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.     

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.     

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.     

Application of the higher‐order Cauchy–Born rule in mesh‐free continuum and multiscale simulation of carbon nanotubes

This paper investigates the application of a recently proposed higher‐order Cauchy–Born rule in the continuum simulation and multiscale analysis of carbon nanotubes (CNTs). A mesh‐free computational framework is developed to implement the numerical computation of the hyper‐elastic constitutive model that is derived from the higher‐order Cauchy–Born rule. The numerical computation reveals that the buckling pattern of a single‐walled carbon nanotube (SWCNT) can be accurately displayed by taking into consideration the second‐order deformation gradient, and fewer mesh‐free nodes can provide a good simulation of homogeneous deformation. The bridging domain method is employed to couple the developed mesh‐free method and the atomistic simulation. The coupling method is used to simulate the bending buckling of an SWCNT and the tensile failure of an SWCNT with a single‐atom vacancy defect, and good computational results are obtained. Copyright © 2008 John Wiley & Sons, Ltd.     

Temperature-dependent multi-scale modeling of surface effects on nano-materials

In this paper, a novel temperature-dependent multi-scale method is developed to investigate the role of temperature on surface effects in the analysis of nano-scale materials. In order to evaluate the temperature effect in the micro-scale (atomic) level, the temperature related Cauchy–Born hypothesis is implemented by employing the Helmholtz free energy, as the energy density of equivalent continua relating to the inter-atomic potential. The multi-scale technique is applied in atomistic level (nano-scale) to exhibit the temperature related characteristics. The first Piola–Kirchhoff stress and tangential stiffness tensor are computed, as the first and second derivatives of the free energy density to the deformation gradient, which are transferred to the macro-scale level. The Lagrangian finite element formulation is incorporated into the heat transfer analysis to develop the thermo-mechanical finite element model, and an intrinsic function is employed to model the surface and temperature effects in macro-scale level. The stress and tangential stiffness tensors are derived at each quadrature point by interpolating the data from nearby representative atom. The boundary Cauchy–Born (BCB) elements are introduced to capture the surface, edge and corner effects. Finally, the numerical simulation of proposed model together with the direct comparison with fully atomistic model illustrates that the technique provides promising results for facile modeling of boundary effect on thermo-mechanical behavior of metallic nano-scale devices.     

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.     

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.     

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.     

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.     

A stochastic second‐order and two‐scale thermo‐mechanical model for strength prediction of concrete materials

A stochastic thermo‐mechanical model for strength prediction of concrete is developed, based on the two‐scale asymptotic expressions, which involves both the macroscale and the mesoscale of concrete materials. The mesoscale of concrete is characterized by a periodic layout of unit cells of matrix‐aggregate composite materials, consisting of randomly distributed aggregate grains and cement matrix. The stochastic second‐order and two‐scale computational formulae are proposed in detail, and the maximum normal stress is assumed as the strength criterion for the aggregates, and the cement paste, in view of their brittle characteristics. Numerical results for the strength of concrete obtained from the proposed model are compared with those from known experiments. The comparison shows that the proposed method is validated for strength prediction of concrete. The proposed thermo‐mechanical model is also employed to investigate the influence of different volume fraction of the aggregates on the strength of concrete. Copyright © 2016 John Wiley & Sons, Ltd.     

Coupled multi‐scale cohesive modeling of failure in heterogeneous adhesives

A multi‐scale cohesive numerical framework is proposed to simulate the failure of heterogeneous adhesively bonded systems. This multi‐scale scheme is based on Hill's variational principle of energy equivalence between the higher and lower level scales. It provides an easy way to obtain accurate homogenized macroscopic properties while capturing the physics of failure processes at the micro‐scale in sufficient detail. We use an isotropic rate‐dependent damage model to mimic the failure response of the constituents of heterogeneous adhesives. The finite element method is used to solve the equilibrium equation at each scale. A nested iterative scheme inspired by the return mapping algorithm used in computational inelasticity is implemented. We propose a computationally attractive technique to couple the macro‐ and micro‐scales for rate‐dependent constitutive laws. We introduce an adhesive patch test to study the numerical performance, including spatial and temporal convergence of the multi‐scale scheme. We compare the solution of the multi‐scale cohesive scheme with a direct numerical simulation. Finally, we solve mode I and mode II fracture problems to demonstrate failure at the macro‐scale. Copyright © 2010 John Wiley & Sons, Ltd.     

