Micromechanics based fatigue life prediction of a polycrystalline metal applying crystal plasticity

The fatigue-life of a polycrystalline superalloy under symmetrical cyclic strain controlled loading at a temperature of 650 °C is investigated by numerical simulations on the micro-level, focusing on the inhomogeneous evolution of plastic deformation in a polycrystalline aggregate. A methodology (Zhang et al., 2011, 2013) to predict the low-cycle fatigue life by micro-level simulations along with statistical analysis is applied following the steps: (1) A statistically representative volume element (RVE) consisting of a number of crystal grains is constructed by Voronoi tessellation. Stresses and plastic strains are calculated by a crystal plasticity model including nonlinear kinematic hardening. (2) The RVE is subjected to repeated symmetric tensile-compressive loading. (3) The inhomogeneous stress and strain fields are statistically analyzed during the load cycles. (4) Failure by LCF is strain controlled and occurs if either of the quantities, standard deviation of longitudinal strain in tensile direction, maximum or statistical average of first principal strains in the RVE at the tension peak of cyclic loading reaches a respective critical value. (5) Using the present methodology, a family of failure curves for fatigue lives under different strain amplitudes can be predicted by varying the critical values. Finally, appropriate critical values can be identified by a respective cyclic experiment with only one strain amplitude.     

The polycrystalline plasticity due to slip and twinning during magnesium alloy forming

In order to simulate the magnesium alloy-forming processes accurately, it is necessary to consider the plastic anisotropy. In this paper, a new rate-independent constitutive model for polycrystalline plastic deformation by slip and twinning has been formulated, and then introduced into a FEM program. Metal flow is assumed to occur by crystallographic slip on given slip and twinning systems within each crystal. Each integration point represents a single crystal. Then uniaxial compression and cup drawing of Mg alloy are studied by using a rate-independent polycrystalline plasticity finite element analysis. In this paper, the ear distributions of the polycrystal are predicted for different typical initial orientation cases. The values of the twinning factors associated with slip system deformation are deduced. It is found that the twinning factors vary with the value of the stress. The basal slip and twinning system plays the dominant role in the deformation of magnesium alloy, which might be the most important contribution to strain hardening.     

Studie zum Einfluß des plastisch anisotropen Werkstoffverhaltens auf technische Umformprozesse

Study about the influence of anisotropic plastic material behaviour on technical forming processes Because of the process of manufacturing sheets used in technical forming processes show usually a microstructure in which the grains of the polycrystalline material own the same orientation. The consequence is anisotropic material properties depending on the direction. The influence of that anisotropy on the plastic deformation of metals is investigated by Finite Element (FE) simulations. At first a model describing the plastic anisotropy by different yield stresses in different directions is proposed. The different plastic behaviour for tensile and compressive stress is also taken into account as well as plastic volume change. A simple example of application is a sheet under a biaxial balanced load. In FE simulations the beginning of plastification is shown by the curves of plastic strains for different directions. At the end the study concerning plastic anisotropy is completed by the simulation of the sheet bending in V-forging die. The computations here are concentrated on the punch force at the beginning of plastification.     

Multiscale finite element modeling of superelasticity in Nitinol polycrystals

An efficient method is proposed for modeling superelastic polycrystalline NiTi by solving a two-scale problem. The RVE size of the fine scale is determined using a statistics-based approach. Both problems are discretized in space using the finite element method and their communication is effected using MPI. Representative simulations illustrate the modeling capabilities of the proposed approach.     

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.     

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.     

Effects of grain sizes, shapes, and distribution on minimum sizes of representative volume elements of cubic polycrystals

In the literature the concept of representative volume element (RVE) was introduced to correlate the effective or macroscopic properties of materials with the properties of the microscopic constituents and microscopic structures of the materials. However, to date little quantitative knowledge is available about minimum RVE sizes of various engineering materials. In our recent paper [J. Mech. Phys. Solids 50 (2002) 881], a new definition of minimum RVE size was introduced based on the concept of nominal modulus. Numerical experiments using the finite element method (FEM) were then carried out for determining the minimum RVE sizes of more than 500 cubic polycrystals in the plane stress problem, under the assumption that all grains in a polycrystal have the same square shape––called the simple polycrystal model. The major finding is that the minimum RVE sizes for effective elastic moduli have a roughly linear dependence on crystal anisotropy degrees. The present paper takes into account the effect of grain sizes, shapes, and distribution on the minimum RVE sizes for real cubic polycrystals that are formed by crystallization processes. Similar roughly linear dependence is found again, with the slope about 19% lower than that in the simple polycrystal model. This finding is interesting and useful because numerical experiments on minimum RVE sizes for a large number of crystals are quite time-consuming and the simple polycrystal model reduces significantly the FEM pre- and post-processing works. This should be particularly true in numerically testing minimum RVE sizes for three-dimensional polycrystals and for nonelastic properties in future works. With a maximum relative error 5%, all the polycrystals tested have a minimum RVE size of 16 or less times the grain size.     

