Multi‐scale modeling of surface effects in nano‐materials with temperature‐related Cauchy‐Born hypothesis via the modified boundary Cauchy‐Born model

In nano‐structures, the influence of surface effects on the properties of material is highly important because the ratio of surface to volume at the nano‐scale level is much higher than that of the macro‐scale level. In this paper, a novel temperature‐dependent multi‐scale model is presented based on the modified boundary Cauchy‐Born (MBCB) technique to model the surface, edge, and corner effects in nano‐scale materials. The Lagrangian finite element formulation is incorporated into the heat transfer analysis to develop the thermo‐mechanical finite element model. The temperature‐related Cauchy‐Born hypothesis is implemented by using the Helmholtz free energy to evaluate the temperature effect in the atomistic level. The thermo‐mechanical multi‐scale model is applied to determine the temperature related characteristics at the nano‐scale level. The first and second derivatives of free energy density are computed using the first Piola‐Kirchhoff stress and tangential stiffness tensor at the macro‐scale level. The concept of MBCB is introduced to capture the surface, edge, and corner effects. The salient point of MBCB model is the definition of radial quadrature used at the surface, edge, and corner elements as an indicator of material behavior. The characteristics of quadrature are derived by interpolating the data from the atomic level laid in a circular support around the quadrature in a least‐square approach. Finally, numerical examples are modeled using the proposed computational algorithm, and the results are compared with the fully atomistic model to illustrate the performance of MBCB multi‐scale model in the thermo‐mechanical analysis of metallic nano‐scale devices. Copyright © 2013 John Wiley & Sons, Ltd.     

A multi-scale modeling of surface effect via the modified boundary Cauchy–Born model

In this paper, a new multi-scale approach is presented based on the modified boundary Cauchy–Born (MBCB) technique to model the surface effects of nano-structures. The salient point of the MBCB model is the definition of radial quadrature used in the surface elements which is an indicator of material behavior. The characteristics of quadrature are derived by interpolating data from atoms laid in a circular support around the quadrature, in a least-square scene. The total-Lagrangian formulation is derived for the equivalent continua by employing the Cauchy–Born hypothesis for calculating the strain energy density function of the continua. The numerical results of the proposed method are compared with direct atomistic and finite element simulation results to indicate that the proposed technique provides promising results for modeling surface effects of nano-structures.     

An extended finite element/level set method to study surface effects on the mechanical behavior and properties of nanomaterials

We present a new approach based on coupling the extended finite element method (XFEM) and level sets to study surface and interface effects on the mechanical behavior of nanostructures. The coupled XFEM‐level set approach enables a continuum solution to nanomechanical boundary value problems in which discontinuities in both strain and displacement due to surfaces and interfaces are easily handled, while simultaneously accounting for critical nanoscale surface effects, including surface energy, stress, elasticity and interface decohesion. We validate the proposed approach by studying the surface‐stress‐driven relaxation of homogeneous and bi‐layer nanoplates as well as the contribution from the surface elasticity to the effective stiffness of nanobeams. For each case, we compare the numerical results with new analytical solutions that we have derived for these simple problems; for the problem involving the surface‐stress‐driven relaxation of a homogeneous nanoplate, we further validate the proposed approach by comparing the results with those obtained from both fully atomistic simulations and previous multiscale calculations based upon the surface Cauchy–Born model. These numerical results show that the proposed method can be used to gain critical insights into how surface effects impact the mechanical behavior and properties of homogeneous and composite nanobeams under generalized mechanical deformation. Copyright © 2010 John Wiley & Sons, Ltd.     

