Molecular Dynamic Finite Element Method (MDFEM)
... (a) p-P, p-As với các nguyên tử nằm trên hai mặt phẳng song song (b) p-Bi, p-Sb với các nguyên tử nằm trên 4 mặt phẳng song song (c) cấu trúc không gian của p-P, p-As Hiện tại, màng vât liệu hai chiều được ứng dụng trong nhiều lĩnh vực khác nhau như: y học [17], thiết bị năng lượng [18], thiết bị điện tử [19], ... Do đó, việc tính toán, mô phỏng xác định thông số cơ học của chúng sẽ là cơ sở để ứng dụng chúng trong thực tế. Phương pháp phần tử hữu hạn nguyên tử cho kết quả đáng tin cậy, đảm bảo độ chính xác (nghiên cứu [20][21][22][23]). Phương pháp này sử dụng nhiều dạng hàm thế khác nhau: hàm thế Stillinger-Weber [24], hàm thế Tersoff [25], hàm thế điều hòa [26], ... ...
... Chuyển vị của nguyên tử được tính toán khi chúng ta giải hệ phương trình (9). Ở đây, sử dụng phương pháp lặp Newton-Raphson để giải hệ phương trình (9), cách sử dụng phương pháp này đã thể hiện rõ trong [20][21][22][23], khi đó phương trình có dạng: ...
Cơ tính (mô đun đàn hồi, hệ số Poisson, ứng suất kéo đứt và biến dạng kéo đứt) của 4 vật liệu hai chiều cấutrúc nếp gấp (two-dimensional puckered hexagonal materials) gồm black phosphorus (p-P), p-arsenene (p-As),p-antimonene (p-Sb), p-bismuthene (p-Bi) được xác định bằng phương pháp phần tử hữu hạn nguyên tử vớihàm thế Stillinger-Weber. Mô đun đàn hồi hai chiều của 4 vật liệu trên khi kéo theo phương armchair có giá trị trong khoảng 10,2 - 23,6 N/m và 26,2 - 89,3 N/m khi kéo theo phương zigzag. Hệ số Poisson có giá trị trong khoảng từ 0,003 đến 0,58 khi kéo theo cả hai phương. Ứng suất hai chiều lớn nhất trong khoảng 2,35 - 4,11 N/m khi kéo theo phương armchair, trong khoảng 4,24 – 7,0 N/m khi kéo theo phương zigzag. Kết quả đó là cơ sở để sử dụng các vật liệu này trong thực tế.
... Experimentally obtained measurement data of the curing process is used to compare and validate the presented numerical crosslinking method with respect to the crosslinking procedure and the resulting network structure. The molecular modelling method is incorporated into the Molecular Dynamic Finite Element Method (MDFEM) [33][34][35] framework that can be used for deriving physical material properties [36,37]. ...
... The simulations are done using the Molecular Dynamic Finite Element Method (MDFEM) [34,35], which incorporates the MD method into a FEM (Finite Element Method) framework. In principle, the MDFEM is an implementation of the MD method into a finite element (FE) framework. ...
Reliable simulation of polymers on an atomistic length scale requires a realistic representation of the cured material. A molecular modeling method for the curing of epoxy systems is presented, which is developed with respect to efficiency while maintaining a well equilibrated system. The main criterion for bond formation is the distance between reactive groups and no specific reaction probability is prescribed. The molecular modeling is studied for three different mixing ratios with respect to the curing evolution of reactive groups and the final curing stage. For the first time, the evolution of reactive groups during the curing process predicted by the molecular modeling is validated with near-infrared spectroscopy data, showing a good agreement between simulation results and experimental measurements. With the proposed method, deeper insights into the curing mechanism of epoxy systems can be gained and it allows us to provide reliable input data for molecular dynamics simulations of material properties.
... Experimentally obtained measurement data of the curing process is used to compare and validate the presented numerical crosslinking method with respect to the crosslinking procedure and the resulting network structure. The molecular modelling method is incorporated into the Molecular Dynamic Finite Element Method (MDFEM) [33][34][35] framework that can be used for deriving physical material properties [36,37]. ...
