Assessment of crystal plasticity based calculation of the lattice spin of polycrystalline metals for FE implementation

When texture is incorporated in the finite element simulation of a metal forming process, much computer time can be saved by replacing continuous texture and corresponding yield locus updates by intermittent updates after strain intervals of e.g. 20%. The hypothesis that the evolution of the anisotropic properties of a polycrystalline material during such finite interval of plastic deformation can be modelled by just rotating the initial texture instead of continuously updating it by means of a polycrystal deformation model is tested in this work. Two spins for rotating the frame have been assessed: the classical rigid body spin and a crystal plasticity based “Mandel spin” (calculated from the rotated initial texture) which is the average of the spins of all the crystal lattices of the polycrystal. Each of these methods was used to study the evolution of the yield locus and the r-value distribution during the 20% strain interval. The results were compared to those obtained by simulating the texture evolution continuously using a polycrystal deformation model. When the texture was not updated during deformation, it was found that for most initial textures the Mandel spin does not perform better than the rigid body spin, except for some special initial textures for which the Mandel spin is much better. The latter ones are textures which are almost stable for the corresponding strain mode. When the texture was updated after each strain interval of e.g. 20% the Mandel spin performed much better than the rigid body spin.     

Elasto-plastic material parameter identification by inverse methods: Calculation of the sensitivity matrix

Inverse methods offer a powerful tool for the identification of elasto-plastic material properties of metals. The basic principle of the inverse method we are studying, is to compare an experimentally measured strain field with a strain field computed by a finite element (FE) model. The material parameters in the FE model are iteratively tuned in such a way that the numerically computed strain field matches the experimentally measured field as closely as possible. One of the building blocks in this identification procedure is the optimization algorithm for the material parameters in the numerical model. The key problem of this optimization algorithm is the determination of a sensitivity matrix, which expresses the sensitivities of the strains with respect to the material parameters. This paper presents an analytical method for the calculation of this sensitivity matrix in the case of a tensile test with non-rotating principal axes of strain.     

Modeling the effects of material non-linearity using moving window micromechanics

In the analysis of materials with random heterogeneous microstructure the assumption is often made that material behavior can be represented by homogenized or effective properties. While this assumption yields accurate results for the bulk behavior of composite materials, it ignores the effects of the random microstructure. The spatial variations in these microstructures can focus, initiate and propagate localized non-linear behavior, subsequent damage and failure. In previous work a computational method, moving window micromechanics (MW), was used to capture microstructural detail and characterize the variability of the local and global elastic response. Digital images of material microstructure described the microstructure and a local micromechanical analysis was used to generate spatially varying material property fields. The strengths of this approach are that the material property fields can be consistently developed from digital images of real microstructures, they are easy to import into finite element models (FE) using regular grids, and their statistical characterizations can provide the basis for simulations further characterizing stochastic response. In this work, the moving window micromechanics technique was used to generate material property fields characterizing the non-linear behavior of random materials under plastic yielding; specifically yield stress and hardening slope, post yield. The complete set of material property fields were input into FE models of uniaxial loading. Global stress strain curves from the FE–MW model were compared to a more traditional micromechanics model, the generalized method of cells. Local plastic strain and local stress fields were produced which correlate well to the microstructure. The FE–MW method qualitatively captures the inelastic behavior, based on a non-linear flow rule, of the sample continuous fiber composites in transverse uniaxial loading.     

Modelling and simulation of process-integrated powder coating by radial axial rolling of rings

In this paper, we present finite element (FE) process simulations of a new production method used to coat ring-shaped work pieces with functional layers. Similarly to the conventionally applied hot isostatic pressing (HIP), this new coating method is based on powder metallurgy. It is expected to overcome some important drawbacks by integrating the consolidation of the powdery layer material into the hot rolling of the substrate ring. This makes HIP dispensable. Nevertheless several difficulties arise through the process integration. E.g., the presence of the compactable layer requires a different handling of the rolling stage compared with classical ring rolling. FE simulations shall support the design of this new process in order to investigate critical process parameters. For this purpose, new finite element modules have to be developed. A crucial point is the adequate modelling of the layer material. In this regard, we present a rate-dependent finite strain material model that describes the consolidation of the layer material in a thermodynamically consistent way. Moreover, FE process simulations of the new production method are presented and compared with experimental results.     

