In-plane Elastic Constants Research Articles

Simultaneous optimization of elastic constants of laminated composites using artificial bee colony algorithm

Multifunctional application of laminated composites requires multi-objective optimization of their characteristics. In this paper, the ply angles of laminated composites are determined in order to maximize the effective in-plane elastic constants simultaneously. These constants are determined by considering a representative small element of the laminated composite and imposing the conditions of uniformity of out-of-plane stresses and in-plane strains at orthotropic layer interfaces. Then, by combining the artificial bee colony algorithm and various multi-objective optimization methods, the optimal ply angles and the corresponding co-optimized constants are determined. The correctness and accuracy of the method is verified not only by providing a comparison with the existing results but also by solving a known problem. The results are presented and discussed, considering different multi-objective optimization problems. The results show that the increasing number of layers in the problem of simultaneous optimization of Young’s moduli not only does result in finding more Pareto optimal solutions in the feasible objective region but also shifts the Pareto frontier toward the utopia point. However, when the shear modulus optimization is engaged in the problem, using more than two layers only leads to obtaining more Pareto optimal answers.

Phase stabilities and vibrational analysis of hydrogenated diamondized bilayer graphenes: A first principles investigation

The phase stabilities as well as some intrinsic properties of hydrogenated diamondized bilayer graphenes, 2-dimensional materials adopting the crystal structure of diamond and of lonsdaleite, are investigated using a first-principles approach. Our simulations demonstrate that hydrogenated diamondized bilayer graphenes are thermodynamically stable with respect to bilayer graphene and hydrogen molecule even at 0 GPa, and additionally they are found to withstand the physical change in structure up to at least 1000 K, ensuring their dynamical and thermal stabilities. The studied hydrogenated diamondized bilayer graphenes are predicted not only to behave as direct and wide band gap semiconductors, but also to have a remarkably high resistance to in-plane plastic deformation induced by indentation as implied by their high in-plane elastic constants comparable to those of diamond and of lonsdaleite. The mechanical stability of the materials is confirmed though the fulfilment of the Born stability criteria. Detailed analysis of phonon vibrational frequencies of hydrogenated diamondized bilayer graphenes reveals possible Raman active and IR active modes, which are found to be distinctly different from those of hydrogenated diamond-like amorphous carbon and defective graphene and thus could be used as a fingerprint for future experimental characterization of the materials.

In-plane elastic anisotropic constants (IEACs) measurement of rolling aluminum plate at different depth using ultrasonic LCR wave

The inhomogeneous in-plane elastic property can be described by the in-plane elastic anisotropic constants (IEACs) at different depths (such as the rolling plate), which have significant impact on the in-plane plastic behavior. The IEACs should get more attention in the design and the service of structural components. To measure IEACs of aluminum plate at different depths, an ultrasonic approach using the longitudinal critical refracted (LCR) wave is proposed. The relation between IEACs and LRC wave velocity is described analytically. So, the IEACs can be calculated by measuring the LCR wave velocities along different directions. Experimentally, this approach was applied to measure the IEACs of rolling 2219-T6 aluminum alloy plate at different depths. The distribution of C11 and C22 of the same depth indicated that the aluminum plate exhibited homogeneous property at the same layer. As for IEACs at different depths, aluminum plate showed transverse symmetry at the surface layer and orthotropic symmetry at the inner layer near the median plate plane.

Geometric analysis and constrained optimization of woven z-pinned composites for maximization of elastic properties

In this study, an optimization problem is carried out for maximizing the out-of-plane elastic constants of woven z-pinned composites, subjected to some constraints on their in-plane elastic constants and geometric parameters. A unit-cell model is used to obtain the elastic constants as functions of geometric parameters including the pin radius and distribution density, the resin-rich zone length, the unit-cell length, the yarn thickness and width, and the yarn waviness created by z-pinning or weaving. A parametric study on the unit-cell geometry is conducted to evaluate the optimization variables and determine whether they are effective enough on the elastic properties or not. Using the key results of the parameter study, which are the equality of the yarn width and the unit-cell length as well as selecting the lowest practical yarn thickness, the constrained optimization problem is defined and solved using a modified artificial bee colony algorithm. Significant improvements in the out-of-plane constant against controlled reductions in the in-plane ones are observed. The more flexible the constraints on the in-plain constants, the better the out-of-plane constants. The optimum out-of-plane elastic modulus is greatly dependent on the pin distribution density and the ratio of the yarn width to the pin radius.

