Article

# Stochastic low-velocity impact analysis of sandwich plates including the effects of obliqueness and twist

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## Abstract

This paper quantifies the influence of uncertainty in the low-velocity impact responses of sandwich plates with composite face sheets considering the effects of obliqueness in impact angle and twist in the plate geometry. The stochastic impact analysis is conducted by using finite element (FE) modelling based on an eight nodded isoparametric quadratic plate bending element coupled with multivariate adaptive regression spline (MARS) in order to achieve computational efficiency. The modified Hertzian contact law is employed to model contact force and other impact parameters. Newmark's time integration scheme is used to solve the time-dependent equations. Comprehensive deterministic as well as probabilistic results are presented by considering the effects of location of impact, ply orientation angle, impactor velocity, impact angle, face-sheet material property, twist angle, plate thickness and mass of impactor. The relative importance of various input parameters is determined by conducting a sensitivity analysis. The results presented in this paper reveal that the impact responses of sandwich plates are significantly affected by the effect of source-uncertainty that in turn establishes the importance of adopting an inclusive stochastic design approach for impact modelling in sandwich plates.

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... Computational analyses of polymer composites often encounter uncertainties because of the variations in the properties of the material, measurement uncertainty, limitations in the test set-up, operating environment and inaccurate geometrical features [102][103][104][105][106]. Uncertainty in parametric inputs, initial conditions and the boundary conditions, computational and numerical uncertainties arising from the unavoidable assumptions and approximations along with the inherent inaccuracy of the model result in major deviation from the deterministic values or the expected material behavior, altering the overall performance of composites. ...
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The superior multi-functional properties of polymer composites have made them an ideal choice for aerospace, automobile, marine, civil, and many other technologically demanding industries. The increasing demand of these composites calls for an extensive investigation of their physical, chemical and mechanical behavior under different exposure conditions. Machine learning (ML) has been recognized as a powerful predictive tool for data-driven multi-physical modeling, leading to unprecedented insights and exploration of the system properties beyond the capability of traditional computational and experimental analyses. Here we aim to abridge the findings of the large volume of relevant literature and highlight the broad spectrum potential of ML in applications like prediction, optimization, feature identification, uncertainty quantification, reliability and sensitivity analysis along with the framework of different ML algorithms concerning polymer composites. Challenges like the curse of dimensionality, overfitting, noise and mixed variable problems are discussed, including the latest advancements in ML that have the potential to be integrated in the field of polymer composites. Based on the extensive literature survey, a few recommendations on the exploitation of various ML algorithms for addressing different critical problems concerning polymer composites are provided along with insightful perspectives on the potential directions of future research.
... In case of layerwise sensitivity analysis, it may further be noticed that the sensitivity of material properties is higher in the outer layers. Such trends are in good agreement with the literature of free vibration [60], buckling [94], failure [42] and low-velocity impact [93] analyses. However, note that the laminate configurations considered in these references are different from the current study. ...
Article
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The present paper proposes a surrogate-assisted moment-independent stochastic sensitivity analysis of laminated composite plates for establishing a unified measure in the case of multi-objective performances. With the advancements in artificially engineered structural systems spanning across different length scales, it has become more common to design composite structures for multi-objective performances like the criteria of deflection, buckling and vibration of multiple modes, different impact parameters, and failure. Normally sensitivity analysis is carried out separately and individually for different such performance parameters. This paradigm is no more suitable for advanced multi-functional structures like composite laminates. In this article, we propose an efficient unified sensitivity analysis approach based on weighted relative importance of different performance parameters by introducing the notion of engineering judgment. A moment-independent sensitivity analysis is proposed here based on finite element modeling of composites in conjunction with the Least Angle Regression assisted Polynomial Chaos Expansion (PCE) to achieve computational efficiency without compromising the outcome. Such surrogate-assisted finite element approaches are particularly crucial for computationally intensive multi-objective systems like composites. The layer-wise unified sensitivity quantification of laminated composites considering multi-functional objectives, as presented here, would lead to more optimized designs and better quality control while manufacturing the complex advanced structural systems.
... In the above descriptions,  represents a Monte Carlo simulation symbolic operator,  represents degree of stochasticity. We have considered layer-wise random variable based stochasticity [55][56][57][58][59][60][61][62] for modelling uncertainty in the hybrid structures, to represent which the ply orientation angle (for the laminated composites part) and other material properties are shown as a function of the thickness (t). In the present study we have considered uncertainty in the input parameters as depicted above, Table 9(ad). ...
Article
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Functionally graded materials, sandwich structures and composite laminates have been widely investigated individually in last few decades for their respective set of advantages over conventional monolithic structures. However, the tremendous recent advances in manufacturing processes have opened up new frontiers of research specifically to form hybrid structures for multifunctional applications, where the advantages of each of the constituting components could potentially be exploited in a single structure. Such complex hybrid structures are often susceptible to manufacturing uncertainties and different forms of variability. To characterize the stochastic dynamic behavior of these hybrid structural forms in a comprehensive, practically-relevant, yet efficient framework, here we present a multi-physical probabilistic vibration analysis based on Gaussian Process Regression (GPR) assisted finite element (FE) approach coupled with Monte Carlo Simulation. Integration of the GPR based machine learning model with the physics based simulation model leads to a significant level of computational efficiency in the quantification of system uncertainty. In the stochastic approach for various hybrid structural configurations, the compound effects of source-uncertainties are considered in order to assess the effect of different critical multi-physical parameters systematically involving material, geometric and physical aspects. The numerical results categorically indicate that the source-uncertainty of hybrid shells with different geometries has a significant effect on dynamic behaviour of the structure, which makes it imperative to take into account such probabilistic deviations to ensure adequate operational safety and serviceability.