... The simulations are done using the Molecular Dynamic Finite Element Method (MDFEM) [34,35], which incorporates the MD method into a FEM (Finite Element Method) framework. In principle, the MDFEM is an implementation of the MD method into a finite element (FE) framework. ...
Reliable simulation of polymers on an atomistic length scale requires a realistic representation of the cured material. A molecular modelling method for the curing of epoxy systems is presented, which is developed with respect to efficiency while maintaining a well equilibrated system. The main criterion for bond formation is the distance between reactive groups and no specific reaction probability is prescribed. The molecular modelling is studied for three different mixing ratios with respect to the curing evolution of reactive groups and the final curing stage. For the first time, the evolution of reactive groups during the curing process predicted by the molecular modelling is validated with near-infrared spectroscopy data, showing a good agreement between simulation results and experimental measurements. With the proposed method, deeper insights into the curing mechanism of epoxy systems can be gained and it allows us to provide reliable input data for molecular dynamics simulations of material properties.
... The MD method has increasingly been incorporated in the Finite Element Method (FEM) framework [6][7][8][9][10][11] as the equilibrium equations of MD and FEM may be expressed in equivalent forms. The resulting Atomistic Finite Element Method (AFEM) [8], also named Molecular Dynamic Finite Element Method (MDFEM) [12], is both computationally more favourable than MD [8], and offers a significant increase in compatibility and integrability with larger scale continuum FEM simulations. Several comprehensive presentations and reviews of AFEM/MDFEM and its implementations are available [8,[12][13][14]. ...
... The resulting Atomistic Finite Element Method (AFEM) [8], also named Molecular Dynamic Finite Element Method (MDFEM) [12], is both computationally more favourable than MD [8], and offers a significant increase in compatibility and integrability with larger scale continuum FEM simulations. Several comprehensive presentations and reviews of AFEM/MDFEM and its implementations are available [8,[12][13][14]. ...
The recent rise of 2D materials, such as graphene, has expanded the interest in nanoelectromechanical systems (NEMS). The increasing ability of synthesizing more exotic NEMS architectures, creates a growing need for a cost-effective, yet accurate nano-scale simulation method. Established methodologies like Molecular Dynamics (MD) trail behind synthesis capabilities because the computational effort scales quadratically. The equilibrium equations of MD are equivalent with those of the computationally more favourable Finite Element Method (FEM). However, current implementations exploiting this equivalence remain limited due to the FEM iterative solvers requiring a large number of lengthy force field derivatives
and specifically tailored element topologies. This paper proposes a merged Molecular Dynamic Finite Element Method (MDFEM) which does not require the manual derivation of these derivatives. Hence, implementing MDFEM-specific element topologies is straightforward and thus, different non-linear MD force field potentials can be solved exactly within the FEM, at reduced computational costs. The proposed multi-scale and multi-physics compatible MDFEM is equivalent to the MD, as demonstrated firstly by an example of brittle fracture in Carbon Nanotubes (CNT), and secondly by conformational analyses on Non-Equilibrium initial meshes of Pillared Graphene Structures (PGS).
... The MD enriched continuum method by Belytschko and Xiao (2003) and Xiao and Belytschko (2003) was also another pioneering approach to couple a potential energy Hamiltonian calculation conducted on a fine scale MD domain with a Lagrangian calculation on a coarse scale continuum domain with an overlapped bridging domain among the two representations. Recently, an implementation of interatomic potential laws within a displacement-based finite element (FE) formulation has also been proposed in Nasdala et al. (2010), with a rigorous implicit solution scheme, aiming at generating models where non-linear discrete and continuous systems can be suitably combined. ...
