In-plane Elastic Moduli Research Articles

Experimental and numerical characterization of thin woven composites used in printed circuit boards for high frequency applications

The elastic orthotropic behavior of thin woven composites is studied combining a numerical and experimental strategy. For thin materials, in-plane properties are measured by classical tensile tests and digital image correlation. The out-of-plane properties are derived performing finite element simulations at the level of the internal structure of the laminate. Indeed, the glass fiber arrangement in the yarn and the weaving pattern are defined based on microtomography and SEM observations. So a representative unit cell is found. A statistical approach is further proposed to derive the behavior of the warp and fill yarns, since the fiber position may fluctuate between yarns. For the considered laminate, the matrix (resin and ceramic inclusion) behavior is unknown and difficult to measure. Therefore an inverse method is proposed. By comparing with measured in-plane elastic moduli, behaviors of the matrix, of yarns and of the laminate are defined. The present homogenization strategy is exemplified by laminates used in printed circuit boards for high frequency applications. This approach has also been applied to investigate the evolution of the elastic moduli of the laminate with temperature. Those information, usually not available in the literature, are important when dealing with reliability of printed circuit boards during thermal cycles.

Effective In-Plane Elastic Modulus of a Periodic Regular Hexagonal Honeycomb Core with Thick Walls

AbstractThe effective in-plane elastic moduli of different types of honeycomb cores with thick walls are studied using analytical and numerical approaches. Analytical equations derived in this pape...

Extended Tersoff potential for boron nitride: Energetics and elastic properties of pristine and defective h -BN

We present an extended Tersoff potential for boron nitride (BN-ExTeP) for application in large scale atomistic simulations. BN-ExTeP accurately describes the main low energy B, N, and BN structures and yields quantitatively correct trends in the bonding as a function of coordination. The proposed extension of the bond order, added to improve the dependence of bonding on the chemical environment, leads to an accurate description of point defects in hexagonal BN $(h$-BN) and cubic BN $(c$-BN). We have implemented this potential in the molecular dynamics LAMMPS code and used it to determine some basic properties of pristine 2D $h$-BN and the elastic properties of defective $h$-BN as a function of defect density at zero temperature. Our results show that there is a strong correlation between the size of the static corrugation induced by the defects and the weakening of the in-plane elastic moduli.

Numerical Evaluation of the Size-Dependent Elastic Properties of Cellular Polymers

Several experimental studies have revealed that the size-dependent deformation in polymers at nano- to micro-meter length scales is significantly associated with elastic deformation. Such size-dependent deformation in polymers is expected to affect the in-plane macroscopic elastic properties of cellular polymers with micrometer-sized cells. A finite element (FE) formulation of a higher-order elasticity theory is applied to evaluate the in-plane macroscopic elastic properties of different polymer cellular geometries by varying the cell size from the macroscopic to micron length scale. For a given relative density of the cellular solid, a reduction in the cell size from the macroscopic to micron length scale resulted in geometry-specific variations in the in-plane macroscopic elastic moduli and Poisson's ratios. Furthermore, an increase in the relative density for a given cell size revealed variations in the size dependence of the elastic properties. The size dependence of elastic properties is interpreted based on the influence of rotation gradients with varying cell size of the cellular solid. Also, the evaluated size-dependent elastic properties are compared with the available analytical solutions from the literature.

Comparative study on measurement of elastic constants of wood-based panels using modal testing: choice of boundary conditions and calculation methods

Modal testing based on the theory of transverse vibration of orthotropic plate has shown great potentials in measuring elastic constants of panel products. Boundary condition (BC) and corresponding calculation method are key in affecting its practical application in terms of setup implementation, frequency identification, accuracy and calculation efforts. To evaluate different BCs for non-destructive testing of wood-based panels, three BCs with corresponding calculation methods were investigated for measuring their elastic constants, namely in-plane elastic moduli (E x , E y ) and shear modulus (G xy ). As a demonstration of the concept, the products used in this study were oriented strand board (OSB) and medium density fiberboard (MDF). The BCs and corresponding calculated methods investigated were, (a) all sides free (FFFF) with one-term Rayleigh frequency equation and finite element modeling, (b) one side simply supported and the other three free (SFFF) with one-term Rayleigh frequency equation, (c) a pair of opposite sides along minor strength direction simply supported and the other pair along major strength direction free (SFSF) with improved three-term Rayleigh frequency equation. Differences between modal and static results for different BCs were analyzed for each case. Results showed that all three modal testing approaches could be applied for evaluation of the elastic constants of wood-based panels with different accuracy levels compared with standard static test methods. Modal testing on full-size panels is recommended for developing design properties of structural panels as it can provide global properties.

