A two‐scale failure model for heterogeneous materials: numerical implementation based on the finite element method

In the first part of this contribution, a brief theoretical revision of the mechanical and variational foundations of a Failure‐Oriented Multiscale Formulation devised for modeling failure in heterogeneous materials is described. The proposed model considers two well separated physical length scales, namely: (i) the macroscale where nucleation and evolution of a cohesive surface is considered as a medium to characterize the degradation phenomenon occurring at the lower length scale, and (ii) the microscale where some mechanical processes that lead to the material failure are taking place, such as strain localization, damage, shear band formation, and so on. These processes are modeled using the concept of Representative Volume Element (RVE). On the macroscale, the traction separation response, characterizing the mechanical behavior of the cohesive interface, is a result of the failure processes simulated in the microscale. The traction separation response is obtained by a particular homogenization technique applied on specific RVE sub‐domains. Standard, as well as, Non‐Standard boundary conditions are consistently derived in order to preserve objectivity of the homogenized response with respect to the micro‐cell size. In the second part of the paper, and as an original contribution, the detailed numerical implementation of the two‐scale model based on the finite element method is presented. Special attention is devoted to the topics, which are distinctive of the Failure‐Oriented Multiscale Formulation, such as: (i) the finite element technologies adopted in each scale along with their corresponding algorithmic expressions, (ii) the generalized treatment given to the kinematical boundary conditions in the RVE, and (iii) how these kinematical restrictions affect the capturing of macroscopic material instability modes and the posterior evolution of failure at the RVE level. Finally, a set of numerical simulations is performed in order to show the potentialities of the proposed methodology, as well as, to compare and validate the numerical solutions furnished by the two‐scale model with respect to a direct numerical simulation approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Scale transition and enforcement of RVE boundary conditions in second‐order computational homogenization

Formulation of the scale transition equations coupling the microscopic and macroscopic variables in the second‐order computational homogenization of heterogeneous materials and the enforcement of generalized boundary conditions for the representative volume element (RVE) are considered. The proposed formulation builds on current approaches by allowing any type of RVE boundary conditions (e.g. displacement, traction, periodic) and arbitrary shapes of RVE to be applied in a unified manner. The formulation offers a useful geometric interpretation for the assumptions associated with the microstructural displacement fluctuation field within the RVE, which is here extended to second‐order computational homogenization. A unified approach to the enforcement of the boundary conditions has been undertaken using multiple constraint projection matrices. The results of an illustrative shear layer model problem indicate that the displacement and traction RVE boundary conditions provide the upper and lower bounds of the response determined via second‐order computational homogenization, and the solution associated with the periodic RVE boundary conditions lies between them. Copyright © 2007 John Wiley & Sons, Ltd.     

A study of the influence of REV variability in double‐scale FEM ×DEM analysis

In this work, the consequences of using several different discrete element granular assemblies for the representation of the microscale structure, in the framework of multiscale modeling, have been investigated. The adopted modeling approach couples, through computational homogenization, a macroscale continuum with microscale discrete simulations. Several granular assemblies were used depending on the location in the macroscale finite element mesh. The different assemblies were prepared independently as being representative of the same material, but their geometrical differences imply slight differences in their response to mechanical loading. The role played by the micro‐assemblies, with weaker macroscopic mechanical properties, on the initiation of strain localization in biaxial compression tests is demonstrated and illustrated by numerical modeling of different macroscale configurations. Copyright © 2016 John Wiley & Sons, Ltd.     

变形制度对低碳锰钢组织性能的影响

为了获得细晶铁素体/贝氏体的复相组织,通过控轧控冷工艺研究了低碳锰钢在奥氏体区变形时变形量、终轧温度和卷取温度对组织演变和力学性能的影响规律.研究表明,增加变形量(对应道次间隔时间缩短)可以细化铁素体晶粒,但当终轧温度降低到800℃时,变形量的增加以及开冷温度的降低不利于贝氏体组织的获得.通过调整变形量、终轧温度、可开冷温度并适当降低卷取温度,可使实验钢获得晶粒尺寸约为5μm的铁素体和10%~20%的贝氏体组织,低碳锰钢强塑性能良好.     