All the previous discrete systems and probably many others analyzed by the scientific Community share the common features of complex systems where the overall properties emerge from the non-local non-linear interactions between their basic network components and can only be predicted by numerically simulating the system response and the dynamic evolution of defects (cracks, node removal, link removal). Failure of the system is generally the result of a percolation of defects at different scales, which leads to complex redistribution of internal forces/flows. Understanding how the system is able to withstand perturbations, i.e., its resilience or flaw-tolerance, is a problem common to all of the disciplines mentioned in the previous section.
Using atomic finite element method with valence force field model, we estimated elastic mechanical properties of 34 transition metal dichalcogenides monolayers (ScO2, ScS2, ScSe2, ScTe2, TiTe2, VO2, VS2, VSe2, VTe2, CrO2, CrS2, CrSe2, CrTe2, MnO2, FeO2, FeS2, FeSe2, FeTe2, CoTe2, NiS2, NiSe2, NiTe2, NbS2, NbSe2, MoO2, MoS2, MoSe2, MoTe2, TaS2, TaSe2, WO2, WS2, WSe2, WTe2). While WO2 is the hardest sheet, ScTe2 is the softness one with the value of Young’s modulus as about 245 N/m and 30 N/m in the zigzag direction, respectively. For TMDs monolayers with the same transition metal M, Young’s modulus and shear modulus of MO2 monolayers are always maximum. While those of MTe2 ones are minimum. Besides, one of them was chosen for investigating the size effect on Young’s modulus, Poisson’s ratio and shear modulus. The Young’s modulus in the armchair direction increases while that of the zigzag one decreases. Both tend to be size-independent and isotropic for large sizes. The findings from this work are useful in applications using these above monolayers.KeywordsTransition metal dichalcogenidesElastic propertiesAtomic finite element method
This work investigates size effects on mechanical properties of single layer molybdenum disulfide (SLMoS2) nanoribbon under uniaxial tension using molecular dynamics finite element method with Stillinger-Weber potential. For the square shaped nanoribbon of increasing size, Young’s modulus of the armchair nanoribbon increases while that of the zigzag one decreases. Both of them tend to be size-independent and isotropic for large sizes. Fracture stress of square shaped armchair nanoribbon is nearly independent of size but that of zigzag one reduces as size increases. Poisson’s ratio of them goes up as size increases but the larger nanoribbons have a slower increase. For the rectangle shaped nanoribbons, the aspect ratio has no influence on SLMoS2 nanoribbons with fixed width but significantly affected on those with fixed length. Young’s modulus, Poisson’s ratio, fracture stress and strain of SLMoS2 zigzag nanoribbons with fixed length decrease as width increases. With SLMoS2 armchair nanoribbon of fixed length, narrower one has lower Young’s modulus and higher Poisson’s ratio but fracture stress and strain are almost unchanged.
The standard molecular mechanics (MM) method with the DREIDING force field (see Mayo et al. The Journal of Physical Chemistry, 1990, 94: 8897–8909) and the molecular structural mechanics (MSM) method with Bernoulli–Euler beam elements are used to study the quasi-static nonlinear buckling and post-buckling behavior of a compressed nearly square single layer graphene sheet (SLGS) for different types of boundary conditions. The novelty of this study is the finding that well-calibrated parameter sets provide similar values of buckling forces/modes and similar post-buckling deformations for stable equilibrium configurations of SLGSs in simulations using the standard MM and MSM methods. In addition, the effect of accounting for non-bonded van der Waals (vdW) forces on the advanced post-buckling deformation modes of compressed SLGSs was studied for the first time. It is shown that the column-like post-buckling deformation modes of compressed SLGSs obtained without the inclusion of vdW interatomic forces are similar to the well-known ones for the planar Euler elastica, and the advanced post-buckling modes obtained with the inclusion of interatomic vdW forces are qualitatively different from the corresponding modes for the planar Euler elastica. In addition, the study shows the theoretical possibility of the existence of post-buckling out-of-plane equilibrium configurations whose stability is provided by attractive vdW forces.
