Micropolar hyper‐elastoplasticity: constitutive model,consistent linearization,and simulation of 3D scale effects

A computational model for micropolar hyperelastic‐based finite elastoplasticity that incorporates isotropic hardening is developed. The basic concepts of the non‐linear micropolar kinematic framework are reviewed, and a thermodynamically consistent constitutive model that features Neo‐Hooke‐type elasticity and generalized von Mises plasticity is described. The integration of the constitutive initial value problem is carried out by means of an elastic‐predictor/plastic‐corrector algorithm, which retains plastic incompressibility. The solution procedure is developed carefully and described in detail. The consistent material tangent is derived. The micropolar constitutive model is implemented in an implicit finite element framework. The numerical example of a notched cylindrical bar subjected to large axial displacements and large twist angles is presented. The results of the finite element simulations demonstrate (i) that the methodology is capable of capturing the size effect in three‐dimensional elastoplastic solids in the finite strain regime, (ii) that the formulation possesses a regularizing effect in the presence of strain localization, and (iii) that asymptotically quadratic convergence rates of the Newton–Raphson procedure are achieved. Throughout this paper, effort is made to present the developments as a direct extension of standard finite deformation computational plasticity. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiscale prediction of mechanical behavior of ferrite–pearlite steel with numerical material testing

Multiscale mechanical behaviors of ferrite–pearlite steel were predicted using numerical material testing (NMT) based on the finite element method. The microstructure of ferrite–pearlite steel is regarded as a two‐component aggregate of ferrite crystal grains and pearlite colonies. In NMT, the macroscopic stress–strain curve and the deformation state of the microstructure were examined by means of a two‐scale finite element analysis method based on the framework of the mathematical homogenization theory. The microstructure of ferrite–pearlite steel was modeled with finite elements, and constitutive models for ferrite crystal grains and pearlite colonies were prepared to describe their anisotropic mechanical behavior at the microscale level. While the anisotropic linear elasticity and the single crystal plasticity based on representative characteristic length have been employed for the ferrite crystal grains, the constitutive model of a pearlite colony was newly developed in this study. For that reason, the constitutive behavior of the pearlite colony was investigated using NMT on a smaller scale than the scale of the ferrite–pearlite microstructure, with the microstructure of the pearlite colony modeled as a lamellar structure of ferrite and cementite phases with finite elements. On the basis of the numerical results, the anisotropic constitutive model of the pearlite colony was formulated based on the normal vector of the lamella. The components of the anisotropic elasticity were estimated with NMT based on the finite element method, where the elasticity of the cementite phase was numerically evaluated with a first‐principles calculation. Also, an anisotropic plastic constitutive model for the pearlite colony was formulated with two‐surface plasticity consisting of yield functions for the interlamellar shear mode and yielding of the overall lamellar structure. After addressing the microscopic modeling of ferrite–pearlite steel, NMT was performed with the finite element models of the ferrite–pearlite microstructure and with the microscopic constitutive models for each of the components. Finally, the results were compared with the corresponding experimental results on both the macroscopic response and the microscopic deformation state to ascertain the validity of the numerical modeling. Copyright © 2011 John Wiley & Sons, Ltd.     

Derivation of 3D masonry properties using numerical homogenization technique

Lots of research work has been conducted on homogenization technique, which derives global homogenized properties of masonry from the behaviour of the constitutive materials (brick and mortar). Such a technique mainly focused on two‐dimensional media in the previous studies with the out‐of‐plane properties of masonry material neglected. In this paper, homogenization technique and damage mechanics theory are used to model a three‐dimensional masonry basic cell to numerically derive the equivalent elastic properties, strength envelope, and failure characteristics of masonry material. The basic cell is modelled with distinctive consideration of non‐linear material properties of mortar and brick. Various displacement boundaries are applied on the basic cell surfaces in the numerical simulation. The detailed material properties of mortar and brick are modelled in a finite element program in the numerical analysis. The stress–strain relations of masonry material under various conditions are obtained from the simulation. The homogenized elastic properties and failure characteristics of masonry material are derived from the simulation results. The homogenized 3D model is then utilized to analyse the response of a masonry panel to airblast loads. The same panel is also analysed with distinctive material modelling. The efficiency and accuracy of the homogenized model are demonstrated. The homogenized material properties and failure model can be used to model large‐scale masonry structure response. Copyright © 2006 John Wiley & Sons, Ltd.     