... In this subsection, we will first explore the prediction accuracy and efficiency of the four machine learning models (linear regression, regression trees, support vector machines, and gaussian process regression) considered in this study following the methodology described in figure 2. In this context it may be noted that different other machine learning algorithms have been exploited for efficient prediction in a multitude of engineering problems, wherein the fundamental steps are similar [59][60][61][62][63][64][65][66][67][68][69][70]. To determine the prediction accuracy of an individual model for the current analysis, This can be explained by the inherent behavior of regression tree approach of finding locally optimal solution especially for the cases where multiple feature variables are involved [43]. ...
Article
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Despite the tremendous capabilities of Molecular dynamics (MD) simulations, they suffer from the limitation of computationally intensive and time-consuming nature. This hinders the seamless discovery of nanomaterials with adequate computational insights. Over the last decade, graphene has received widespread attention from the scientific community due to its extra-ordinary multi-physical properties, primarily focusing on the fundamental physics and chemistry along with the notion of scalable synthesis. However, the recent advances in machine learning have opened new frontiers in the research of such exceptional two-dimensional materials where the boundaries of different multi-physical properties could be pushed further efficiently with the assistance of deep computational intelligence. Here we propose a coupled machine learning (ML) based approach to investigate the critical mechanical responses of graphene by capturing the underlying physics of the system through an (D-)optimally minimum number of MD simulations. We have investigated five different internal and external controlling input features like temperature, strain rate, nanostructural defects, doping, and chirality, which influence the critical mechanical properties of graphene such as the constitutive behaviour including fracture strength, failure strain and Young’s modulus. Even though the aspect of computational intensiveness in molecular dynamics simulations is addressed through the coupled ML based approach, in a data-driven comprehensive analysis, it is often difficult to get complete information about the concerning input parameters including their statistical distributions and occurrence bounds. Thus, it becomes difficult to carry out computational investigations based on powerful techniques like Monte Carlo simulation. To mitigate this lacuna, here we have exploited the ML-assisted approach for developing a level-based fuzzy framework at nano-scale under sparse input descriptions. In this article, we first demonstrate the computational efficiency achieved through the proposed ML based framework without compromising quality of the analysis, followed by data-intensive correlation analysis, sensitivity and uncertainty quantification considering various levels of the influencing system parameters, revealing detailed computational insights on mechanical properties of graphene including the coupled interactive influence of intrinsic defects and doping.
... All the subsequent results in this paper are obtained using PC-Kriging trained with 512 training samples. It can be noted in this context that stochastic analysis of composite structures leading to the uncertainty quantification of different global responses have recently received significant attention from the scientific community [69][70][71][72][73][74][75][76][77][78][79]. However, most of these studies consider a single machine learning algorithm to map the stochastic input-output domain. ...
Article
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Due to the absence of adequate control at different stages of complex manufacturing process, material and geometric properties of composite structures are often uncertain. For a secure and safe design, tracking the impact of these uncertainties on the structural responses is of utmost significance. Composite materials, commonly adopted in various modern aerospace, marine, automobile and civil structures, are often susceptible to low-velocity impact caused by various external agents. Here, along with a critical review, we present machine learning based probabilistic and non-probabilistic (fuzzy) low-velocity impact analyses of composite laminates including a detailed deterministic characterization to systematically investigate the consequences of source-uncertainty. While probabilistic analysis can be performed only when complete statistical description about the input variables are available, the non-probabilistic analysis can be executed even in the presence of incomplete statistical input descriptions with sparse data. In this study, the stochastic effects of stacking sequence, twist angle, oblique impact, plate thickness, velocity of impactor and density of impactor are investigated on the crucial impact response parameters such as contact force, plate displacement, and impactor displacement. For efficient and accurate computation, a hybrid polynomial chaos based Kriging (PC-Kriging) approach is coupled with in-house finite element codes for uncertainty propagation in both the probabilistic and non-probabilistic analyses. The essence of this paper is a critical review on the hybrid machine learning algorithms followed by detailed numerical investigation in the probabilistic and non-probabilistic regimes to access the performance of such hybrid algorithms in comparison to individual algorithms from the viewpoint of accuracy and computational efficiency.
... Computational analyses of polymer composites often encounter uncertainties because of the variations in the properties of the material, measurement uncertainty, limitations in the test set-up, operating environment and inaccurate geometrical features[102][103][104][105][106]. Uncertainty in parametric inputs, initial conditions and the boundary conditions, computational and numerical uncertainties arising from the unavoidable assumptions and approximations along with the inherent inaccuracy of the model result in major deviation from the deterministic values or the expected material behavior, altering the overall performance of composites. ...
Article
Full-text available
The superior multi-functional properties of polymer composites have made them an ideal choice for aerospace, automobile, marine, civil, and many other technologically demanding industries. The increasing demand of these composites calls for an extensive investigation of their physical, chemical and mechanical behavior under different exposure conditions. Machine learning (ML) has been recognized as a powerful predictive tool for data-driven multi-physical modeling, leading to unprecedented insights and exploration of the system properties beyond the capability of traditional computational and experimental analyses. Here we aim to abridge the findings of the large volume of relevant literature and highlight the broad spectrum potential of ML in applications like prediction, optimization, feature identification, uncertainty quantification, reliability and sensitivity analysis along with the framework of different ML algorithms concerning polymer composites. Challenges like the curse of dimensionality, overfitting, noise and mixed variable problems are discussed, including the latest advancements in ML that have the potential to be integrated in the field of polymer composites. Based on the extensive literature survey, a few recommendations on the exploitation of various ML algorithms for addressing different critical problems concerning polymer composites are provided along with insightful perspectives on the potential directions of future research.
... Refined LWTs, also called zigzag theories, are more computational than the discrete LWT because of the independence of the number of unknowns concerning the number of layers [29]. Stochastic-based studies are also available in literature for the analysis of sandwich structures [30][31][32]. ...
