Continuum Shell Model Research Articles

A hybrid path-following approach for constrained nonlinear-buckling optimization of variable stiffness composite shells with shape imperfections

Nonlinear buckling optimization of variable stiffness (VS) composite shells is computationally expensive due to frequent nonlinear analyses. Although the generalized path-following method can serve to reduce the redundant computation, it is weakened by two kinds of discontinuity. First, the design variables may not always be searched continuously in the constrained optimization. Second, the critical buckling states may not be continuous at singular points on the search path. To address this issue, a hybrid path-following approach is proposed to trace the critical buckling states in the nonlinear-buckling optimization with constraints on the realizability of the lamination parameters and the amplitude range of geometric imperfections. Firstly, the optimal search direction is determined with the sensitivity analysis of the critical buckling equations in the isogeometric continuum shell model. Then, a continuous following strategy is adopted to provide a proper initial estimate for the Newton iterations at a disconnected point. Moreover, an investigation into the singularity of the critical buckling equations is conducted to identify the discontinuous points of the critical states in a line-search. Upon detection of the singularity, a segment of path in state space is inserted to recover the continuity. Numerical experiments show that the continuity of the critical states in the optimization is well maintained, which leads to a more efficient optimization for the VS shell with shape imperfections.

β −p and β−α decay of the 11Be neutron halo ground state

Beta-delayed proton emission from the neutron halo ground state of 11Be raised much attention due to the unusually high decay rate. It was argued that this may be due to the existence of a resonance just above the proton decay threshold. In this letter, we use the lenses of real-energy continuum shell model to describe several observables including the Gamow–Teller rates for the β −-delayed α and proton decays, and argue that, within our model, the large β −p branching ratio cannot be reconciled with other data.

A geometrically nonlinear variable-kinematics continuum shell element for the analyses of laminated composites

To facilitate further gains in structural efficiency, the use of composite materials in engineering structures is on the rise. Simultaneously, a drive for thinner components is leading to structural behaviour that is governed by elastic nonlinearities such as large deflections and instabilities. For efficient and reliable design, numerical models must predict the nonlinear displacements, as well as the corresponding stress and strain responses, both accurately and at minimal computational cost. In this work, we present a novel tensor-based variable kinematics continuum shell (VKCS) formulation that is geometrically nonlinear in a total Lagrangian sense. The key contribution is the development and validation of a nonlinear continuum shell model that is completely general in terms of its geometric and kinematic descriptions. The governing equations are derived and presented in tensorial form, which enables a straightforward spatial mapping for models with complex curvatures. The ‘variable-kinematics’ capability means that the element field variables can be refined in a hierarchical and orthotropic manner, i.e. the in-plane and through-thickness displacements can be independently discretised using any polynomial functions with arbitrary orders of expansion. With this feature, the model configurations can be tailored for specific nonlinear problems, whilst also achieving fast solution convergence rate through the use of higher-order basis functions. For validation, the VKCS model has been benchmarked against existing nonlinear problems in the literature that feature large displacements with complex equilibrium paths. In addition, we have proposed two new benchmarks to investigate the 3D Cauchy stress in a snapped shallow roof, and the postbuckling behaviour of a wind turbine blade section. The VKCS formulation is shown to be a versatile tool that allows the user to easily switch between a multitude of model configurations, and can thus accommodate the varying fidelity of analyses required across different design stages. Furthermore, our benchmarks have demonstrated that the variable-kinematics model requires fewer degrees of freedom and run time to track complex 3D stresses when compared to conventional low-order continuum elements. • A novel hierarchical finite element formulation for analysis of laminated structures. • Easily switch between model configurations required across different design stages. • Shell element with completely general geometric and kinematic discretisation. • 3D stress tracking in snapped roof and post-buckling analysis of wind turbine blades.

A computationally efficient methodology to simulate hybrid bolted joints including thermal effects

Carbon-aluminum bolted assemblies are difficult to simulate because of the complex phenomenology involved (contact, friction, preload and thermal expansion). Therefore, accurate but computationally feasible methodologies are necessary. We propose two simplified methodologies, one based on continuum shell elements and the other on conventional shells, and compare them with a full 3D solids model. The two cases explored are a single-lap shear coupon with one bolt, and a hybrid wingbox subcomponent with 46 bolts. The effect of temperature jumps on the bolt preloads are explored. Results show that the continuum shell model presents the best tradeoff between accuracy and computational cost.

Spectroscopic factors in dripline nuclei

Single-nucleon knockout reaction studies of the proton-dripline nuclei $^{9}\mathrm{C}$ and $^{13}\mathrm{O}$ suggest an appreciable suppression of spectroscopic factors. In this work, we calculate the one-neutron and one-proton spectroscopic factors for the mirror pair $^{9}\mathrm{C}\text{\ensuremath{-}}^{9}\mathrm{Li}$ and $^{13}\mathrm{O}$ using two variants of the continuum shell model: the complex-energy Gamow shell model and the real-energy shell model embedded in the continuum. Our results indicate that the continuum effects strongly suppress the spectroscopic factors of well-bound orbits in the dripline systems but have less impact on the spectroscopic factors of weakly bound states.

