moduli in SW-fGr/TPU has been confirmed by using atomic density profile, fractional free-volume calculation and Connolly surface analyses. Some of these MD simulation results are in good agreement with those from previous investigations.

Linear and non-linear in-plane behaviour of a modified re-entrant core cell

In this paper, a work to improve to the auxetic re-entrant cell is reported. This work aims to conserve the negative or zero Poisson's ratio feature of the classical re-entrant cell with no degradation while amplifying the in-plane elastic moduli. To develop and control the stiffness of the cell a new cell was created using a circular wall element in the middle of the classical re-entrant cell. The analytical expressions of the new re-entrant cell were derived and validated by finite element models. Derived analytical expressions of this work can be used to model the classical cell. Using this feature, validation of analytical and numerical models of this work is also provided with a classical cell's experimental works reported in the literature. In the linear region, both the analytical and the finite element models match very closely with the literature experiments. Because of the highly non-linearity of the re-entrant structures, after the linear examination, this work was expanded to model the non-linear behaviour of both the classical and new cells numerically. Literature experiments also verify the non-linear finite element models of this work in means of the classical re-entrant cell's results. Detailed design graphs are given for tuning the rigidity and negative Poisson’s ratios. The presented new design in this paper grants an extra rigidity while providing the same negative Poisson's ratios with those of the classical cells.

In-plane elasticity of regular hexagonal honeycombs with three different joints: A comparative study

This paper is focused on the in-plane linear elastic properties of regular hexagonal honeycombs with three different joint geometries: hexagonal joint, circular joint and triangular joint. A combination of theoretical and finite element (FE) methods was adopted to investigate their in-plane elastic moduli (Young's modulus, shear modulus and Poisson's ratio), and a good agreement between the two methods was obtained. The influences of the geometric parameters on the elastic moduli, such as ρ*/ρS and r/l, were fully discussed. Interestingly, a special relationship can exist among the three joint types, that is, the circular joint can be considered as a minimum circumscribed circle of the hexagonal and triangular joints. Based on this, a comparison among the honeycombs with three different types of joints was conducted. Compared to the conventional regular hexagonal honeycomb, the Young's modulus of the circular joint, hexagonal joint, and triangular joint honeycombs is enhanced by 61%, 80% and 431%, respectively; while the shear modulus is improved by 101%, 133% and 469%, respectively. Consequently, the triangular joint honeycomb was shown to be more successful in micro-structural layout compared with the other two types of honeycombs. This work could be a good guide for the design of novel cellular structures.

Unraveling the Elastic Properties of (Quasi)Two-Dimensional Hybrid Perovskites: A Joint Experimental and Theoretical Study.

The unique properties of hybrid organic-inorganic perovskites (HOIPs) promise to open doors to next-generation flexible optoelectronic devices. Before such advances are realized, a fundamental understanding of the mechanical properties of HOIPs is required. Here, we combine ab initio density functional theory (DFT) modeling with a diverse set of experiments to study the elastic properties of (quasi)2D HOIPs. Specifically, we focus on (quasi)2D single crystals of phenethylammonium methylammonium lead iodide, (PEA)2PbI4(MAPbI3)n-1, and their 3D counterpart, MAPbI3. We used nanoindentation (both Hertzian and Oliver-Pharr analyses) in combination with elastic buckling instability experiments to establish the out-of-plane and in-plane elastic moduli. The effect of Van der Waals (vdW) forces, different interlayer interactions, and finite temperature are combined with DFT calculations to accurately model the system. Our results reveal a nonmonotonic dependence of both the in-plane and out-of plane elastic moduli on the number of inorganic layers (n) rationalized by first-principles calculations. We discuss how the presence of defects in as-grown crystals and macroscopic interlayer deformations affect the mechanical response of (quasi)2D HOIPs. Comparing the in- and out-of-plane experimental results with the theory reveals that perturbations to the covalent and ionic bonds (which hold a 2D layer together) is responsible for the relative out-of-plane stiffness of these materials. In contrast, we conjecture that the in-plane softness originates from macroscopic or mesoscopic motions between 2D layers during buckling experiments. Additionally, we learn how dispersion and π interactions in organic bilayers can have a determining role in the elastic response of the materials, especially in the out-of-plane direction. The understanding gained by comparing ab initio and experimental techniques paves the way for rational design of layered HOIPs with mechanical properties favorable for strain-intensive applications. Combined with filters for other favorable criteria, e.g., thermal or moisture stability, one can systematically screen viable (quasi)2D HOIPs for a variety of flexible optoelectronic applications.

