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.     

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.     

一种考虑尺寸效应的颗粒材料流变模型及其验证

经典连续介质理论的粘塑性本构关系缺乏材料尺度的相关性,难以表征颗粒材料流变的尺寸效应,而Cosserat连续体中的内禀特征长度为刻画材料的尺寸效应提供了一种可能途径。该文旨在Cosserat连续体的理论框架下发展Perzyna粘塑性模型,以探讨颗粒材料流变的尺寸效应与影响机制。首先基于Drucker-Prager屈服准则导出了Cosserat连续体粘塑性模型的一致性算法,获得了过应力本构方程积分算法与一致切向模量的封闭形式,并在ABAQUS二次平台上采用用户自定义单元(UEL)予以程序实现。有限元数值算例模拟了软岩试样的三轴压缩蠕变和两种堆石料试样在常规三轴条件下的蠕变和应力松弛,数值预测结果与相应试验结果具有较好的一致性,表明该流变模型的适应性。同时,将颗粒的球型指数、圆度和平均粒径作为表征颗粒材料内禀特征长度的一种度量,以反映颗粒材料的试样尺寸及其颗粒粒径与形状对流变过程中的轴向应变、偏应变和偏应力的影响关系,表明所发展的流变模型可以捕捉颗粒材料流变行为的压力相关性和尺寸效应。     

Computational continua for linear elastic heterogeneous solids on unstructured finite element meshes

The computational continua framework, which is a variant of higher‐order computational homogenization theories that is free of scale separation, does not require higher‐order finite element continuity, and is free of higher‐order boundary conditions, has been generalized to unstructured meshes. The salient features of the proposed generalization are (i) a nonlocal quadrature scheme for distorted elements that accounts for unit cell distortion in the parent element domain and (ii) an approximate variant of the nonlocal quadrature that eliminates the cost of computing positions of the quadrature points in the preprocessing stage. The performance of the computational continua framework on unstructured meshes has been compared to the first‐order homogenization theory and the direct numerical simulation.     

Modeling arbitrary crack propagation in coupled shell/solid structures with X‐FEM

In this paper, the extended finite element method (X‐FEM) formulation for the modeling of arbitrary crack propagation in coupled shell/solid structures is developed based on the large deformation continuum‐based (CB) shell theory. The main features of the new method are as follows: (1) different kinematic equations are derived for different fibers in CB shell elements, including the fibers enriched by shifted jump function or crack tip functions and the fibers cut into two segments by the crack surface or connecting with solid elements. So the crack tip can locate inside the element, and the crack surface is not necessarily perpendicular to the middle surface. (2) The enhanced CB shell element is developed to realize the seamless transition of crack propagation between shell and solid structures. (3) A revised interaction integral is used to calculate the stress intensity factor (SIF) for shells, which avoids that the auxiliary fields for cracks in Mindlin–Reissner plates cannot satisfy exactly the equilibrium equations. Several numerical examples, including the calculation of SIF for the cracked plate under uniform bending and crack propagation between solid and shell structures are presented to demonstrate the performance of the developed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Recovery of strong equilibrium from displacement‐based finite element models of Reissner‐Mindlin plates

In this paper, we extend to Reissner‐Mindlin plate bending problems a technique, originally proposed in the context of two‐dimensional and three‐dimensional continua, for recovering fully equilibrated stresses from the solution of a compatible finite element model. The technique involves partition of unity functions and the analyses of overlapping star patches modelled with hybrid equilibrium plate elements. The patches are subjected to balanced systems of loads composed of partitioned and fictitious loads, where the latter are derived from the stresses of the compatible solution. The special case of assumed linear displacement fields of both deflection and rotation for the compatible model is included. This case requires additional fields of stress resultants to correct possible rotational imbalances of star patches, and these are derived elementwise. Other cases of nonconforming elements are briefly considered. Numerical examples are presented to illustrate the effectiveness of these techniques in terms of the deviation of the recovery, which compares the complementary strain energy of a recovered solution with that obtained by a global equilibrated analysis based on the same stress approximations.     

