Variational Asymptotic Method Research Articles

Equivalent single-layer model for hierarchical diamond honeycomb sandwich panels using variational asymptotic method

Sandwich panels with a hierarchical core facilitate the optimal utilization of limited materials to enhance bearing capacity in strategic locations. Nevertheless, the complex sub-structural cells have hindered extensive investigation into their static and modal characteristics. The objective of this study is to obtain the effective plate properties by homogenizing the representative unit cell and subsequently applying them to a two-dimensional equivalent single layer model (2D-ESL) based on the variational asymptotic method (VAM). Furthermore, three-point bending test of 3D-printed specimen and 3D detailed finite element results are employed to confirm the predictions of 2D-ESL under various load and boundary conditions. The effects of material factors (such as fiber volume fraction and layup configuration) and structural parameters (including aspect ratio, hierarchy ratio, and slenderness ratios of ribs and ligaments) on the equivalent stiffness and modal characteristics are also examined. Moreover, sandwich panels with different hierarchical configurations (including 2, 3 and infinite segments within the inclined ribs) are investigated to provide valuable insights for selecting appropriate hierarchies based on specific conditions. Notably, the transition from polyline to arc-shaped segments has been proven to effectively mitigate local stress concentration, leading to a more uniform distribution of local displacement.

3D equivalent Cauchy model for serrated re-entrant auxetic honeycombs based on variational asymptotic method

The presence of serrated ligaments in the re-entrant auxetic honeycomb (abbreviated as serrated RAH) effectively enhances the stiffness while maintaining auxeticity in the re-entrant direction. To investigate its auxetic characteristics, a 3D equivalent Cauchy model (3D-ECM) was established using the variational asymptotic method. The unique aspect is that the engineering constants are obtained by homogenizing the representative unit-cell and then utilized in the 3D-ECM for global analysis. Subsequently, the resulting global responses are fed into the recovery relationships for localized field analysis. The accuracy of the 3D-ECM and the recovered local field distributions were confirmed by comparing them with the results obtained from the three-dimensional FE model (3D-FEM) and the experimental data collected from the 3D-printed specimen. The effects of the geometric parameters on the negative Poisson’s ratio and engineering constants were in-depth discussed. Specifically, an optimal value of 0.5 for the re-entrant ratio and d-value (corresponding to the re-entrant angle of 90 degree) yield maximum auxeticity and comparatively higher elastic moduli. In addition, the 3D-ECM greatly enhanced the computational efficiency without compromising the accuracy in predicting global behaviors and local field distributions. The unit-cell tailorability in the proposed model offers valuable guidance for optimizing the design of serrated RAHs.

Effect of damage evolution on the auxetic behavior of 2D and 3D re-entrant type geometries

In this work, a mathematical formulation based on variational asymptotic method (VAM) has been proposed to determine the effect of damage on the auxetic properties of two-dimensional (2D) and three-dimensional (3D) re-entrant geometries. The influence of damage progression on the auxetic behavior was captured using a geometrically exact one-dimensional beam theory and an isotropic damage law, implemented in a nonlinear finite element framework. The effect of material degradation on the macroscale effective elastic properties such as the elastic modulus and Poisson’s ratio for the two-dimensional and three-dimensional re-entrant auxetic geometries was quantified. The mechanical behavior as predicted by the in-house Python-based implementation of the proposed VAM-based formulation is verified with the results from the commercial finite element solver Abaqus, wherein the user material subroutine was used to capture damage evolution. The numerical examples presented in this paper reveal that the macroscale auxetic behavior of the geometries was affected significantly by damage progression. The results of this research will provide insights into the design and analysis of auxetic materials for applications that warrant consideration of damage evolution.

Asymptotically correct isoenergetic formulation of geometrically nonlinear anisotropic plates

In the present work, a new displacement-based Equivalent Single Layer (ESL) plate theory has been developed using the Variational Asymptotic Method (VAM) and the principle of isoenergetics. In contrast to the existing assumption-based ESL plate theories, the proposed VAM-based plate theory is devoid of any presuppositions. The VAM decouples the 3D plate problem into a 1D through the thickness analysis and a 2D planar problem. Closed form solutions have been obtained for the 1D problem in terms of 2D displacement variables and their higher-order derivatives. The complexity involving higher-order derivatives of 2D displacement variables has been simplified through the isoenergetic principle ensuring a similar rate of convergence for strains as well as displacements. Solutions for various field variables have been compared against the benchmark problems available in the literature and 3D FEA. These comparisons demonstrate the accuracy, efficiency, and versatility of the present methodology. Key contributions of the present work are (a) First principles-based derivation of the reduced-order 2D plate model from the 3D model energy. (b) The plane stress condition as well as the quadratic variation of transverse shear stress and strain are natural outcomes of the present mathematical framework. (c) This leads to zero tangential traction boundary conditions on the plate surface. (d) The effectiveness of capturing higher-order behavior at lower-order results in simplified analysis, reduced computational complexity and increased efficiency.