Probing the frequency-dependent elastic moduli of lattice materials

An insightful mechanics-based concept is developed for probing the frequency-dependence in in-plane elastic moduli of microstructured lattice materials. Closed-form expressions for the complex elastic moduli are derived as a function of frequency by employing the dynamic stiffness matrix of beam elements, which can exactly capture the sub-wavelength scale dynamics. It is observed that the two Poisson's ratios are not dependent on the frequency of vibration, while the amplitude of two Young's moduli and shear modulus increase significantly with the increase of frequency. The variation of frequency-dependent phase of the complex elastic moduli is studied in terms of damping factors of the intrinsic material. The tunable frequency-dependent behaviour of elastic moduli in lattice materials could be exploited in the pseudo-static design of advanced engineering structures which are often operated in a vibrating environment. The generic concepts presented in this paper introduce new exploitable dimensions in the research of engineered materials for potential applications in various vibrating devices and structures across different length-scales.

Noise reduction capability of hybrid flax fabric-reinforced polypropylene-based composites

In recent years, natural fiber polymer composites have demonstrated great potential for reducing noise. This study reports sound transmission loss (STL) of hybrid flax fabric-reinforced polypropylene composites. The STL values were determined by the BSWA impedance tube and evaluated by ISO 10534 standard test procedure in the frequency range of 64 to 1600 Hz. Measurements show that the STL per areal density of flax/polypropylene is two times higher compared to aluminum. A noteworthy result is that upon hybridization with 13% carbon fibers (carbon-flax/polypropylene), the STL per areal density increases by 2.4 times. This study also presents the finite element (FE) simulation of sound transmission through the composites. Good agreement between FE predicted and experimentally measured STL is obtained. Moreover, parametric sensitivity studies are performed using the developed FE model to examine the effect of possible variation in the mechanical properties of the natural fiber composites on the STL performance. It is found that within 15% variation, no significant effect on STL is obtained and that STL is only sensitive to the in-plane elastic moduli.

In-plane mechanics of a novel cellular structure for multiple morphing applications

Cellular structures are potential candidates as underlying supports for flexible skins due to their outstanding characteristics such as light weight, zero Poisson’s ratio and low effective moduli. This paper develops a novel zero Poisson’s ratio cellular structure composed of semi-periodic sinusoidal beams with light weight, low in-plane moduli and high strain capability. The equivalent in-plane elastic and shear moduli of the structure are studied through a combination of theoretical and finite element analysis. Then the maximum global strains are investigated to show the morphing capability of the structure. Results show that the in-plane moduli of the cellular structures are several orders of magnitude lower than the raw material. The global strains of the cellular structures could be over 10 times greater than the raw material. The results also demonstrate the potential of the proposed structure for uniaxial, shear or hybrid in-plane morphing.

Evaluation methods for predicting the elastic constants of C/Epoxy plain-weave composites by considering the realistic mesostructures

Variabilities of mesostructures existing in textile composites can affect their mechanical properties. Most of the deterministic mechanical models are based on the assumptions of ideal Representative Volume Element, which cannot predict the mechanical properties accurately. Two analytical models predicting the elastic constants of C/Epoxy plain-weave composites by considering the realistic mesostructures are presented in this paper. These models utilize the variable metric stochastic theory to introduce the fluctuations of yarn feature parameters (yarn path and elliptical cross-section parameters) into the model of macro elastic properties. C/Epoxy plain-weave composite is taken as an example to quantify the influences of realistic yarn feature parameters on the elastic constants of the composite. The predicted elastic constants by analytical models and finite element method are verified by the results of mechanical experiments. It can be concluded that for C/Epoxy plain-weave composite the stochastic fluctuations of yarn feature parameters reduce in-plane elastic moduli by a maximum of 4%, and increase the in-plane shear modulus and Poisson’s ratio by a maximum of 15% and 33%, respectively.