Article
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In the present work, free vibration and buckling analyses of sandwich plates with various functionally graded foam cores are carried out. Foam cores are assumed to be made of metal, and three different configurations of the porous distribution in the core layer are taken into consideration. To carry out a comparative study between the distributions of pores in the core foam, the mass of foam in all three cases is kept the same. The vibration and buckling behaviors of skew plates are also analyzed as a part of the current investigation. The principle of minimization of potential energy and Hamilton’s principle are used for the derivation of the governing equations, while a C-0 finite element-based higher-order zigzag formulation is developed to solve the free vibration and buckling problems. The influences of gradation laws, boundary conditions, skew angle and geometry of plates are studied in detail for the dynamic and stability characteristics. It is found that both the non-dimensional natural frequency and buckling load decrease with the increase in the thickness of the metal foam cores, while they show an increasing trend as the skew angle of the plate increases
... Atahan et al. [25] mainly studied the low velocity oblique impact behavior of the single lap joint made by adhesive bonding. Kumar et al. [26] quantified the influence of uncertainty in the low-velocity impact responses of sandwich plates with composite face sheets considering the effects of obliqueness in impact angle and twist in the plate geometry. Chen et al. [27] investigated and optimized a composite sandwich structure under low-velocity impact with the impact angle of 0° and 45° by a new method. ...
Article
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The mechanical response of CFRP laminates under low-velocity oblique impact was investigated using finite element simulation and experimental test in this paper. Based on the 3D Hashin failure criterion, bilinear cohesive constitutive model and progressive failure theory, a finite element model of CFRP laminate with hemispherical cylindrical impactor under low-velocity oblique impact was established, by using finite element simulation software ABAQUS. A clamp for multi-angle impact was designed and manufactured. A low-velocity oblique test platform was built based on the clamp. Several groups of experiments with different impact angles and impact energy were carried out. It was found that the damage morphology and energy absorption caused by normal impact and oblique impact is quite different. For normal impact, the main damage mode on the front of the laminate is indentation, while for oblique impact, the main damage mode on the front surface is scratch due to the sliding of the impactor on the laminate. The damage caused by normal impact on the back of the laminate is always more significant than that caused by oblique impact. The simulation results of the contact force and the residual velocity of impactor are compared with the experimental results, and the good agreement verifies the reliability of the simulation model. The mechanical response is studied by simulation, and it is found that the mechanism of normal impact is different from that of oblique impact. The contact force and residual kinetic energy of the impactor decrease with the increase of impact angle, while the contact time and the energy consumed by friction increase are the opposite. The energy absorbed by the laminate is not negatively related to the impact angle.
... Compared to the intrusive SFEMs, the deterministic FE codes can be directly used without any modifications in the non-intrusive SFEMs. Among the existing nonintrusive SFEMs, the Monte Carlo simulation (MCS)-based SFEM [40,41] is a popular versatile method, while it needs a great computational effort to implement an amount of expensive finite element simulations. Thus, the MCS has limited practical applicability, especially for complex structural systems, which is usually taken as benchmark method for simple cases. ...
Article
Currently, several intrusive stochastic finite element methods (SFEMs) such as perturbation method and spectral SFEM are widely applied for stochastic response analysis of continuous structures. However, the intrusive SFEMs need to modify conventional finite element formulations to establish the stochastic stiffness matrix, and cannot calculate the probability density function of structural response and the reliability straightforwardly. This paper proposes a new non-intrusive SFEM for efficiently computing stochastic responses and reliabilities of plates in a unified way. Firstly, the direct probability integral method (DPIM) is developed to obtain the probability density function of stochastic response by solving probability density integral equation (PDIE). Secondly, the non-intrusive SFEM based on DPIM decouples the deterministic finite element analysis and PDIE to calculate the stochastic responses and reliabilities of uncertain plate structures, and the discretization and quantification of random fields of elastic modulus and thickness are implemented through Karhunen-Loève expansion. Finally, several examples of uncertain Kirchhoff and Mindlin plates demonstrate the efficiency and versatility of the proposed non-intrusive method by comparing with the results from Monte Carlo simulation and literature. The effects of correlation length, mean and variability of random field on the probability distribution of responses and the reliabilities of plates are revealed. For Gaussian random thickness, the linearly elastic plate yields non-Gaussian distributed responses. Increasing the correlation length and variability of random field reduces the reliabilities of plates.
... Thus, it is essential to consider the material and geometric uncertainty in order to accommodate above mentioned environmental changes as well as other inaccuracies occurring during design and fabrication of the sandwich plate. In most of the scientific literatures, researchers have concentrated on deterministic natural frequency analysis [9][10][11][12][13], however some researchers [14][15][16][17][18][19][20][21][22][23] have considered the stochastic behaviour in their study. No study has been carried out for uncertainty quantification of natural frequency for laminated sandwich plate by employing FE coupled PNN surrogate model approach. ...
Article
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In this paper, uncertainty quantification in natural frequencies for laminated soft core sandwich plates is presented by employing finite element (FE) coupled polynomial neural network (PNN) approach. The computational efficiency and accuracy is achieved by using PNN as surrogate model. Latin hypercube sampling method is employed for training of data in PNN model. The stochastic first three natural frequencies of sandwich plates are studied for individual variation in input parameters. The stochasticity in individual input parameters are considered in order to assess their influence on global response of the structure. The algorithm discussed in this article is observed to be converging with the previously published literature (for deterministic case) and validated with full scale Monte Carlo simulation (MCS) i.e. original finite element approach (for stochastic case). The computational time and cost reduced significantly by employing the present surrogate based FE approach compared to that of conventional Monte Carlo simulation approach.