Three‐dimensional measurement and cohesive element modelling of deformation and damage in a 2.5‐dimensional woven ceramic matrix composite

A cohesive element numerical model, which reproduces the three‐dimensional microstructure of a 2.5‐dimensional silicon‐nitrogen‐oxide fibre/fabric‐reinforced boron nitride ceramic matrix composite (SiNO/BN) is applied to simulate the failure of specimens that are observed in situ during diametral compression testing. Measurements of deformation by image correlation of two‐dimensional optical surface observations and three‐dimensional X‐ray computed tomographs are used to fit the simulation's elastic properties for the matrix and fibre tows. The observed patterns of damage nucleation and propagation are correctly simulated using a local tensile strain criterion.     

Infrared thermography for void mapping of a graphene/epoxy composite and its full‐field thermal simulation

Important challenges are faced during the manufacturing of graphene nanoplatelet (GNP)/polymer composites, associated with material quality and how to eliminate or reduce fabrication‐induced defects in the effort to improve performance. In the present work, infrared thermography (IRT) is used to measure void content and map void distribution, formed during fabrication of GNP/epoxy nanocomposites. Taking into consideration the size of each pixel (~100 μm), this method enables the non‐destructive detection of flaws with a size of approximately 200 μm. Their effect on thermal conductivity of the nanocomposite is studied by a 3D multiscale finite element analysis. Generic and full‐field comparisons demonstrate a good agreement between measurements and numerical predictions, validating assumptions and simplifications made in the proposed model.     

EBSD study on the formation of fine ferrite grains in plain carbon steel during warm deformation

Torsion testing was carried out on a plain carbon steel to study ferrite grain refinement during warm deformation within two-phase (α + γ) region. Fine ferrite grains development was analyzed by using optical microscope and EBSD technique. Microstructural analysis shows that with increasing strain the new fine equiaxed ferrite grains surrounded by high angle boundaries start generating at the initial boundaries and the volume fraction of fine grains is increased and that of work hardened grains decreased. It was seen that there is no evidence of discontinuous dynamic recrystallization. Thus, it is suggested that the occurrence of continuous dynamic recrystallization is responsible for the formation of new fine ferrite grains.     

High‐throughput,high‐accuracy determination of coefficient of thermal expansion of carbon fibre–epoxy composites using digital image correlation

High‐throughput, high‐accuracy determination of thermal deformation and coefficient of thermal expansion (CTE) of carbon fibre–epoxy composites using 2D‐digital image correlation (2D‐DIC) is described. With the aid of a specially designed ultra‐stable and high fidelity imaging system, which integrates a high‐quality bilateral telecentric lens with monochromatic blue light illumination, surface images of multiple samples heated by a heating furnace can be captured simultaneously. The images of these samples at different temperatures are processed by advanced DIC algorithm to extract the thermal strains and the CTE of isotropic Al alloy, anisotropic unidirectional, and bidirectional carbon fibre–epoxy composites. Pure thermal expansions of these samples obtained after removing the small rigid‐body rotations clearly indicate the isotropic and anisotropic expansions of these samples. The well‐agreed results with literature values demonstrate the effectiveness and practicality of the proposed method for high‐throughput and high‐accuracy CTE measurements.     

In‐situ microscopic analysis of ferritic ductile iron during tensile loading: Relation between matrix heterogeneities and damage mechanisms

An in‐situ microscopic analysis of the damage mechanisms of ferritic ductile iron under uniaxial tensile testing is carried out in this work. The experimental methodology combines specialized metallography techniques, in‐situ optical microscopy observation during loading, and digital image correlation analysis to obtain the strain values at the microscopic level. The results show that the crack initiation is preferentially located at matrix‐nodule interface and ferritic grain boundaries. The propagation of multiple cracks across the internodular ligaments that later coalesce into a single dominant crack is responsible for the final fracture. Noticeably, despite the heterogeneous nature of the Last‐to‐Freeze zones, the cracks propagate avoiding these microsegregated areas. The results provide new insights for the better understanding of the influence of ferritic ductile iron heterogeneities on the fracture process for quasistatic uniaxial loading.     

