Second-order Perturbation Technique Research Articles

Effect of random variation in input parameter on cracked orthotropic plate using extended isogeometric analysis (XIGA) under thermomechanical loading

Abstract This research introduces the Stochastic Extended Isogeometric Analysis (XIGA) method to investigate fracture behavior of isotropic and orthotropic materials under mechanical, thermal, and thermomechanical loads. Employing knot spans from Isogeometric Analysis (IGA) for domain discretization, the study utilizes identical basis functions for geometry construction and solution discretization. Utilizing Extended Finite Element Method (XFEM) enrichment functions, accurate crack face displacement discontinuity and tip singularity within the stress field are characterized. Additionally, employing a second-order perturbation technique within XIGA framework, the research derives mean and coefficient of variance values for mixed-mode Stress Intensity Factors (SIF). Stochastic variations in material elastic properties, crack length, and crack angle are considered in this computation. Credibility and robustness of the study are confirmed through comparative analyses against available literatures and Monte Carlo Simulations (MCS). The observed exceptional agreement validates the precision and reliability of the proposed stochastic XIGA method for fracture analysis in orthotropic material systems under thermomechanical loading conditions.

Effective correlation analysis algorithms for uncertain structures based on multidimensional parallelepiped model

The correlation analysis of the uncertain structures with multi-responses is carried out based on the multidimensional parallelepiped model in this research. Firstly, the multidimensional parallelepiped model is introduced to quantify the uncertainty and correlation of input parameters. Then, to deal with the large uncertainty of input parameters, the sub-parallelepiped perturbation analysis algorithm is designed to calculate the marginal intervals of output responses. Next, based on the Monte Carlo simulation and the second-order perturbation technique, the Monte Carlo correlation analysis algorithm and the second-order perturbation correlation analysis algorithm are respectively designed to obtain the correlation coefficients of output responses. Subsequently, two uncertainty domain analysis algorithms based on the marginal intervals analysis and the correlation coefficients analysis, are presented to establish the uncertainty domains of output responses. Finally, the effectiveness of proposed algorithms is verified by three numerical examples.

Instability and Parametric Amplification of a Piezoelectric Energy Harvester Periodically Plucked by a Rotating Magnet

AbstractMagnetic plucking is an enabling technique to harvest energy from a rotary host as it converts the low-frequency excitation of rotational energy sources to high-frequency excitation that leads to resonance of small-scale piezoelectric energy harvesters. Traditional nonlinear analysis of the plucking phenomenon has relied on numerical integration methods. In this work, a semi-analytical method is developed to investigate the stability and bifurcation behaviors of rotary magnetic plucking, which integrates a second-order perturbation technique and discrete Fourier transform. Analysis through this method unfolds that the oscillatory response of the beam can lose stability through the saddle-node bifurcation and Hopf bifurcation, which eventually causes the beam to collide with the rotary host. Further, the influence of the magnetic gap and rotational speed on the stability is discussed. The study also reveals that the nonlinearity of the magnetic force can amplify the electrical power at primary resonance. As a result, the traditional impedance matching approach that neglects the nonlinearity of the magnetic force fails to predict the optimal electrical resistance. Finally, a finite element analysis shows that the instability is sensitive to damping, and the traditional single-mode approximation can lead to considerable error.

Probabilistic Progressive Failure of Multiwall Carbon Nanotube-Reinforced Composite Plate Under Transverse Patch Loading