Stability and size-dependency of temperature-related Cauchy–Born hypothesis

In continuum mechanics, the constitutive models are usually based on the Cauchy–Born (CB) hypothesis which seeks the intrinsic characteristics of the material via the atomistic information and it is valid in small deformation. The main purpose of this paper is to investigate the temperature effect on the stability and size-dependency of Cauchy–Born hypothesis. Three-dimensional temperature-related Cauchy–Born formulations are developed for crystalline structure and the stability and size-dependency of temperature-related Cauchy–Born hypothesis are investigated by means of direct comparison between atomistic and continuous mediums. In order to control the temperature effect, the Nose–Hoover thermostat is employed. Since the Helmholtz free energy is temperature dependent; the first Piola–Kirchhoff stresses are explicitly computed as the first derivative of the Helmholtz free energy density to the deformation gradient. It is numerically shown that the validity surfaces become smaller at higher temperature, which is significant in larger specimen. It is also presented that the material stability decreases with increasing the ambient temperature.     

Multi‐scale modeling of surface effect via the boundary Cauchy–Born method

In this paper, a novel multi‐scale approach is developed for modeling of the surface effect in crystalline nano‐structures. The technique is based on the Cauchy–Born hypothesis in which the strain energy density of the equivalent continua is calculated by means of inter‐atomic potentials. The notion of introducing the surface effect in the finite element method is based on the intrinsic function of quadratures, called as an indicator of material behavior. The information of quadratures is derived by interpolating the data from probable representative atoms in their proximity. The technique is implemented by the definition of reference boundary CB elements, which enable to capture not only the surface but also the edge and corner effects. As the surface effect is important in small‐scale simulation, the relative number of boundary CB elements increases which leads to predomination of boundary effects in the model. In order to implement the equivalent continua in boundary value problems, the updated‐Lagrangian formulation of nonlinear finite element is derived. The numerical simulation of the proposed model together with the direct comparison with fully atomistic model indicates that the technique provides promising results for facile modeling of boundary effects and investigating its effect on the mechanical response of metallic nano‐scale devices. Copyright © 2010 John Wiley & Sons, Ltd.     

Multi-scale modeling of mechanical behavior of polycrystalline materials

A systematic procedure for multi-scale computational modeling is presented to simulate polycrystalline material behaviors for structural application. The modeling procedure bridged different length scales, from macro-scale to nano-scale, by proper flow of information from one scale to the next scale. The multi-scale model could include material characteristics at different length scales like nano-, micro-, meso-, and macro-scale. Both finite element analysis and molecular dynamics were used in the multi-scale analysis. An example of a large plate with a center hole is given to demonstrate the multi-scale modeling procedure.     

Finite element methods for the non‐linear mechanics of crystalline sheets and nanotubes

The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy–Born rule. The constitutive model for a two‐dimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyper‐elastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thin‐shell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multi‐million systems illustrate the computational savings which can be achieved. Copyright © 2003 John Wiley & Sons, Ltd.     

Buckling analysis of carbon nanotubes by a mixed atomistic and continuum model

A mixed atomistic and continuum model is applied to carbon nanotubes, in order to study their buckling behavior. Herein, the term “atomistic” refers to the underlying constitutive model that is formulated on the basis of interatomic potentials, whereas “continuum” means the application of the Cauchy–Born rule, which links the bond vectors before and after deformation via the deformation gradient of the continuum. Because the bond vectors are not infinitesimal and the continuum is modeled as surface, the Cauchy–Born rule has to be appropriately adapted to crystalline sheets. This is done via an exponential mapping in a new and surprisingly simple form such that in the analysis the current configuration has never to be left. The numerical buckling analysis of carbon nanotubes using the mixed atomistic and continuum model is carried out by means of the finite element method. For this purpose, the linearization of the equilibrium equations is provided.     