A methodology for measuring and modeling crystallographic texture gradients in processed alloys

The explicit representation of internal material structure in alloy processing and in-service performance simulations is becoming increasingly prevalent. This paper presents a methodology for characterizing and representing a spatially-varying orientation distribution function (ODF) that can be used in processing and performance simulations for alloys containing texture gradients. We use thick AA 7050 aluminum plate, which is known to contain texture gradients, as a case study to demonstrate the methodology, which employs a finite element representation of the ODF initialized using individual lattice orientation measurements taken using the electron backscatter pattern (EBSP) technique. As expected, we find that the texture varies significantly through the plate thickness. We use the ODF to examine the effect of the varying texture on the resulting yield strength distribution as embodied by the average Taylor factor. We find that the predicted yield strength anisotropy is different at different locations through the thickness of the plate. We examine the optimal number of orientation measurements necessary for determining the ODF in the presence of this texture gradient. We find that as we increase the number of orientations, the ODF quickly becomes stable but eventually starts to change under the influence of the texture gradient. We also investigate spatial interpolation of the ODF using the finite element representation. We find that, as with finite element representations of other fields, interpolation accuracy depends on the variation of the field variable and the discretization of the domain. In this case, gradients in both physical space and orientation space affect the accuracy of the interpolation. Finally, the effects of the texture gradient on the mechanical response of the material is demonstrated by employing the ODFs taken from various locations through the thickness of the plate in polycrystal plasticity simulations of uniaxial tension and plane strain compression.     

Using texture components in crystal plasticity finite element simulations

We discuss methods to map crystallographic textures in crystal plasticity finite element simulations. Fourier-type series expansion methods which use spherical harmonic library functions as well as the direct pole figure inversion methods are not well suited to reproduce texture information in a sufficiently localized spherical form onto finite element grids. Mathematically compact Gauss-shaped spherical texture components represent a better approach for including textures in finite element models since they represent an excellent compromise between discreteness (spherical localization), compactness (simple functions), mathematical precision (very good approximation also of complex orientation distribution functions already with small sets of texture components), scalability (the number of used texture components can be systematically varied according to the desired exactness of the texture fit), conceptual simplicity (simple mathematical handling), and physical significance (texture components can be directly linked to characteristic metallurgical mechanisms). The use of texture component functions has also advantages over the use of large sets of discrete single orientations with equal scatter and height since they are more compact, practical, and provide better physical insight into microstructural mechanisms and composition sensitive effects. The article presents a new approach for the mathematical reproduction of such crystallographic texture components in crystal plasticity finite element simulations. It explains in some detail why they are particularly suited for this purpose and how they can be used to map and recover textures in/from plasticity simulations.     

Anisotropic response of high-purity α-titanium: Experimental characterization and constitutive modeling

This paper presents a comprehensive experimental and theoretical investigation of the deformation behavior of high-purity, polycrystalline α-titanium under quasi-static conditions at room temperature. The initial material in this study was a cross-rolled plate with a strong basal texture. To quantify the plastic anisotropy and the tension–compression asymmetry of this material, monotonic tensile and compressive tests were conducted, on samples cut along different directions of the plate. A new anisotropic elastic/plastic model was developed to describe the quasi-static macroscopic response of the aggregate. Key in its formulation is the use of an anisotropic yield criterion that captures strength-differential effects and an anisotropic hardening rule that accounts for texture evolution associated to twinning. A very good agreement between FE simulations using the model developed and uniaxial data was obtained.     