... Some other theories, which have been developed and used in special cases, are described in Wang et al. [28]. Various contact laws have been utilized in complete modeling such as Liu and Swaddiwudhipong [29], Sburlati [30], Rossikhin and Shitikova [31] and Kumar et al. [32]. For collision problems, the Hertzian contact law has been used to describe the contact between the two elastic deformable bodies at the macroscopic scale [33]. ...
Article
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In this paper, the dynamic behavior of the nanoparticle low-velocity impact on viscoelastic laminated composite nanoplate has been investigated. To determine the impact force between nanoparticle and laminated composite nanoplate, the van der Waals interaction force is considered based on the description of Lennard-Jones. The material properties of each layer of laminated composite nanoplate are supposed to be viscoelastic based on the Kelvin–Voigt model. The governing equations of the system are derived based on the first-order shear deformation plate theory and the nonlocal strain gradient theory by employing Hamilton’s principle. Galerkin’s method is employed to solve differential equations of nanoplate with different boundary conditions. Afterward, the system of time-dependent equations is solved by applying the Newmark’s method. The parametric study is presented to examine the effect of particle initial velocity, particle radius, viscoelastic modulus, fiber orientation, nonlocal parameter, length scale parameter, and different boundary conditions on the impact response of nanoplate.
... Therefore, surrogate models are integrated with the MCS technique to overcome this problem and the computationally efficient overall process. Some literature [54][55][56][57][58][59][60][61] showcased utilising MCS and MARS in the stochastic domain to analyse the natural frequency of FGM cantilever plates at different temperatures. For the present study, MARS is integrated with the finite element framework. ...
Article
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Purpose: This paper presents the sensitivity analysis (SA) of the random natural frequency responses of hybrid multi-functionally graded sandwich (HMGS) shells for establishing a unified measure in the case of multi-objective performances. The functionally graded materials, laminated composites, and sandwich cores are employed to develop such novel structures to tailor the benefits of each component in a single structure. Methods: A novel MARS-based sensitivity analysis of these hybrid multi-functionally graded sandwich shells is developed to achieve computational efficiency without compromising the outcome. Such surrogate-assisted FE approaches are crucial for computationally intensive multi-objective systems. The basic governing equations of random natural frequency are framed based on finite element formulation. The variabilities of major influencing random input parameters (here, geometric and material properties) are carried out by employing Monte Carlo simulation (MCS). The multivariate adaptive regression spline (MARS) is adopted as a surrogate model to increase computational efficiency. Results and Conclusion: The results are portrayed to showcase the significant effects of variable input parameters (sensitivity) on random frequency responses of such novel HMGS shells. Hence, it provides the predominant random input parameters and their relative degree of importance while designing such multi-dimensional structural systems. Thus, the contribution of this article lies in both the development of a computationally efficient sensitivity analysis approach and the insightful numerical results for hybrid structures presented thereafter. The comprehensive and collective sensitivity quantification considering multi-functional objectives, as presented in this article, would lead to efficient computational modeling of complex structural systems for more optimized designs and better quality control during manufacturing.
... These values are then further compared to understand the differences in each type of joint and also get an idea about the type of joint that can be used based on the requirements of the different industries. Probabilistic analysis can be carried out in order to incorporate the Material and Geometric Uncertainties [8][9][10][11][12][13] in the analysis. ...
Article
In recent years, adhesively bonded joints are extensively used in a multitude of technological applications , mainly in industries such as aerospace and automobile. Adhesive bonding is often chosen over other joining techniques like bolting, riveting as well as welding, due to the reduction of stress concentrations , easy manufacturing and reduced weight penalty, amongst other issues. For the investigation of the behavior of adhesive bonded joints, the stress-strain behavior of the structural adhesive is necessary. The present study aims to investigate the strength of various types of adhesively bonded lap joints by using the commercial finite element tool ANSYS. It is seen from the analysis that the double lap joint is the type of joint which is least affected by the forces acting on it followed by the single step lap joint.
... By using the similar idea of RVE in conjunction with the FE numerical methods, many researchers (Yang and Qin, 2001;Sun and Tzeng, 2002;Berger et al., 2005;Kari et al., 2007;Rodríguez-Ramos et al., 2013;Tian et al., 2015;Martínez-Ayuso et al., 2017) used such a powerful method to predict effective properties including elastic, piezoelectric as well as dielectric constants considering the concept of single RVE. Several authors (Chandra et al., 2012;Patil et al., 2017;Mirabedini et al., 2020;Kumar et al., 2019) investigated the response of the composite materials with multiscale approach. The overall properties of the composites which are estimated through the experimentation and are well fitted with the investigations performed through the micro-/macro-mechanical analysis. ...
Article
This article presents different micromechanical modelling techniques based on analytical and numerical approaches to determine the effective elastic and piezoelectric (piezoelastic) properties of graphene-based composite materials. Different types, orientations and shapes as well as different geometrical parameters of fiber reinforcement are considered for estimating the effective properties. The effective properties of composite are predicted with and without considering the strong covalent bond which provides interaction and in-plane stability of 2Dcrystalline graphene or strong van der Wall forces formed between graphene layers and the matrix. It is revealed that the axial, transverse and shear effective piezoelastic properties of graphene reinforced piezoelectric composite (GRPC) are significantly enriched due to the incorporation of graphene into the epoxy matrix. The importance of incorporating graphene as nanofillers/interphase into the conventional epoxy matrix to form an advanced composite and its effective properties are illustrated while these results show excellent agreement with previously reported experimental estimates. These results reveal that due to incorporation of graphene nanofillers, there is a significant enhancement in effective properties of composite. The results would also help to recognize the most important material properties with respect to different shapes and orientation of reinforcements which influences the performance of system significantly. To confirm safety, robustness and sustainability of the structure, it is the most prior requirement to determine the effective properties of composites considering different parameters for the different static and structural analyses.