Effective in-plane elastic moduli of quasi-random spatially irregular hexagonal lattices

An analytical framework is developed for predicting the effective in-plane elastic moduli (longitudinal and transverse Young’s modulus, Poisson’s ratios and shear modulus) of irregular hexagonal lattices with generalized form of spatially random structural geometry. On the basis of a mechanics based bottom-up multi-step approach, computationally efficient closed-form formulae are derived in this paper. As a special case when there is no irregularity, the derived analytical expressions reduce to the respective well known formulae of regular honeycombs available in literature. Previous analytical investigations include the derivation of effective in-plane elastic moduli for hexagonal lattices with spatially random variation of cell angles, which is a special case of the generalized form of irregularity in material and structural attributes considered in this paper. The present study also includes development of a highly generalized finite element code for obtaining equivalent elastic properties of random lattices, which is employed to validate the proposed analytical formulae. The statistical results of elastic moduli obtained using the developed analytical expressions and using direct finite element simulations are noticed to be in good agreement affirming the accuracy and validity of the proposed analytical framework. All the in-plane elastic moduli are found to be significantly influenced by spatially random irregularity resulting in a decrease of the mean values for the two Young’s moduli and two Poisson’s ratios, while an increase of the mean value for the shear modulus.

A Projected Finite Element Update Method for Inverse Identification of Material Constitutive Parameters in transversely Isotropic Laminates

In this paper, a novel application of Finite Element Update Method (FEUM) is proposed for the inverse identification of material constitutive parameters in transversely isotropic laminates. Two-dimensional Digital Image Correlation (2D–DIC) is used for full-field measurements which is required for the identification process. Instead of measuring the in-plane displacements, which is a well-known application of 2D–DIC, we seek to measure the pseudo-displacements resulting from out-of-plane (towards camera) deflection of plate under a point load. These pseudo-displacements are basically the perspective projection of the three dimensional displacement fields on the image-plane of the image acquisition system. The cost function in this method is defined in terms of these projections instead of the true displacements – and hence the name Projected Finite Element Update Method (PFEUM). In this article, identification of in-plane elastic moduli of Carbon Fiber Reinforced Plastic (CFRP) plate has been performed using plate bending experiments which show pre-dominantly out-of-plane deflection with little contribution from the in-plane displacements. Identification results are validated by direct experimental measurements of the unknown elastic constants as well as theoretical estimates based on volume ratio of constituents. The results show good conformance between estimated and target values for at least three material parameters namely E1, E2 and G12. Effects of experimental noise on parameter estimates has also been evaluated to explain the observed deviation in estimated parameters with current test configuration.

Statistical Mechanics of Thin Spherical Shells

We explore how thermal fluctuations affect the mechanics of thin amorphous spherical shells. In flat membranes with a shear modulus, thermal fluctuations increase the bending rigidity and reduce the in-plane elastic moduli in a scale-dependent fashion. This is still true for spherical shells. However, the additional coupling between the shell curvature, the local in-plane stretching modes, and the local out-of-plane undulations leads to novel phenomena. In spherical shells, thermal fluctuations produce a radius-dependent negative effective surface tension, equivalent to applying an inward external pressure. By adapting renormalization group calculations to allow for a spherical background curvature, we show that while small spherical shells are stable, sufficiently large shells are crushed by this thermally generated “pressure.” Such shells can be stabilized by an outward osmotic pressure, but the effective shell size grows nonlinearly with increasing outward pressure, with the same universal power-law exponent that characterizes the response of fluctuating flat membranes to a uniform tension.Received 20 June 2016DOI:https://doi.org/10.1103/PhysRevX.7.011002Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCell mechanicsClassical statistical mechanicsContinuum mechanicsEquilibrium fluctuationsMembranesTechniquesRenormalization groupStatistical PhysicsPolymers & Soft MatterBiological PhysicsInterdisciplinary PhysicsCondensed Matter, Materials & Applied Physics

Stochastic mechanics of metamaterials

The effect of stochasticity in mechanical behaviour of metamaterials is quantified in a probabilistic framework. The stochasticity has been accounted in the form of random material distribution and structural irregularity, which are often encountered due to manufacturing and operational uncertainties. An analytical framework has been developed for analysing the effective stochastic in-plane elastic properties of irregular hexagonal structural forms with spatially random variations of cell angles and intrinsic material properties. Probabilistic distributions of the in-plane elastic moduli have been presented considering both randomly homogeneous and randomly inhomogeneous stochasticity in the system, followed by an insightful comparative discussion. The ergodic behaviour in spatially irregular lattices is investigated as a part of this study. It is found that the effect of random micro-structural variability in structural and material distribution has considerable influence on mechanical behaviour of metamaterials.