Application of Moiré Interferometry to the Characterization of Orthotropic Materials in the Antisymmetric Configuration using the Virtual Fields Method

Moire interferometry is an effective full-field deformation measurement technique and has been utilized for mechanical behavior analysis of materials and structures. For isotropic materials, Moire patterns can be obtained by performing standard tests, such as, tensile and bending tests, to calculate the displacement and strain. Then, the mechanical properties can be characterized. However, standard tests are not sufficient to characterize the mechanical parameters of anisotropic materials due to the complexity of their material properties. Thus, in this work, Moire interferometry was combined with the Virtual Fields Method to obtain the four in-plane elastic constants (Q11, Q22, Q12, and Q66) of orthotropic materials in the form of a diametrically compressed disk. Firstly, according to finite element method simulation results, optimized parameters can be achieved when the principal direction of the material does not coincide with the loading direction, making the loading configuration antisymmetric. Therefore, Moire interferometry experiment was simulated to demonstrate the feasibility of measurement in the antisymmetric configuration. Finally, the Q11, Q22, Q12 and Q66 values of a unidirectional carbon fiber composite were measured in a real Moire interferometry experiment using the proposed method, yielding results that agreed closely with those obtained using the strain gauges.

Atlas for the properties of elemental two-dimensional metals

Common two-dimensional (2D) materials have a layered 3D structure with covalently bonded, atomically thin layers held together by weak van der Waals forces. However, in a recent transmission electron microscopy experiment, atomically thin 2D patches of iron were discovered inside a graphene nanopore. Motivated by this discovery, we perform a systematic density-functional study on atomically thin elemental 2D metal films, using 45 metals in three lattice structures. Cohesive energies, equilibrium distances, and bulk moduli in 2D are found to be linearly correlated to the corresponding 3D bulk properties, enabling the quick estimation of these values for a given 2D metal and lattice structure. In-plane elastic constants show that most 2D metals are stable in hexagonal and honeycomb, but unstable in square 2D structures. Many 2D metals are surprisingly stable against bending.

Evolutions in the Elastic Constants of Ca-Montmorillonites with H2O/CO2 Mixture under Supercritical Carbon Dioxide Conditions

Clay mineral plays a crucial role in determining permeability and affecting mechanical properties of caprock formations in geologic carbon sequestration (GCS) and enhanced oil recovery (EOR) engineering. In the corresponding environment, CO2–H2O intercalation not only causes swelling or shrinkage of clay mineral but also results in affecting mechanical properties of mineral. However, influence of quantitative CO2/H2O adsorption on mechanical properties of clay still remains unknown. This work investigated evolutions of elastic properties of Ca-MMT with variable CO2–H2O component using molecular dynamic method. Elastic coefficients Cij, bulk modulus K, shear modulus G, Young’s modulus Eii, and Poisson’s ratio Eij were correlated to qualified adsorption of CO2–H2O mixture. We obtained the following conclusions: (1) in-plane elastic constants, bulk modulus, shear modulus, and Young’s modulus E33 had distinct correlation with the profile of d001 spacing as a function of H2O content at fixed CO2 content; (2) n...

Prediction of Elastic Constants of Carbon Fabric/Polyurethane Composites

The purpose of this work is to study a new composite material consisting of polyurethane (PU) resin and carbon fiber fabric. This PU resin is superior in impact, viscosity, low curing temperature, and short curing time. If this resin is combined with fiber fabric by vacuum assisted resin transfer method, the fabrication time will be short. Since it is a braided composite, it’s important to have a model to predict the elastic constants for different braid angels. To predict the elastic constants including Young’s modulus, shear modulus, and Poisson’s ratio, a finite element model is established. In this model a braided layer is treated as two uni-directional layers. Then, the elastic constants of this composite with different braid angels are estimated. After that, the composites with different braid angels are fabricated and tested to obtain the elastic constants, and the comparison with the finite element results is made. The results indicate that the agreement is very good for the Young’s modulus. For the Poisson’s ratio, the difference between the prediction and the measurement is reasonable. From the comparison, it can be concluded that the finite element model is good. Then, this model is used to predict all in-plane elastic constants for arbitrary braid angles.

On the derivation of the elastic properties of lattice nanostructures: The case of graphene sheets

Abstract Several nanomechanics formulations based on common two and three-body bonding potentials are compared. Numerical simulations and analytical approaches are used to investigate the not negligible differences among the predictions of the in-plane elastic constants of graphene sheets in the literature, separately exploring the role of the potentials and that of the structural descriptions (beams and trusses) of the original molecular mechanics (MM) model. The energetic differences between the structural models and the MM model are highlighted through exact discrete homogenization procedures. In so doing, some theoretical expressions of the graphene elastic constants available in the literature are recovered and supported by the numerical experimentation. The results provide also an assessment of the accuracy of some potentials largely employed in the literature with respect to several ab-initio reference solutions and the experimental measurements available. Some suggestions towards a reparametrization of the modified Morse potential are consequently formulated.