Boehmite nanoparticles show great potential in improving mechanical properties of fiber reinforced polymers. In order to predict the properties of nanocomposites, knowledge about the material parameters of the constituent phases, including the boehmite particles, is crucial. In this study, theAtomic force microscopy mechanical behavior of boehmite is investigated using Atomic Force Microscopy (AFM) experiments and Molecular Dynamic Finite Element Method
Molecular dynamic finite element method (MDFEM) simulations. The Young’s modulus of the perfect crystalline boehmite nanoparticlesBoehmite nanoparticle is derived from numerical AFM simulations. Results of AFM experiments on boehmite nanoparticles deviate significantly. Possible causes are identified by experiments on complementary types of boehmite, i.e. geological and hydrothermally synthesized samples, and further simulations of imperfect crystals and combined boehmite/epoxy models. Under certain circumstances, the mechanical behavior of boehmite was found to be dominated by inelastic effects that are discussed in detail in the present work. The studies are substantiated with accompanying X-ray DiffractionX-ray diffraction and RamanRaman spectroscopy experiments.
The accurate prediction of the complex material response of nanoparticle/epoxy nanocomposites for thermomechanical load cases is of great interest for engineering applications. In the present work, three main contributions with respect to multi-scale modelling of the viscoelastic damage behaviour of nanocomposites are presented. Firstly, a constitutive model for the viscoelastic damage behaviour at finite temperatures below the glass-transition temperature is proposed. The constitutive model captures the main characteristics of the material response including the non-linear hyperelasticity, softening behaviour and the effect of temperature. Secondly, the material model is calibrated using purely experimental results to evaluate the best capability of the model in reproducing the stress–strain response at different strain rates and temperatures. The calibrated model predicts the material behaviour across a range of nanoparticle weight fractions with good agreement with experimental results. Finally, a combined approach of experimental testing and molecular simulations is proposed to identify the parameters of the constitutive model. This study shows that the proposed simulation-based framework can be used to significantly reduce the number of experimental tests required for identification of material parameters without a significant loss of accuracy in the material response prediction. The predictive capability of the atomistically calibrated constitutive model is validated, with additional experimental results not used within the parameter identification, in terms of an accurate representation of the viscoelastic damage behaviour of nanoparticle/epoxy nanocomposites at finite temperatures. The present study underlines the capabilities of numerical molecular simulations intended for the characterisation of material properties with respect to physically based constitutive modelling and multi-scale approaches.
The precise knowledge of the temperature-dependent non-linear viscoelastic material behaviour of polymers is of great importance for engineering applications. The present work is a contribution to meet the challenge of bridging the inherently different time scales of molecular dynamics (MD) and experiments by providing a consistent comparison and assessment of viscoelastic theories. For this reason, the physically motivated theories for viscoelasticity of Eyring and Argon as well as the Cooperative model are evaluated with regard to their predictive capability for the characterisation of the viscous behaviour over a broad range of temperatures and strain rates. MD simulations of tensile tests are performed and the effect of strain rate and temperature on the yield stress is examined. The distinctive feature of this study is to demonstrate that viscoelastic theories can be successfully calibrated using only MD results. For a comparison to experimental data, we conduct tensile tests at three different strain rates and at three temperatures in the glassy regime. Experimental validation confirms the predictive capability of the Argon model, which can provide an accurate formulation of epoxy viscoelasticity for physically motivated constitutive models. The present study not only underlines the ability of MD simulations for identifying and characterising physical phenomena on the molecular level, but also shows that molecular simulations can substitute experimental tests for the characterisation of the viscoelastic material behaviour of polymers.
A mathematically rigorous methodology for embedding the governing equations of molecular dynamics in the formalism of the finite element method is presented. Only one generalized finite element type is needed to cover all different types of existing interatomic potentials. The finite element type is simply specified by two parameters characterizing the type of the interatomic potential to be considered. Built on this formulation a partitioned Runge–Kutta method—summarizing a wide range of explicit and implicit, single- and multi-stage, lower and higher order time integration schemes—is embedded in a unified manner. The required finite element residual vector and the related Jacobian matrix are stated explicitly. The related FE-mesh coincides with the neighborhood lists used in standard molecular dynamics enabling the use of common tools. The range, versatility and performance of the proposed finite element formulation have been demonstrated by means of several numerical examples.