Molecular analysis of out-plane displacement effect on temperature dependent properties of carbon nanotubes

This paper presents a modified ‘stick-spiral’ model based on molecular mechanics to study the effect of thermal environment on elastic properties of carbon nanotubes (CNTs). At the same time, a closed form formulation is derived for Young's modulus as a function of the length of CC bonds and the out-plane displacement. The results show that the Young's modulus of both armchair and zigzag CNTs decreases with the increase of temperature, but the Poisson's ratio is independent of temperature. The temperature-dependent Young's modulus increases with the increase in tube diameter. It's also found that the out-place displacement attributed by temperature will lead to the surface Young's modulus decreases when temperature rises. According to the principle of elasticity theory, a temperature-dependent continuum shell model of strain energy is also established.

Electromagnetic Transitions in Near-threshold Resonances

Near-threshold collectivization of continuum shell model eigenstates is investigated in $^{20}$O on the example of B(E$\lambda$) decays of 4$^+$ states in the vicinity of elastic and inelastic neutron threshold. Changes of the electromagnetic transition probabilities as a function of the continuum-coupling strength are explained by the corresponding evolution of the double poles of the scattering matrix.

Comparing quantum, molecular and continuum models for graphene at large deformations

In this paper, the validity and accuracy of three interatomic potentials and the continuum shell model of Ghaffari and Sauer [1] are investigated. The mechanical behavior of single-layered graphene sheets (SLGSs) under uniaxial stretching, biaxial stretching and pure bending is studied for this comparison. The validity of the molecular and continuum models is assessed by direct comparison with density functional theory (DFT) data available in the literature. The molecular simulations are carried out employing the MM3, Tersoff and REBO + LJ potentials. The continuum formulation uses an anisotropic hyperelastic material model in the framework of the geometrically exact Kirchhoff-Love shell theory and isogeometric finite elements. Results from the continuum model are in good agreement with those from DFT. The results from the MM3 potential agree well up to the point of material instability, whereas those from the REBO + LJ and Tersoff potentials agree only for small deformations. Only the Tersoff potential is found to yield auxetic response in SLGSs under uniaxial stretch. Additionally, the transverse vibration frequencies of a pre-stretched graphene sheet and a carbon nanocone are obtained using the continuum model and molecular simulations with the MM3 potential. The variations of the frequencies from these approaches agree within an error of ≈5%.

Coupled atomistic‐continuum simulation of the mechanical properties of single‐layered graphene sheets

AbstractThe purpose of this work is the multiscale modeling of a single‐layered graphene sheet. The model is divided into three parts. One is an atomistic domain which is simulated with the atomic‐scale finite element method (AFEM). Another is a continuum domain. In this domain, the mechanical properties are investigated by using a finite element based on a nonlocal continuum shell model with a high order strain gradient. To be exact, it is a 4‐node 60‐generalized degree of freedom (DOF) Mindlin–Reissner finite shell element with a second order strain gradient. In the third part, a new transitional finite element is developed for smoothing the transition between the atomistic domain and continuum domain.

Thermal environment and strain energy related micro-model for properties of carbon nanotubes

In this paper, a molecular structural mechanics model is proposed to study the effect of thermal environment on elastic properties of carbon nanotubes. The covalent bonds are traded as a truss in this new model. The influence of the temperature on Young’s modulus and shear modulus of both armchair and zigzag carbon nanotubes are investigated. According to the principle of elastic theory, a temperature-dependent continuum shell model of strain energy density is also established. The strain energy density of single-walled carbon nanotubes (SWCNTs) decreases with the increase of temperature. The Young's modulus of SWCNTs increases with the increase of tube diameter, but decreases with the increase of temperature. The temperature-dependent modulus of SWCNTs has a similar trend of change with multi-walled carbon nanotubes (MWCNTs), but MWCNTs are more sensitive to the wall thickness.

Environment and strain energy related micromechanics analysis for properties of carbon nanotubes

AbstractIn the present study, a theoretical prediction of the effect of thermal environment on elastic properties of single‐walled carbon nanotubes (SWCNTs) is reported, and a molecular structural mechanics model is proposed in which the covalent bonds are traded as a truss. The influences of the temperature and the coefficient of thermal expansion (CTE) on Young's modulus and shear modulus of both armchair and zigzag SWCNTs are investigated. When CTE is and remains constant, the surface Young's modulus and the normal Young's modulus are reduced, respectively, as compared with the normal temperature. With the change of CTE, its effect on the elastic constant is far greater than the temperature effect itself. According to the principle of elastic theory, a temperature‐dependent continuum shell model of strain energy is also established. The strain energy of single‐walled carbon nanotubes decreases with the increase of the temperature.

