Higher-order Deformation Research Articles

A layerwise finite element formulation for vibration and damping analysis of sandwich plate with moderately thick viscoelastic core

Most previous studies of viscoelastic sandwich plates were based on the assumption that damping was only resulting from shear deformation in the viscoelastic core. However, extensive and compressive deformations in the viscoelastic core should also be considered especially for sandwich plates with moderately thick viscoelastic core. This paper presents a finite element formulation for vibration and damping analysis of sandwich plates with moderately thick viscoelastic core based on a mixed layerwise theory. The face layers satisfy the Kirchhoff theory while the viscoelastic core meets a general high-order deformation theory. The viscoelastic core is modeled as a quasi-three-dimensional solid where other types of deformation such as longitudinal extension and transverse compression are also considered. To better describe the distribution of the displacement fields, auxiliary points located across the depth of the sandwich plate are introduced. And based on the auxiliary points, the longitudinal and transverse displacements of the viscoelastic core are interpolated independently by Lagrange interpolation functions. Quadrilateral finite elements are developed and dynamic equations are derived by Hamilton’s principle in the variation form. Allowing for the frequency-dependent characteristics of the viscoelastic material, an iterative procedure is introduced to solve the nonlinear eigenvalue problem. The comparison with results in the open literature validates the remarkable accuracy of the present model for sandwich plates with moderately thick viscoelastic core, and demonstrates the importance of the higher-order variation of the transverse displacement along the thickness of the viscoelastic core for the improvement of the analysis accuracy. Moreover, the influence of the thickness and stiffness ratios of the viscoelastic core to the face layers on the damping characteristics of the viscoelastic sandwich plate is discussed.

Optimization of Laminates for Maximum Buckling Load

Objective of this work is to optimize the structural configuration of laminates in terms of design for the maximization of buckling resistance of laminates under the effect of constraints. The predominant design parameter is the orientation of the fibers in the layers of the laminate. High-order deformation theory is used in the mathematical formulation in the buckling analysis of stratified composite plates. For this purpose, we will use the optimization technique genetic algorithm taking into account the various factors: number of fibers, geometric ratio, load ratio and boundary conditions.

High-order crack propagation in compressed sandwich panels

The response and the debonding mechanisms in axially compressed sandwich panels with an interfacial delamination are investigated using a nonlinear model. The mathematical model combines the extended high-order sandwich panel theory with a cohesive interface modeling. It includes the first-order shear deformation kinematic assumptions for the face sheets and high-order small deformations kinematic assumptions that account for out-of-plane compressibility for the core. The interfaces bond the face sheets and the core by means of traction–displacement gap laws. These interfacial laws can describe a diversity of physical conditions. In particular, interfacial debonding nucleation and propagation are described using cohesive laws that introduce the interfacial nonlinearity into the model. Geometrical nonlinearity of the face sheets is introduced in order to capture the instability associated with the buckling of the delaminated face sheet. The cohesive interfaces and others parameters are calibrated to match experimental results taken from the literature for a sandwich specimen subjected to an end-shortening compression. The instabilities due to the in-plane compression, together with the existence of delaminated regions and their tendency to grow, prompt buckling of the delaminated face sheet as well as nucleation and propagation of the interfacial debonding. The theoretical quantification of this complex mechanism compares well with the experimental results in terms of the physical response, the nucleation and propagation of the interfacial crack, and the evolution of local/global geometrical instabilities. In addition, the analysis explores debonding mechanisms that are beyond the capabilities of the experimental technique. Finally, the sensitivity of the response and the associated geometrical and interfacial instabilities to the boundary conditions are investigated.

On the Identification of Sectional Deformation Modes of Thin-Walled Structures with Doubly Symmetric Cross-Sections Based on the Shell-Like Deformation

In this paper, a new approach is proposed to identify sectional deformation modes of the doubly symmetric thin-walled cross-section, which are to be employed in formulating a one-dimensional model of thin-walled structures. The approach considers the three-dimensional displacement field of the structure as the linear superposition of a set of sectional deformation modes. To retrieve these modes, the modal analysis of a thin-walled structure is carried out based on shell/plate theory, with the shell-like deformation shapes extracted. The components of classical modes are removed from these shapes based on a novel criterion, with residual deformation shapes left. By introducing benchmark points, these shapes are further classified into several deformation patterns, and within each pattern, higher-order deformation modes are derived by removing the components of identified ones. Considering the doubly symmetric cross-section, these modes are approximated with shape functions applying the interpolation method. The identified modes are finally used to deduce the governing equations of the thin-walled structure, applying Hamilton’s principle. Numerical examples are also presented to validate the accuracy and efficiency of the new model in reproducing three-dimensional behaviors of thin-walled structures.