Ductile fracture by multiple void growth and interaction ahead of a notch tip in polycrystalline plastic solids

The objectives of this paper are to study the effects of plastic anisotropy and evolution in crystallographic texture with deformation on the ductile fracture behaviour of polycrystalline solids. To this end, numerical simulations of multiple void growth and interaction ahead of a notch tip are performed under mode I, plane strain, small scale yielding conditions using two approaches. The first approach is based on the Hill yield theory, while the second employs crystal plasticity constitutive equations and a Taylor-type homogenization in order to represent the ductile polycrystalline solid. The initial textures pertaining to continuous cast Al–Mg AA5754 sheets in recrystallized and cold rolled conditions are considered. The former is nearly-isotropic, while the latter displays pronounced anisotropy. The results indicate distinct changes in texture in the ligaments bridging the voids ahead of the notch tip with increase in load level which gives rise to retardation in porosity evolution and increase in tearing resistance for both materials.     

On the adaptive use of the Taylor assumption in computational homogenization of thin metal sheets

A strategy for macroscale modeling adaptivity in fully nested two-scale computational (first order) homog- enization based on assumed scale separation is proposed. The Representative Volume Element (RVE) for a quite complex substructure pertinent to Duplex Stainless Steel (DSS) is considered, whereby crystal plasticity is adopted for the subscale material modeling. The choice of the pertinent prolongation condition defining the deformation mapping from the macro- to the subscale is the sole source of model error discussed here. Two common choices are (in hierarchical order): (1) ”simplified” model based on homogeneous (macroscale) deformation within the RVE, i.e., the Taylor assumption, (2) ”reference” model employing Dirichlet boundary conditions on the RVE.     

Computational aspects of tangent moduli tensors in rate-independent crystal elastoplasticity

The computational efficiencies of the continuum and consistent (algorithmic) tangent moduli tensors in rate-independent crystal elastoplasticity are examined in conjunction with the available implicit state update algorithms. It is, in this context, shown that the consistent tangent moduli associated with the state update algorithm with the exponential mapping coincide with the continuum tangent moduli. After verifying the reported performance of the exponential mapping algorithm in preserving the incompressibility of plastic deformation in a single crystal grain, we carry out numerical experiments to understand the convergence trends of the global Newton–Raphson iterative procedure with different kinds of tangent moduli tensors. Having done this, we are concerned with the performance of those tangent moduli tensors for the micro-scale analysis of a polycrystalline aggregate, which is regarded as a representative volume element, subjected to macro-scale uniform deformation in the context of the two-scale homogenization method.     

A method of two-scale thermo-mechanical analysis for porous solids with micro-scale heat transfer

A two-scale thermo-mechanical model for porous solids is derived and is implemented into a multi-scale multi-physics analysis method. The model is derived based on the mathematical homogenization method and can account for the scale effect of unit cells, which is our particular interest in this paper, on macroscopic thermal behavior and, by extension, on macroscopic deformation due to thermal expansion/contraction. The scale effect is thought to be the result of microscopic heat transfer, the amount of which depends on the micro-scale pore size of porous solids. We first formulate a two-scale model by applying the method of asymptotic expansions for homogenization and, by using a simple numerical model, verify the validity and relevancy of the proposed two-scale model by comparing it with a corresponding single-scale direct analysis with detailed numerical models.     

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 simulations of crack tip fields in polycrystalline plastic solids

In this paper, modes I and II crack tip fields in polycrystalline plastic solids are studied under plane strain, small scale yielding conditions. Two different initial textures of an Al-Mg alloy, viz., continuous cast AA5754 sheets in the recrystallized and cold rolled conditions, are considered. The former is nearly-isotropic, while the latter displays distinct anisotropy. Finite element simulations are performed by employing crystal plasticity constitutive equations along with a Taylor-type homogenization as well as by using the Hill quadratic yield theory. It is found that significant texture evolution occurs close to the notch tip which profoundly influences the stress and plastic strain distributions. Also, the cold rolling texture gives rise to higher magnitude of plastic strain near the tip.     