Nanomechanics model for properties of carbon nanotubes under a thermal environment

This paper proposes a modified molecular mechanics model in which the covalent bond is treated as a beam element, and a theoretical prediction of the thermal environment effect on the elastic properties of single-walled carbon nanotubes (SWCNTs) is reported. The influence of the temperature on Young’s modulus of both armchair and zigzag SWCNTs is investigated. The study shows that the moduli decrease with the increase in temperature, but Poisson’s ratio is not dependent on the temperature. The temperature-dependent Young’s modulus increases with the increase in tube diameter. According to the principle of elasticity theory and energy consistent theory, a temperature-dependent continuum shell model of strain energy is also established. It is found that the deviation between the potential energy of carbon nanotubes and the strain energy of a continuum shell is significantly reduced based on the molecular mechanics model proposed in this study.

A material point method for thin shells with frictional contact

We present a novel method for simulation of thin shells with frictional contact using a combination of the Material Point Method (MPM) and subdivision finite elements. The shell kinematics are assumed to follow a continuum shell model which is decomposed into a Kirchhoff-Love motion that rotates the mid-surface normals followed by shearing and compression/extension of the material along the mid-surface normal. We use this decomposition to design an elastoplastic constitutive model to resolve frictional contact by decoupling resistance to contact and shearing from the bending resistance components of stress. We show that by resolving frictional contact with a continuum approach, our hybrid Lagrangian/Eulerian approach is capable of simulating challenging shell contact scenarios with hundreds of thousands to millions of degrees of freedom. Without the need for collision detection or resolution, our method runs in a few minutes per frame in these high resolution examples. Furthermore we show that our technique naturally couples with other traditional MPM methods for simulating granular and related materials.

An efficient size-dependent shear deformable shell model and molecular dynamics simulation for axial instability analysis of silicon nanoshells

Understanding the size-dependent behavior of structures at nanoscale is essential in order to have an effective design of nanosystems. In the current investigation, the surface elasticity theory is extended to study the nonlinear buckling and postbuckling response of axially loaded silicon cylindrical naoshells. Thereby, an efficient size-dependent shear deformable shell model is developed including the size effect of surface free energy. A boundary layer theory of shell buckling in conjunction with a perturbation-based solution methodology is employed to predict the size dependency in the buckling loads and postbuckling behavior of silicon nanoshells having various thicknesses. After that, on the basis of the Tersoff empirical potential, a molecular dynamics (MD) simulation is performed for a silicon cylindrical nanoshell with thickness of four times of silicon lattice constant, the critical buckling load and critical shortening of which are extracted and compared with those of the developed non-classical shell model. It is demonstrated that by taking the effects of surface free energy into account, a very good agreement is achieved between the results of the developed size-dependent continuum shell model and those of MD simulation.

Thermo-mechanical vibration of a single-layer graphene sheet and a single-walled carbon nanotube on a substrate

The thermo-mechanical vibration of a single-layer graphene sheet (SLGP) and a single-walled carbon nanotube (SWCNT) on a substrate is studied by using a nonlocal elastic plate model and two nonlocal elastic beam models (including Timoshenko-beam model and Euler-beam model) with quantum effects, respectively. The effect of the van der Waals (vdW) interactions between the SLGP (or the SWCNT) and the substrate on the vibration is obtained. Checking against our molecular dynamics simulations shows that the present models are reasonable. In particular, the radial vibration of the SWCNT on the substrate with quantum effects is further derived through the continuum shell model due to the different vdW forces on each point of the SWCNT circumference. The present models show that the normalized transverse frequency decreases and the normalized radial frequency increases with increasing SWCNT radius, respectively. The radial amplitude of the SWCNT (or the amplitude of the SLGP) nonlinearly increases and the transverse amplitude of the SWCNT linearly increases with increasing temperature for a given distance, respectively. The obtained analytical solution should be of great importance for understanding the thermo-mechanical vibration of nanoelectronic devices on a substrate.