Second-order statistics of progressive failure response of multiwall carbon nanotube-reinforced composite (MWCNTRC) plate subjected to the transverse loading is presented. The mathematical model is based on the higher-order shear deformation theory (HSDT) and von Karman nonlinear kinematics through the mechanics of the material approach. Second-order perturbation technique (SOPT) and polynomial chaos (PC) methods are used for the uncertainty quantification. The properties of the MWCNTRC plate are obtained by using the Halpin–Tsai model. Epoxy-matrix reinforced with multiwall carbon nanotube (MWCNT) is considered as a new matrix and boron fiber is reinforced with this new matrix in an orthotropic way to get MWCNTRC. The distribution of volume fraction of carbon nanotube (CNT) in the epoxy matrix is considered in the functionally graded (FG) manner for thermo-mechanical loading. Stochastic progressive failure analysis (PFA) along with the probability of failure of MWCNTRC plate is performed by using the progressive damage model incorporated in the ply discount method and considering randomness in the material properties, the geometry of plate and loading. The conclusions are based on the effect of the parameters like stacking sequence, fiber volume fraction, temperature, CNT distribution, fiber material and number of CNT on the mean of first and last ply failure loads (FPFL and LPFL) and coefficient of variance (COV) of MWCNTRC plate. Stochastic failure analysis of MWCNTRC plate up to last ply failure by considering patch loading, nonlinearity and FG distribution of CNT is unique in this work.

A Novel Second-Order Perturbation Technique for Improving the Signal Processing in MIMO-NOMA Based Cognitive Radio Networks

A Novel Second-Order Perturbation Technique for Improving the Signal Processing in MIMO-NOMA Based Cognitive Radio Networks

Stochastic MMSIF of multiple edge cracks FGMs plates subjected to combined loading using XFEM

Abstract The second order statistics of multiple edge crack functionally graded materials (FGMs) under tensile, shear and combined loading assuming uncertain system parameters is presented in this paper. The uncertain parameters used under the present study are the material properties, and crack parameters such as crack length and crack angle. In this present analysis extended finite element method (XFEM) is used. The stochastic analysis is carried out using second order perturbation technique (SOPT) for the evaluation of mean and coefficient of variance (COV) of mixed mode stress intensity factor (MMSIF).

Stochastic mixed mode stress intensity factor of center cracks FGM plates using XFEM

Extended finite element method (XFEM) and second-order perturbation technique (SOPT) were combinedly utilized using interaction integral (M-integral) through partition of unity method to find out the mean and variance of mixed mode stress intensity factor (MMSIF). Uncertain system parameters are considered in material properties, crack length, crack orientation, gradient coefficients in the present study. MMSIF in a numerical example with center crack is computed to validate the accuracy of the presented model. Finally, typical numerical results are presented to examine the different modulus ratios, crack angle, crack length, position of crack and tensile, shear and combined loadings with uncertain system properties on the MMSIF.

Analytic computation of the secular effects of encounters on a binary: features arising from second-order perturbation theory

AbstractBinary–single interactions play a crucial role in the evolution of dense stellar systems such as globular clusters. In addition, they are believed to drive black hole (BH) binary mergers in these systems. A subset of binary–single interactions are secular encounters, for which the third body approaches the binary on a relatively wide orbit, and such that it is justified to average the equations of motion over the binary’s orbital phase. Previous works used first-order (FO) perturbation theory to compute the effects of such secular encounters on the binary. However, this approach can break down for highly eccentric binaries, which are important for BH binary mergers and gravitational wave sources. Here, we present an analytic computation using second-order perturbation techniques, valid to the quadrupole-order approximation. In our calculation, we take into account the instantaneous back reaction of the binary to the third body, and compute corrections to previous FO results. Using singly averaged and direct three-body integrations, we demonstrate the validity of our expressions. In particular, we show that the eccentricity change for highly eccentric binaries can reach a plateau, associated with a large inclination change, and can even reverse sign. These effects are not captured by previous FO results. We provide a simple script to conveniently evaluate our analytic expressions, including routines for numerical integration and verification.

Stochastic XFEM based fracture behavior and crack growth analysis of a plate with a hole emanating cracks under biaxial loading

In the paper, the stochastic fracture response of an isotropic finite plate with a circular hole emanating radial cracks, having different angles, under biaxial loading with different biaxiality ratio (β=0,1 and −1) is examined. In this direction, the mean values and corresponding coefficient of variations (COVs) of mixed mode stress intensity factor (MMSIFs) using the second order perturbation technique (SOPT) combined with a finite difference method through extended finite element method (XFEM) are evaluated in MATLAB environment. The random system parameters, e.g., normalized crack length, crack angle, the normalized radius of the hole, are assumed as uncorrelated input random variables. The effects, of these input parameters and biaxial load factors, on the statistics of MMSIFs with input individual and combined random system parameters, are analyzed. Also, the stochastic fracture behavior is predicted with the increment in the radius of the hole and with the change in the position of the discontinuities (a hole emanating cracks) in the plate using uncorrelated combined randomness with different input variables. Further, the static crack propagation behavior of the plate with hole emanating cracks under biaxial loading with different biaxiality ratio is analyzed.