Constitutive description of Bulk Metallic Glass composites at high homologous temperatures

Experimental stress–strain responses of La-based in situ Bulk Metallic Glass (BMG) composites within the supercooled liquid region reveal initial post-yield hardening, followed by softening and subsequent strain-hardening. This behavior contrasts with that of monolithic La-based BMGs, which reach a steady stress level after an initial overshoot. XRD analysis of BMG composites shows the formation of intermetallic compounds during compressive deformation. These intermetallic compound formation/interactions are associated with storage of energy in the material and affect the stress–strain response. In this study, an elastic–viscoplastic, three-dimensional, finite deformation constitutive model is also established to describe the behavior of a recently developed La-based in situ BMG (La–Al–Cu–Ni) composite, within the supercooled liquid region, at ambient pressure and a range of strain rates. The constitutive model is incorporated into a finite element program (ABAQUS/Explicit) via a user-defined material subroutine. Numerical predictions are compared with compression test results on BMG composites cast in-house. The comparison shows that the model is able to describe the material behavior observed.     

Nonlinear fracture mechanics and plasticity of the split cylinder test

The split cylinder test is subjected to an analysis combining nonlinear fracture mechanics and plasticity. The fictitious crack model is applied for the analysis of splitting tensile fracture, and the Mohr-Coulomb yield criterion is adopted for modelling the compressive crushing/sliding failure. Two models are presented, a simple semi-analytical model based on analytical solutions for the crack propagation in a rectangular prismatic body, and a finite element model including plasticity in bulk material as well as crack propagation in interface elements. A numerical study applying these models demonstrates the influence of varying geometry or constitutive properties. For a split cylinder test in load control it is shown how the ultimate load is either plasticity dominated or fracture mechanics dominated. The transition between the two modes is related to changes in geometry or constitutive properties. This implies that the linear elastic interpretation of the ultimate splitting force in term of the uniaxial tensile strength of the material is only valid for special situations, e.g. for very large cylinders. Furthermore, the numerical analysis suggests that the split cylinder test is not well suited for determining the tensile strength of early age or fibre reinforced concrete.     

Computational applications of a coupled plasticity-damage constitutive model for simulating plain concrete fracture

A coupled plasticity-damage model for plain concrete is presented in this paper. Based on continuum damage mechanics (CDM), an isotropic and anisotropic damage model coupled with a plasticity model is proposed in order to effectively predict and simulate plain concrete fracture. Two different damage evolution laws for both tension and compression are formulated for a more accurate prediction of the plain concrete behavior. In order to derive the constitutive equations and for the easiness in the numerical implementation, in the CDM framework the strain equivalence hypothesis is adopted such that the strain in the effective (undamaged) configuration is equivalent to the strain in the nominal (damaged) configuration. The proposed constitutive model has been shown to satisfy the thermodynamics requirements. Detailed numerical algorithms are developed for the finite element implementation of the proposed coupled plasticity-damage model. The numerical algorithm is coded using the user subroutine UMAT and then implemented in the commercial finite element analysis program Abaqus. Special emphasis is placed on identifying the plasticity and damage model material parameters from loading-unloading uniaxial test results. The overall performance of the proposed model is verified by comparing the model predictions to various experimental data, such as monotonic uniaxial tension and compression tests, monotonic biaxial compression test, loading-unloading uniaxial tensile and compressive tests, and mixed-mode fracture tests.     

