Consistent micro–macro transitions at large objective strains in curvilinear convective coordinates

This paper presents a computational homogenization scheme that is of particular interest for problems formulated in curvilinear coordinates. The main goal of this contribution is to generalize the computational homogenization scheme to a formulation of micro–macro transitions in curvilinear convective coordinates, where different physical spaces are considered at the homogenized macro‐continuum and at the locally attached representative micro‐structures. The deformation and the coordinate system of the micro‐structure are assumed to be coupled with the local deformation and the local coordinate system at a corresponding point of the macro‐continuum. For the consistent formulation of micro–macro transitions, the operations scale‐up and scale‐down are introduced, considering the rotated representation of tensor variables at the different physical reference frames of micro‐ and macro‐structure. The second goal of this paper is to use objective strain measures like the Green–Lagrange strain tensor for the solution of boundary value problems on the micro‐ and macro‐scale by providing the required transformations for the work‐conjugate stress, strain and tangent tensors into variables admissible for the considered micro–macro transitions and satisfying the averaging theorem. Copyright © 2007 John Wiley & Sons, Ltd.     

A Methodology for Accurately Measuring Mechanical Properties on the Micro‐Scale

Abstract: A methodology has been developed for accurately measuring the mechanical properties of materials used on the micro‐scale. The direct tension test method using a dog bone‐type specimen has been employed, as it is the most effective and straightforward method to obtain results including a full stress–strain curve. The goal of this investigation was to develop a universal, yet simple and reliable, methodology to be used for accurate characterisation of mechanical properties for a wide variety of materials. Specimens from single crystal silicon were fabricated using photolithography by means of deep reactive ion etching. This material was chosen as it is expected that on both the micro‐ and macro‐scales, Young's modulus will have the same value. Hence, the accuracy of the methodology may be unambiguously examined. The test set‐up includes a small test machine containing a load cell whose maximum capacity is 5 N and is capable of direct gripping and displacement control. The specimens were found to have a trapezoidal cross‐section that was accurately measured using a scanning electron microscope. The strains were obtained by means of digital image correlation using images obtained via optical microscopy. The quantities measured include Young's modulus E, the fracture strength σ_f and the fracture strain ε_f. The average value of E obtained in the micro‐tests agrees well with the reference value obtained on the macro‐scale.     

A new multi‐scale scheme for modeling heterogeneous incompressible hyperelastic materials

A computational homogenization scheme is developed to model heterogeneous hyperelastic materials undergoing large deformations. The homogenization scheme is based on a so‐called computational continua formulation in which the macro‐scale model is assumed to consist of disjoint unit cells. This formulation adds no higher‐order boundary conditions and extra degrees of freedom to the problem. A computational procedure is presented to calculate the macroscopic quantities from the solution of the representative volume element boundary value problem. The proposed homogenization scheme is verified against a direct numerical simulation. It is also shown that the computational cost of the proposed model is lower than that of standard homogenization schemes. Copyright © 2015 John Wiley & Sons, Ltd.     

A novel contact interaction formulation for voxel‐based micro‐finite‐element models of bone

Voxel‐based micro‐finite‐element (μFE) models are used extensively in bone mechanics research. A major disadvantage of voxel‐based μFE models is that voxel surface jaggedness causes distortion of contact‐induced stresses. Past efforts in resolving this problem have only been partially successful, ie, mesh smoothing failed to preserve uniformity of the stiffness matrix, resulting in (excessively) larger solution times, whereas reducing contact to a bonded interface introduced spurious tensile stresses at the contact surface. This paper introduces a novel “smooth” contact formulation that defines gap distances based on an artificial smooth surface representation while using the conventional penalty contact framework. Detailed analyses of a sphere under compression demonstrated that the smooth formulation predicts contact‐induced stresses more accurately than the bonded contact formulation. When applied to a realistic bone contact problem, errors in the smooth contact result were under 2%, whereas errors in the bonded contact result were up to 42.2%. We conclude that the novel smooth contact formulation presents a memory‐efficient method for contact problems in voxel‐based μFE models. It presents the first method that allows modeling finite slip in large‐scale voxel meshes common to high‐resolution image‐based models of bone while keeping the benefits of a fast and efficient voxel‐based solution scheme.     

