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.
This article presents a probabilistic framework to characterize the dynamic and stability parameters of composite laminates with spatially varying micro and macro-mechanical system properties. A ...novel approach of stochastic representative volume element (SRVE) is developed in the context of two dimensional plate-like structures for accounting the correlated spatially varying properties. The physically relevant random field based uncertainty modelling approach with spatial correlation is adopted in this paper on the basis of Karhunen-Loève expansion. An efficient coupled HDMR and DMORPH based stochastic algorithm is developed for composite laminates to quantify the probabilistic characteristics in global responses. Convergence of the algorithm for probabilistic dynamics and stability analysis of the structure is verified and validated with respect to direct Monte Carlo simulation (MCS) based on finite element method. The significance of considering higher buckling modes in a stochastic analysis is highlighted. Sensitivity analysis is performed to ascertain the relative importance of different macromechanical and micromechanical properties. The importance of incorporating source-uncertainty in spatially varying micromechanical material properties is demonstrated numerically. The results reveal that stochasticity (/system irregularity) in material and structural attributes influences the system performance significantly depending on the type of analysis and the adopted uncertainty modelling approach, affirming the necessity to consider different forms of source-uncertainties during the analysis to ensure adequate safety, sustainability and robustness of the structure.
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.
•Equivalent elastic moduli of irregular auxetic honeycombs are analysed.•Closed-form analytical formulae are presented.•Spatially random cell angles are considered.
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.
This paper presents a concise state-of-the-art review along with an exhaustive comparative investigation on surrogate models for critical comparative assessment of uncertainty in natural frequencies ...of composite plates on the basis of computational efficiency and accuracy. Both individual and combined variations of input parameters have been considered to account for the effect of low and high dimensional input parameter spaces in the surrogate based uncertainty quantification algorithms including the rate of convergence. Probabilistic characterization of the first three stochastic natural frequencies is carried out by using a finite element model that includes the effects of transverse shear deformation based on Mindlin’s theory in conjunction with a layer-wise random variable approach. The results obtained by different metamodels have been compared with the results of traditional Monte Carlo simulation (MCS) method for high fidelity uncertainty quantification. The crucial issue regarding influence of sampling techniques on the performance of metamodel based uncertainty quantification has been addressed as an integral part of this article.
Two-dimensional lattices are ideal candidate for developing artificially engineered materials and structures across different length-scales, leading to unprecedented multi-functional mechanical ...properties which can not be achieved in naturally occurring materials and systems. Characterization of effective elastic properties of these lattices is essential for their adoption as structural elements of various devices and systems. An enormous amount of research has been conducted on different geometry of lattices to identify and characterize various parameters which affect the elastic properties. However, till date we can not control the elastic properties actively for a lattice microstructure, meaning that the elastic properties of such lattices are not truly programmable. All the parameters that control the effective elastic properties are passive in nature. After manufacturing the lattice structure with a certain set of geometric or material-based parameters, there is no room to modulate the properties further. In this article, we propose a hybrid lattice micro-structure by integrating piezo-electric materials with the members of the lattice for active voltage-dependent modulation of elastic properties. A bottom-up multi-physics based analytical framework leading to closed-form formulae is derived for hexagonal lattices to demonstrate the concept of active lattices. It is noticed that the Young’s moduli are voltage-dependent, while the shear modulus and the Poisson’s ratios are not functions of the applied voltage. Thus, the compound mechanics of deformation induced by external mechanical stresses and electric field lead to an active control over the Young’s moduli as a function of voltage. Interestingly, it turns out that a programmable state-transition of the Young’s moduli from positive to negative values with a wide range can be achieved in such hybrid lattices. The physics-based analytical framework for active modulation of voltage-dependent elastic properties on the basis of operational demands provide the necessary physical insights and confidence for potential practical exploitation of the proposed concept in various futuristic multi-functional structural systems and devices across different length-scales.
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This paper presents a stochastic approach to study the natural frequencies of thin-walled laminated composite beams with spatially varying matrix cracking damage in a multi-scale ...framework. A novel concept of stochastic representative volume element (SRVE) is introduced for this purpose. An efficient radial basis function (RBF) based uncertainty quantification algorithm is developed to quantify the probabilistic variability in free vibration responses of the structure due to spatially random stochasticity in the micro-mechanical and geometric properties. The convergence of the proposed algorithm for stochastic natural frequency analysis of damaged thin-walled composite beam is verified and validated with original finite element method (FEM) along with traditional Monte Carlo simulation (MCS). Sensitivity analysis is carried out to ascertain the relative influence of different stochastic input parameters on the natural frequencies. Subsequently the influence of noise is investigated on radial basis function based uncertainty quantification algorithm to account for the inevitable variability for practical field applications. The study reveals that stochasticity/ system irregularity in structural and material attributes affects the system performance significantly. To ensure robustness, safety and sustainability of the structure, it is very crucial to consider such forms of uncertainties during the analysis.