Modeling bamboo as a functionally graded material: lessons for the analysis of affordable materials

Natural fibers are promising for engineering applications due to their low cost. They are abundantly available in tropical and subtropical regions of the world, and they can be employed as construction materials. Among natural fibers, bamboo has been widely used for housing construction around the world. Bamboo is an optimized composite that exploits the concept of Functionally Graded Material (FGM). Biological structures such as bamboo have complicated microstructural shapes and material distribution, and thus the use of numerical methods such as the finite element method, and multiscale methods such as homogenization, can help to further understanding of the mechanical behavior of these materials. The objective of this work is to explore techniques such as the finite element method and homogenization to investigate the structural behavior of bamboo. The finite element formulation uses graded finite elements to capture the varying material distribution through the bamboo wall. To observe bamboo behavior under applied loads, simulations are conducted under multiple considerations such as a spatially varying Young’s modulus, an averaged Young’s modulus, and orthotropic constitutive properties obtained from homogenization theory. The homogenization procedure uses effective, axisymmetric properties estimated from the spatially varying bamboo composite. Three-dimensional models of bamboo cells were built and simulated under tension, torsion, and bending load cases.     

Modelling of overall plastic deformation in rubber-toughened polymers

Summary. In this paper, we provide a constitutive model for overall (macroscopic) plastic deformation behavior in a rubber-toughened polymer blend. A porous plasticity theory is employed as a basis for the constitutive modeling. In our investigation, the polycarbonate (PC) is chosen as a matrix material of polymer blend. First, the true uniaxial stress-strain relation for PC, which is an important part of the constitutive model, is carefully measured. Secondly, finite element analyses of neck propagation in a tensile specimen of PC are performed to test the efficiency of the introduction of the accurately measured true stress-strain relation into the model. Then, in order to investigate local and average deformation behavior of the matrix material (PC) around cavitated rubber particles in polymer blend, an axisymmetric unit cell analysis is carried out. Finally, finite element analyses of the neck propagation in a tensile specimen of a rubber-toughened PC are performed, and the numerical results are compared to experimental results. It is revealed that the present constitutive model has the ability to well reproduce the behavior of a rubber-toughened polymer blend with rather small volume fraction of rubber particles, which is up to about 10%. However, for blends with larger volume fraction of the rubber particles, the discrepancy between the computational and the experimental results increases. Several possibilities of enhancing the model are discussed.     

Multiscale finite element modeling of failure process of composite laminates

A multiscale nonlinear finite element modeling technique is developed in this paper to predict the progressive failure process for composite laminates. A micromechanical elastic–plastic bridging constitutive model, which considers the nonlinear material properties of the constituent fiber and matrix materials and their interaction and the damage and failure in fibrous composites at the fiber and matrix level, is proposed to represent the material behavior of fiber-reinforced composite laminates. The micromechanics constitutive model is employed in the macroscale finite element analysis of structural behavior especially progressive failure process of the fiber-reinforced composites based on a 4-node 24-DOF shear-locking free rectangular composite plate element.     

Concepts for Integrating Plastic Anisotropy into Metal Forming Simulations

Modern metal forming and crash simulations are usually based on the finite element method. Aims of such simulations are typically the prediction of the material shape, failure, and mechanical properties during deformation. Further goals lie in the computer assisted lay‐out of manufacturing tools used for intricate processing steps. Any such simulation requires that the material under investigation is specified in terms of its respective constitutive behavior. Modern finite element simulations typically use three sets of material input data, covering hardening, forming limits, and anisotropy. The current article is about the latter aspect. It reviews different empirical and physically based concepts for the integration of the elastic‐plastic anisotropy into metal forming finite element simulations. Particular pronunciation is placed on the discussion of the crystallographic anisotropy of polycrystalline material rather than on aspects associated with topological or morphological microstructure anisotropy. The reviewed anisotropy concepts are empirical yield surface approximations, yield surface formulations based on crystallographic homogenization theory, combinations of finite element and homogenization approaches, the crystal plasticity finite element method, and the recently introduced texture component crystal plasticity finite element method. The paper presents the basic physical approaches behind the different methods and discusses engineering aspects such as scalability, flexibility, and texture update in the course of a forming simulation.     