A three‐dimensional atomistic‐based process zone model simulation of fragmentation in polycrystalline solids

A three‐dimensional atomistic‐based process zone model (APZM) is used to simulate high‐speed impact induced dynamic fracture process such as fragmentation and spall fracture. This multiscale simulation model combines the Cauchy–Born rule, colloidal crystal process model, and micromechanics homogenization technique to construct constitutive relations in both grains and grain boundary at mesoscale. The proposed APZM has some inherent advantages to describe mechanical behaviors of polycrystalline solids. First, in contrast to macroscale phenomenological constitutive models, the APZM takes into account atomistic binding energy and atomistic lattice structure. In particular, the electron density related embedded atom method (EAM) potential has been adopted to describe interatomistic interactions of metallic polycrystalline solids in bulk elements; second, a mixed type of EAM potential and colloidal crystal depletion potential is constructed to describe heterogeneous microstructure in the process zone; third, the atomistic potential in both bulk material and process zone has the same atomistic origin, and hence, the bulk and process potentials are self‐consistent. The simulation of dynamic fracture process of a cylinder made of aluminum powder metallurgy (P/M) alloy during high‐speed impact/penetration is carried out, and numerical results demonstrate that APZM finite element method has remarkable ability to accurately capture complex three‐dimensional fragmentation formation and damage morphology. Copyright © 2012 John Wiley & Sons, Ltd.     

点蚀损伤下海洋平台结构剩余强度的多尺度分析方法

均匀腐蚀损伤影响结构承载力的研究已趋于成熟,但点蚀损伤影响整体结构承载力的规律及其局部损伤的演化行为仍不明确。该文采用均匀分布的圆柱体点蚀损伤模型,探讨在宏观结构尺度中点蚀损伤的描述、多尺度区域的划分和跨尺度界面的连接以及结构非线性的求解方法,提出了研究考虑细观尺度点蚀损伤的海洋平台结构多尺度模型的构建和计算方法。计算结果表明:该文提出的多尺度模型构建策略有效地解决了细观尺度点蚀损伤在宏观结构尺度中的描述问题,且所建多尺度模型能够体现点蚀损伤削弱结构极限承载力的规律,为评估在役海洋平台结构的剩余强度提供了可行的方法。     

Elastic crack propagation model for crystalline solids using a self-consistent coupled atomistic–continuum framework

Deformation and failure processes of crystalline materials are governed by complex phenomena at multiple scales. It is necessary to couple these scales for physics-based modeling of these phenomena, while overcoming limitations of modeling at individual scales. To address this issue, this paper develops self-consistent elastic constitutive and crack propagation relations of crystalline materials containing atomic scale cracks, from observations made in a concurrent multi-scale simulation system coupling atomistic and continuum domain models. The concurrent multi-scale model incorporates a finite temperature atomistic region containing the crack, a continuum region represented by a self-consistent crystal elasticity constitutive model, and a handshaking interphase region. Atomistic modeling is done by the molecular dynamics code LAMMPS, while continuum modeling is conducted by the finite element method. For single crystal nickel a nonlinear and nonlocal crystal elasticity constitutive relation is derived, consistent with the atomic potential function. An efficient, staggered solution scheme with parallel implementation is designed for the coupled problem. The atomistic–continuum coupling is achieved by enforcing geometric compatibility and force equilibrium in the interphase region. Quantitative analyses of the crack propagation process focuses on size dependence, strain energy release rate, crack propagation rate and degradation of the local stiffness. The self-consistent constitutive and crack propagation relations, derived from the concurrent model simulation results are validated by comparing results from the concurrent and full FE models. Excellent accuracy and enhanced efficiency are observed in comparison with pure MD and concurrent model results.     

Thermo-Oxidative Induced Damage in Polymer Composites: Microstructure Image-Based Multi-Scale Modeling and Experimental Validation