Bending sound in graphene: Origin and manifestation

It is proved that the acoustic-type dispersion of bending mode in graphene is generated by the fluctuation interaction between in-plane and out-of-plane terms in the free energy arising with account of non-linear components in the graphene strain tensor. In doing so we use an original adiabatic approximation based on the alleged (confirmed a posteriori) significant difference of sound speeds for in-plane and bending modes. The explicit expression for the bending sound speed depending only on the graphene mass density, in-plane elastic constants and temperature is deduced as well as the characteristics of the microscopic corrugations of graphene. The obtained results are in good quantitative agreement with the data of real experiments and computer simulations.

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.

Optimal design of a honeycomb core composite sandwich panel using evolutionary optimization algorithms

The present paper deals with the optimal design of a composite sandwich panel with honeycomb core structure using particle swarm optimization (PSO) technique. The face sheets of sandwich panel are considered to be thin and the sandwich panel is subjected to a uniformly distributed normal load. The Navier type solution and the closed-form expressions for the macroscopic in-plane elastic constants of the honeycombs are used to predict the deflection of the panel. Also, the maximum stress in the composite sandwich panel is determined using the macroscopic stress–strain relation of the honeycomb core and the plate. Niching Memetic PSO (NMPSO) and Locally Informed Particle Swarm (LIPS) variants of PSO are examined to minimize the weight of the panel. Numerical results showed that the optimal geometry of the honeycomb cell has this property that its radius and thickness converge to their lower bounds while its length converges to its upper bound. This means that to have an optimal panel, each cell should have the allowable minimum cross section (highest number of cells) and maximum allowable length. Moreover the effects of panel aspect ratio, cell width and applied load were examined. Also, the numerical results confirm the efficiency and effectiveness of the NMPSO in finding optimal solution on the constrained and unconstrained objective functions.

Determination of in-plane elastic constants of angle ply laminates from tensile measurements

This paper presents a new method to completely characterise the in-plane elastic properties of an important type of angle ply laminates using only unidirectional tests. The angle ply laminates having the same number of identical plies are considered. Some new results for orthotropic laminates are found using this technique, namely, the conditions of existence of a particular direction in which there is no shear extension coupling. The laminates are subjected to three tensile tests: two in the orthotropic axes and the third one in a particular direction. The results allow a complete semiexperimental characterisation of the angle ply laminate, which do not need a priori knowledge of the elastic properties of the constitutive layer. Comparisons are made between numerical predictions and experimental results. Finally, the computational results were compared to the experimental results, verifying the capability of the method in order to predict the in-plane elastic properties of composite laminates.

Equivalent Mechanical Properties of Graphene Predicted by an Improved Molecular Structural Mechanics Model

Based on molecular mechanics and the stick-spiral model, this paper first presents the analytical analysis of the effective in-plane mechanical properties of both zigzag and armchair monolayer graphene sheets. We find that the equivalent in-plane elastic constants of monolayer graphene sheets are the same in the two principal directions of graphene. The effective in-plane mechanical properties of graphene are then evaluated numerically using an improved molecular structural mechanics (MSM) model, in which the flexible connections are used to characterize the bond angle variations of graphene. Furthermore, the effective bending rigidity of the beam representing a C-C bond in this improved MSM model is determined from the energy equivalence over the basic cell of graphene and the force constants given by molecular mechanics. A rigidly connected frame model with the bending stiffness of the equivalent beams for C-C bonds different from the existing structural mechanics model is also used to evaluate the mechanical properties of graphene. The flexibly connected frame model gives very good results of Youngs modulus and Poisson ratio of monolayer graphene sheet. The new rigidly connected frame model presented here also gives improved results than the existing rigidly connected frame model of graphene.

Through-thickness elastic constants of composite laminates

This article presents relationships for the through-thickness elastic constants of composite laminates, needed for modelling three-dimensional stress problems. Using the individual ply elastic constants together with the in-plane laminate elastic constants (from laminate theory) results in explicit relationships for, ν13, ν23, E3, G13 and G23. The Poisson’s ratio and Young’s modulus expressions apply to all laminate types. The shear modulus relationships apply to balanced laminates, but are extendable to other laminate types. Relationships are developed here for angle-ply, cross-ply and quasi-isotropic laminates. Results are presented for glass/epoxy and carbon/epoxy laminates, above and below the resin Tg. The importance of coupling stresses and their influence on ν13, ν23 and E3 is underlined. It is shown, for instance, that equating the laminate through-thickness modulus to the individual ply value can result in a significant under-estimate.