Insight into nuclear midshell structures by studying K isomers in rare-earth neutron-rich nuclei

Inspired by the newly discovered isomeric states in the rare-earth neutron-rich nuclei, high-$K$ isomeric states in neutron-rich samarium and gadolinium isotopes are investigated within the framework of the cranked shell model (CSM) with pairing correlation treated by a particle-number-conserving (PNC) method. The experimental multi-particle state energies and moments of inertia are reproduced quite well by the PNC-CSM calculations. A remarkable effect from the high-order deformation $\varepsilon_{6}$ is demonstrated. Based on the occupation probabilities, the configurations are assigned to the observed high-$K$ isomeric states. The lower $5^-$ isomeric state in $^{158}$Sm is preferred as the two-proton state with configuration $\pi\frac{5}{2}^{+}[413]\otimes\pi\frac{5}{2}^{-}[532]$. More low-lying two-particle states are predicted. The systematics of the electronic quadrupole transition probabilities, $B(E2)$ values along the neodymium, samarium, gadolinium and dysprosium isotopes and $N=96,98,100,102$ isotones chains is investigated to reveal the midshell collectivities.

Systematic investigation of the high- K isomers and the high-spin rotational bands in the neutron-rich Nd and Sm isotopes by a particle-number conserving method

The rotational properties of the neutron rich Nd and Sm isotopes with mass number $A\approx150$ are systematically investigated using the cranked shell model with pairing correlations treated by a particle-number conserving method, in which the Pauli blocking effects are taken into account exactly. The 2-quasiparticle states in even-even Nd and Sm isotopes with excitation energies lower than 2.5~MeV are systematically calculated. The available data can be well reproduced and some possible 2 and 4-quasiparticle isomers are also suggested for future experiments. The experimentally observed rotational frequency variations of moments of inertia for the even-even and odd-$A$ nuclei are reproduced very well by the calculations. The effects of high-order deformation $\varepsilon_6$ on the 2-quasiparticle excitation energies and moments of inertia of the ground state bands in even-even Nd and Sm isotopes are analyzed in detail. By analyzing the occupation probability $n_\mu$ of each cranked Nilsson orbitals near the Fermi surface and the contribution of each major shell to the angular momentum alignments, the alignment mechanism in these nuclei is understood clearly.

Nonlinear metric learning for kNN and SVMs through geometric transformations

In recent years, research efforts to extend linear metric learning models to handle nonlinear structures have attracted great interests. In this paper, we propose a novel nonlinear solution through the utilization of deformable geometric models to learn spatially varying metrics, and apply the strategy to boost the performance of both kNN and SVM classifiers. Thin-plate splines (TPS) are chosen as the geometric model due to their remarkable versatility and representation power in accounting for high-order deformations. By transforming the input space through TPS, we can pull same-class neighbors closer while pushing different-class points farther away in kNN, as well as make the input data points more linearly separable in SVMs. Improvements in the performance of kNN classification are demonstrated through experiments on synthetic and real world datasets, with comparisons made with several state-of-the-art metric learning solutions. Our SVM-based models also achieve significant improvements over traditional linear and kernel SVMs with the same datasets.

Nonlinear model of human descending thoracic aortic segments with residual stresses.

The nonlinear static deformation of human descending thoracic aortic segments is investigated. The aorta segments are modeled as straight axisymmetric circular cylindrical shells with three hyperelastic anisotropic layers and residual stresses by using an advanced nonlinear shell theory with higher-order thickness deformation not available in commercial finite element codes. The residual stresses are evaluated in the closed configuration in an original way making use of the multiplicative decomposition. The model was initially validated through comparison with published numerical and experimental data for artery and aorta segments. Then, two different cases of healthy thoracic descending aorta segments were numerically simulated. Material data and residual stresses used in the models came from published layer-specific experiments for human aortas. The material model adopted in the study is the mechanically based Gasser-Ogden-Holzapfel, which takes into account collagen fiber dispersion. Numerical results present a difference between systolic and diastolic inner radii close to the data available in literature from in vivo measurements for the corresponding age groups. Constant length of the aortic segment between systolic and diastolic pressures was obtained for the material model that takes the dispersion of the fiber orientations into account.