Stochastic bending characteristics of finite element modeled Nano-composite plates

This study reported, the effect of random variation in system properties on bending response of single wall carbon nanotube reinforced composite (SWCNTRC) plates subjected to transverse uniform loading is examined. System parameters such as the SWCNT armchair, material properties, plate thickness and volume fraction of SWCNT are modelled as basic random variables. The basic formulation is based on higher order shear deformation theory to model the system behavior of the SWCNTRC composite plate. A C0 finite element method in conjunction with the first order perturbation technique procedure developed earlier by the authors for the plate subjected to lateral loading is employed to obtain the mean and variance of the transverse deflection of the plate. The performance of the stochastic SWCNTRC composite model is demonstrated through a comparison of mean transverse central deflection with those results available in the literature and standard deviation of the deflection with an independent First Order perturbation Technique (FOPT), Second Order perturbation Technique (SOPT) and Monte Carlo simulation.

A note on anomalous band-gap variations in semiconductors with temperature

An attempt is made to theoretically study the band-gap variations (∆Eg) in semiconductors with temperature following the works, did by Fan and O’Donnell et al. based on thermodynamic functions. The semiconductor band-gap reflects the bonding energy. An increase in temperature changes the chemical bondings, and electrons are promoted from valence band to conduction band. In their analyses, they made several approximations with respect to temperature and other fitting parameters leading to real values of band-gap variations with linear temperature dependences. In the present communication, we have tried to re-analyse the works, specially did by Fan, and derived an analytical model for ∆Eg(T). Because, it was based on the second-order perturbation technique of thermodynamic functions. Our analyses are made without any approximations with respect to temperatures and other fitting parameters mentioned in the text, leading to a complex functions followed by an oscillating nature of the variations of ∆Eg. In support of the existence of the oscillating energy band-gap variations with temperature in a semiconductor, possible physical explanations are provided to justify the experimental observation for various materials.

Stochastic nonlinear bending response of elastically supported nanotube-reinforced composite beam in thermal environment

This paper presents the second-order statistics of the nonlinear bending response of elastically supported single-wall carbon nanotube reinforced composite (CNTRC) beam composed of uniformly distributed (UD) and functionally graded (FG) reinforcements in thermal environments with uncertain system properties. The uncertain system properties such as material properties of matrix and SWCNTs, foundation parameters, thermal expansion coefficients are be modeled as uncorrelated Gaussian random variables. The material properties of FG-CNTRCs are assumed to be graded in the beam thickness direction and are evaluated through a micromechanical model. The higher order shear deformation theories (HSDT) with von-Karman nonlinear strain kinematics are used for the mathematical formulation of CNTRCs beam. The thermal effects are also included in the material properties of CNTRCs which are assumed to be temperature dependent and independent. The second-order perturbation technique (SOPT) and Monte Carlo simulation (MCS) via [Formula: see text] nonlinear finite element method (FEM) through Newton–Raphason method are proposed to examine the mean, COV and probability density function (PDF) of transverse deflection of the beam. Typical numerical results are presented for the different volume fraction of carbon nanotube, slenderness ratios, boundary conditions, foundation parameters, load parameters, CNTRC distribution, temperature dependent and independent material properties with random system properties. The present outlined approach is validated with the results available in the literature and by employing MCS.