Equivalent-oriented model for sandwich panels with ZPR accordion honeycomb

The core layer of accordion honeycomb sandwich panel (accordion HSP) exhibits exceptional affinity for one-dimensional curvature deformations due to its special zero Poisson’s ratio effect. This study thoroughly examines its static and dynamic characteristics by decomposing the analysis into unit-cell constitutive modeling and a two-dimensional equivalent-oriented model (2D-EOM) through the variational asymptotic method. The unit-cell constitutive modeling enables determining equivalent stiffness properties for the 2D-EOM. To validate the accuracy and efficiency of the proposed 2D-EOM, its results are compared against those obtained from a detailed 3D finite element model, including in-plane and out-of-plane behaviors of the core layer and sandwich panel, as well as free and random vibration under diverse excitation conditions. The comparison results demonstrate a significant three-fold increase in the equivalent stiffness of the accordion honeycomb in the direction of the added ligaments, as compared to the re-entrant honeycomb. Simultaneously, the acceleration amplitude of the accordion HSP is effectively reduced by 4% due to the involvement of lower high-order frequencies, rendering it exceptionally suited for shockproof meta-structural applications. This equivalent model not only fulfills accuracy requirements but also improves computational efficiency by 15 times, providing an effective approach for preliminary design of honeycomb sandwich panels.

Prediction of thermal conduction in a four-phase composite with lowly-conducting interfaces based on variational asymptotic method

Due to the complex physical and chemical reactions, surface contamination and debonding, the interfaces within the four-phase composite, represented by hollow-glass-bead reinforced cement based material (HGB-CBM), cannot be assumed to be perfect. In order to consider the effect of imperfect interfaces, a novel thermal conduct model is developed. An imperfect interface is used to replace the pre-existing transition layer in the composite material, and the discontinuous temperature field proportional to the normal heat flow is preserved by using Hadamard’s relation and Taylor’s expansion. Based on the principle of minimum functional and variational asymptotic method, the governing variational statement for thermal conduct model is derived, and validated through comparison with RVE-based FEM and experimental results. It can be concluded that HGB-CBM with lowly-conducting imperfect interfaces has lower effective thermal conductivity, smaller heat flux within HGB and more significant changes at imperfect interfaces, which may be due to longer and more complex heat flux paths along the imperfect interfaces. HGB-CBM with larger volume fraction of HGB and smaller thermal conductivity of bead wall can provide better thermal insulation effect, which could provide reference for using HGB-CBM as thermal-insulating material in building engineering.

Exploiting nonlinearities through geometric engineering to enhance the auxetic behaviour in re-entrant honeycomb metamaterials

Classical approaches to enhance auxeticity quite often involve exploring or designing newer architectures. In this work, simple geometrical features at the member level are engineered to exploit non-classical nonlinearities and improve the auxetic behaviour. The structural elements of the auxetic unit cell are here represented by thin strip-like beams, or thin-walled tubular beams. The resulting nonlinear stiffness enhances the auxeticity of the lattices, especially under large deformations. To quantify the influence of the proposed structural features on the resulting Poisson’s ratio, we use here variational asymptotic method (VAM) and geometrically exact beam theory. The numerical examples reveal that 2D re-entrant type micro-structures made of thin strips exhibit an improvement in terms of auxetic behaviour under compression. For the auxetic unit cell with thin circular tubes as members, Brazier’s effect associated with cross-sectional ovalisation improves the auxetic behaviour under tension; the enhancement is even more significant for the 3D re-entrant geometry. Thin strip-based auxetic unit cells were additively manufactured and tested under compression to verify the numerical observations. The experimentally measured values of the negative Poisson’s ratio are in close agreement with the numerical results, revealing a 66% increase due to the nonlinearity. Simulation results showcase these alternative approaches to improve the auxetic behaviour through simple geometric engineering of the lattice ribs.