In-plane elastic constants and strengths of circular cell honeycombs

Abstract The in-plane elastic modulus, Poisson’s ratio, brittle crushing strength and plastic yielding strength of honeycombs with hexagonally packed circular cells are analyzed theoretically. The resulting theoretical expressions are compared with the numerical results obtained from a series of finite element analyses for circular cell honeycombs with various relative densities, leading to a good correlation. It is also found that the in-plane mechanical properties of circular cell honeycombs are significantly affected by the ratio of cell-wall thickness to cell radius. Though the elastic constants along the two principal directions of circular cell honeycombs are the same, the brittle crushing strength and plastic yielding strength along the two principal directions are not identical. Furthermore, the in-plane mechanical properties of circular cell honeycombs are compared to those of regular hexagonal honeycombs with straight and uniform-thickness cell walls to evaluate their microstructural efficiency.

Investigation of the uncertainty of the in-plane mechanical properties of composite laminates

The uncertainty of the in-plane mechanical properties of the laminate used in an aircraft wing structure is investigated. Global sensitivity analysis is used to identify the source of the uncertainties of the response performance. Due to the limitations of the existing global sensitivity analysis method for nonlinear models with correlated input variables, a new one using nonlinear regression is proposed. Furthermore, a contribution matrix is defined for engineering convenience. Two nonlinear numerical examples are employed in this article to demonstrate the ability of the proposed global sensitivity analysis method. After applying the proposed global sensitivity analysis method to the laminate model, the contribution matrices are obtained; from these matrices, researchers can identify the dominant variance contributions that contribute the most to the response variance. Factor analysis is then employed to analyze the global sensitivity analysis results and determine the most efficient methods to decrease the variances of the in-plane elastic constants. Monte Carlo simulation is used to demonstrate the efficiency of the methods in decreasing the variances.

Determination of elastic properties of surface layers and coatings by resonant ultrasound spectroscopy

This paper deals with determination of in-plane elastic constants of thin layers deposited on substrates. Modified resonant ultrasound spectroscopy is used to measure resonant spectra before and after layer deposition . These two spectra are compared and changes in the position of the resonant peaks are associated with layer properties. It is shown that for thin layers either the elastic moduli or the surface mass density can be determined, providing the complementary information (the surface mass density for the determination of the moduli, the elastic moduli for the determination of the surface mass density) is known. As an experimental demonstration of this approach, the elastic moduli of diamond-like-carbon film deposited on a silicon substrate and the surface mass density of a thin spray paint on a silicon substrate are determined.

The in-plane linear elastic constants and out-of-plane bending of 3-coordinated ligament and cylinder-ligament honeycombs

Four novel cylinder-ligament honeycombs are described, where each cylinder has 3 tangentially-attached ligaments to form either a hexagonal or re-entrant hexagonal cellular network. The re-entrant cylinder-ligament honeycombs are reported for the first time. The in-plane linear elastic constants and out-of-plane bending response of these honeycombs are predicted using finite element (FE) modelling and comparison made with hexagonal and re-entrant hexagonal honeycombs without cylinders. A laser-crafted re-entrant cylinder-ligament honeycomb is manufactured and characterized to verify the FE model. The re-entrant honeycombs display negative Poisson’s ratios and synclastic curvature upon out-of-plane bending. The hexagonal and ‘trichiral’ honeycombs possess positive Poisson’s ratios and anticlastic curvature. The ‘anti-trichiral’ honeycomb (short ligament limit) displays negative Poisson’s ratios when loaded in the plane of the honeycomb, but positive Poisson’s ratio behaviour (anticlastic curvature) under out-of-plane bending. These responses are understood qualitatively through considering deformation occurs via direct ligament flexure and cylinder rotation-induced ligament flexure.

Elastic constants of 3-, 4- and 6-connected chiral and anti-chiral honeycombs subject to uniaxial in-plane loading

Finite element models are developed for the in-plane linear elastic constants of a family of honeycombs comprising arrays of cylinders connected by ligaments. Honeycombs having cylinders with 3, 4 and 6 ligaments attached to them are considered, with two possible configurations explored for each of the 3- (trichiral and anti-trichiral) and 4- (tetrachiral and anti-tetrachiral) connected systems. Honeycombs for each configuration have been manufactured using rapid prototyping and subsequently characterised for mechanical properties through in-plane uniaxial loading to verify the models. An interesting consequence of the family of ‘chiral’ honeycombs presented here is the ability to produce negative Poisson’s ratio (auxetic) response. The deformation mechanisms responsible for auxetic functionality in such honeycombs are discussed.