Static and Dynamic Response of FG-CNT-Reinforced Rhombic Laminates

The present study focuses on the static and dynamic response of functionally graded carbon nanotube (FG-CNT)-reinforced rhombic laminates. The cubic variation of thickness coordinate in the displacement field is considered in terms of Taylor’s series expansion, which represents the higher-order transverse cross-sectional deformation modes. The condition of zero-transverse shear strain at upper and lower surface of FG-CNT-reinforced rhombic laminates is imposed in the present formulation. The present two-dimensional model is formulated in a finite element, with the C0 element consisting of seven nodal unknowns per node. The final material properties of FG-CNT-reinforced rhombic laminates are estimated using the rule of mixture. The obtained numerical are compared with the results available in the literature to verify the reliability of the present model. The present study investigates the effect of CNT distribution, loading pattern, volume fraction, and various combinations of boundary constraints by developing a finite element code in FORTRAN.

On the nonlinear extension-twist coupling of beams

Abstract In many applications, beams carry large axial forces, such as centrifugal forces, for instance, and the resulting axial stresses affect their torsional behavior. To capture this important coupling effect within the framework of beam models, a nonlinear theory must be developed that takes into account higher-order strain deformations. The proposed development starts with a three-dimensional elasticity model of beam-like structures. While strain components are assumed to remain small, the strain energy expression is expanded to retain strain components up to cubic order. This expansion of the three-dimensional model leads to material and geometric stiffness matrices. For linear problems, the three-dimensional warping fields induced by each of the six unit sectional strain components can be obtained. These warping fields are used to reduce the three-dimensional elasticity model to one-dimensional beam equations. Byproducts of this reduction process include the sectional stiffness matrix and six nonlinear stiffness matrices representing the geometric stiffness induced by unit sectional strains. It is shown that the nonlinear extension-twist coupling is captured by these geometric stiffness matrices. Comparison of the predictions of the proposed approach with analytical and experimental results demonstrate their effectiveness and accuracy.

Simulating stress wave with flat-top partition of unity based high-order discontinuous deformation analysis

Modelling the stress wave propagation in rock mass is a critical step to accurately assess the stability and damage of underground rock structures under dynamic loading. In this paper, the recently developed flat-top partition of unity (PU) based high-order approximation is coupled with Discontinuous Deformation Analysis (DDA) method to simulate stress wave propagation. In this method, the continuous blocks in DDA method are discretized by flat-top PU mesh, which is generated independently of the problem domain. High-order polynomials are employed as local displacement approximation to improve the numerical accuracy. Furthermore, non-reflective boundary is incorporated into this method to model the infinite problem domain. Numerical examples are presented and analyzed to verify the accuracy and efficiency of this method.

A Ritz type solution with exponential trial functions for laminated composite beams based on the modified couple stress theory

This paper proposes novel Ritz functions for the size-dependent analysis of micro laminated composite beams with arbitrary lay-ups. Displacement field is based on a higher-order deformation beam theory and size effect is captured by the modified couple stress theory. Lagrange’s equations are used to obtain the governing equations of motion. The present beam model, which can recover the classical one by neglecting the material length scale parameter, is used to predict the size-dependent responses of micro composite beams. The results indicate that the present study is efficient for bending, vibration and buckling problems of micro composite beams. Some new results are given to serve as benchmarks for future studies.

Higher-Order Moving Mesh Methods for PDE-Constrained Shape Optimization

We present a new approach to discretizing shape optimization problems that generalizes standard moving mesh methods to higher-order mesh deformations and that is naturally compatible with higher-order finite element discretizations of PDE-constraints. This shape optimization method is based on discretized deformation diffeomorphisms and allows for arbitrarily high resolution of shapes with arbitrary smoothness. Numerical experiments show that this method allows the solution of PDE-constrained shape optimization problems to high accuracy.