Consistent coupling of beam and shell models for thermo‐elastic analysis

In this paper, the finite element formulation of a transition element for consistent coupling between shell and beam finite element models of thin‐walled beam‐like structures in thermo‐elastic problems is presented. Thin‐walled beam‐like structures modelled only with beam elements cannot be used to study local stress concentrations or to provide local mechanical or thermal boundary conditions. For this purpose, the structure has to be modelled using shell elements. However, computations using shell elements are a lot more expensive as compared to beam elements. The finite element model can be more efficient when the shell elements are used only in regions where the local effects are to be studied or local boundary conditions have to be provided. The remaining part of the structure can be modelled with beam elements. To couple these two models (i.e. shell and beam models) at transitional cross‐sections, transition elements are derived here for thermo‐elastic problems. The formulation encloses large displacement and rotational behaviour, which is important in case of thin‐walled beam‐like structures. Copyright © 2004 John Wiley & Sons, Ltd.     

A modified scaled boundary finite element method for three‐dimensional dynamic soil‐structure interaction in layered soil

This paper is devoted to the analysis of elastodynamic problems in 3D‐layered systems which are unbounded in the horizontal direction. For this purpose, a finite element model of the near field is coupled to a scaled boundary finite element model (SBFEM) of the far field. The SBFEM is originally based on describing the geometry of a half‐space or full‐space domain by scaling the geometry of the near field / far field interface using a radial coordinate. A modified form of the SBFEM for waves in a 2D layer is also available. None of these existing formulations can be used to describe a 3D‐layered medium. In this paper, a modified SBFEM for the analysis of 3D‐layered continua is derived. Based on the use of a scaling line instead of a scaling centre, a suitable scaled boundary transformation is proposed. The derivation of the corresponding scaled boundary finite element (SBFE) equations in displacement and stiffness is presented in detail. The latter is a nonlinear differential equation with respect to the radial coordinate, which has to be solved numerically for each excitation frequency considered in the analysis. Various numerical examples demonstrate the accuracy of the new method and its correct implementation. These include rigid circular and square foundations embedded in or resting on the surface of layered homogeneous or inhomogeneous 3D soil deposits over rigid bedrock. Hysteretic damping is assumed in some cases. The dynamic stiffness coefficients calculated using the proposed method are compared with analytical solutions or existing highly accurate numerical results. Copyright © 2011 John Wiley & Sons, Ltd.     

Predictive multiscale theory for design of heterogeneous materials

A general multiscale theory for modeling heterogeneous materials is derived via a nested domain based virtual power decomposition. Three variations on the theory are proposed; a concurrent approach, a simplified hierarchical approach and a statistical power equivalence approach. Deformation at each scale of analysis is solved either (a) by direct numerical simulation (DNS) of the microstructure or (b) by higher order homogenization of the microstructure. If the latter approach is chosen, a set of multiscale homogenized constitutive relations must be derived. This is demonstrated using a computational cell modeling technique for a four scale metal alloy. In each variation of the theory, a transfer of information occurs between the scales giving a coupled formulation. The concurrent approach achieves a more comprehensive coupling than the hierarchical approach, making it more accurate for dynamic fast time scale simulations. The power equivalence approach is strongly coupled and is useful for performing larger scale simulations as the expensive multiple scale DNS boundary value problems are replaced with statistical higher order continua. Furthermore, these continua may be solved on a single spatial discretisation using an extended finite element framework, making the theory applicable within existing high performance computing codes. Submitted to a Special Issue of Computational Mechanics in Honor of Professor Ladeveze’s 60th Birthday.     

Localized remeshing techniques for three‐dimensional metal forming simulations with linear tetrahedral elements

The localized remeshing technique for three‐dimensional metal forming simulations is proposed based on a mixed finite element formulation with linear tetrahedral elements in the present study. The numerical algorithm to generate linear tetrahedral elements is developed for finite element analyses using the advancing front technique with local optimization method which keeps the advancing fronts smooth. The surface mesh generation using mesh manipulations of the boundary elements of the old mesh system was made to improve mesh quality of the boundary surface elements, resulting in reduction of volume change in forming simulations. The mesh quality generated was compared with that obtained from the commercial CAD package for the complex geometry like lumbar. The simulation results of backward extrusion and bevel gear and spider forgings indicate that the currently developed simulation technique with the localized remeshing can be used effectively to simulate the three‐dimensional forming processes with a reduced computation time. Copyright © 2006 John Wiley & Sons, Ltd.     