Molecular mechanics/molecular dynamics (MM/MD) methods are widely used in computer simulations of deformation (including buckling, vibration, and fracture) of low-dimensional carbon nanostructures (single-layer graphene sheets (SLGSs), single-walled nanotubes, fullerenes, etc). In MM/MD simulations, the interactions between carbon atoms in these nanostructures are modeled using force fields (e.g., AIREBO, DREIDING, MM3/MM4). The objective of the present study is to fit the DREIDING force field parameters (see Mayo et al. J Phys Chem 94:8897–8909, 1990) to most closely reproduce the mechanical parameters of graphene (Young’s modulus, Poisson’s ratio, bending rigidity modulus, and intrinsic strength) known from experimental studies and quantum mechanics simulations since the standard set of the DREIDING force field parameters (see Mayo et al. 1990) leads to unsatisfactory values of the mechanical parameters of graphene. The values of these parameters are fitted using primitive unit cells of graphene acted upon by forces that reproduce the homogeneous deformation of this material in tension/compression, bending, and fracture. (Different sets of primitive unit cells are used for different types of deformation, taking into account the anisotropic properties of graphene in states close to failure.) The MM method is used to determine the dependence of the mechanical moduli of graphene (Young’s modulus, Poisson’s ratio, and bending rigidity modulus) on the scale factor. Computer simulation has shown that for large linear dimensions of SLGSs, the mechanical parameters of these sheets are close to those of graphene. In addition, computer simulation has shown that accounting for in-layer van der Waals forces has a small effect on the value of the mechanical moduli of graphene.
Effects of vacancies and Stone-Wales defects on the mechanical properties of silicene are investigated through molecular dynamic finite element method with Tersoff potential. Young's modulus, Poisson's ratio and uniaxial tensile stress-strain curves are considered in the armchair and zigzag directions. It is found that pristine and lowly defective silicene sheets exhibit almost the same elastic nature up to fracture points. However, a single defect weakens significantly the silicene sheet, resulting in a considerable reduction in the fracture strength. One 2-atom vacancy in the sheet's center reduces 18-20 % in fracture stress and 33-35 % in fracture strain. The weakening effects of Stone-Wales defects vary with the tensile direction and the orientation of these defects.
A new Molecular Dynamics Finite Element Method (MDFEM) with a coupled mechanical‐charge/dipole formulation is proposed. The equilibrium equations of Molecular Dynamics (MD) are embedded exactly within the computationally more favourable Finite Element Method (FEM). This MDFEM can readily implement any force field because the constitutive relations are explicitly uncoupled from the corresponding geometric element topologies. This formal uncoupling allows to differentiate between chemical‐constitutive, geometric and mixed‐mode instabilities. Different force fields, including bond‐order reactive and polarisable fluctuating charge–dipole potentials, are implemented exactly in both explicit and implicit dynamic commercial finite element code. The implicit formulation allows for larger length and time scales and more varied eigenvalue‐based solution strategies.
The proposed multi‐physics and multi‐scale compatible MDFEM is shown to be equivalent to MD, as demonstrated by examples of fracture in carbon nanotubes (CNT), and electric charge distribution in graphene, but at a considerably reduced computational cost. The proposed MDFEM is shown to scale linearly, with concurrent continuum FEM multi‐scale couplings allowing for further computational savings. Moreover, novel conformational analyses of pillared graphene structures (PGS) are produced. The proposed model finds potential applications in the parametric topology and numerical design studies of nano‐structures for desired electro‐mechanical properties (e.g. stiffness, toughness and electric field induced vibrational/electron‐emission properties). Copyright © 2014 John Wiley & Sons, Ltd.
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