Chapter
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The present chapter investigates the moment-independent sensitivity analysis for hybrid sandwich structures (having cylindrical shell geometry) subjected to low-velocity impact. These hybrid structures are extensively used in lightweight applications where thermal exposure/resistance is of prime importance. Here, the FG facesheet is placed at the upper layer of core whereas laminated composite facesheet is kept at lower layer so that the structure can sustain high-temperature exposure at reduced weight because of laminated composite facesheet at the inner layer of the core. The probabilistic study is performed for the transient impact response of the structure which in turn utilized to assess the sensitiveness of the parameters. The computational efficiency is achieved by implementing polynomial chaos expansion (PCE) metamodel in conjunction with Monte Carlo simulation (MCS). The results illustrate the parameters which significantly affect the transient impact response of the structure.
Article
The stability of functionally graded porous plates with graphene platelets reinforcement (FGP-GPLs) is investigated in this paper. Combining with a new metamodeling approach, namely the Kriging enhanced Neural Network, a stochastic isogeometric analysis (SIGA) framework is proposed for assessing the structural stability. The uncertainties of material properties of both FGP matrix and graphene platelets are considered in the form of random fields and variables. Karhunen-Loève expansion based Nyström method is applied to random field discretization. The Dagum function is adopted as a new kernel function to further improve the performance of the proposed approach. Statistical information including but not limited to statistical moments, probability density function (PDF), and cumulative distribution function (CDF) of the critical buckling load of the plate structure can be effectively estimated through a non-intrusive fashion. In order to illustrate the effectiveness and applicability of the proposed stochastic computational analysis, both theoretical and real-life engineering examples have been investigated in this study.
Chapter
The manufacturing and fabrication of complex polymer sandwich composite plates involve various processes and parameters, and the lack of control over them causes uncertain system parameters. It is essential to consider randomness in varying parameters to analyse polymer sandwich composite plates. The present study portrays uncertainty quantification in structural responses (such as natural frequencies) of polymer sandwich composite plates using the surrogate model. The comparative study of artificial neural network (ANN) and polynomial neural network (PNN) for uncertain structural responses of the sandwich plate is presented. The proposed ANN as well as PNN algorithm is found to be convergent with intensive Monte Carlo simulation (MCS) for uncertain vibration responses. The predictability of PNN is observed to be more efficient than that of ANN. Typical material properties, skew angle, fibre orientation angle, number of laminate and core thickness are randomly varied to quantify the uncertainties. The use of both the surrogate models (PNN and ANN) results in a significant saving of computational time and cost compared to that of full-scale intensive finite element-based MCS approach.
Chapter
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This study presents the effect of skewness in natural frequency responses of sandwich plates. The free vibration analysis is carried out by using higher order zigzag theory (HOZT) considering random input parameters. It satisfies the transverse shear stress continuity condition and the transverse flexibility effect. The in-plane displacement throughout the thickness is assumed to vary cubically while transverse displacement is considered to vary quadratically within the core and constant at top and bottom plates. An efficient stochastic finite element approach is developed for the implementation of proposed plate theory in the random variable surrounding. Compound stochastic effect of all input parameters is presented for the different degrees of skewness in sandwich plates. Intensive Monte Carlo simulation (MCS) is employed for solving the stochastic-free vibration equations and statistical analysis is conducted for illustration of the results. The present algorithm for sandwich plate is validated with previous literatures and it is found to be in good agreement.
Research
Microstructural image based convolutional neural networks for efficient prediction of full-field stress maps in short fiber polymer composites Abstract The increased demand for superior materials has highlighted the need of investigating the mechanical properties of composites to achieve enhanced constitutive relationships. Fiber-reinforced polymer composites have emerged as an integral part of materials development with tailored mechanical properties. However, the complexity and heterogeneity of such composites make it considerably more challenging to have precise quantification of properties and attain an optimal design of structures through experimental and computational approaches. In order to avoid the complex, cumbersome, and labor-intensive experimental and numerical modeling approaches, a machine learning (ML) model is proposed here such that it takes the microstructural image as input with a different range of Young's modulus of carbon fibers and neat epoxy, and obtains output as visualization of the stress component S 11 (principal stress in the x-direction). For obtaining the training data of the ML model, a short carbon fiber-filled specimen under quasi-static tension is modeled based on 2D Representative Area Element (RAE) using finite element analysis. The composite is inclusive of short carbon fibers with an aspect ratio of 7.5 that are infilled in the epoxy systems at various random orientations and positions generated using the Simple Sequential Inhibition (SSI) process. The study reveals that the pix2pix deep learning Convolutional Neural Network (CNN) model is robust enough to predict the stress fields in the composite for a given arrangement of short fibers filled in epoxy over the specified range of Young's modulus with high accuracy. The CNN model achieves a correlation score of about 0.999 and L2 norm of less than 0.005 for a majority of the samples in the design spectrum, indicating excellent prediction capability. In this paper, we have focused on the stage-wise chronological development of the CNN model with optimized performance for predicting the full-field stress maps of the fiber-reinforced composite specimens. The development of such a robust and efficient algorithm would significantly reduce the amount of time and cost required to study and design new composite materials through the elimination of numerical inputs by direct microstructural images.
Chapter
In the present chapter, multivariate adaptive regression splines (MARS) is explored as a surrogate model in conjunction to Monte Carlo simulation (MCS) to analyse the random first-ply failure loads of graphite–epoxy laminated composite plates. The five failure criteria, namely maximum strain theory, maximum stress theory, Tsai–Hill theory, Tsai–Wu theory, and Hoffman theory, are considered. The numerical validation of deterministic failure load is presented first. Thereafter, a concise investigation is carried out to examine the capability of MARS model for efficiently predicting the first-ply failure loads. Comparative results are presented using scatter plots and probability density function plots to access the prediction capability with respect to direct MCS. The current results portray the successful application of MARS as the surrogate model to achieve computational efficiency and analyse the first-ply failure loads.