A semi‐analytical curved element for linear elasticity based on the scaled boundary finite element method

This work introduces a semi‐analytical formulation for the simulation and modeling of curved structures based on the scaled boundary finite element method (SBFEM). This approach adapts the fundamental idea of the SBFEM concept to scale a boundary to describe a geometry. Until now, scaling in SBFEM has exclusively been performed along a straight coordinate that enlarges, shrinks, or shifts a given boundary. In this novel approach, scaling is based on a polar or cylindrical coordinate system such that a boundary is shifted along a curved scaling direction. The derived formulations are used to compute the static and dynamic stiffness matrices of homogeneous curved structures. The resulting elements can be coupled to general SBFEM or FEM domains. For elastodynamic problems, computations are performed in the frequency domain. Results of this work are validated using the global matrix method and standard finite element analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Stress recovery and error estimation for the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. This paper develops a stress recovery procedure based on a modal interpretation of the scaled boundary finite‐element method solution process, using the superconvergent patch recovery technique. The recovered stresses are superconvergent, and are used to calculate a recovery‐type error estimator. A key feature of the procedure is the compatibility of the error estimator with the standard recovery‐type finite element estimator, allowing the scaled boundary finite‐element method to be compared directly with the finite element method for the first time. A plane strain problem for which an exact solution is available is presented, both to establish the accuracy of the proposed procedures, and to demonstrate the effectiveness of the scaled boundary finite‐element method. The scaled boundary finite‐element estimator is shown to predict the true error more closely than the equivalent finite element error estimator. Unlike their finite element counterparts, the stress recovery and error estimation techniques work well with unbounded domains and stress singularities. Copyright © 2002 John Wiley & Sons, Ltd.     

A simple finite element‐based framework for the analysis of elastic pseudo‐rigid bodies

This article advocates a general procedure for the numerical investigation of pseudo‐rigid bodies. The equations of motion for pseudo‐rigid bodies are shown to be mathematically equivalent to those corresponding to certain constant‐strain finite element approximations for general deformable continua. A straightforward algorithmic implementation is achieved in a classical finite element framework. Also, a penalty formulation is suggested for modelling contact between pseudo‐rigid bodies. Representative planar simulations using a non‐linear elastic model demonstrate the predictive capacity of the pseudo‐rigid theory, as well as the robustness of the proposed computational procedure. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical simulation of Lamb wave scattering in semi‐infinite plates

A finite element formulation is applied to study Lamb wave scattering in homogeneous and sandwich isotropic plates. Dispersion curves are calculated in a simple and automatic way by solving a quadratic eigenproblem. A meshing criterion to obtain accurate results with linear and quadratic elements is provided. An absorbing boundary condition for semi‐infinite plates is derived from this formulation by means of a truncated normal mode expansion technique, where the finite element eigenvectors are used instead of the analytical expressions for the normal modes. This non‐reflecting boundary condition is directly applicable to study Lamb wave reflection by simple obstacles such as a flat edge. In order to tackle Lamb wave diffraction problems by defects with more complex geometries, a hybrid boundary element‐finite element formulation is developed. The validity and accuracy of both formulations are checked thoroughly with a series of test problems studied by other researchers. Copyright © 2001 John Wiley & Sons, Ltd.     

A physically and geometrically nonlinear scaled‐boundary‐based finite element formulation for fracture in elastomers

This paper is devoted to the formulation of a plane scaled boundary finite element with initially constant thickness for physically and geometrically nonlinear material behavior. Special two‐dimensional element shape functions are derived by using the analytical displacement solution of the standard scaled boundary finite element method, which is originally based on linear material behavior and small strains. These 2D shape functions can be constructed for an arbitrary number of element nodes and allow to capture singularities (e.g., at a plane crack tip) analytically, without extensive mesh refinement. Mapping these proposed 2D shape functions to the 3D case, a formulation that is compatible with standard finite elements is obtained. The resulting physically and geometrically nonlinear scaled boundary finite element formulation is implemented into the framework of the finite element method for bounded plane domains with and without geometrical singularities. The numerical realization is shown in detail. To represent the physically and geometrically nonlinear material and structural behavior of elastomer specimens, the extended tube model and the Yeoh model are used. Numerical studies on the convergence behavior and comparisons with standard Q1P0 finite elements demonstrate the correct implementation and the advantages of the developed scaled boundary finite element. Copyright © 2014 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.     

A mortar‐finite element formulation for frictional contact problems

In this paper a finite element formulation is developed for the solution of frictional contact problems. The novelty of the proposed formulation involves discretizing the contact interface with mortar elements, originally proposed for domain decomposition problems. The mortar element method provides a linear transformation of the displacement field for each boundary of the contacting continua to an intermediate mortar surface. On the mortar surface, contact kinematics are easily evaluated on a single discretized space. The procedure provides variationally consistent contact pressures and assures the contact surface integrals can be evaluated exactly. Copyright © 2000 John Wiley & Sons, Ltd.