横截面参数化网格变形算法及耐撞性优化应用

薄壁梁结构是汽车等运载工具的主要承载构件,提高该类结构的耐撞性对乘员安全具有重要意义。然而,形状优化设计要求多组有限元模型与仿真分析,因此需要特定的建模技术或人工交互。本文提出了一种基于横截面形状的参数化网格变形方法,以实现已有有限元模型的有效重用。以给定有限元模型为输入,采用基于各向异性径向基函数网格变形方法,并结合骨架内嵌空间,可快速生成适用于仿真分析的有限元模型变体。以S形梁轴向冲击耐撞性设计为例,采用所提方法改变构件塑性铰区域的横截面形状,可快速（低于4 s）获取100组局部变形有限元模型,并采用代理模型技术和多目标遗传算法优化结构耐撞性。数值结果显示,构件耐撞性获显著提高,验证了所提参数化变形方法的有效性,展示了与一般形状优化框架的可集成性。     

Contact laws between solid particles

This paper presents a comprehensive study for the contact laws between solid particles taking into account the effects of plasticity, strain hardening and very large deformation. The study takes advantage of the development of a so-called material point method (MPM) which requires neither remeshing for large deformation problems, nor iterative schemes to satisfy the contact boundary conditions. The numerical results show that the contact law is sensitive to impact velocity and material properties. The contact laws currently used in the discrete element simulations often ignore these factors and are therefore over-simplistic. For spherical particles made of elastic perfectly plastic material, the study shows that the contact law can be fully determined by knowing the relative impact velocity and the ratio between the effective elastic modulus and yield stress. For particles with strain hardening, the study shows that it is difficult to develop an analytical contact law. The same difficulty exists when dealing with particles of irregular shapes or made of heterogeneous materials. The problem can be overcome by using numerical contact laws which can be easily obtained using the material point method.     

Analysis of quantum-dot-induced strain and electric fields in piezoelectric semiconductors of general anisotropy

Characteristics of the self-organized quantum dots (QDs) such as electron and hole energy levels and wave functions are dependent to the state of strain and electric field produced during the growing process of QDs in a semiconductor substrate. The calculation of the strain and electric field is one of the most challenging components in the QDs simulation process. It involves material anisotropy induced coupling between the elastic and electric fields and it must include the full three-dimensional and usually intricate shapes of the QDs. Numerical simulations are often performed by finite difference, finite element, or atomistic techniques, all require substantial computational time and memory. In this paper, we present a new Green’s function approach which takes into account QDs of arbitrary shape and semiconductor substrates with the most general class of anisotropy and piezoelectricity. Following the literature of micromechanics, the problem is formulated as an Eshelby inclusion problem of which the solution can be expressed by a volume-integral equation that involves the Green’s functions and the equivalent body-force of eiegenstrain. The volume integral is subsequently reduced to a line integral based on exploiting a unique structure of the Green’s functions. The final equations are cast in a form that most of the computational results can be repeatedly used for QDs at different locations—a very attractive feature for simulating large systems of QD arrays. The proposed algorithm has been implemented and validated by comparison with analytical solutions. Numerical simulations are presented for pyramidal QDs in the substrates of gallium arsenide (GaAs) (0 0 1).     

Identification of Mechanical Material Behavior Through Inverse Modeling and DIC

Inverse methods offer a powerful tool for the identification of the elasto-plastic material parameters. One of the advantages with respect to classical material testing is the fact that those inverse methods are able to deal with heterogeneous deformation fields. The basic principle of the inverse method that is presented in this paper, is the comparison between experimentally measured strain fields and those computed by the finite element (FE) method. The unknown material parameters in the FE model are iteratively tuned so as to match the experimentally measured and the numerically computed strain fields as closely as possible. This paper describes the application of an inverse method for the identification of the hardening behavior and the yield locus of DC06 steel, based on a biaxial tensile test on a perforated cruciform specimen. The hardening behavior is described by a Swift type hardening law and the yield locus is modeled with a Hill 1948 yield surface.     