Continuum coupled cohesive zone elements for analysis of fracture in solid bodies

Embedding cohesive surfaces into finite element models is a widely used technique for the numerical simulation of material separation (i.e. crack propagation). Typically, a traction-separation law is specified that relates the magnitude of the cohesive traction to the distance between the separating surfaces. Thus the characterization of fracture in such models is not directly coupled to the bulk constitutive response, in the sense that the cohesive traction does not explicitly depend on material stretching in the plane of the fracture surface. In this work, an initially-rigid cohesive-traction formulation that is coupled to the surrounding continuum is introduced as a further development of the cohesive zone idea. In this model, the traction-separation law - and therefore the fracture phenomenology - derives directly from the bulk constitutive law. The immediate goal is an improved cohesive zone framework that naturally and logically initiates cohesive separation behavior, and couples its evolution to the material state in the region of the crack tip. A cohesive element based on this model is implemented in an explicit three-dimensional finite element code. Proof-of-concept analyses using both linear elastic and Gurson void growth constitutive relations are presented. A three-point bend simulation is found to give good agreement with experimental results.     

Quantitative re-examination of Taylor model for FCC polycrystals

The quantitative adequacy of the Taylor model for representing the behaviors of FCC polycrystals is discussed through comparison with crystal plasticity analysis using the homogenization-based finite method. The key element of the crystal plasticity theory is the constitutive relation for single crystals. The most classical way to apply it to polycrystals is the Taylor model. This model assumes that all crystal grains in a crystal aggregate are subjected to the same strain under macroscopically uniform deformation. This assumption provides a solution satisfying the continuity of displacement between crystal grains. The effect and evolution of the crystallographic texture can easily be taken into account. However, the assumption of uniform strain, the main idea in the Taylor model, has never been validated quantitatively. On the other hand, the homogenization-based finite element method can represent arbitrary microscopic deformations, i.e., each crystal grain may have nonuniform deformation, and can provide a material response under more realistic boundary conditions. In this paper, we first determine the appropriate size for the representative volume element (RVE) in the homogenization-based finite element method that can represent the macroscopic polycrystalline behavior of FCC. After that, the polycrystalline behaviors obtained using the Taylor model are compared with those obtained using the homogenization-based finite element method. Finally, the quantitative adequacy of the Taylor model is discussed. It is clarified that the Taylor model is qualitatively consistent with the homogenization-based finite element method and can be used as a practical model of polycrystalline FCC metals for a first-order approximation, although it is not quantitatively reasonable even for FCC metals.     

Numerical simulation of the plastic flow properties of iron single crystals

Summary The present paper deals with the numerical simulation of the plastic flow properties of iron single crystals as well as their influence on the macroscopic elastic-plastic deformation and localization behavior affected by superimposed hydrostatic pressure. Based on experimental observations the onset of plastic yielding on the microscale is described by an extended microscopic yield condition taking into account various microscopic stress components acting on the respective slip systems. In addition, to be able to compute inelastic deformations from a plastic potential, the latter is expressed in terms of workconjugate microscopic stress and strain measures which leads to a non-associated flow rule for the macroscopic plastic strain rate. On the numerical side, generalized functions for constitutive parameters will be used to be able to simulate the single crystal's microscopic deformation behavior observed in experiments. Estimates of the current microscopic stresses and strains are obtained via an efficient and remarkably stable plastic predictor-elastic corrector technique which is incorporated into a nonlinear finite element program. Numerical simulations of uniaxial tests show quantitatively the influence of hydrostatic pressure on current material data. Further numerical studies on the additional constitutive non-Schmid terms elucidate their effect on iron single crystal's macroscopic deformation and localization behavior.     

