A smoothed inverse eigenstrain method for reconstruction of the regularized residual fields

A smoothed inverse eigenstrain method is developed for reconstruction of residual field from limited strain measurements. A framework for appropriate choice of shape functions based on the prior knowledge of expected residual distribution is presented which results in stabilized numerical behavior. The analytical method is successfully applied to three case studies where residual stresses are introduced by inelastic beam bending, laser-forming and shot peening. The well-rehearsed advantage of the proposed eigenstrain-based formulation is that it not only minimizes the deviation of measurements from its approximations but also will result in an inverse solution satisfying a full range of continuum mechanics requirements. The smoothed inverse eigenstrain approach allows suppressing fluctuations that are contrary to the physics of the problem. Furthermore, a comprehensive discussion is performed on regularity of the asymptotic solution in the Tikhonov scheme and the regularization parameter is then exactly determined utilizing Morozov discrepancy principle. Gradient iterative regularization method is also examined and shown to have an excellent convergence to the Tikhonov–Morozov regularization results.     

The principle of equivalent eigenstrain for inhomogeneous inclusion problems

In this paper, based on the principle of virtual work, we formulate the equivalent eigenstrain approach for inhomogeneous inclusions. It allows calculating the elastic deformation of an arbitrarily connected and shaped inhomogeneous inclusion, by replacing it with an equivalent homogeneous inclusion problem, whose eigenstrain distribution is determined by an integral equation. The equivalent homogeneous inclusion problem has an explicit solution in terms of a definite integral. The approach allows solving the problems about inclusions of arbitrary shape, multiple inclusion problems, and lends itself to residual stress analysis in non-uniform, heterogeneous media. The fundamental formulation introduced here will find application in the mechanics of composites, inclusions, phase transformation analysis, plasticity, fracture mechanics, etc.     

Boundary Effects in the Eigenstrain Method

We present a comprehensive study of the effects of internal boundaries on the accuracy of residual stress values obtained from the eigenstrain method. In the experimental part of this effort, a composite specimen, consisting of an aluminum cylinder sandwiched between steel cylinders of the same diameter, was uniformly heated under axial displacement constraint. During the experiment, the sample temperature and the reaction stresses in the load frame in response to changes in sample temperature were monitored. In addition, the local (elastic) lattice strain distribution within the specimen was measured using neutron diffraction. The eigenstrain method, utilizing finite element modeling, was then used to predict the stress field existing within the sample in response to the constraint imposed by the load frame against axial thermal expansion. Our comparison of the computed and measured stress distributions showed that, while the eigenstrain method predicted acceptable stress values away from the cylinder interfaces, its predictions did not match experimentally measured values near them. These observations indicate that the eigenstrain method is not valid for sample geometries with this type of internal boundaries.     

Evaluation of residual stresses and strains using the Eigenstrain Reconstruction Method

The study of residual stress has long been an important research field in science and engineering, due to the fact that uncontrolled residual stresses are detrimental to the performance of products. Numerous research contributions have been devoted to the quantification of residual stress states for the purpose of designing engineering components and predicting their lifetime and failure in service. For the purposes of the present study these can be broadly classified into two main approaches, namely, the interpretation of experimental measurements and process modelling. In this paper, a novel approach to residual stress analysis is developed, called here the Eigenstrain Reconstruction Method (ERM). This is a semi-empirical approach that combines experimental characterisation, specifically, residual elastic strain measurement by diffraction, with subsequent analysis and interpretation based on the eigenstrain theory. Three essential components of the ERM, i.e. the residual strain measurement, the solution of the inverse problem of eigenstrain theory, and the Simple Triangle (SIMTRI) method, are described. The ERM allows an approximate reconstruction of the complete residual strain and stress state in the entire engineering component. This is a significant improvement compared to the experimentally obtained limited knowledge of stress components at a selected number of measurement points, or to the simple interpolation between these points.     

任意形状热夹杂位移场的三角形单元离散算法

平面夹杂模型在纤维增强型复合材料中有广泛应用.复合材料内部通常含有不规则形状夹杂,而夹杂物的存在能严重影响材料的机械力学性能,往往导致应力集中及裂纹萌生等失效先兆.先前关于多边形夹杂的研究大多数关注受均匀本征应变下的应力/应变解,而对位移的分析较少. 基于格林函数方法和围道积分,本文给出了平面热夹杂边界线单元的封闭解析解,可方便应用于受任意分布本征应变的任意形状平面热夹杂位移场的数值计算.当夹杂受均匀本征应变时, 只需将该夹杂边界进行一维离散,因而本文方法可直接得出受均匀分布热本征应变的任意多边形夹杂位移场的封闭解析解.当夹杂区域存在非均匀分布本征应变时,可将该区域划分为足够小的三角形单元进行数值计算. 众所周知,应力应变场在多边形夹杂顶点处具有奇异性,容易导致数值计算上的处理困难及相应的数值稳定性问题; 然而本文工作表明,在多边形顶点处位移场是连续有界的, 因而数值稳定性较好.本文算法可以便捷高效地通过计算机编程实现. 文中给出的验证算例,均体现了本文离散方法的高精度、以及计算编程的鲁棒性.      