Size dependent response of large shell elements under in-plane tensile loading

Large-scale thin-walled structures with a low weight-to-stiffness ratio provide the means for cost and energy efficiency in structural design. However, the design of such structures for crash and impact resistance requires reliable FE simulations. Large shell elements are used in those simulations. Simulations require the knowledge of the true stress–strain response of the material until fracture initiation. Because of the size effects, local material relation determined with experiments is not applicable to large shell elements. Therefore, a numerical method is outlined to determine the effect of element size on the macroscopic response of large structural shell elements until fracture initiation. Macroscopic response is determined by introducing averaging unit into the numerical model over which volume averaged equivalent stress and plastic strain are evaluated. Three different stress states are considered in this investigation: uniaxial, plane strain and equi-biaxial tension. The results demonstrate that fracture strain is highly sensitive to size effects in uniaxial tension whereas in plane strain or equi-biaxial tension size effects are much weaker. In uniaxial and plane strain tension the fracture strain for large shell elements approaches the Swift diffuse necking condition.     

An analytical method for the determination of stress and strain concentrations in textile-reinforced GF/PP composites with elliptical cutout and a finite outer boundary and its numerical verification

Stress concentrations in the vicinity of cutouts can often be regarded to be the limiting factor for a whole structure. As a further development of prior research at the Institute of Lightweight Engineering and Polymer Technology, an analytical method for the determination of the whole stress-strain fields in the vicinity of holes in multilayered textile-reinforced composites has been developed, which takes into consideration the influences of a finite outer boundary of the specimen. The analytical method is based on the classical laminate theory and the use of complex-valued potential functions. To account for the shape of the specimen, the method of conformal mappings is applied for the inner boundary, while a combination of boundary collocation and least squares method is used for the outer boundary. The method allows a layer-by-layer analysis of stress concentrations. For the verification of the developed calculation model, extensive experimental and numerical finite-element (FE) studies have been carried out on multilayered GF/PP plates with different laminate layups, notches, and specimen dimensions. The comparison of the experimentally or numerically determined results with the analytically calculated ones shows a very good correlation, of which the numerical studies are presented here for the first time. In a second step, the applicable boundary conditions on the outer boundary have been extended in such a way that varying stress and moment resultants can be applied, so that the calculation method can be used as an analytical sub-model in combination with FE techniques.     

建筑织物膜材双轴剪切试验与分析

为了研究建筑织物膜材的剪切力学性能,提出一种新的剪切测试方法。采用中心区域宽度和四臂长度均为16cm的十字形试件,试件纱线的经纬向与加载方向呈45°角。根据膜材变形和应力关系,推导了剪应力和剪应变的计算方法。定义了使试件中心区域产生三个循环剪应力场的加载谱,循环产生正负交替的剪切应力。试验采用表面抛光的(Polyvinylidene Fluoride,聚偏氟乙烯)涂层膜材,测量x、y两个方向的应力和应变,通过计算得到剪切应力应变曲线,并对试验结果进行分析。结果表明,新的剪切测试方法能够反映建筑织物膜材剪切力学性能。最后通过有限元方法模拟材料受剪状态下的应力和应变,与试验得到的应力值和应变值相近。     

Microscale prediction of deformation in an austenitic stainless steel under uniaxial loading

In this study, the deformation behaviour of polycrystalline austenitic 316H stainless steel under uniaxial loading is investigated by means of in-situ neutron diffraction (ND) measurement and crystal plasticity-based finite element (FE) modelling. Data have been obtained for the macroscopic stress–strain response and the lattice strain evolution in the longitudinal and transverse direction relative to the uniaxial loading axis. Comparison between the model predictions and the ND measurements suggests that in most cases the FE model can predict the lattice strain evolution at the microscale and capture the general trends observed in the experiments. Both ND measurements and FE modelling simulations identify little effect of micromorphology effect on the longitudinal lattice strain evolution, while the transverse lattice strain response appears to be sensitive to the microstructure, in particular the initial crystallographic orientation of the material.     