Effective in-plane elastic properties of auxetic honeycombs with spatial irregularity

An analytical framework has been developed for predicting the equivalent in-plane elastic moduli (longitudinal and transverse Young’s modulus, shear modulus, Poisson’s ratios) of irregular auxetic honeycombs with spatially random variations in cell angles. Employing a bottom up multi-scale based approach, computationally efficient closed-form expressions have been derived in this article. This study also includes development of a highly generalized finite element code capable of accepting number of cells in two perpendicular directions, random structural geometry and material properties of irregular auxetic honeycomb and thereby obtaining five in-plane elastic moduli of the structure. The elastic moduli obtained for different degree of randomness following the analytical formulae have been compared with the results of direct finite element simulations and they are found to be in good agreement corroborating the validity and accuracy of the proposed approach. The transverse Young’s modulus, shear modulus and Poisson’s ratio for loading in transverse direction (effecting the auxetic property) have been found to be highly influenced by the structural irregularity in auxetic honeycombs.

Quantification of effects of stochastic feature parameters of yarn on elastic properties of plain-weave composite – Part 2: Statistical predictions vs. mechanical experiments

This paper presents a linear discretized theoretical model on the basis of the ideal theoretical model to evaluate elastic constants of plain-weave composite by using the statistics of the feature parameters of yarn measured from Micro CT data. A finite element method is utilized to calculate the elastic constants of the composite using the modified and global mean feature parameters of yarn, respectively. Uniaxial tensile and in-plane shear experiments are then completed to measure in-plane elastic constants of the composite. Finally, comparisons among the predictions of two theoretical models, FEM and experimental results are conducted. The results show that the stochastic fluctuations of yarn feature parameters decrease the in-plane elastic moduli and increase the in-plane shear moduli and Poisson’s ratios of the plain-weave composite. The discretized theoretical model with taking account of real yarn stochastic features can predict more accurate elastic constants of the composite than deterministic models.

Scaling Behavior and Strain Dependence of In-Plane Elastic Properties of Graphene.

We show by atomistic simulations that, in the thermodynamic limit, the in-plane elastic moduli of graphene at finite temperature vanish with system size L as a power law L(-η(u)) with η(u)≃0.325, in agreement with the membrane theory. We provide explicit expressions for the size and strain dependence of graphene's elastic moduli, allowing comparison to experimental data. Our results explain the recently experimentally observed increase of the Young modulus by more than a factor of 2 for a tensile strain of only a few per mill. The difference of a factor of 2 between the measured asymptotic value of the Young modulus for tensilely strained systems and the value from ab initio calculations remains, however, unsolved. We also discuss the asymptotic behavior of the Poisson ratio, for which our simulations disagree with the predictions of the self-consistent screening approximation.

Equivalent in-plane elastic properties of irregular honeycombs: An analytical approach

An analytical formulation has been developed in this article for predicting the equivalent elastic properties of irregular honeycombs with spatially random variations in cell angles. Employing unit-cell based approaches, closed-form expressions of equivalent elastic properties of regular honeycombs are available. Closed-form expressions for equivalent elastic properties of irregular honeycombs are very scarce in available literature. In general, direct numerical simulation based methods are prevalent for this case. This paper proposes a novel analytical framework for predicting equivalent in-plane elastic moduli of irregular honeycombs using a representative unit cell element (RUCE) approach. Using this approach, closed-form expressions of equivalent in-plane elastic moduli (longitudinal and transverse Young’s modulus, shear modulus, Poisson’s ratios) have been derived. The expressions of longitudinal Young’s modulus, transverse Young’s modulus, and shear modulus are functions of both structural geometry and material properties of irregular honeycombs, while the Poisson’s ratios depend only on structural geometry of irregular honeycombs. The elastic moduli obtained for different degree of randomness following the proposed analytical approach are found to have close proximity to direct finite element simulation results.