Optimal design of composite sandwich panel with auxetic reentrant honeycomb using asymptotic equivalent model and PSO algorithm

The composite honeycomb sandwich with auxetic reentrant honeycomb (CSP-ARH) is a typical negative Poisson’s ratio structure, which has better mechanical properties than composite structure with positive Poisson’s ratio. In this study, a 2D asymptotic equivalent model (AEM) based on the variational asymptotic method (VAM) is proposed and used to analyze the vibration and buckling problems of CSP-ARH incorporating CFRP facesheets and an aluminum honeycomb core. After validation with 3D finite element results, a particle swarm optimization (PSO) algorithm is applied to optimize the mesoscopic size of composite facesheets and honeycomb core. The primary objectives of the optimization process involve maximizing the buckling load and natural frequency, minimizing the mass while satisfying specific mechanical properties requirements. Moreover, a few additional optimization iterations are performed to find the optimal shape coefficients of the curved fibers in the CFRP facesheets for variable stiffness composite sandwich panels. The shape of the curved fiber is defined using a cubic polynomial function f(x,y) and is described by a 20 × 20 grid of discrete elements. The results indicate that CSP-ARH with optimal curved fibers shapes exhibit superior performance compared to those with straight fibers, with buckling load and natural frequency increased by 8.18% and 7.09%, respectively.

Asymptotically correct 3D displacement of the Mooney–Rivlin model using VAM

The current work focuses on the nonlinear analytical analysis (dimensional reduction and recovery relations) of a hyperelastic plate governed by the compressible Mooney–Rivlin model using the Variational Asymptotic Method (VAM). The geometric nonlinearity is accommodated through finite deformations, and generalized 3D warping functions, while material nonlinearity through the hyperelastic material model. VAM mathematically splits the 3D nonlinear elastic problem into the 1D through the thickness analysis and 2D nonlinear plate analysis, using the inherent small parameters. These are the geometric small parameter (ratio of thickness to the characteristic dimension), and the physical small parameter (moderate strains). This work results include the derivation of closed-form analytical expressions of 3D warping functions, 2D nonlinear constitutive relation, and recovery relation to express the 3D displacement field for a plate. The 2D nonlinear constitutive relation is given as an input to the in-house developed 2D nonlinear finite element analysis of the reference surface to determine the 2D displacements and 2D strains. In order to validate the current theory, standard test cases are solved and compared with 3D nonlinear Finite Element Analysis (FEA).

Dimension reduction model for sandwich panels with novel triangular chiral honeycomb using variational asymptotic method

The triangular chiral (Tri-Chi) honeycomb is a novel negative Poisson’s ratio structure, featuring fully triangular cells inspired by the anti-tetra chiral design. This work aims to reveal the deformation mechanisms of the Tri-Chi honeycomb and employs the variational asymptotic method to determine the effective plate properties of sandwich panels with Tri-Chi honeycombs (i.e., SP-TCH). Based on this, a 2D dimension reduction model (2D-DRM) is developed, and the warping functions are solved through asymptotic analysis of the leading terms in the energy functional. Comparative analysis with the 3D FE model demonstrates that the 2D-DRM exhibited maximum errors of only 6.83% and 2.93% respectively in analyzing tension-bending coupling behavior and natural frequency, with the computational efficiencies being improved by approximate 55-fold and 32-fold, respectively. Furthermore, the parameter analysis indicate that the structural stiffness and natural frequencies of SP-TCH were higher than those of stiffened triangle honeycomb sandwich panel with the same dimensions and materials. The facesheet layup design had a significant impact on the equivalent stiffness and vibration characteristics, whereas the included angle had a significant influence on the Poisson’s ratio of the Tri-Chi core layer, leading to a reversal from a negative to a positive ratio as the included angle increased.

Closed form Expressions of Shear Correction Factor for Functionally Graded Beams

Abstract In this work closed-form expressions of Shear Correction Factor (SCF) have been derived for beams with Functionally Graded Material (FGM), through Variational Asymptotic Method (VAM). An energy equivalence approach has been adopted between VAM and Timoshenko model, for estimating the SCF. A planar FGM beam has been considered and the calculation for SCF has been carried out. The formulation has been derived in a functional form that permits solutions for a large class of gradation models of FGM. In the limiting case when the material becomes homogeneous the estimated SCF matches exactly with that of the literature, thus validating the solution. A detailed discussion has been carried out on the results and conclusions have been drawn.