A stabilized finite element formulation for monolithic thermo‐hydro‐mechanical simulations at finite strain

An adaptively stabilized monolithic finite element model is proposed to simulate the fully coupled thermo‐hydro‐mechanical behavior of porous media undergoing large deformation. We first formulate a finite‐deformation thermo‐hydro‐mechanics field theory for non‐isothermal porous media. Projection‐based stabilization procedure is derived to eliminate spurious pore pressure and temperature modes due to the lack of the two‐fold inf‐sup condition of the equal‐order finite element. To avoid volumetric locking due to the incompressibility of solid skeleton, we introduce a modified assumed deformation gradient in the formulation for non‐isothermal porous solids. Finally, numerical examples are given to demonstrate the versatility and efficiency of this thermo‐hydro‐mechanical model. Copyright © 2015 John Wiley & Sons, Ltd.     

A three‐dimensional model describing stress‐temperature induced solid phase transformations: solution algorithm and boundary value problems

An always increasing knowledge on material properties as well as a progressively more sophisticated production technology make shape memory alloys (SMA) extremely interesting for the industrial world. At the same time, SMA devices are typically characterized by complex multi‐axial stress states as well as non‐homogeneous and non‐isothermal conditions both in space and time. This aspect suggests the finite element method as a useful tool to help and improve application design and realization. With this aim, we focus on a three‐dimensional macroscopic thermo‐mechanical model able to reproduce the most significant SMA features (Int. J. Numer. Methods Eng. 2002; 55 : 1255–1264), proposing a simple modification of such a model. However, the suggested modification allows the development of a time‐discrete solution algorithm, which is more effective and robust than the one previously discussed in the literature. We verify the computational tool ability to simulate realistic mechanical boundary value problems with prescribed temperature dependence, studying three SMA applications: a spring actuator, a self‐expanding stent, a coupling device for vacuum tightness. The effectiveness of the model to solve thermo‐mechanical coupled problems will be discussed in a forthcoming work. Copyright © 2004 John Wiley & Sons, Ltd.     

Propagation of material and surface profile uncertainties on MEMS micro‐resonators using a stochastic second‐order computational multi‐scale approach

This paper aims at accounting for the uncertainties because of material structure and surface topology of micro‐beams in a stochastic multi‐scale model. For micro‐resonators made of anisotropic polycrystalline materials, micro‐scale uncertainties exist because of the grain size, grain orientation, and the surface profile. First, micro‐scale realizations of stochastic volume elements are obtained based on experimental measurements. To account for the surface roughness, the stochastic volume elements are defined as a volume element having the same thickness as the microelectromechanical system (MEMS), with a view to the use of a plate model at the structural scale. The uncertainties are then propagated up to an intermediate scale, the meso‐scale, through a second‐order homogenization procedure. From the meso‐scale plate‐resultant material property realizations, a spatially correlated random field of the in‐plane, out‐of‐plane, and cross‐resultant material tensors can be characterized. Owing to this characterized random field, realizations of MEMS‐scale problems can be defined on a plate finite element model. Samples of the macro‐scale quantity of interest can then be computed by relying on a Monte Carlo simulation procedure. As a case study, the resonance frequency of MEMS micro‐beams is investigated for different uncertainty cases, such as grain‐preferred orientations and surface roughness effects. Copyright © 2016 John Wiley & Sons, Ltd.     