航空薄壁弧形结构件喷丸强化变形预测

喷丸强化是一种有效提高工件抗疲劳性能的表面处理工艺,经喷丸强化处理后的航空薄壁弧形结构件发生变形是航空制造领域的共性难题．本文基于变截面曲梁平面弯曲理论,建立了弧形件喷丸诱导应力与变形方程,将得到的构件最大变形与有限元方法计算结果进行对比,所建立理论模型能有效预测弧形件喷丸强化后的变形．讨论了喷丸压力和喷丸区域对喷丸变形的影响,结果表明：喷丸强化工艺仅作用于弧形件腹板表面时,弧形件变形以径向收缩为主;喷丸强化工艺仅作用于弧形件筋板表面时,弧形件变形以径向扩张为主;与局部喷丸相比,对弧形件所有表面喷丸强化时,弧形件弯曲变形减小,轴向伸长增大．     

Size dependent,non-uniform elastic field inside a nano-scale spherical inclusion due to interface stress

The primary objective of the present paper is to analyze the influence of interface stress on the elastic field within a nano-scale inclusion. Special attention is focused on the case of non-hydrostatic eigenstrain. From the viewpoint of practicality, it is assumed that the inclusion is spherically shaped and embedded into an infinite solid, within which an axisymmetric eigenstrain is prescribed. Following Goodier’s work, the elastic fields inside and outside the inclusion are obtained analytically. It is found that the presence of interface stress leads to conclusion that the elastic field in the inclusion is not only dependent on inclusion size but also on non-uniformity. The result is in strong contrast to Eshelby’s solution based on classical elasticity, and it is helpful in the understanding of relevant physical phenomena in nano-structured solids.     

Development of plasticity and damage in vibrating structural elements performing guided rigid-body motions

Summary  A numerical algorithm for studying the development of plastic and damaged zones in a vibrating structural element with a large, guided rigid-body motion is presented. Beam-type elements vibrating in the small-strain regime are considered. A machine element performing rotatory motions, similar to an element of a slider-crank mechanism, is treated as a benchmark problem. Microstructural changes in the deforming material are described by the mesolevel variables of plastic strain and damage, which are consistently included into a macroscopic analysis of the overall beam motion. The method is based on an eigenstrain formulation, considering plastic strain and damage to contribute to an eigenstrain loading of a linear elastic background structure. Rigid-body coordinates are incorporated into this beam-type structural formulation, and an implicit numerical scheme is presented for iterative computation of the eigenstrains from the mesolevel constitutive behavior. Owing to the eigenstrain formulation, any of the existing constitutive models with internal variables could in principle be implemented. Linear elastic/perfectly plastic behavior is exemplarily treated in a numerical study, where plastic strain is connected to the Kachanov damage parameter by a simple damage law. Inelastic effects like plastic shakedown and damage-induced low-cycle rupture are shown to occur in the examplary problems. Received 1 September 1999; accepted for publication 9 March 2000     

The determination of the elastic field of a polygonal star shaped inclusion

Recently we found that the elastic field is uniform in a pentagonal star (five-pointed star inclusion) [1], and in a triangular inclusion [2], when an eigenstrain is distributed uniformly in these inclusions. This result is similar to the famous result of Eshelby (1957) that the elastic field is uniform in an ellipsoidal inclusion in an infinitely body when an eigenstrain is distributed uniformly in the ellipsoidal inclusion. We also found that for a Jewish star (Star of David or six points star) or a rectangular inclusion subjected to a uniform eigenstrain, the stress field is not uniform in these inclusions. These results also hold for two dimensional plane strain cases. Furthermore these analytical results are confirmed experimentally by photoelasticity method. In this paper, we investigate a more general inclusion of an m-pointed polygonal inclusion subjected to the uniform eigenstrain. We conclude that the stress field is uniform when m is odd number. This conclusion agrees with the speculation made by B. Boley after the author's talk at Shizuoka [2].     

Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity

In traditional continuum mechanics, the effect of surface energy is ignored as it is small compared to the bulk energy. For nanoscale materials and structures, however, the surface effects become significant due to the high surface/volume ratio. In this paper, two-dimensional elastic field of a nanoscale elliptical inhomogeneity embedded in an infinite matrix under arbitrary remote loading and a uniform eigenstrain in the inhomogeneity is investigated. The Gurtin–Murdoch surface/interface elasticity model is applied to take into account the surface/interface stress effects. By using the complex variable technique of Muskhelishvili, the analytic potential functions are obtained in the form of an infinite series. Selected numerical results are presented to study the size-dependency of the elastic field and the effects of surface elastic moduli and residual surface stress. It is found that the elastic field of an elliptic inhomogeneity under uniform eigenstrain is no longer uniform when the interfacial stress effects are taken into account.     

Inclusion of arbitrary polygon with graded eigenstrain in an anisotropic piezoelectric full plane

In this paper, an exact closed-form solution for the Eshelby problem of a polygonal inclusion with a graded eigenstrain in an anisotropic piezoelectric full plane is presented. For this electromechanical coupling problem, by virtue of Green’s function solutions, the induced elastic and piezoelectric fields are first expressed in terms of line integrals on the boundary of the inclusion. Using the line-source Green’s function, the line integral is then carried out analytically for the linear eigenstrain case, with the final expression involving only elementary functions. Finally, the solution is applied to the semiconductor quantum wire (QWR) of square, triangle, circle and ellipse shapes within the GaAs (0 0 1) substrate. It is demonstrated that there exists significant difference between the induced field by the uniform eigenstrain and that by the linear eigenstrain. Since the misfit eigenstrain in most QWR structures is actually non-uniform, the present solution should be particularly appealing to nanoscale QWR structure analysis where strain and electric fields are coupled and are affected by the non-uniform misfit strain.     

Plasticity Effects in the Hole-Drilling Residual Stress Measurement in Peened Surfaces

The incremental hole-drilling technique (IHD) is a widely established and accepted technique to determine residual stresses in peened surfaces. However, high residual stresses can lead to local yielding, due to the stress concentration around the drilled hole, affecting the standard residual stress evaluation, which is based on linear elastic equations. This so-called plasticity effect can be quantified by means of a plasticity factor, which measures the residual stress magnitude with respect to the approximate onset of plasticity. The observed resultant overestimation of IHD residual stresses depends on various factors, such as the residual stress state, the stress gradients and the material’s strain hardening. In peened surfaces, equibiaxial stresses are often found. For this case, the combined effect of the local yielding and stress gradients is numerically and experimentally analyzed in detail in this work. In addition, a new plasticity factor is proposed for the evaluation of the onset of yielding around drilled holes in peened surface layers. This new factor is able to explain the agreement and disagreement found between the IHD residual stresses and those determined by X-ray diffraction in shot-peened steel surfaces.     

基于等效特征应变原理的复合材料有效模量细观力学分析

基于等效特征应变原理,提出了一种新的复合材料有效模量细观力学分析方法。首先,在等效特征应变原理基础上提出平均等效特征应变原理,它可用于解决有限体下任意形状(无论是凸或凹形)的单个夹杂或多个夹杂的弹性变形问题。其次,将平均等效特征应变原理与细观力学直接均匀法相结合,来分析确定复合材料的有效模量。最后利用复合材料纤维与基体的力学性能参数及纤维的体分比,借助MATLAB编程方法,预测其有效模量。通过将理论预测值与已有的的试验值、其它理论预测值进行对比,验证了新分析方法的合理性和分析精度。     

Variational formulation of the equivalent eigenstrain method with an application to a problem with radial eigenstrains

In this paper a variational formulation of the equivalent eigenstrain method is established. A functional of the Hashin–Shtrikman type is proposed such that the solution of the equivalent eigenstrain equation is a unique minimizer of the functional. Moreover, it is also shown that the equivalent eigenstrain equation is the Euler–Lagrange equation of the potential energy of the inclusions. An approximate solution of the equivalent eigenstrain equation is then found as a minimizer of the functional on a finite dimensional span of basic eigenstrains. Special attention is paid to possible symmetries of the problem. The variational formulation is illustrated by determination of effective linear elastic properties. In particular, material with a simple cubic microstructure is considered in detail. A solution for the polynomial radial basic eigenstrains approximation is found. In particular, for the homogeneous eigenstrain approximation, the effective moduli are derived in an exact closed form.     