Computational evaluation of absorption characteristics of ceramic-based auxetic materials in X-band frequency range

This research study investigates the absorption capabilities of ceramic-based auxetic metamaterials within the X-band frequency range, emphasising their potential application in stealth technology. Four distinct auxetic topologies have been chosen for this purpose: star, re-entrant, anti-tetrachiral, and missing-rib/cross-chiral while maintaining an equal cross-sectional area for comparison analysis. A computationally efficient homogenisation scheme based on the variational asymptotic method is used to evaluate the effective properties of these auxetics. The absorption spectra are then obtained by evaluating scattering matrices using these effective properties. The influence of auxetics out-of-plane thickness, incidence and polarisation angles on the proposed ceramic absorber’s absorption spectra is evaluated. One of the interesting observations is the identical absorption capabilities of star and missing-rib/cross-chiral geometries despite their distinct architectures. The star and missing-rib/cross-chiral based absorbers achieved a maximum absorption of 99.99% or a minimum reflection loss (RL) of −40 dB with a thickness of 3.50 mm. The RL is less than −10 dB (the standard for an electromagnetic (EM) absorber) for all incidence angles less than 70∘. The findings of this study hold significant potential for the advancement of ceramic-based auxetic metamaterials in EM absorption applications within the aerospace industry.

Dynamic Characteristics and Effective Stiffness Properties of Sandwich Panels with Hierarchical Hexagonal Honeycomb.

The dynamic characteristics of sandwich panels with a hierarchical hexagonal honeycomb (SP-HHHs) show significant improvements due to their distinct hierarchy configurations. However, this also increases the complexity of structural analysis. To address this issue, the variational asymptotic method was utilized to homogenize the unit cell of the SP-HHH and obtain the equivalent stiffness, establishing a two-dimensional equivalent plate model (2D-EPM). The accuracy and effectiveness of the 2D-EPM were then verified through comparisons with the results from a detailed 3D FE model in terms of the free vibration and frequency- and time-domain forced vibration, as well as through local field recovery analysis at peak and trough times. Furthermore, the tailorability of the typical unit cell was utilized to perform a parametric analysis of the effects of the length and thickness ratios of the first-order hierarchy on the dynamic characteristics of the SP-HHH under periodic loading. The results reveal that the vertices serve as weak points in the SP-HHH, while the vertex cell pattern significantly influences the specific stiffness and stiffness characteristics of the panel. The SP-HHH with hexagonal vertex cells has superior specific stiffness compared to panels with circular and rectangular vertex cells, resulting in a more lightweight design and enhanced stiffness.

Asymptotic analysis of Timoshenko-like orthotropic beam with elliptical cross-section

This work presents the asymptotically exact analysis of homogeneous-orthotropic prismatic beam with elliptic cross-section. Variational Asymptotic Method (VAM) has been used to systematically reduce the 3D elastic problem of beam analysis into 2D cross-sectional analysis and 1D along the length of beam analysis. This reduction simplifies beam analysis to a great extent. The 2D analysis supplies cross-sectional stiffness constants to the 1D beam problem. This analysis provides closed-form expression of cross-sectional stiffness constants, 3D displacement and strain fields in terms of 1D strain measures. Present work does not make any a priori ad-hoc assumptions about the deformation as it uses VAM and ensures asymptotic correctness of the solution. The results are obtained for Classical and Timoshenko-like beam models loaded with most general loads. These results are validated by comparing them with existing results in the literature and Finite Element Analysis (FEA) results. Current results are in close agreement with FEA results.

Effective Mechanical Properties of Auxetic Materials: Numerical Predictions Using Variational Asymptotic Method Based Homogenization

Abstract In this work, the variational asymptotic method (VAM) based homogenization framework is used for the first time to determine the equivalent elastic stiffness tensor of auxetic materials. The proposed method allows the structural elements of the auxetic unit cell to naturally incorporate rotational degrees-of-freedom, without any ad-hoc assumptions. The overall macroscale homogenized response of the unit cells is considered to be fully anisotropic; specific possible responses, representative of orthotropy or transverse isotropy naturally emerge from the VAM-based homogenization, due to the arrangements of the structural elements making up the unit cell. For all the auxetic unit cell geometries considered in this study, the predictions obtained from the in-house python-based implementation of the VAM-based homogenization framework are validated using commercial finite element software (abaqus) and open literature. The results demonstrate the versatility and the computational efficiency of the VAM-based homogenization framework to describe auxetic metamaterials.