High-K Isomers in Light Superheavy Nuclei by PNC-CSM method

The high-K isomeric states in light superheavy nuclei around A = 250 mass region are investigated by the Cranked Shell Model (CSM) with pairing treated by a Particle-Number Conserving (PNC) method. With including the higher-order deformation ε6, both of the high–K multi-particle state energies and the rotational bands in 254No and N = 150 isotone are reproduced well. The isomeric state energies and the microscopic mechanism of kinematic moment of inertia variations versus rotational frequency are discussed. The irregularity of the two-neutron Kπ = 8- state band at ħω ≈ 0:17 in 252No is caused by the configuration mixing with the two-proton Kπ = 8- band. .

Thermal effect on the dynamic behavior of nanobeam resonator assuming size-dependent higher-order strain gradient theory

The temperature change and size scale dependent effects on the dynamical behaviors of the Carbon nanotube (CNT)-based nano-resonator are investigated in this work. The equation of motion of the CNT-based nano-resonator incorporating the higher-order strain gradient deformation, the geometric nonlinearity due to von Karman nonlinear strain, and the thermal effects is derived. A Galerkin based modal decomposition is used to investigate both the free vibration and forced vibration of the nano-resonator. The temperature variation effect is analyzed by assuming both low and high temperature environments. The results show that when assuming a high temperature in the post buckling regime, the nano-resonator natural frequencies dispersion graph shows possibilities of modes-veering and modes-crossing variation due to the initial buckling of the nano-resonator. From the forced-vibration analysis, the results demonstrate that taking into consideration the higher-order strain gradient deformation in modeling the CNT based nano-resonator completely changes the frequency response of the CNT-based nanoresonator.

A generalized micromorphic approach accounting for variation and dispersion of preferred material directions

Materials exhibiting a heterogeneous and non-uniform composition in terms of elastic and anisotropic properties such as biological tissues require special efforts to accurately describe their constitutive behavior. In contrast to classical models, micromorphic formulations can predict the macroscopically observable material response as originated from distinct scale-dependent micro-structural deformation mechanisms. This is facilitated by additional independent degrees of freedom and associated additional strain and stress quantities. Here, a generalized continuum is mathematically constructed from a macro-continuum and a micro-continuum which are both adequately coupled on kinematics and constitutive levels as well as by micro-boundary conditions. In view of biomechanical modeling, the potential of the formulation is studied for a number of academic examples characterized by an anisotropic material composition to elucidate the micromorphic material response as compared with the one obtained using a classical continuum mechanics approach. The results demonstrate the ability of the generalized continuum approach to address non-affine elastic reorientation of the preferred material direction in the macro-space and its dispersion in the micro-space as affecting deformation, strain and stress on the macroscopic level. In particular, if the anisotropy in the micromorphic formulation is solely linked to the extra degrees of freedom and associated strain and stress measures, the deformation for small and large strains is shown to be distinctly different to the classical response. Together with the ability to implicitly account for scale-dependent higher-order deformation effects in the constitutive law the proposed generalized micromorphic formulation provides an advanced description, especially for fibrous biological materials.

Fusion dynamics of stable and weakly bound colliding systems

This work systematically analyzed the fusion dynamics of the projectile-target combinations involving stable and loosely bound systems within the view of the energy-dependent Woods–Saxon potential model (EDWSP model) and the coupled channel approach. The different projectiles are bombarded onto series of Sm-isotopes, which possess the dominance of the different kinds of the nuclear structure degrees of freedom and with the increase of the neutron richness, the Sm-isotopes gradually shift from spherical shape to a statically deformed shape. In the fusion of [Formula: see text] reaction, the impacts of vibrational degrees of freedom of the colliding nuclei are dominant while in the case of [Formula: see text] systems, the rotational states of the deformed target isotopes have a strong impression on the below-barrier fusion data. The heavier target isotopes ([Formula: see text] also exhibit the higher order deformation such as [Formula: see text], [Formula: see text]-deformation parameter in its ground state and couplings to such channels must be incorporated in theoretical calculations in order to achieve close agreement with the sub-barrier fusion data. However, in the case of the loosely bound systems, the projectile breakup channel significantly affects the fusion excitation functions in the domain of the Coulomb barrier. To ensure the role of the projectile breakup channel, the fusion of the different loosely bound projectiles ([Formula: see text] and [Formula: see text] with Sm-isotopes are investigated, wherein the above-barrier fusion data of these reactions are suppressed with reference to the coupled channel calculations. This hindrance is the result of the projectile breakup effects that occur as a consequence of the breakup of the projectile before reaching the fusion barrier due to its low binding energy. However, in the EDWSP model calculations the magnitude of the hindrance of the above-barrier fusion data of [Formula: see text] and [Formula: see text] reactions is reduced by a factor varying from 7% to 13% with respect to a value reported in the literature. In contrast to this, the sub-barrier fusion enhancement of [Formula: see text] and [Formula: see text] reactions is the result of the dominance of the nuclear structure degrees of freedom of the colliding systems.