Relativity versus exchange currents inO16(e,e′p)

The 16O(e,e'p) reaction in the quasi-elastic region has been studied in several experiments to determine spectroscopic factors, hence, the degree to which $^{16}$O looks like a closed shell. By varying the kinematics, experimentalists are able to extract response functions which comprise the cross section. However, analysis of the response functions separately produces very different spectroscopic factors. Two calculations led to different conclusions as to whether exchange currents can eliminate the discrepancies. Neither calculation considered relativistic corrections. The purpose of the article is to investigate the disagreement as to whether exchange currents are the solution to obtaining consistent spectroscopic factors, and to show that relativistic corrections have a much greater influence on providing this consistency. This calculation employs the recoil corrected continuum shell model, a model that uses a realistic interaction, and produces non-spurious scatterings states that are solutions to the coupled channels problems. Pionic and pair contributions to the exchange currents were calculated as developed by Dubach, Koch, and Donnelly. Relativistic effects are included by use of the direct Pauli reduction. Contributions of the exchange currents are shown to be insufficient to provide consistent spectroscopic factors. However, the inclusion of relativistic corrections, that can be large, lead to more consistent spectroscopic factors and cross sections. The influence of channel coupling is also shown to be significant. Tests of current conservation show that inclusion of the direct Pauli reduction produces small increases in its violation. Response functions which depend on the transverse current are sensitive to the lower component of relativistic wave functions, and hence, would provide a measure of the appropriateness of any relativistic model.

A Study on Equivalent Spherical Structure of Buckyball-C 60 Based on Continuum Shell Model

The main goal of this research is to extract a suitable continuum modeling of buckyball-C60. For this purpose, firstly the lattice structure of buckyball-C60 is modelled and subsequently a spherical structure equivalent to fullerene structure is considered. The fullerene structure modeled with shell elements is under internal pressure and in the continuum shell modeling process. The results of simulation demonstrate that the fullerene structure can be modelled using spherical structure. The comparison between strain energies of the equivalent fullerene spherical model and molecular mechanics model under radial displacement, shows that C60-fullerene spherical structures can be modeled using a shell with 0.665 A thickness, 5.07 TPa elastic modulus and 0.165 Poisson's ratio or a shell with 0.75 A thickness, 4.84 TPa elastic modulus and 0.19 Poisson's ratio. Moreover, the applied elliptical strain is used to demonstrate that the performance of the continuum spherical shell model of C60 is faultless.

Analysis on mass sensing characteristics of SWCNT-based nano-mechanical resonators using continuum mechanics based finite element analysis

In this study, we used the Finite element method (FEM) based on continuum mechanics, instead of molecular dynamics, to design and analyze an ultra-small mass nanomechanical sensor using Single wall carbon nanotube (SWCNT). We tested the validity of the FEM modeling technique based on continuum mechanics by comparing its analysis results with those of molecular dynamics modeling. After developing an elastic continuum shell model having an effective thickness, Young’s modulus, and Poisson’s ratio, we proved the superiority of the implemented analysis technique by comparing its resonance frequency with that of the molecular dynamics model. Additionally, using the developed model, we established various design parameters for a resonator and observed the parameter-dependent changes in frequency characteristics of the resonator.

Torsional properties of hexagonal boron nitride nanotubes, carbon nanotubes and their hybrid structures: A molecular dynamics study

The torsional mechanical properties of hexagonal single-walled boron nitride nanotubes (SWBNNTs), single-walled carbon nanotubes (SWCNTs), and their hybrid structures (SWBN-CNTs) are investigated using molecular dynamics (MD) simulation. Two approaches - force approach and energy approach, are adopted to calculate the shear moduli of SWBNNTs and SWCNTs, the discrepancy between two approaches is analyzed. The results show that the shear moduli of single-walled nanotubes (SWNTs), including SWBNNTs and SWCNTs are dependent on the diameter, especially for armchair SWNTs. The armchair SWNTs show the better ability of resistance the twisting comparable to the zigzag SWNTs. The effects of diameter and length on the critical values of torque of SWNTs are obtained by comparing the torsional behaviors of SWNTs with different diameters and different lengths. It is observed that the MD results of the effect of diameter and length on the critical values of torque agrees well with the prediction of continuum shell model. The shear modulus of SWBN-CNT has a significant dependence on the percentages of SWCNT and the hybrid style has also an influence on shear modulus. The critical values of torque of SWBN-CNTs increase with the increase of the percentages of SWCNT. This phenomenon can be interpreted by the function relationship between the torque of different bonds (B-N-X, C-C-X, C-B-X, C-N-X) and the angles of bonds.

A continuum shell model including van derWaals interaction for free vibrations of double-walled carbon nanotubes

This paper proposes the free vibration analysis of Double-Walled Carbon Nano Tubes (DWCNTs). A continuum elastic three-dimensional shell model is used for natural frequency investigation of simply supported DWCNTs. The 3D shell method is compared with beam analyses to show the applicability limits of 1D beam models. The effect of van der Waals interaction between the two cylinders is shown for different Carbon Nano Tube (CNT) lengths and vibration modes. Results give the van der Waals interaction effect in terms of frequency values. In order to apply the 3D shell continuum model, DWCNTs are defined as two concentric isotropic cylinders (with an equivalent thickness and Young modulus) which can be linked by means of the interlaminar continuity conditions or by means of an infinitesimal fictitious layer which represents the van der Waals interaction.