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The stability of slope is usually characterized by many parameters which are mostly uncertain in nature. Deterministic approach is usually followed to calculate the factor of safety of a slope, but it does not depict the true state of the slope. Hence, a probabilistic approach is a better alternative, which can quantify the probability of failure of a slope under uncertain input parameters. In the present study, slope stability assessment of Pakhi landside is carried out using Finite Element Modeling (FEM) to ascertain the stability conditions of the multilayer slope under both deterministic and probabilistic framework. The multilayer configurations of the profiles are established from the Electrical Resistivity Tomography (ERT). The Shear Strength Reduction (SSR) method is employed to determine the critical strength reduction factor of the slope considering four random variables namely cohesion (c), angle of internal friction (ϕ), Poisson’s ratio (ν) and Elastic modulus (E) of each individual layers. The deterministic factor of safety values along two considered profiles namely section X-X’ and Y-Y’ are calculated as 1.41 and 1.25, respectively. A data driven machine learning algorithm is used to build a computationally efficient surrogate model to perform Monte Carlo Simulations (MCS). MCS are performed for two different values of coefficients of variation i.e., 5% and 15% for all the four random variables of all the layers. The proposed method has no idealization regarding the layering configuration and the failure surface. Probabilistic analysis has been made exhaustive and computationally efficient. The probabilistic analysis indicates good adherence with the recent landslide incident in the field. Further, the analysis indicates that the proposed methodology is favorable and useful tool for the system reliability analysis of landslide slopes.
Chapter
This paper presents the effect of thickness on probabilistic buckling load of sandwich plate using radial basis function (RBF) approach. Due to inevitable material anisotropy and manufacturing inaccuracy such type of structure are always subjected to spatial variability. So, it is required to consider the stochastic effect which makes computational approach more challenging. Using C⁰ stochastic finite element method (FEM) model and higher order zigzag theory, the mathematical formulation has been developed. Cost-effective and computationally efficient RBF-based surrogate is used to get the result without compromising with accuracy in comparison with direct Monte Carlo simulation technique. Skyline technique is used to store global stiffness matrix in a single array and it is solved by subspace iteration.
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First-order shear deformation theory (FSDT) is less accurate compared to higher-order theories like higher-order zigzag theory (HOZT).In case of large-scale simulation-based analyses like uncertainty quantification and optimization using FSDT, such errors propagate and accumulate over multiple realizations, leading to significantly erroneous results. Consideration of higher-order theories results in significantly increased computational expenses, even though these theories are more accurate. The aspect of computational efficiency becomes more critical when thousands of realizations are necessary for the analyses. Here we propose to exploit Gaussian process-based machine learning for creating a computational bridging between FSDT and HOZT, wherein the accuracy of HOZT can be achieved while having the low computational expenses of FSDT. The machine learning augmented FSDT algorithm is referred to here as modified FSDT (mFSDT), based on which extensive deterministic results and Monte Carlo simulation-assisted probabilistic results are presented for the free vibration analysis of shear deformation sensitive structures like laminated composite and sandwich plates considering various configurations. The proposed algorithm of bridging different laminate theories is generic in nature and it can be utilized further in a range of other static and dynamic analyses concerning composite plates and shells for accurate, yet efficient results.
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In this paper, a multi-objective multiparameter optimization procedure is developed by combining rigorously developed metamodels with an evolutionary search algorithm-Genetic Algorithm (GA). Response surface methodology (RSM) is used for developing the metamodels to replace the tedious finite element analyses. A nine-node isoparametric plate bending element is used for conducting the finite element simulations. Highly accurate numerical data from an author compiled FORTRAN finite element program is first used by the RSM to develop second-order mathematical relations. Four material parameters-E1/E2, G12/E2, G23/E2 and ν12 are considered as the independent variables while simultaneously maximizing fundamental frequency, λ1 and frequency separation between the 1st two natural modes, λ21. The optimal material combination for maximizing λ1 and λ21 is predicted by using a multi-objective GA. A general sensitivity analysis is conducted to understand the effect of each parameter on the desired response parameters.
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The truss and honeycomb sandwich plates with transverse crack and delamination in the facesheets are studied, and an Extended Layerwise/Solid-Element (XLW/SE) method is developed. The governing equations of laminated composite facesheets and cores (truss or honeycomb) are established based on the Extended Layerwise Method (XLWM) and 3D solid elements, respectively. The XLW/SE method can obtain the local stress and displacement fields accurately, considering complicated cores without any assumptions. In the numerical examples, the results obtained by the proposed method are compared with those obtained by the 3D elasticity models, and the good agreements are achieved.
Article
Sandwich structure offer more advantage in bringing flexural stiffness and energy absorption capabilities in the application of automobile and aerospace components. This paper presents comparison study and analysis of two types of composite sandwich structures, one having Jute Epoxy skins with rubber core and the other having Glass Epoxy skins with rubber core subjected to low velocity normal impact loading. The behaviour of sandwich structure with various parameters such as energy absorption, peak load developed, deformation and von Mises stress and strain, are analyzed using commercially available analysis software. The results confirm that sandwich composite with jute epoxy skin absorbs approximately 20% more energy than glass epoxy skin. The contact force developed in jute epoxy skin is approximately 2.3 times less when compared to glass epoxy skin. von Mises stress developed is less in case of jute epoxy. The sandwich with jute epoxy skin deforms approximately 1.6 times more than that of same geometry of sandwich with glass epoxy skin. Thus exhibiting its elastic nature and making it potential candidate for low velocity impact application.