A rate-dependent constitutive equation for 5052 aluminum diaphragms

In this article, both experimental and numerical approaches are conducted to present a constitutive equation for 5052 aluminum diaphragms under quasi-static strain rate loadings. For this purpose the stress–strain curves at different strain rates are obtained using tensile tests. Brittle behavior during tensile tests is observed due to samples thin thicknesses. Employing Johnson–Cook constitutive equation no yields in reasonable agreement with these tensile tests results. Therefore, developing a more suitable constitutive equation for aluminum diaphragms is taken into consideration. This equation is then implemented into the commercial finite element software, ABAQUS, via a developed user material (UMAT) subroutine utilizing von Mises plasticity theory and an own solution algorithm. A single-element pathological test method is adopted to show the well-development of the UMAT subroutine. In order to verify the proposed constitutive equation for precision predicting of mechanical behavior, a bulge test is performed in which demonstrates a good agreement between experimental and numerical results.     

Non-Gaussian simulation of local material properties based on a moving-window technique

In this paper, a moving-window micromechanics technique, Monte Carlo simulation, and finite element analysis are used to assess the effects of microstructural randomness on the local stress response of composite materials. The randomly varying elastic properties are characterized in terms of a field of local effective elastic constitutive matrices using a moving-window technique based on a finite element model of a given digitized composite material microstructure. Once the fields are generated, estimates of the random properties are obtained for use as input to a simulation algorithm that was developed to retain spectral, correlation, and non-Gaussian probabilistic characteristics. Rapidly generated Monte Carlo simulations of the constitutive matrix fields are used in a finite element analysis to create a series of local stress fields associated with the random material sample under uniaxial tension. This series allows estimation of the statistical variability in the local stress response for the random composite. The identification of localized extreme stress deviations from those of the aggregate or effective properties approach highlight the importance of modeling the stochastic variability of the microstructure.     

Finite heterogeneous element method using sliced microstructures for linear elastic analysis

A new finite heterogeneous element consisting of sliced microstructures (FHES) is applied in a multi?scale technique. The FHES represents a heterogeneous material with microscopic constituents without homogenization or microscopic finite element analysis. A representative volume element extracted from a heterogeneous structure is thinly sliced. Each slice is modeled as a combined spring to calculate properties of the FHES. Each FHES has the same number of nodes as an ordinary finite element, and the macroscopic analysis cost is the same as that for ordinary finite element analysis. However, the FHES retains information about the microscopic material layout (i.e., the distribution of a material's property) in itself that is lost during homogenization. In the proposed approach, materials are not homogenized. The FHES does not have a constant (homogenized) material property and can ‘change stiffness’ depending on its deformation behavior. This reduces error due to coarse?graining and allows us to calculate the macroscopic deformation behavior with sufficient accuracy even if a large gradient of strain is generated in the macroscopic field. The novelty of the research is the development of rational heterogeneous finite elements. The paper presents the theory behind the FHES and its practical application to a linear elastic problem. Copyright © 2014 John Wiley & Sons, Ltd.     

三维四向编织CMCs拉伸性能及损伤演化数值预测

发展了一种能够预测三维四向编织陶瓷基复合材料(3D-B-CMCs)拉伸模量与强度以及损伤演化过程的数值计算方法.首先,利用复合圆柱(CCA)和全局载荷分担(GLS)两种模型预测了纤维束的弹性模量和拉伸强度;然后,利用微焦点CT技术建立了能够反映3D-B-CMCs真实编织几何结构的胞元模型;其次,采用Hashin纤维束失效模型以及考虑单元尺寸的各向异性损伤力学本构模型,编制了ABAQUS/UMAT子程序,对3D-B-CMCs材料宏观拉伸的整个过程进行了计算模拟,预测了宏观拉伸应力-应变曲线,并与试验结果相吻合,证明了所建立方法的合理性和UMAT程序的有效性.同时,研究和讨论了拉伸过程中材料内部不同的损伤破坏模式对复合材料整体力学性能的影响,为材料的疲劳和蠕变等力学行为的内部损伤演化提供了依据.     