Computational homogenization for heat conduction in heterogeneous solids

In this paper, a multi‐scale analysis method for heat transfer in heterogeneous solids is presented. The principles of the method rely on a two‐scale computational homogenization approach which is applied successfully for the stress analysis of multi‐phase solids under purely mechanical loading. The present paper extends this methodology to heat conduction problems. The flexibility of the method permits one to take into account local microstructural heterogeneities and thermal anisotropy, including non‐linearities which might arise at some stage of the thermal loading history. The resulting complex microstructural response is transferred back to the macro level in a consistent manner. A proper macro to micro transition is established in terms of the applied boundary conditions and likewise a micro to macro transition is formulated in the form of consistent averaging relations. Imposition of boundary conditions and extraction of macroscopic quantities are elaborated in detail. A nested finite element solution procedure is outlined, and the effectiveness of the approach is demonstrated by some illustrative example problems. Copyright © 2007 John Wiley & Sons, Ltd.     

A domain decomposition method for concurrent coupling of multiscale models

Motivated by atomistic‐to‐continuum coupling, we consider a fine‐scale problem defined on a small region embedded in a much larger coarse‐scale domain and propose an efficient solution technique on the basis of the domain decomposition framework. Specifically, we develop a nonoverlapping Schwarz method with two important features: (i) the use of an efficient approximation of the Dirichlet‐to‐Neumann map for the interface conditions; and (ii) the utilization of the inherent scale separation in the solution. The paper includes a detailed formulation of the proposed interface condition, along with the illustration of its effectiveness by using simple but representative numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Micro and macro contact mechanics for interacting asperities

Contact of rough surfaces at micro and macro scales is studied in this paper. The asperities at micro scale are characterised by small radius of curvature whereas the waviness is characterised by large radius of curvature. When two rough surfaces come in contact, on the micro scale, of asperities contacts in a very small area leave large gaps between the surfaces; whereas on the macro scale the surfaces conform to each other under the application of load without gaps. Contact at micro scale is modelled by superposition of Hertzian stress fields of individual asperity contacts and the waviness at macro scale is modelled as a mixed boundary problem of rough punch indentation where displacements of uneven profile are prescribed along the region of contact. In both the cases for simplification the roughness is assigned to one surface making the other surface perfectly flat an assumption often made in contact mechanics of rough bodies. The motivation for modelling the asperities at micro scales comes from the preliminary results obtained from photoelastic experiments. Numerical results are presented based on the analytical results available for Hertzian contacts. The motivation for modelling the asperities at macro scales comes from the results available in literature for flat contacts from solving mixed boundary elasticity problems. A condition of full stick is assumed along the contact which is a common assumption made for rough contacts. The numerical results are presented for both the cases of rough contact at micro and macro scales.     

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.     

An efficient coarse‐grained parallel algorithm for global–local multiscale computations on massively parallel systems

The existing global–local multiscale computational methods, using finite element discretization at both the macro‐scale and micro‐scale, are intensive both in terms of computational time and memory requirements and their parallelization using domain decomposition methods incur substantial communication overhead, limiting their application. We are interested in a class of explicit global–local multiscale methods whose architecture significantly reduces this communication overhead on massively parallel machines. However, a naïve task decomposition based on distributing individual macro‐scale integration points to a single group of processors is not optimal and leads to communication overheads and idling of processors. To overcome this problem, we have developed a novel coarse‐grained parallel algorithm in which groups of macro‐scale integration points are distributed to a layer of processors. Each processor in this layer communicates locally with a group of processors that are responsible for the micro‐scale computations. The overlapping groups of processors are shown to achieve optimal concurrency at significantly reduced communication overhead. Several example problems are presented to demonstrate the efficiency of the proposed algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Fabrication of Millimeter‐Scale,Single‐Crystal One‐Third‐Hydrogenated Graphene with Anisotropic Electronic Properties