Influence of sticking vs non-sticking limits of moment of inertia and higher order deformations in the decay of 214,216Rn⁎ compound systems

The dynamical cluster decay model (DCM) is employed to explore the relative effect of sticking (IS) and non-sticking (INS) limits of moment of inertia (MOI) in the decay of hot and rotating 214,216Rn⁎ compound nuclei, formed in 16,18O + 198Pt reactions. Beside this, the nuclear deformation effects i.e. quadrupole β2 (static and dynamic) and higher order static deformations up to hexadecapole (β4) are duly incorporated and studied within DCM. The influence of both ‘INS/IS’ addressing rotational energy component and ‘deformations’ is gauged through the barrier characteristics, preformation factor and barrier lowering effects. The experimentally given ER and ff data is addressed by optimizing the neck-length ΔR, that strongly depends on the limiting angular momentum, which in turn depends on the sticking or non-sticking limits of interaction. In addition to this, the influence of increase in energy and neutron number is probed in reference to ER survival probability of Rn compound nucleus. Finally, the ff cross-sections of 214,216Rn⁎ nuclei are predicted within sticking limit of moment of inertia as the same seems to be more suitable for such decay paths.

Size-dependent behavior of slacked carbon nanotube actuator based on the higher-order strain gradient theory

This paper proposes to investigate the nonlinear size dependent behavior of electrically actuated carbon nanotube (CNT) based nano-actuator while including the higher-order strain gradient deformation, the geometric nonlinearity due to the von Karman nonlinear strain as well as the slack effect, and the temperature gradient effects. The assumed non-classical beam model adopts some internal material size scale parameters related to the material nanostructures and is capable of interpreting the size effect that the classical continuum beam model is unable to pronounce. The higher-order governing equations of motion and boundary conditions are derived using the so-called extended Hamilton principle. A Galerkin based reduced-order model (ROM) modal decomposition is developed to prescribe the non-classical nanotube mode shape as well as its static behavior under any applied DC actuation load. Results of the static analysis is compared with those obtained by both classical elasticity continuum and strain gradient theories. A Jacobian method is utilized to determine the variation of the natural frequencies of the nanobeam with the DC load as well as the slack level. A thorough parametric study is conducted to study the influences of the size scale dependent parameters, the geometric nonlinearity, the initial curvature, the gate voltage, and the temperature gradient effect on structural behavior of the CNT-based nano-actuator. It is found that the size effect based on the strain gradient deformation has significant influence on the fundamental nanotube natural frequency dispersion. Also, varying this size effect have revealed the offering of numerous possibilities of modes veering and crossing, all shown to be dependent of the strain gradient parameters as well as the CNT slack level.

Higher-order thin-walled beam analysis for axially varying generally shaped cross sections with straight cross-section edges

A higher-order beam theory is proposed for the analysis of a thin-walled beam with a generally shaped cross section, which consists of straight cross-section edges and is non-uniform along the axial direction. To derive cross-sectional shape functions for the higher-order deformation modes, a new approach is introduced using a set of beam frame models. The distortions with inextensional cross-sectional walls are determined by solving an eigenvalue problem of a beam frame model under inextensional wall constraints. Subsequently, the distortions with extensional cross-sectional walls are evaluated by considering orthogonality with respect to the inextensional distortions. Moreover, the extensional distortions due to the Poisson effect, which is generated due to the uniform axial strain of the rigid-body cross-sectional deformations, are considered. Warpings induced by the inextensional and extensional distortions are consistently defined based on the orders of the tangential displacements of their corresponding distortions. To deal with the varying cross section, three-dimensional displacements at an arbitrary point are interpolated using those at the cross sections of the nodes, where the beam frame analyses are performed. The proposed method is validated by performing static and vibration analyses of beams with varying single- and multi-cell cross sections.