基于弹塑性损伤本构模型的复合材料层合板破坏荷载预测

在连续损伤力学和塑性力学框架内,建立一个同时考虑塑性效应和损伤累积导致材料属性退化的复合材料弹塑性损伤本构模型。基于最近点投影回映算法,开发本构模型的应变驱动隐式积分算法以更新应力及与解答相关的状态变量,并推导与所开发算法相应的数值一致性切线刚度矩阵,保证有限元分析采用NewtonRaphson迭代法解答非线性问题的计算效率。采用断裂带模型对已开发的本构模型软化段进行规则化,以减轻有限元分析结果的网格相关性问题。对损伤变量进行粘滞规则化,并推导出相应的粘滞规则化数值一致性切线刚度张量,解决了在有限元隐式计算程序中采用含应变软化段本构关系的数值分析由于计算困难而提前终止的问题。开发包含数值积分算法的用户材料子程序UMAT,并嵌于有限元程序Abaqus v6.14中。通过对力学行为展现显著塑性效应的AS4/3501-6V型开口复合材料层合板的渐进失效分析,验证本文提出的材料本构模型的有效性。结果显示,预测结果与已报道的试验结果吻合良好,并且预测精度高于其他已有弹性损伤模型。表明已建立的弹塑性损伤本构模型能够准确预测力学行为,展现显著塑性效应的复合材料层合板的破坏荷载,为其构件和结构设计提供一种有效的分析方法。     

Multilayered shell finite element with interlaminar continuous shear stresses: a refinement of the Reissner–Mindlin formulation

A finite element formulation for refined linear analysis of multilayered shell structures of moderate thickness is presented. An underlying shell model is a direct extension of the first‐order shear‐deformation theory of Reissner–Mindlin type. A refined theory with seven unknown kinematic fields is developed: (i) by introducing an assumption of a zig‐zag (i.e. layer‐wise linear) variation of displacement field through the thickness, and (ii) by assuming an independent transverse shear stress fields in each layer in the framework of Reissner's mixed variational principle. The introduced transverse shear stress unknowns are eliminated on the cross‐section level. At this process, the interlaminar equilibrium conditions (i.e. the interlaminar shear stress continuity conditions) are imposed. As a result, the weak form of constitutive equations (the so‐called weak form of Hooke's law) is obtained for the transverse strains–transverse stress resultants relation. A finite element approximation is based on the four‐noded isoparametric element. To eliminate the shear locking effect, the assumed strain variational concept is used. Performance of the derived finite element is illustrated with some numerical examples. The results are compared with the exact three‐dimensional solutions, as well as with the analytical and numerical solutions obtained by the classical, the first‐order and some representative refined models. Copyright © 2000 John Wiley & Sons, Ltd.     

Industrial challenges for finite element and multiscale methods for material modeling

The automotive industry promotes lightweight design to reduce the CO2-emission and enhances the passenger’s safety using high strength steel grades. One limiting factor to the accuracy of modern stamping simulation are the empirical constitutive models. In particular for high strength multiphase steels the modelling techniques like multi-scale methods are becoming more interesting. However they should meet the industrial needs. Not only the accuracy but also features like time, costs and complexity are rapidly increasing. The challenge is the development of finite element technologies and multi-scale methods in an appropriate framework for industrial projects. The crystal plasticity finite element method bridges the gap between the micro level and macroscopic mechanical properties that opens the way for more profound consideration of metal anisotropy in stamping process simulation. Nevertheless new empirical constitutive models are favourable for spring back prediction in forming simulations, even if the number of material parameters and the amount of tests for their identification increases. In this paper the application of crystal plasticity FEM within the concept of virtual material testing with a representative volume element (RVE) is demonstrated.