Periodically hydrogenated graphene is predicted to form new kinds of crystalline 2D materials such as graphane, graphone, and 2D C_xH_y, which exhibit unique electronic properties. Controlled synthesis of periodically hydrogenated graphene is needed for fundamental research and possible electronic applications. Only small patches of such materials have been grown so far, while the experimental fabrication of large‐scale, periodically hydrogenated graphene has remained challenging. In the present work, large‐scale, periodically hydrogenated graphene is fabricated on Ru(0001). The as‐fabricated hydrogenated graphene is highly ordered, with a √3 × √3/R30° period relative to the pristine graphene. As the ratio of hydrogen and carbon is 1:3, the periodically hydrogenated graphene is named “one‐third‐hydrogenated graphene” (OTHG). The area of OTHG is up to 16 mm². Density functional theory calculations demonstrate that the OTHG has two deformed Dirac cones along one high‐symmetry direction and a finite energy gap along the other directions at the Fermi energy, indicating strong anisotropic electrical properties. An efficient method is thus provided to produce large‐scale crystalline functionalized graphene with specially desired properties.     

Wavelet-based homogenization of unidirectional multiscale composites

The main aim is to present a homogenization algorithm for the multiscale heterogeneous (composite) materials, which is based on the wavelet representation of material properties and the relevant multiscale reduction. It is shown that classical homogenization method used before for two-scale composites (with micro and macro scales) is a special case of general multiresolutional strategy, where a single scale parameter tends to 0. The approach presented is applied to unidirectional wavelet-based homogenization of linear elasticity heterogeneous problem and to wave propagation, which may be applied in conjunction with various discrete numerical methods for efficient modeling of heterogeneous solids, fluids and multiphase media.     

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.     

3D‐FEM formulations of limit analysis methods for porous pressure‐sensitive materials

The first purpose of this paper is the numerical formulation of the three general limit analysis methods for problems involving pressure‐sensitive materials, that is, the static, classic, and mixed kinematic methods applied to problems with Drucker–Prager, Mises–Schleicher, or Green materials. In each case, quadratic or rotated quadratic cone programming is considered to solve the final optimization problems, leading to original and efficient numerical formulations. As a second purpose, the resulting codes are applied to non‐classic 3D problems, that is, the Gurson‐like hollow sphere problem with these materials as matrices. To this end are first presented the 3D finite element implementations of the static and kinematic classic methods of limit analysis together with a mixed method formulated to give also a purely kinematic result. Discontinuous stress and velocity fields are included in the analysis. The static and the two kinematic approaches are compared afterwards in the hydrostatic loading case whose exact solution is known for the three cases of matrix. Then, the static and the mixed approaches are used to assess the available approximate criteria for porous Drucker–Prager, Mises–Schleicher, and Green materials. Copyright © 2013 John Wiley & Sons, Ltd.     

A continuous–discontinuous approach to simulate physical degradation processes in porous media

A macroscopic framework for the simulation of physical degradation processes in quasi‐brittle porous materials is proposed. The framework employs the partition of unity (PU) concept and introduces a cohesive zone model, capturing the entire failure process starting from the growth and coalescence of micro‐defects until the formation of macro‐cracks. The framework incorporates the interaction between the failure process and the heat and mass transfer in the porous medium. As an example, physical degradation of an outside render is studied. The analysis illustrates that both material and interface failure can be investigated with this formulation. Depending on the boundary conditions, either one dominant crack or a network of small cracks is formed. Copyright © 2010 John Wiley & Sons, Ltd.     