Lattice-based artificial microstructures have been receiving significant attention from the scientific community over the past decade due to the possibility of developing materials with tailored ...multifunctional capabilities that are not achievable in naturally occurring materials. Such lattice materials can be conceptualized as a network of beams with different periodic architectures, wherein the common practice is to adopt straight beams. While a large set of mechanical properties can be simultaneously modulated by adopting an appropriate network architecture in the conventional periodic lattices, the prospect of on-demand global specific stiffness and flexibility modulation has become rather saturated lately due to intense investigation in this field. Thus there exists a strong rationale for innovative design at a more elementary level in order to break the conventional bounds of specific stiffness that can be obtained only by lattice-level geometries. Here we propose a novel concept of anti-curvature in the design of lattice materials, which reveals a dramatic capability in terms of enhancing the effective elastic moduli in the nonlinear regime while keeping the relative density unaltered. A semi-analytical bottom-up framework is developed for estimating effective elastic moduli of honeycomb lattices with the anti-curvature effect in cell walls considering geometric nonlinearity under large deformation. We propose to consider the deformed shapes corresponding to compressive or tensile modes of the honeycomb cell walls as the initial beam-level configuration. A substantially increased resistance against deformation can be realized when such a lattice is subjected to the opposite mode, leading to increased effective elastic moduli. Moreover, unlike conventional materials, we demonstrate that it is possible to achieve non-invariant elastic moduli under tension and compression. Within the framework of a unit cell based approach, the cell walls with initial curvature are modeled as curved beams including nonlinear bending and axial deformation, wherein the governing equation is derived using variational energy principle through the Ritz method. The developed physically insightful semi-analytical model captures nonlinearity in elastic moduli as a function of the degree of anti-curvature and applied stress along with conventional parameters related to unit cell geometry and intrinsic material property. The concept of anti-curvature in lattice materials proposed in the present article introduces novel exploitable dimensions in mode-dependent effective elastic property modulation, leading to an expanded design space including more generic scopes of nonlinear large deformation analysis.
•We propose a novel concept of anti-curvature in the design of lattice materials, which reveals a dramatic capability in terms of modulating the elastic properties in the generic nonlinear regime while keeping the relative density unaltered.•The numerical results reveal that a variation up to 43% and 25% for Young’s moduli and Poisson’s ratios can be obtained based on the application-specific structural demands of stiffness and flexibility.•Within the framework of unit cell based approach, initially curved lattice cell walls are modeled as programmed curved beams under large deformation in the body-embedded curvilinear frame considering the combined effect of bending, stretching and shear deformations.•The physically insightful semi-analytical model captures nonlinearity in the elastic properties of anti-curvature lattices as a function of the degree of curvature and applied stress together with conventional microstructural and intrinsic material properties.•The proposed concept of anti-curvature in lattice materials introduces novel exploitable dimensions in the enhancement of mode-dependent elastic properties, leading to an expanded design space including more generic scopes of nonlinear large deformation.
•Irregular lattices are analysed considering the compound effect of viscoelasticity and stochasticity.•A practically relevant stochastic modelling approach is developed in conjunction with ...quasi-periodic lattices to consider spatially correlated structural and material attributes.•Computationally efficient and physically insightful closed-form analytical formulae are developed for analysing viscoelastic properties of spatially irregular lattices in frequency domain.
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An analytical framework is developed for investigating the effect of viscoelasticity on irregular hexagonal lattices. At room temperature many polymers are found to be near their glass temperature. Elastic moduli of honeycombs made of such materials are not constant, but changes in the time or frequency domain. Thus consideration of viscoelastic properties are essential for such honeycombs. Irregularity in lattice structures being inevitable from practical point of view, analysis of the compound effect considering both irregularity and viscoelasticity is crucial for such structural forms. On the basis of a mechanics based bottom-up approach, computationally efficient closed-form formulae are derived in frequency domain. The spatially correlated structural and material attributes are obtained based on Karhunen–Loève expansion, which is integrated with the developed analytical approach to quantify the viscoelastic effect for irregular lattices. Consideration of such spatially correlated behaviour can simulate the practical stochastic system more closely. The two effective complex Young’s moduli and shear modulus are found to be dependent on the viscoelastic parameters, while the two in-plane effective Poisson’s ratios are found to be independent of viscoelastic parameters and frequency. Results are presented in both deterministic and stochastic regime, wherein it is observed that the amplitude of Young’s moduli and shear modulus are significantly amplified in the frequency domain. The response bounds are quantified considering two different forms of irregularity, randomly inhomogeneous irregularity and randomly homogeneous irregularity. The computationally efficient analytical approach presented in this study can be quite attractive for practical purposes to analyse and design lattices with predominantly viscoelastic behaviour along with consideration of structural and material irregularity.
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