J0440204 共振法による植物の葉の面内弾性定数計測([J044-02]ソフトマター・イノベーション(2),機械材料・材料加工部門)

In this paper, we developed a resonance measurement system for plant leaves. Square shaped specimens (10 mm×10 mm) were cut off from Pothos leaves and their in-plane elastic moduli were measured. Half part of specimens are clamped and resonance oscillations of the remaining free part of the specimens were excited by a piezoelectric actuator and detected by a laser displacement sensor. We observed resonance frequency of leaf specimens related to their vein and calculated their in-plane elastic moduli using the finite element method. In-plane elastic modulus parallel and perpendicular to vein are E_<//>= 62.2±5.6 MPa and E_⊥= 42.7±1.3 MPa, respectively, and the anisotoropy parameter is E_<//>/E_⊥=1.46±0.14.

In-plane elastic moduli of covalently functionalized single-wall carbon nanotubes

Effective utilization of single-wall carbon nanotubes (SWCNTs) as reinforcements in composites necessitates their strong interfacial bonding with the surrounding matrix. The covalent functionalization of SWCNTs is an effective method to enhance this bonding. However, covalent bonds introduced by a functional group may alter the pristine structure of the SWCNT and affect its mechanical properties. Thus it is important to delineate effects of covalent functionalization on elastic properties of a SWCNT. We study here effects of covalent functionalization on Young’s modulus, Poisson’s ratio and shear modulus of a SWCNT in the graphitic plane. We consider hydrogen (–H), hydroxyl (–OH), carboxyl (–COOH), and amine (–NH2) as model functional groups in this work. We use molecular mechanics (MM) simulations with the MM3 potential and the software TINKER to analyze simple tension and torsional deformations of pristine and functionalized SWCNTs. As is commonly assumed, we hypothesize that the response of a SWCNT to these deformations is the same as that of an energetically and geometrically equivalent continuum cylinder of wall thickness 3.4Å. From curves of the strain energy density of deformation versus the axial strain and the shear strain for each functionalization group, values of Young’s modulus and the shear modulus, respectively, are deduced. From results of the tension tests on a pristine and a SWCNT fully functionalized with hydrogen (–H), Poisson’s ratio is computed. It is found that for each functional group studied and 20% functionalization, Young’s modulus and the shear modulus decrease by about 34% and 43%, respectively, and Poisson’s ratio of the functionalized SWCNT is more than that of the pristine SWCNT. These results should help in determining mechanical properties of SWCNT reinforced nanocomposites by using a micromechanics approach.

Square‐Micrometer‐Sized, Free‐Standing Organometallic Sheets and Their Square‐Centimeter‐Sized Multilayers on Solid Substrates

Oligofunctional terpyridine-based monomers are spread at an air/water interface, where they are connected with transition metal salts such as Fe(II) into mechanically coherent monolayer sheets of macroscopic dimension. The conversions of these processes are determined by XPS for several monomer/metal ion combinations. The sheets are transferred onto TEM grids, the 20 × 20 square micrometer sized holes of which can be spanned. AFM indentation experiments provide in-plane elastic moduli which are compared with naturally occurring sheets such as graphene. The new organometallic sheets are also used to create multilayer assemblies on square centimeter length scales on solid substrates. Finally some directions are provided where this research can lead to in future and where its application potential lies.

Compatible model for herringbone bond masonry: Linear elastic homogenization, failure surfaces and structural implementation

A simplified kinematic procedure at a cell level is proposed to obtain in-plane elastic moduli and macroscopic masonry strength domains in the case of herringbone masonry. The model is constituted by two central bricks interacting with their neighbors by means of either elastic or rigid-plastic interfaces with friction, representing mortar joints. The herringbone pattern is geometrically described and the internal law of composition of the periodic cell is defined.A sub-class of possible elementary deformations is a-priori chosen to describe joints cracking under in-plane loads. Suitable internal macroscopic actions are applied on the Representative Element of Volume (REV) and the power expended within the 3D bricks assemblage is equated to that expended in the macroscopic 2D Cauchy continuum. The elastic and limit analysis problem at a cell level are solved by means of a quadratic and linear programming approach, respectively.To assess elastic results, a standard FEM homogenization is also performed and a sensitivity analysis regarding two different orientations of the pattern, the thickness of the mortar joints and the ratio between block and mortar Young moduli is conducted. In this way, the reliability of the numerical model is critically evaluated under service loads.When dealing with the limit analysis approach, several computations are performed investigating the role played by (1) the direction of the load with respect to herringbone bond orientation, (2) masonry texture and (3) mechanical properties adopted for joints.At a structural level, a FE homogenized limit analysis is performed on a masonry dome built in herringbone bond. In order to assess limit analysis results, additional non-linear FE analyses are performed, including a full 3D numerical expensive heterogeneous approach and models where masonry is substituted with an equivalent macroscopic material with orthotropic behavior and possible softening. Reliable predictions of collapse loads and failure mechanisms are obtained, meaning that the approach proposed may be used by practitioners for a fast evaluation of the effectiveness of herringbone bond orientation.