考虑孔隙及微裂纹影响的混凝土宏观力学特性研究

混凝土是一种典型的多孔介质材料,孔隙分布错综复杂,孔径尺寸跨越微观尺度和宏观尺度,对混凝土弹性模量及强度等力学参数产生巨大影响.认为混凝土是由骨料、孔隙及砂浆基质组成的三相复合材料,采用Monte Carlo 法将孔隙、微裂纹及微缺陷与骨料颗粒随机投放在砂浆基质中.根据三相球模型及中空圆柱形杆件模型得到含孔材料的有效力学性质,并推导得到含孔材料的等效本构模型.建立含孔隙混凝土试件的细观单元等效化力学模型,对二级配含孔隙混凝土试件在单轴拉伸及压缩条件下的反应进行了非线性分析.结果表明:孔隙、微裂纹的存在对混凝土宏观弹性模量、强度及残余强度等力学性质都有很大影响,在对混凝土宏观力学特性分析及研究混凝土损伤断裂时不应忽略其影响.     

Spontaneous Assembly of Extremely Long,Horizontally‐Aligned,Conductive Gold Micro‐Wires in a Langmuir Monolayer Template

“Bottom‐up” technologies are based upon the premise that organized systems – from the nano‐scale up to the macro‐scale – can be assembled spontaneously from basic building blocks in solution. We demonstrate a simple strategy for the generation of extremely long (up to several centi­meters), horizontally‐aligned gold micro‐wires, produced through a surfactant monolayer template deposited from gold thiocyanate [Au(SCN)₄⁻] aqueous solution. Specifically, we show that the surfactant, octyl‐maleimide (OM), spontaneously forms oriented micro‐wires at the air/water interface, which constitute a template for deposition of metallic gold through binding and crystallization of the soluble gold complex. The Au micro‐wires can be subsequently transferred onto solid substrates, and following plasma treatment and gold enhancement exhibit excellent conductivity even at electrode spacings of several centimeters. Importantly, the micro‐wire alignment determines the direction of electrical current, demonstrating that long‐range ordering of the micro‐wires can be accomplished, significantly affecting the physical properties of the system. The new approach is simple, robust, and can be readily exploited for bottom‐up fabrication of micro‐wire assemblies and transparent conductive electrodes.     

Multi‐time‐step and two‐scale domain decomposition method for non‐linear structural dynamics

In this paper we propose a method to improve the means of taking into account the specific time‐scale and space‐scale characteristics in time‐dependent non‐linear problems. This method enables the use of arbitrary time steps in each subdomain: these can be coupled by prescribing continuous velocities at the interfaces, which are modelled using a dual Schur formulation. For certain subdomains, in space, we adopt a two‐scale resolution technique inspired by the multigrid methods in order to obtain the part of the solution related to small variation lengths on a refined scale and the part corresponding to large variation lengths on a coarse scale. For non‐linear problems, we propose an algorithm with a single iteration level to deal with both the non‐linear equilibrium and the two space scales thanks to a two‐grid method in which the relaxation steps are performed using a non‐linear, preconditioned conjugate gradient algorithm. Finally, we present an example which demonstrates the feasibility of the method. Copyright © 2003 John Wiley & Sons, Ltd.     

Subspace recycling accelerates the parametric macro‐modeling of MEMS

A fast computational technique that speeds up the process of parametric macro‐model extraction is proposed. An efficient starting point is the technique of parametric model order reduction (PMOR). The key step in PMOR is the computation of a projection matrix V, which requires the computation of multiple moment matrices of the underlying system. In turn, for each moment matrix, a linear system with multiple right‐hand sides has to be solved. Usually, a considerable number of linear systems must be solved when the system includes more than two free parameters. If the original system is of very large size, the linear solution step is computationally expensive. In this paper, the subspace recycling algorithm outer generalized conjugate residual method combined with generalized minimal residual method with deflated restarting (GCRO‐DR), is considered as a basis to solve the sequence of linear systems. In particular, two more efficient recycling algorithms, G‐DRvar1 and G‐DRvar2, are proposed. Theoretical analysis and simulation results show that both the GCRO‐DR method and its variants G‐DRvar1 and G‐DRvar2 are very efficient when compared with the standard solvers. Furthermore, the presented algorithms overcome the bottleneck of a recently proposed subspace recycling method the modified Krylov recycling generalized minimal residual method. From these subspace recycling algorithms, a PMOR process for macro‐model extraction can be significantly accelerated. Copyright © 2013 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.