Multi‐scale modeling of moving interface problems with flux and field jumps: Application to oxidative degradation of ceramic matrix composites

Problems involving reaction and species diffusion involve field and flux jumps at a moving reaction front. In multi‐scale problems such as carbon fiber composite oxidation, these effects need to be tracked at the microscopic scale of individual carbon fibers. A multi‐scale model is derived in this paper for predicting species distribution in such problems using a fully coupled multi‐scale homogenization approach. The homogenized fluxes from the micro‐scale are derived using Hill's macro‐homogeneity condition accounting for both flux jumps and species density field jumps at the reacting interface in the micro‐scale unit cell. At the macro‐scale, the competition between the transport of reacting species (oxygen) and the reaction product (carbon dioxide) is modeled using homogenized mass conservation equations. The moving reaction front in carbon fibers at the micro‐scale is tracked using level set method and an adaptive meshing strategy. The macroscopic weight loss of the composite when exposed to oxygen is simulated as a function of time using a coupled finite element methodology at various locations in a validated macroscopic model. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimization of thermo‐elasto‐plastic processes using Eulerian sensitivity analysis

A computational scheme for the analysis and optimization of quasi‐static thermo‐mechanical processes is presented in this paper. In order to obtain desirable mechanical transformations in a workpiece using a thermal treatment process, the optimal control parameters need to be determined. The problem is addressed by posing the process as a decoupled thermo‐mechanical finite element problem and performing an optimization using gradient methods. The forward problem is solved using the Eulerian formulation since it is computationally more efficient compared to an equivalent Lagrangian formulation. The design sensitivities required for the optimization are developed analytically using direct differentiation. This systematic design approach is applied to optimize a laser forming process. The objective is to maximize the angular distortion of a specimen subject to the constraint that the phase transition temperature is not exceeded at any point in the model. Copyright © 2003 John Wiley & Sons, Ltd.     

A simple dynamical scale‐coupling method for concurrent simulation of hybridized atomistic/coarse‐grained‐particle system

We propose a simple method for dynamical coupling of two sub‐systems with different characteristic scales described with different theoretical models, such as the fine‐scale sub‐system with the atomistic model (AM) such as the empirical inter‐atomic potential and the coarse‐scale sub‐system with the coarse‐grained particle (CGP) method, in a concurrent hybrid simulation scheme. Naive coupling of the different‐scale sub‐systems results in reflection of high wavenumber waves at the interface because of the differences in the phonon Brillouin‐zone and in the dispersion relation. To solve the problem, the present scale‐coupling method introduces (virtual) extra atoms and particles for the AM and the CGP sub‐systems, respectively, beyond the atom–particle interface, and uses the extra atoms and the particles to mutually transfer information of the waves between the two sub‐systems and to suppress the artificial reflection of the incident wave in the whole wavenumber range. As the algorithm in the present scale‐coupling method is local in time and space, it is applicable to hybrid systems with any interface shape at low computation and memory requirement. Accuracy of the present scale‐coupling method is compared with that of the existing methods for a simple model system. The hybrid AM‐CGP simulation of indentation of a graphene nano‐drum using the present scale‐coupling method is performed to demonstrate its accuracy and usefulness through its comparison with the fully atomistic results. Copyright © 2010 John Wiley & Sons, Ltd.     

A coupled thermomechanical nonordinary state‐based peridynamics for thermally induced cracking of rocks

A novel coupled thermo‐mechanical nonordinary state‐based peridynamics is proposed to study thermally induced damage in rocks. The thermal expansion characteristics of solid material are introduced into the coupled thermomechanical model to consider the influence of temperature. The deformation gradient tensor is obtained by the temperature fields, which is solved by peridynamic heat conduction theory. By introducing the deformation gradient tensor into the force state function of the nonordinary state‐based peridynamics, the coupling of thermal and mechanical is realized. A failure criterion is developed to investigate the thermally induced cracking of rocks. Then, the validity of the coupled thermo‐mechanical model is demonstrated by a numerical simulation. The correctness of the coupled model is validated by a benchmark example with analytic solution. Moreover, the thermal cracking progress in rocks is simulated using the proposed coupled nonordinary state‐based peridynamic model, and it is found that the numerical results are in good agreement with the previous experimental observations.