A multi-scale modeling approach is presented to simulate and validate thermo-oxidation shrinkage and cracking damage of a high temperature polymer composite. The multi-scale approach investigates coupled transient diffusion-reaction and static structural at macro- to micro-scale. The micro-scale shrinkage deformation and cracking damage are simulated and validated using 2D and 3D simulations. Localized shrinkage displacement boundary conditions for the micro-scale simulations are determined from the respective meso- and macro-scale simulations, conducted for a cross-ply laminate. The meso-scale geometrical domain and the micro-scale geometry and mesh are developed using the object oriented finite element (OOF). The macro-scale shrinkage and weight loss are measured using unidirectional coupons and used to build the macro-shrinkage model. The cross-ply coupons are used to validate the macro-shrinkage model by the shrinkage profiles acquired using scanning electron images at the cracked surface. The macro-shrinkage model deformation shows a discrepancy when the micro-scale image-based cracking is computed. The local maximum shrinkage strain is assumed to be 13 times the maximum macro-shrinkage strain of 2.5 × 10^?5, upon which the discrepancy is minimized. The microcrack damage of the composite is modeled using a static elastic analysis with extended finite element and cohesive surfaces by considering the modulus spatial evolution. The 3D shrinkage displacements are fed to the model using node-wise boundary/domain conditions of the respective oxidized region. Microcrack simulation results: length, meander, and opening are closely matched to the crack in the area of interest for the scanning electron images.     

Loss behavior of viscoelastic sandwich structures: A statistical-continuum multi-scale approach

This work presents a multi-scale model of viscoelastic constrained layer damping treatments for vibrating plates/beams. The approach integrates a finite element (FE) model of macro-scale vibrations and a statistical-continuum homogenization model to include effects of micro-scale structure and properties. The statistical-continuum homogenization model makes the micro- to macro-scale transition to approximate the effective behavior of the heterogeneous core by using n-point probability functions. A simple sound transmission model is used to show the effect of material microstructure on the sound transmission loss of the sandwich structure. The damping behavior resulting from the presence of voids and negative stiffness regions in the core material is modeled. This study clearly shows that, it is of high interest to research either material structures or processing techniques which lead to negative stiffness behavior. The results also poignantly show that the proposed multi-scale model yields insight on heterogeneous material behavior leading to increased damping properties and ultimately enhances the ability to design sandwich beam/plates.     

A surface Cauchy–Born model for nanoscale materials

We present an energy‐based continuum model for the analysis of nanoscale materials where surface effects are expected to contribute significantly to the mechanical response. The approach adopts principles utilized in Cauchy–Born constitutive modelling in that the strain energy density of the continuum is derived from an underlying crystal structure and interatomic potential. The key to the success of the proposed method lies in decomposing the potential energy of the material into bulk (volumetric) and surface area components. In doing so, the method naturally satisfies a variational formulation in which the bulk volume and surface area contribute independently to the overall system energy. Because the surface area to volume ratio increases as the length scale of a body decreases, the variational form naturally allows the surface energy to become important at small length scales; this feature allows the accurate representation of size and surface effects on the mechanical response. Finite element simulations utilizing the proposed approach are compared against fully atomistic simulations for verification and validation. Copyright © 2006 John Wiley & Sons, Ltd.     

Multiscale analysis of stiffness and fracture of nanoparticle-reinforced composites using micromechanics and global–local finite element models

A combined micromechanics analysis and global–local finite element method is proposed to study the interaction of particles and matrix at the nano-scale near a crack tip. An analytical model is used to obtain the effective elastic modulus of nanoparticle-reinforced composites, then a global–local multi-scale finite element model with effective homogeneous material properties is used to study the fracture of a compact tension sample. For SiO₂ particle-reinforced epoxy composites with various volume fractions, the simulation results for effective elastic modulus, fracture toughness, and critical strain energy release rate show good agreement with previously published experimental data. It is demonstrated that the proposed parametric multi-scale model can be used to efficiently study the toughness mechanisms at both the macro and nano-scale.     