Stochastic Fracture Response and Crack Growth Analysis of Laminated Composite Edge Crack Beams Using Extended Finite Element Method

This paper presents a stochastic fracture response and crack growth analysis of mixed-mode stress intensity factors (MSIFs) for edge cracked laminated composite beams subjected to uniaxial, uniform tensile, shear and combined stresses with random system properties. The randomness in material properties of the composite material, lamination angle, laminate thickness, the crack length and the crack angle are modeled as both input uncorrelated and correlated random variables. An extended finite element method (XFEM) through the so-called M-interaction approach combined with the second-order perturbation technique (SOPT) and Monte Carlo simulation (MCS) is used to obtain the statistics in terms of the mean and coefficient of variation (COV) of MSIFs for edge cracked laminated composite beams. The effect of crack propagation on the MSIFs in the presence of tensile, shear and combined stresses using a global tracking algorithm is also investigated. The results using the present approach are compared with the available published results. A good agreement is seen whenever alternative results are available.

Nonlinear free vibration analysis of elastically supported carbon nanotube-reinforced composite beam with the thermal environment in non-deterministic framework

AbstractThis paper deals with the investigation of nonlinear free vibration behavior of elastically supported carbon nanotube reinforced composite (CNTRC) beam subjected to thermal loading with random system properties. Material properties of each constituent’s material, volume fraction exponent and foundation parameters are considered as uncorrelated Gaussian random input variables. The beam is supported by a Pasternak foundation with Winkler cubic nonlinearity. The higher order shear deformation theory (HSDT) with von-Karman non-linearity is used to formulate the governing equation using Hamilton principle. Convergence and validation study is carried out through the comparison with the available results in the literature for authenticity and accuracy of the present approach used in the analysis. First order perturbation technique (FOPT),Second order perturbation technique (SOPT) and Monte Carlo simulation (MCS) methods are employed to investigate the effect of geometric configuration, volume fraction exponent, foundation parameters, distribution of reinforcement and thermal loading on nonlinear vibration characteristics CNTRC beam.The present work signifies the accurate analysis of vibrational behaviour influences by different random variables. Results are presented in terms of mean, variance (COV) and probability density function (PDF) for various aforementioned parameters.

Finite element based nonlinear dynamic response of elastically supported piezoelectric functionally graded beam subjected to moving load in thermal environment with random system properties

The second order statistics in terms of mean and standard deviation (SD) of normalized nonlinear transverse dynamic central deflection (NTDCD) response of un-damped elastically supported functionally graded materials (FGMs) beam with surface-bonded piezoelectric layers under the action of moving load are investigated in this paper. The random system properties such as Young's modulus, Poisson's ratio, density, thermal expansion coefficients, piezoelectric materials, volume fraction exponent and external loading are modeled as uncorrelated random variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics combined with Newton–Raphson technique through Newmark's time integrating scheme using finite element method (FEM). The non-uniform temperature distribution with temperature dependent material properties is taken into consideration for consideration of thermal loading. The one parameter Pasternak elastic foundation with Winkler cubic nonlinearity is considered as an elastic foundation. The stochastic based second order perturbation technique (SOPT) and direct Monte Carlo simulation (MCS) are adopted for the solution of nonlinear dynamic governing equation. The influences of volume fraction exponents, temperature increments, moving loads and velocity, nonlinearity, slenderness ratios, foundation parameters and external loadings with random system properties on the NTDCD are examined. The capability of present stochastic model in predicting the NTDCD statistics are compared by studying their convergence with the existing results those available in the literature.

Stochastic hygro-thermo-mechanically induced nonlinear static analysis of piezoelectric elastically support sandwich plate using secant function based shear deformation theory (SFSDT)

The second order statistics of transverse nonlinear central deflection of elastically supported piezoelectric laminated composite sandwich plate (ESPLCSP) subjected to hygro-thermo-mechanical loading using micromechanical approach is evaluated in this paper. System randomness as micro-level material properties of fiber and matrix, material properties of piezoelectric, laminate thickness, lamination angle, foundation parameters, and load intensity are taken as independent random variables. The mechanical loading is taken as uniformly distributed and sinusoidal loadings. The secant function based shear deformation theory (SFSDT) with von-Karman nonlinearity is used for basic formulation. The elastic and hygrothermal properties of the composite material are considered to be dependent on temperature and moisture concentration have been evaluated utilized micromechanical modeling. A Newton–Raphson method based on [Formula: see text] nonlinear finite element method combined with mean centered second order perturbation technique (SOPT) proposed by present authors for the composite plate is extended for sandwich composite plate. The effect of random system properties with changing the plate geometry, stacking sequences, support conditions, foundation parameters, piezoelectric layers, fiber volume fraction and temperature, and moisture distribution on ESPLCSP is presented. The performance of proposed approach is validated through comparison with those available in the literature and independent Monte Carlo simulation (MCS).