Free and random vibration analyses of hourglass lattice sandwich panel using an equivalent model based on variational asymptotic method

The hourglass lattice sandwich panel (hourglass LSP) represents a novel type of lightweight and high-strength structure, showcasing new random vibration characteristics related to its distinct geometric configuration. In this paper, a two-dimensional equivalent plate model (2D-EPM) is constructed based on the equivalent stiffness obtained by the variational asymptotic homogenization of the unit cell. The accuracy of 2D-EPM in analyzing the random vibration characteristics (power spectral density (PSD) and root mean square (RMS) responses, as well as effective mass fraction) is validated by the three-dimensional direct numerical simulation and experiments, and its computational efficiency is greatly improved due to the reduced degree of freedoms. On this basis, the effects of structural parameters on the free and random vibration characteristics are systematically analyzed by using the novelty of unit-cell tailorability. The investigation revealed direct correlations between the displacement PSD and RMS with the aspect ratio and lattice rod radius, whereas the inclination angle and face-to-core thickness ratio exhibited inverse correlations. Furthermore, similarities in the low-frequency behaviors of hourglass, pyramid and auxetic LSPs were evident. Specifically, the displacement–RMS curve of the auxetic LSP was comparatively lower, and the hourglass LSP exhibited a similar response, with the second-lowest RMS curve implying correspondingly strong deformation resistance.

An asymptotically exact first-order shear deformation theory for functionally graded plates

An asymptotically exact first-order shear deformation theory for functionally graded elastic plates is derived using the variational-asymptotic method. As an application, an analytical solution to the problem of wave propagation in a sandwich plate is found in accordance with this refined theory. Comparison between the dispersion curves obtained by 2-D plate theory and 3-D elasticity theory reveals that the former is accurate up to the order of h2/l2, where h is the plate thickness and l the wavelength.

Continuum modeling of nonlinear buckling behavior of CNT using variational asymptotic method and nonlinear FEA

Carbon nanotubes (CNTs) exhibit a unique buckling behavior due to their slender tubular geometry and thin-walled circular cross section. This study aims to analyze effect of nonlinear cross-sectional deformation on buckling of CNTs. To accomplish this, CNTs are modeled as beam structures, and the analysis is conducted using the Variational Asymptotic Method (VAM) and a geometrically exact beam theory, as well as nonlinear finite element analysis (FEA). The study considers various loading cases, including pure axial compression and combined loading scenarios, such as bending-axial compression and torsion-axial compression. The results of the study indicate that inclusion of cross-sectional deformation-induced nonlinearity reduces the critical buckling load of CNTs. The reduction is 2–5% for pure axial compression and 10–40% for combined loading cases. The results are validated against existing literature and commercial finite element software, ABAQUS®. Additionally, parametric studies with different slenderness and radius-to-thickness ratios were carried out to further understand the impact of these parameters on buckling of CNTs. Finally, the study presents 3D deformed shapes of CNTs during buckling by combining the results of the 1D analysis and the 2D cross section analysis. The findings show that nonlinearity associated with radius-to-thickness ratio has a significant impact on the cross-sectional ovalisation of CNTs and is critical in evaluating their buckling behavior. This aspect of nonlinearity is often overlooked in continuum modeling methods of CNTs, making this study an important contribution to the field.

Application of Green’s function in frequency solution for wind turbine blade modelled as box beam

In order to study the vibration of composite wind turbine blades, a Green’s function method for frequency calculation is proposed. Based on the variational asymptotic method and Hamilton principle, the mechanical model of wind turbine blade, modelled as box beam with composite laminated structure, is established by using circumferential asymmetric stiffness design. The Green function method is used to analyze the frequency of wind turbine blades from the aspect of ply angle, and the numerical examples are compared with other calculation methods. The results show that the Green function method has good accuracy.

Stochastic analysis of a three phase composite using variational asymptotic based homogenization technique coupled with stochastic reduced order model

The current work describes a stochastic framework to investigate the static response characteristics of periodic and random fiber arrays of reinforced three-phase composites with aleatory mechanical and thermal constituent properties. These properties were randomly modeled via Gaussian distribution using the statistical quantities of first and second moments. The framework is based on Variational Asymptotic Method (VAM) based homogenization technique coupled with Stochastic Reduced Order Model (SROM). When compared against the framework of VAM coupled with the traditional Monte Carlo Simulation (MCS) method, the computational cost of the SROM model with VAM was significantly less. Good agreement with solutions from traditional MCS combined with VAM was also achieved. To demonstrate applicability of the framework, a multivariate stochastic analysis was performed to determine the effect of fiber arrangements on the randomness of mechanical and thermal properties in a three-phase composite. Additionally, univariate stochastic analysis was performed, and the sensitivity of each constituent parameter corresponding to the stochastic homogenized response variable was investigated using the bivariate correlation and linear regression analysis methods.