OVERALL ELASTIC PROPERTIES OF POLYSILICON FILMS: A STATISTICAL INVESTIGATION OF THE EFFECTS OF POLYCRYSTAL MORPHOLOGY

In this paper we investigate the effects of polycrystal morphology on the overall properties of polysilicon. Focusing on two-dimensional representative volume elements (RVEs) of textured films, we numerically generate digital polycrystal morphologies through Voronoi tessellations and assume the in-plane orientation of the crystal lattice of silicon grains to be randomly distributed. First, we show how a regularization provision for the Voronoi tessellations, adopted in order to better match the grain boundary (GB) geometry featured by actual polysilicon films, affects the statistics of an internal length-scale which naturally emerges because of the presence of GBs. Second, we provide a numerical homogenization technique to estimate the overall in-plane elastic moduli of the polysilicon film and compare the outcomes with standard Voigt and Reuss bounds. Through this comparison, we furnish a way to also estimate the size of the RVE to get effective results. Third, through Monte Carlo simulations we investigate the effect of microstructural fluctuations on the scattering of the overall elastic moduli of polysilicon. We show that even when the RVE appears to be representative for a single polycrystal realization, the RVE might not be representative if one looks at the extreme values of the aforementioned scattered elastic moduli.

Studies of Physical and Chemical Properties of Two-Dimensional Hexagonal Crystals by First-Principles Calculation

In this study, we implement first-principles calculation to study the physical and chemical properties of two-dimensional (2D) hexagonal crystals. The ab initio results of the crystallographic, elastic, and electronic band-gap parameters of C, Si, Ge, BN, CN, SiN, and SiC monolayers are presented. Similarly to graphene, CN and BN are among compounds with the highest 2D elasticity. The in-plane elastic moduli of Si and Ge monolayers are relative small. C, Si, and Ge monolayers are semimetal. All the four binary 2D crystals are semiconductors with wide band gaps. Two typical 2D hexagonal lattice structures, i.e., sp 2 flat and sp 3 rumpled configurations, are classified. The orbital sp 2 -hybridization in graphene and 2D BN is verified by angular-momentum projected atomic density of state calculation. 2D SiC is basically in sp 2 -hybridization. The orbital hybridization of Si, Ge, CN, and SiN monolayers is of the sp 3 -type on the whole. In view of the structural and chemical features of these monolayers, different methods for the experimental preparation of 2D crystals are suggested.

Correlation of elastic properties of melt infiltrated SiC/SiC composites to in situ properties of constituent phases

The ability to correlate the elastic properties of melt infiltrated SiC/SiC composites to properties of constituent phases using a hybrid Finite Element approach is examined and the influence of material internal features, such as the fabric architecture and intra-tow voids, on such correlation is elucidated. Tensile testing was carried out in air at room temperature and 1204 °C. Through-thickness compressive elastic modulus utilizing the stacked disk method was measured at room temperature. In situ moduli of constituent materials were experimentally evaluated using nano-indentation techniques at room temperature. A consistent relationship is observed between constituent properties and composite properties for in-plane normal and shear moduli and Poisson’s ratio at room temperature. However, experimental data for through-thickness compressive elastic modulus is lower than the calculated value. It is hypothesized that the existence of voids inside the fiber tows and their collapse under compressive loads is the cause of such discrepancy. Estimates for the change in elastic moduli of constituent phases with temperature were obtained from literature and used to calculate the elastic properties of the composites at 1204 °C. A reasonable correlation between the in-plane elastic moduli of the composite and the in situ elastic properties of constituent phases is observed.