A two‐scale approximation of the Schur complement and its use for non‐intrusive coupling

This paper presents a two‐scale approximation of the Schur complement of a subdomain's stiffness matrix, obtained by combining local (i.e. element strips) and global (i.e. homogenized) contributions. This approximation is used in the context of a coupling strategy that is designed to embed local plasticity and geometric details into a small region of a large linear elastic structure; the strategy consists in creating a local model that contains the desired features of the concerned region and then substituting it into the global problem by the means of a non‐intrusive solver coupling technique adapted from domain decomposition methods. Using the two‐scale approximation of the Schur complement as a Robin condition on the local model enables to reach high efficiency. Examples include a large 3D problem provided by our industrial partner Snecma. Copyright © 2011 John Wiley & Sons, Ltd.     

Evaluating the use of digital image correlation for strain measurement in historic tapestries using representative deformation fields

An analysis technique to assess the viability of digital image correlation (DIC) in tracking the full‐field strains across the surface of hanging historic tapestries is presented. Measurement uncertainty related to the use of the inherent tapestry image in tracking displacements is investigated through use of “synthetic” deformation fields. The latter are generated by mapping the details of a given tapestry image into finite element analyses. The combination of self‐weight loading, material non‐linearity, and image specific heterogeneity (related to slit stitching, damage, and patch‐restorations) serve to generate a bespoke deformation field complex enough to assess the reliability of DIC measurements. Accuracy is evaluated by comparing measured results with the original known deformations. The technique demonstrates that the optimum imaging settings and the choice of subset size for DIC analysis are strongly influenced by the tapestry image and the goal of the measurement, they are found using a compromise between conflicting objectives: minimising measurement error while maximising resolution.     

Metallurgical investigations and corrosion behavior of failed weld joint in AISI 1518 low carbon steel pipeline

In this paper, failure analysis was carried out based on the available documents, metallographic studies and corrosion behavior of the welded joint pipe sample made of AISI 1518 low carbon steel. Nondestructive evaluations including penetration test (PT) and radiographic test (RT) were performed on the as-received pipeline and results indicated the presence of micro- and macro-cracks. Optical microscopic images and scanning electron microscopy (SEM) micrographs revealed various microstructures in the base metal (BM), heat affected zone (HAZ) and weld metal (WM). The microstructural variations may result in galvanic feature and lead to failure and rupture of the weld joint during the service. Microhardness measurements showed that hardness value was about 260 HV in the WM, while it declined in the HAZ and BM. Qualitative chemical analyses such as X-ray diffraction pattern (XRD) and SEM equipped with energy dispersive spectroscopy (EDS) confirmed the presence of corrosive media during weld joint rupture. Additionally, SEM and optical investigations indicated that micro-cracks were formed in HAZ due to residual stress as a consequence of improper welding condition. Surface fracture studies showed that the crack initiation, crack growth and finally crack propagation took place in the WM/HAZ interface. Electrochemical studies were conducted on the BM, HAZ and WM to investigate corrosion behavior of the failed joint sample. Finally, a proper corrosion mechanism is proposed based on the failure analyses and electrochemical studies.     

An investigation of the fatigue of CK45 steel in the as‐received state and after pre‐fatigue deformation

Fatigue failure, ratcheting behaviour and influence of pre‐fatigue on fatigue behaviour were investigated under uniaxial cyclic loading for CK45 steel at room temperature. The fatigue life was recorded for various stress ratios, and then, three mean stress models were considered. The Walker model showed an acceptable accuracy in comparison with Smith–Watson–Topper and Park et al. models. The ratcheting strains were measured for various loading conditions in order to evaluate the impact of mean stress, stress amplitude and stress ratio on ratcheting behaviour. The experimental results showed that the ratcheting strain increased with increasing mean stress, stress amplitude and stress ratio. In addition, the results of the post‐ratcheting‐fatigue tests showed that although the fatigue life decreased with increasing pre‐ratcheting strain (the ratcheting strain that is accumulated in pre‐fatigue), the loading condition that pre‐fatigue experiments were conducted has a significant effect on subsequent fatigue behaviour.     