Article
Present investigation deals with the buckling and postbuckling behavior of a sandwich plate with a homogeneous core and graphene-reinforced composite (GRC) face sheets resting on an elastic foundation in thermal environments. The material properties of GRC face sheets are assumed to be piece-wise functionally graded by changing the volume fraction of graphene in the thickness direction. The material properties of both the homogeneous core layer and the GRC face sheets are assumed to be temperature-dependent, and are estimated by the extended Halpin-Tsaia micromechanical model. The higher order shear deformation plate theory and the von Kármán-type kinematic nonlinearity are used to derive the governing equations which account for the plate-foundation interaction and the thermal effects. The buckling loads and the postbuckling equilibrium paths are obtained by using a two-step perturbation technique. The impacts of the distribution type of reinforcements, core-to-face sheet thickness ratio, plate aspect ratio, temperature variation, foundation stiffness and in-plane boundary conditions on the postbuckling behavior of sandwich plates with functionally graded GRC face sheets are studied in detail.
Article
In this study, the uncertain static analysis of Euler-Bernoulli type functionally graded structures with probabilistic parameters is investigated. An effective, yet efficient, computational method is proposed within the framework of the finite element analysis (FEA). Various uncertain systematic parameters, which are including the material properties, dimensions of structural elements, as well as applied forces, can be simultaneously incorporated within the unified analysis framework. By meticulously combining the matrix perturbation theory with Tayler's series expansion, both first and second order statistical characteristics (i.e., mean and variances) of the concerned structural responses can be robustly estimated for practically motivated functionally graded structures. In order to illustrate the applicability, accuracy, as well as efficiency of the proposed computational approach, three distinctive functionally graded engineering structures are thoroughly investigated by comparing the performance of the proposed approach with the simulation based reference method. Furthermore, complementary parametric investigations are also conducted to explore the sensitivity of the Euler-Bernoulli type functionally graded structures against various degrees of uncertainty of each considered uncertain system parameter.
Article
This paper presents a robust non-deterministic free vibration analysis for engineering structures involving hybrid, yet spatially dependent, uncertain system parameters. Distinguished from the conventional hybrid uncertain eigenvalue problem, the concept of interval field is enclosed with random field model such that, both the stochastic and non-stochastic representations of the spatial dependency of the uncertainties are simultaneously incorporated within a unified non-deterministic free vibration analysis. In order to determine the probabilistic characteristics (i.e., means and standard deviations) of the extremities of structural natural frequencies, an extended unified interval stochastic sampling (X-UISS) method is implemented for the purpose of effective hybrid uncertain free vibration analysis. By meticulously blending sharpness-promised interval eigenvalue analysis with stochastic sampling techniques, the stochastic profiles (i.e., probability density functions (PDFs) and the cumulative distribution functions (CDFs)) of the extreme bounds of the structural natural frequencies can be rigorously established by utilizing the adequate statistical inference methods. The applicability and effectiveness of the proposed computational framework are evidently demonstrated through the numerical investigations on various practically motivated engineering structures.
Article
This paper presents an experimental investigation on impact response of sandwich composite panels with thermoplastic and thermoset face-sheet. E-glass reinforced epoxy (thermoset) and polypropylene (thermoplastic) have been used to produce polymer composite face-sheets. PVC foam was used as a core material. Several low velocity impact tests were performed under various impact energies. Besides to the individual impact behavior of the thermoset and thermoplastic sandwich composites, the impact response of sandwich composites having hybrid sequences was also investigated. Along with images of damaged samples, variations of the impact characteristics such as absorbed energy, maximum contact force and maximum deflection of the samples are provided. Most particularly this study showed that sandwich composites must have the harmony between core and the face sheet material. The deformation required for core densification must be able to compensate by the face sheet material.
Article
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.
Article
This experimental study addresses the low velocity impact bending response of sandwich beams with expanded polystyrene (EPS) foam core reinforced by aluminum face-sheets using adhesive bonding technique. The deformation behaviour and the effects of the geometric and mechanical design parameters were investigated to improve the impact energy absorbing capability. The bending impact tests were performed for different foam density, foam and plate thicknesses for three impact energy levels: 4.45 13.05, 26.78 J. The photographs of the permanent deformation geometries of the sandwich beams were presented and the contact force and kinetic energy histories were discussed. The specimens with higher foam core density and thicker foam core resulted in the lowest permanent central deflections and contact force variations. As the foam core density and thickness were increased, the energy absorbing capability was improved. The specimens having thinner face-sheets exhibited higher contact force variations and the lower permanent deflection than those of the specimens with thicker face-sheet even for the lowest impact energy. However, as the impact energy was increased, the specimen with thinner face-sheet exhibited lower contact force levels and higher permanent deflections than those of the specimens having thicker face-sheets. In addition, the plastic dissipation energy was increased in the face-sheets. The face-sheet thickness was more effective on the permanent deflections, whereas the foam core density was more effective on the energy absorbing capability.
Article
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.
Article
This research studied the effect of multi-walled carbon nanotubes (MWCNTs) on the internal and external damages of foam-core sandwich panels with kevlar fiber reinforced epoxy face sheets subjected to a low-velocity impact. The sandwich panels were subjected to six levels of energy. Energy profile diagrams (EPDs) were plotted to determine the rebounding, penetration and perforation thresholds of the sandwich panels. Non-destructive evaluation methods have been employed for detecting and measuring damage size of the sandwich panels using X-ray radiography and active infrared thermography. The results show that MWCNTs can improve the absorbed energy and penetration threshold of the foam-core sandwich panels.