Stochastic extended finite element implementation for fracture analysis of laminated composite plate with a central crack

A stochastic extended finite element method (SXFEM), developed previously by the first two authors, has been extended for the fracture analysis along with reliability analysis of the central cracked laminated composite plate subjected to uni-axial tension with random system properties. The implemented SXFEM approach is based on the M-integral interaction combined with second-order perturbation technique (SOPT) and independent Monte Carlo simulation (MCS) is performed for the evaluation of statistics of mixed mode stress intensity factors (MMSIFs). The random system properties such as material properties, crack length, crack angle, lamination angle, and uni-axial loading, are assumed as input uncorrelated Gaussian random variables. The effect of the different crack angles, crack lengths, lamination angles, and loading on the statistics of MMSIF in terms of mean, standard deviation (SD), probability density function (PDF) and safety factor of cracked laminated composite plate is examined along with the reliability analysis. The effect of crack propagation and its direction along with its effect on the MMSIFs is carried out through global tracking crack growth algorithm. The results obtained by present approach, are compared with an analytical solution and results available in various literature.

Accurate stochastic initial and final failure of laminated plates subjected to hygro-thermo-mechanical loadings using Puck's failure criteria

The second order statistics of first-ply and last-ply failure (FPF and LPF) response of laminated composite plate with random input parameters such as constituent material properties, plate thickness, volume fiber fraction and temperature difference, subjected to hygro-thermo-mechanical loading are investigated in this paper. Stochastic finite element method (SFEM) based on higher order shear deformation theory (HSDT) with material non-linearity via second order perturbation technique (SOPT) is formulated for computing mean and coefficient of variation (COV) of FPF and LPF using Puck’s failure criteria. Failure strength of each ply and corresponding failure mode is predicted by using micromechanics based modeling approach. The effect of different environmental conditions, volume fiber fraction, lamination schemes, lamination angle, side to thickness ratio, different bi-axial loading on FPF and LPF response is investigated. The capability of the present SFEM in predicting FPF and LPF load statistics has been studied, by comparing the results obtained with the independent Monte Carlo simulation (MCS) technique. The predicted hygro-thermo-mechanical strengths of the several laminates are compared with the available experimental data and it shows good agreement.

Stochastic progressive failure analysis of laminated composite plates using Puck's failure criteria

ABSTRACTThe present work deals with second-order statistics of the progressive failure response of laminated composite plates subjected to in-plane uniaxial and bi-axial loadings with random system properties. A stochastic finite element method based on higher-order shear deformation theory combined with first- and second-order perturbation technique is used for solution of random progressive failure equation. A Puck failure criterion is used for the evaluation of first ply and last-ply failure load. The results obtained using the present solution approach are validated with the results available in the literature and Monte Carlo simulation.

Probabilistic fracture investigation of symmetric angle ply laminated composite plates using displacement correlation method

AbstractThe second order statistics of mixed mode stress intensity factors (MSIF) of single edge V-notched angle ply laminated composite plate subjected to uniaxial tensile load with uncertinity in the system properties using displacement correlation method (DCM) is evaluated. The random system properties such as material properties, crack opening and crack length are modelled as combined uncorrelated and correlated random system variables. A C0 finite element method (FEM) based on higher order shear deformation plate theory (HSDT) is used for basic formulation. The Taylor series based first order perturbation technique (FOPT), second order perturbation technique (SOPT) are used and direct Monte Carlo simulation (MCS) is performed to evaluate the statistics (mean and coefficient of variance) of the mixed mode SIFs. The present work signifies the accurate analysis of frature behaviour by influence of different random variables and fibre orientations on the fracture behaviour in angle ply laminates.