Macroscopic theory and nonlinear finite element analysis of micromechanics of single crystals at finite strains

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the macroscopic rate-independent and rate-dependent analysis of micromechanics of metal single crystals undergoing finite elastic-plastic deformations which is based on the assumption that inelastic deformation is solely due to crystallographic slip. The formulation relies on a multiplicative decomposition of the material deformation gradient into incompressible elastic and plastic as well as a scalar valued volumetric part. Furthermore, the crystal deformation is described as arising from two distinct physical mechanisms, elastic deformation due to distortion of the lattice and crystallographic slip due to shearing along certain preferred lattice planes in certain preferred lattice directions. Macro- and microscopic stress measures are related to Green’s macroscopic strains via a hyperelastic constitutive law based on a free energy potential function, whereas plastic potentials expressed in terms of the generalized Schmid stress lead to a normality rule for the macroscopic plastic strain rate. Estimates of the microscopic stress and strain histories are obtained via a highly stable and very accurate semi-implicit scalar integration procedure which employs a plastic predictor followed by an elastic corrector step, and, furthermore, the development of a consistent elastic-plastic tangent operator as well as its implementation into a nonlinear finite element program will also be discussed. Finally, the numerical simulation of finite strain elastic-plastic tension tests is presented to demonstrate the efficiency of the algorithm.     

Multiscale simulations: application to the heat transfer simulation of sliding solids

Molecular dynamics is a powerful tool allowing the simulation of matter behaviour at the atomic scale. Due to computation time, it is clearly not possible to use molecular dynamics to simulate a forming process. However, atomistic simulations can be used to study and understand the physical phenomena that occur during matter deformation. As an example, heat transfer between the contacting solids in forming processes is one of the important physics phenomena that have to be taken into account in order to do realistic simulations. A multiscale analysis of heat transfer is presented. This analysis leads to two kinds of models: a macroscopic model which can be used for the simulation of the process itself and a microscopic model that is used to determine the parameters of the macroscopic model. In this microscopic model, the friction heat generation phenomena has to be described quite accurately. Friction heat is mainly due to plastic and elastic deformation and adhesion. Thus, to understand the underlying friction heat generation phenomena, atomistic simulations using molecular dynamics are carried out. It is shown that friction heat is the transformation of mechanical work given to the system at the macroscopic scale into potential energy during elastic deformation. This potential energy which is stored in the system is finally transformed into atomic kinetic energy (friction heat) during plastic transformation.     

SUS304不锈钢拉伸过程中的声-热像特征

测量了含有Ⅰ型裂纹的SUS304不锈钢试样的单轴拉伸过程中的塑性变形和断裂。分析了裂纹尖端区域的塑性变形和断裂过程。结果表明:SUS304的各向异性在断裂阶段对声发射信号影响较大;红外热图像中的温度分布与塑性应变率有关;通过声发射参数和红外热图像可以从微观和宏观两方面分析裂纹尖端区域的塑性变形和断裂。     

Plastic deformation of high-purity α-titanium: Model development and validation using the Taylor cylinder impact test

In this paper, results of an experimental study on the quasi-static and high-rate plastic deformation due to impact of a high-purity, polycrystalline, α-titanium material are presented. It was found that the material is transversely isotropic and displays strong strength differential effects. Split Hopkinson Pressure Bar tests in tension and compression and Taylor impact tests were conducted. For an impact velocity of 196 m/s, plastic deformation extended to 64% of the length of the deformed specimen, with little radial spreading. A three-dimensional constitutive model was developed. Key in the formulation was the use of a macroscopic yield function that incorporates the specificities of the plastic flow, namely the combined effects of anisotropy and tension–compression asymmetry. Comparison between model predictions and data show the capabilities of the model to describe with accuracy the plastic behavior of the α-titanium material for both quasi-static and high-rate loadings. In particular, the three-dimensional simulations of the Taylor impact test show a very good agreement with data, both the post-test major and minor side profiles and impact interface footprints are very well described.     

Stress intensity factors of microstructurally small crack

The stress intensity factor (SIF) is widely used for evaluating integrity of cracked components. Averaging the anisotropy of each crystal, the macroscopic behavior of polycrystalline materials is isotropic and homogenous in terms of elastic deformation. However, the anisotropic and/or inhomogeneous property influences on the stress field around a crack if the crack size is small in comparison with the grain. Thus, the SIF of the microstructurally small crack may differ from that in the isotropic body. In present study, the effect of anisotropic/inhomogeneous elasticity on the SIF is investigated by using the finite element analysis (FEA). At first, the SIFs of semi-circular crack in a single crystal and a polycrystalline material are calculated. These reveal that the magnitude of SIF is dependent not only on the crystal orientation but also on the deformation constraint by the neighboring crystals. Then, the statistical scatter of SIF due to the random orientation of crystal orientation in a polycrystal is examined by a Monte Carlo simulation.