Effect of thermal exposure on the residual stress relaxation in a hardened cylindrical sample under creep conditions

This paper describes the effect of thermal exposure (high-temperature exposure) (T = 675?C) on the residual creep stress relaxation in a surface hardened solid cylindrical sample made of ZhS6UVI alloy. The analysis is carried out with the use of experimental data for residual stresses after micro-shot peening and exposures to temperatures equal to T = 675?C during 50, 150, and 300 h. The paper presents the technique for solving the boundary-value creep problem for the hardened cylindrical sample with the initial stress–strain state under the condition of thermal exposure. The uniaxial experimental creep curves obtained under constant stresses of 500, 530, 570, and 600 MPa are used to construct the models describing the primary and secondary stages of creep. The calculated and experimental data for the longitudinal (axial) tensor components of residual stresses are compared, and their satisfactory agreement is determined.     

The elastic field caused by a general ellipsoidal inclusion and the application to martensite formation 2

The elastic field throughout an ellipsoidal inclusion in an indefinitely-extended anisotropic material is investigated when an eigenstrain (a stress-free transformation strain) is periodically distributed throughout the inclusion. This is an extention of the results obtained by J.D. Eshelby (1961) for uniform eigenstrains and by R.J. Asaro and D.M. Barnett (1975) for polynomial eigenstrains. The solution is applied to the evaluation of elastic strain energies of a disc-shaped martensite with alternating twins and of a spherical precipitate with a banded structure. The significant amount of the elastic strain energies explains the necessity of the supercooling of austenite steel for the martensitic transformation to occur.     

平面任意形状等剪切模量异质夹杂问题的解析解

本文研究任意形状夹杂域在受到远端均匀荷载和均匀本征应变作用下的弹性场问题,其中基体和夹杂的材料不同但具有相同的剪切模量。利用等效理论将远端均匀荷载引起的扰动转化为等效均匀本征应变的作用,再利用K-M势函数表达扰动场问题的界面连续条件;借助于黎曼映射定理,用洛朗多项式将平面光滑闭合曲线外部区域映射到单位圆外部区域,借助柯西积分公式和Faber多项式求解了等剪切本征应变下夹杂和基体的K-M势函数的显式解析解,其中考虑了夹杂相对于基体的刚体位移。将得到的结果与相关文献的结果进行对比,表明了本论文的方法和结果是有效的和正确的。     

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.     

Multi-Axial Contour Method for Mapping Residual Stresses in Continuously Processed Bodies

This paper describes a method for extending the capability of the contour method to allow for the measurement of spatially varying multi-axial residual stresses in prismatic, continuously processed bodies. Currently, the contour method is used to determine a 2D map of the residual stress normal to a plane. This work uses an approach similar to the contour method to quantify multiple components of eigenstrain in continuously processed bodies, which are used to calculate residual stress. The result of the measurement is an estimate of the full residual stress tensor at every point in the body. The approach is first outlined for a 2D body and the accuracy of the methodology is demonstrated for a representative case using a numerical experiment. Next, an extension to the 3D case is given and the accuracy is demonstrated for representative cases using numerical experiments. Finally, measurements are performed on a thin sheet of Ti-6Al-4V with a band of laser peening down the center (assumed to be 2D) and a thick laser peened plate of 316L stainless steel to show that the approach is valid under real experimental conditions.     

Effective elastic moduli of triangular lattice material with defects

This paper presents an attempt to extend homogenization analysis for the effective elastic moduli of triangular lattice materials with microstructural defects. The proposed homogenization method adopts a process based on homogeneous strain boundary conditions, the micro-scale constitutive law and the micro-to-macro static operator to establish the relationship between the macroscopic properties of a given lattice material to its micro-discrete behaviors and structures. Further, the idea behind Eshelby’s equivalent eigenstrain principle is introduced to replace a defect distribution by an imagining displacement field (eigendisplacement) with the equivalent mechanical effect, and the triangular lattice Green's function technique is developed to solve the eigendisplacement field. The proposed method therefore allows handling of different types of microstructural defects as well as its arbitrary spatial distribution within a general and compact framework. Analytical closed-form estimations are derived, in the case of the dilute limit, for all the effective elastic moduli of stretch-dominated triangular lattices containing fractured cell walls and missing cells, respectively. Comparison with numerical results, the Hashin–Shtrikman upper bounds and uniform strain upper bounds are also presented to illustrate the predictive capability of the proposed method for lattice materials. Based on this work, we propose that not only the effective Young’s and shear moduli but also the effective Poisson’s ratio of triangular lattice materials depend on the number density of fractured cell walls and their spatial arrangements.