Article
The present computational study investigates on stochastic natural frequency analyses of laminated composite curved panels with cutout based on support vector regression (SVR) model. The SVR based uncertainty quantification (UQ) algorithm in conjunction with Latin hypercube sampling is developed to achieve computational efficiency. The convergence of the present algorithm for laminated composite curved panels with cutout is validated with original finite element (FE) analysis along with traditional Monte Carlo simulation (MCS). The variations of input parameters (both individual and combined cases) are studied to portray their relative effect on the output quantity of interest. The performance of the SVR based uncertainty quantification is found to be satisfactory in the domain of input variables in dealing low and high dimensional spaces. The layer-wise variability of geometric and material properties are included considering the effect of twist angle, cutout sizes and geometries (such as cylindrical, spherical, hyperbolic paraboloid and plate). The sensitivities of input parameters in terms of coefficient of variation are enumerated to project the relative importance of different random inputs on natural frequencies. Subsequently, the noise induced effects on SVR based computational algorithm are presented to map the inevitable variability in practical field of applications.
Article
The Layerwise/Solid-Element (LW/SE) method, which was developed based on the layerwise theory and the eight-noded solid element for the laminated composite stiffened plates and sandwich plates in the previous works [1-3], is extended to the static response and free vibration analysis of the composite sandwich structures with multi-layer cores. In the LW/SE method, the layerwise theory is used to model the behavior of the laminated composite facesheets while the eight-noded solid element is employed to discretize the cores. The total governing equations of the composite sandwich plates are assembled by using the compatibility conditions to ensure the compatibility of displacements and the equilibrium conditions to ensure the equilibrium of internal force at the interface between facesheets and cores. In the numerical examples, the static analysis and free vibration analysis of the composite sandwich plates are investigated for the sandwich plates with double-layer honeycombs and pyramidal lattice cores. The proposed method is validated by using the 3D elastic models developed in MSC.Patran/Nastran code. Two kinds of local models of the sandwich plates with double-layer honeycombs are presented for the local response problems to reduce the computational cost in the case of guarantee analysis accuracy of the local response.
Article
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.
Article
This paper presents the effect of noise on surrogate based stochastic natural frequency analysis of composite laminates. Surrogate based uncertainty quantification has gained immense popularity in recent years due to its computational efficiency. On the other hand, noise is an inevitable factor in every real-life design process and structural response monitoring for any practical system. In this study, a novel algorithm is developed to explore the effect of noise in surrogate based uncertainty quantification approaches. The representative results have been presented for stochastic frequency analysis of spherical composite shallow shells considering Kriging based surrogate model. The finite element formulation for laminated composite shells has been developed based on Mindlin’s theory considering transverse shear deformation. The proposed approach for quantifying the effect of noise is general in nature and therefore, it can be extended to explore other surrogates under the influence of noise.
Article
This paper presents an experimental investigation on impact response of sandwich composite panels with different face-sheet thicknesses. A number of low velocity impact tests were performed under various impact energies. The damage process of the sandwich composites consisted of glass/epoxy face-sheets, and foam cores are analyzed from cross-examining some graphs such as load–deflection curves and damaged specimens. The primary damage modes observed are fiber fractures at upper and lower skins, delaminations between adjacent glass-epoxy layers, and core shear fractures.
Article
In this work, inverse hyperbolic zigzag theory and inverse trigonometric zigzag theory, recently developed by the authors, are extended for the free vibration analysis of the laminated composite and sandwich plates. The developed plate models satisfy the condition of transverse shear stress continuity at the layer interfaces and the zero traction boundary conditions on the surfaces of the plate obviating the need of shear correction factor. They consider shear strain shape function assuming the nonlinear distribution of transverse shear stresses. The less numbers of independent variables makes the solution computationally effective. An efficient C0 continuous isoparametric serendipity element is employed for the usual discretization of the plate structure. In order to demonstrate the capabilities of the developed models in predicting the structural free vibration response, some representative results are obtained covering different features of laminated composite and sandwich structures such as boundary conditions, aspect ratios and span-to-thickness ratios and the results are compared with those in the literature wherever available. The excellent agreement of the evaluated results with the three-dimensional exact results conclude that the presented models are not only accurate but also efficient.
Article
In this paper, the polyurethane foam filled pyramidal lattice core sandwich panel is fabricated in order to improve the energy absorption and low velocity impact resistance. Based on the compression tests, a synergistic effect that the foam filled sandwich panels have a greater load carrying capacity compared to the sum of the unfilled specimens and the filled polyurethane block is found. Moreover, the energy absorption efficiency of foam filled sandwich panels with higher relative density (2.58% and 3.17%) lattice cores is lower than that of the unfilled specimens when the compressive strain is small, while it exhibits superior when the compressive strain arrives at about 0.25, and the superiority enlarges as the strain increase. However, the energy absorption of foam filled sandwich panels owning lower relative density (1.83%) lattice cores is inferior to that of the unfilled specimens. During the low velocity impact tests, it is found that the contact duration between the impactor and the sandwich specimens is shorter and the impact peak load has a slight increase for the foam filled specimens.
Article
Although many researches have been attracted to optimization problems of composite sandwich structures, there are rarely special literatures for sensitivity analysis which provides essential gradient information for the optimization. In this paper the linear statics and free vibration sensitivity analysis problems of the composite sandwich plates are studied based on a layerwise/solid-element method (LW/SE) which was developed in our previous work to eliminate or decrease the error induced by the equivalent methods of the core. In the present sensitivity analysis schemes the cores of the sandwich plates are discretized by three models, namely, full model, local model and equivalent model. In the numerical examples, two kinds of sensitivity analysis schemes, the overall finite difference method (OFD) and the semi-analytical method (SAM), are employed to calculate the sensitivity coefficients of displacements, stresses and natural frequencies. The convergence is studied together with the effect of step size on the relative error. The performance of these three methods of modeling the honeycomb in computing displacements and natural frequencies sensitivity coefficients is investigated. At last, the influences of the parameters on the displacements, stresses and natural frequencies are investigated by using the sensitivity analysis scheme based on the local model and SAM.