Atomistic continuum modeling of graphene membranes

The paper deals with the modeling of thin, monolayer graphene membranes, which have significant electrical and physical properties used for nano- or micro-devices, such as resonators and nanotransistors. The membrane is considered as a homogenized graphene monolayer on the macroscopic scale, and a continuum–atomistic multiscale approach is exploited, focusing the Tersoff–Brenner (TB) potential for the interaction between the carbonic bonds. The associated Representative atomistic Unit Lattice (RUL) is thereby considered as a micro-scale quasi-continuum placed in context of computational homogenization. In this development, the Cauchy–Born rule (CBN) is extended by the atomic fluctuation to allow for relaxation in the RUL. The paper discusses the handling of the TB-potential, both in the context of macro–micro homogenization, and in the context of numerical implementation perspectives. In particular, explicit expressions of the homogenized membrane forces and stiffness are expressed in terms of the first and second gradient of the potential, with due consideration to the involved “non-local” pairwise interaction in the model. In addition, the detailed resulting macroscopic non-linear and linearized finite element response is formulated in terms of the relaxed lattice level atomistic response. Numerical results are provided for the lattice response in terms of the apparent anisotropic behavior induced by the graphene atomic structure. An assessment of the convergence of RULs with respect to different deformation states of the lattice membrane is also carried out. Finally, a validation of an experiment of a circular graphene membrane, using atomic force microscopy (AFM) measurements, is provided based on standard TB-parameters available in the literature.     

Multi-scale finite element analysis for tension and ballistic penetration damage characterizations of 2D triaxially braided composite

The multi-scale finite element model is presented to analyze tension and ballistic penetration damage characterizations of 2D triaxially braided composite (2DTBC). At the mesoscopic level, the damage of fiber tows is initiated with 3D Hashin criteria, and the damage initiation of pure matrix is predicted by the modified von Mises. The progressive degradation scheme and energy dissipation method are adopted to capture softening behaviors of tow and matrix. The macro-scale damage model is established by maximum-stress criteria and exponential damage evolution. To simulate interface debonding and inter-ply delamination, a triangle traction–separation law is adopted in each scale. Both scale damage models are verified with available experimental results. Based on numerical predictions, the stress–strain responses and damage developments of 2DTBC under axial and transverse tension loading are studied. For ballistic penetration loading, the meso-scale damage mechanisms of 2DTBC are predicted using 1/4 model, 1/2 model, 1-layer model, 2-layer model and 3-layer model. Then, effects of different model and impactor radius on damage modes are analyzed. Additionally, the macro-scale ballistic penetration behaviors of 2DTBC are simulated and compared with experiment. The prediction results for tension and penetration correlate well with experiment results. Both tension and penetration damage characterizations for tow, matrix within tow, pure matrix, interface and inter-ply delamination are revealed. A comparison of penetration damage between meso- and macro-scale presents a similar crack mechanism between two scales.     

Coupled thermo-mechanical analysis of metal-forming processes through a combined finite element–boundary element approach

This paper presents a new approach for obtaining the distribution of temperature in the dies during thermo-mechanical numerical analysis of metal forming problems. The proposed approach is based on a solution resulting from the combination of the finite element method with the boundary element method. The finite element method is used to perform the numerical modelling of the thermo-mechanical deformation of the workpiece, taking into account the geometrical and material non-linearities as well as the influence of the temperature distribution on the mechanical behaviour of the material. The boundary element method is applied for computing the distribution of temperatures in the dies. The combination of the two numerical methods is made using the finite element solution of the heat flow exchanged across the die–workpiece interface to define the boundary conditions to be applied on the thermal analysis of the dies. A numerical example of compression under plane-strain conditions is included to show the applicability of the proposed approach. © 1998 John Wiley & Sons, Ltd.     

Rate dependent modelling of the forming behaviour of viscous textile composites

A predictive approach to modelling the forming of viscous textile composites has been implemented in two finite element codes; Abaqus Standard™ and Abaqus Explicit™. A multi-scale energy model is used to predict the shear force–shear angle–shear rate behaviour of viscous textile composites, at specified temperatures, using parameters supplied readily by material manufacturers, such as fibre volume fraction, weave architecture and matrix rheology. The predictions of the energy model are fed into finite element simulations to provide the in-plane shear properties of two different macro-scale constitutive models implemented in the finite element codes. The manner of coupling predictions of the multi-scale energy model with the macro-scale models is shown to affect the rate-dependent material response in the simulations. These coupling methods are evaluated using picture frame test simulations.