Finite element modeling of plastic anisotropy induced by texture and strain-path change

Consideration of plastic anisotropy is essential in accurate simulations of metal forming processes. In this study, finite element (FE) simulations have been performed to predict the plastic anisotropy of sheet metals using a texture- and microstructure-based constitutive model. The effect of crystallographic texture is incorporated through the use of an anisotropic plastic potential in strain-rate space, which gives the shape of the yield locus. The effect of dislocation is captured by use of a hardening model with four internal variables, which characterize the position and the size of the yield locus. Two applications are presented to evaluate the accuracy and the efficiency of the model: a cup drawing test and a two-stage pseudo-orthogonal sequential test (biaxial stretching in hydraulic bulging followed by uniaxial tension), using an interstitial-free steel sheet. The experimental results of earing behavior in the cup drawing test, maximum pressure and strain distribution in bulging, and transient hardening in the sequential test are compared against the FE predictions. It is shown that the current model is capable of predicting the plastic anisotropy induced by both the texture and the strain-path change. The relative significance of texture and strain-path change in the predictions is discussed.     

On constitutive modeling of aluminum alloys for tube hydroforming applications

The increased interest in lightweight materials for automotive structures has also lead to a search for efficient forming methods that suit these materials. One attractive concept is to use hydroforming of aluminum tubes. The advantages of this forming method includes better tolerances, decreased number of parts and an increased range of forming options. By using FE simulations, the process can be optimized to reduce the risk for failure, i.e. bursting or wrinkling. However, extruded aluminum is highly anisotropic and it is crucial that the material model used for simulations is able to accurately describe this behavior. Also, tube hydroforming occurs predominantly in a biaxial stress state which should be considered in the material testing, where uniaxial tests are used extensively in the industry today. The present study accentuates the need for improved constitutive models. It is shown that a material model, which accurately describes the anisotropic behavior of aluminum tubes, can be obtained from simple and robust experiments.     

Strain Rates and Material Spins

The material time rate of Lagrangean strain measures, objective corotational rates of Eulerian strain measures and their defining spin tensors are investigated from a general point of view. First, a direct and rigorous method is used to derive a simple formula for the gradient of the tensor-valued function defining a general class of strain measures. By means of this formula and the chain rule as well as Sylvester's formula for eigenprojections, explicit basis-free expressions for the material time rate of an arbitrary Lagrangean strain measure can be derived in terms of the right Cauchy–Green tensor and the material time rate of any chosen Lagrangean strain measure, e.g. Hencky's logarithmic strain measure. These results provide a new derivation of Carlson–Hoger's general gradient formula for an arbitrary generalized strain measure and supply a new, rigorous proof for Carlson–Hoger's conjecture concerning the n-dimensional case. Next, by virtue of the aforementioned gradient formula, a general fact for objective corotational rates and their defining spin tensors is disclosed: Let Ω = ϒ ( B, D, W) be any spin tensor that is continuous with respect to B, where B, D and B are the left Cauchy–Green tensor, the stretching tensor and the vorticity tensor. Then the corotational rate of an Eulerian strain measure defined by Ω is objective iff Ω = W + υ ( B, D), where Υ is isotropic. By means of this fact and certain necessary or reasonable requirements, it is further found that a single antisymmetric function of two positive real variables can be introduced to characterize a general class of spin tensors defining objective corotational rates. A general basis- free expression for all such spin tensors and accordingly a general basis-free expression for a general class of objective corotational rates of an arbitrary Eulerian strain measure are established in terms of the left Cauchy–Green tensor B and the stretching tensor B as well as the introduced antisymmetric function. By choosing several particular forms of the latter, all commonly-known spin tensors and corresponding corotational rates are shown to be incorporated into these general formulas in a natural way. In particular, with the aid of these general formulae it is proved that an objective corotational rate of the Eulerian logarithmic strain measure ln V is identical with the stretching tensor D and moreover that in all possible strain tensor measures only ln V enjoys this property. This revised version was published online in August 2006 with corrections to the Cover Date.