Comparison of the Mechanical Behaviour of Standard and Auxetic Foams by X‐ray Computed Tomography and Digital Volume Correlation

The tensile behaviour of standard and auxetic polyurethane foams are contrasted by digital volume correlation of 3D images collected by in situ X‐ray computed tomography (CT). It was found that subset sizes of 32 and 64 voxels for the auxetic and standard foams were optimal for strain resolutions in the order of 0.1%. For the standard foam, good uniformity of strain was observed at low strains giving a tangent Poisson's ratio of 0.5. Some heterogeneity of strain was observed at higher strains, which may be related to the fixtures. The behaviour of the auxetic foam was totally different, with strain being spatially heterogeneous with transverse strains both positive and negative but giving a negative Poisson's ratio on average. This suggests that the unfolding tendency of some groups of cells was higher than others because of the complex frozen starting microstructure. Further different methods of deriving Poisson's ratio gave different results. Besides revealing interesting microstuctural mechanisms of transverse straining, the study also shows digital volume correlation of tomography sequences to be the perfect tool to study complex mechanical behaviour of cellular materials.     

Finite element simulation of the residual stresses in high strength carbon steel butt weld incorporating solid-state phase transformation

This paper presents a sequentially coupled three-dimensional (3-D) thermal, metallurgical and mechanical finite element (FE) model to simulate welding residual stresses in high strength carbon steel butt weld considering solid-state phase transformation effects. The effects of phase transformation during welding on residual stress evolution are modeled by allowing for volumetric changes and the associated changes in yield stress due to austenitic and martensitic transformations. In the FE model, phase transformation plasticity is also taken into account. Moreover, preheat and inter-pass temperature are included in the modeling process. Based on the FE model, the effects of solid-state phase transformation on welding residual stresses are investigated. The results indicate the importance of incorporating solid-state phase transformation in the simulation of welding residual stresses in high strength carbon steel butt weld.     

Modelling the Effects of Geometric Imperfections on the Buckling and Initial Post‐buckling Behaviour of Flat Plates Under Compression Using Measured Data

Abstract: Recent developments in optical techniques have allowed accurate representations of the geometry of test specimens to be obtained. These enable the nature of geometric imperfections resulting from manufacture or set‐up to be captured in a form which allows them to be incorporated directly into models generated to predict the behaviour of the structure. This study examines the effect of such imperfections on the behaviour of a series of panels, simply supported along all four edges, and subject to uniaxial compressive in‐plane loading. In each case, digital image correlation is used to determine the initial profile and set‐up of the panel and to monitor its behaviour during test. The data are used to automatically generate a series of meshes representative of each of the specimens tested, suitable for finite element analysis. Comparison of the results obtained from these analyses with those found during the experiments modelled shows an improved correlation when compared with standard techniques for assessing imperfection sensitivity. Set‐up is straightforward, and models can be obtained quickly based on the data collected.     

Application of 3D-DIC to characterize the effect of aggregate size and volume on non-uniform shrinkage strain distribution in concrete

To elucidate the effect of aggregate size and volume on the non-uniform strain distribution in concrete, drying shrinkage of mortar and concretes were determined with 3D digital image correlation (3D-DIC). The distribution of shrinkage displacements and strains in mortar and concrete were analyzed. The results show that 3D-DIC makes it possible to measure non-uniform displacement distributions initiated by shrinkage in mortar and concrete. The non-uniformity became more remarkable with drying time. The presence of aggregates larger than 5 mm in concrete have locally changed the displacement and strain fields. Aggregates within 5–25 mm make non-uniform strain of concrete more fluctuant, especially when the aggregate size is larger than 10 mm. The maximum and minimum principal strain distributions became more heterogeneous with decreasing volume of aggregates.     

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.     

传统风格建筑钢结构双梁-柱中节点抗震性能试验研究及有限元分析

设计并制作了4个比例为1:2的传统风格建筑钢结构双梁-柱中节点模型,通过低周反复荷载试验研究其抗震性能,得到了传统风格建筑钢结构双梁-柱中节点的破坏形态、滞回特性、延性等指标。研究表明:传统风格建筑钢结构双梁-柱中节点试件在加载过程中形成了上、下2个小核心区和中间短柱三个区域,其典型破坏形态是节点下核心区在压剪复合应力作用下的剪切破坏。试件变形以下核心区最大,其次为中间短柱。基于试验研究,利用有限元软件ABAQUS对其进行了非线性数值模拟分析,得到的节点破坏形态、应力分布及滞回曲线与试验结果较为吻合。