Analysis of a transversely isotropic rod containing a single cylindrical inclusion with axisymmetric eigenstrains

This paper studies a transversely isotropic rod containing a single cylindrical inclusion with axisymmetric eigenstrains. The analytical elastic solution is obtained for the displacements, stresses and elastic strain energy of the rod. The effects of microstructural parameters and its evolution on the elastic stress and strain fields as well as the strain energy of the rod are quantitatively demonstrated through examples.     

On Structured Surfaces with Defects: Geometry,Strain Incompatibility,Stress Field,and Natural Shapes

In this paper we study the stress and deformation fields generated by nonlinear inclusions with finite eigenstrains in anisotropic solids. In particular, we consider finite eigenstrains in transversely isotropic spherical balls and orthotropic cylindrical bars made of both compressible and incompressible solids. We show that the stress field in a spherical inclusion with uniform pure dilatational eigenstrain in a spherical ball made of an incompressible transversely isotropic solid such that the material preferred direction is radial at any point is uniform and hydrostatic. Similarly, the stress in a cylindrical inclusion contained in an incompressible orthotropic cylindrical bar is uniform hydrostatic if the radial and circumferential eigenstrains are equal and the axial stretch is equal to a value determined by the axial eigenstrain. We also prove that for a compressible isotropic spherical ball and a cylindrical bar containing a spherical and a cylindrical inclusion, respectively, with uniform eigenstrains the stress in the inclusion is uniform (and hydrostatic for the spherical inclusion) if the radial and circumferential eigenstrains are equal. For compressible transversely isotropic and orthotropic solids, we show that the stress field in an inclusion with uniform eigenstrain is not uniform, in general. Nevertheless, in some special cases the material can be designed in order to maintain a uniform stress field in the inclusion. As particular examples to investigate such special cases, we consider compressible Mooney-Rivlin and Blatz-Ko reinforced models and find analytical expressions for the stress field in the inclusion.     

Numerical simulation of solid state sintering

This paper discusses in detail the development of a numerical model capable of simulating microstructural evolution and macroscopic deformation during sintering of complex powder compacts. The model based on the kinetic Monte Carlo (Potts) approach simulates grain growth, vacancy diffusion, and pore annihilation at grain boundaries, which is responsible for densification. Results of 2D simulations for perfect close-packed and random starting configurations are presented and discussed. The microstructural evolution is used to obtain the sintering stress––the macroscopic stress that is equivalent to the microstructural driving force for deformation.     

Circumferentially-symmetric finite eigenstrains in incompressible isotropic nonlinear elastic wedges

Eigenstrains are created as a result of anelastic effects such as defects, temperature changes, bulk growth, etc., and strongly affect the overall response of solids. In this paper, we study the residual stress and deformation fields of an incompressible, isotropic, infinite wedge due to a circumferentially symmetric distribution of finite eigenstrains. In particular, we establish explicit exact solutions for the residual stresses and deformation of a neo-Hookean wedge containing a symmetric inclusion with finite radial and circumferential eigenstrains. In addition, we numerically solve for the residual stress field of a neo-Hookean wedge induced by a symmetric Mooney–Rivlin inhomogeneity with finite eigenstrains.     

On the stress singularities generated by anisotropic eigenstrains and the hydrostatic stress due to annular inhomogeneities

The problems of singularity formation and hydrostatic stress created by an inhomogeneity with eigenstrain in an incompressible isotropic hyperelastic material are considered. For both a spherical ball and a cylindrical bar with a radially symmetric distribution of finite possibly anisotropic eigenstrains, we show that the anisotropy of these eigenstrains at the center (the center of the sphere or the axis of the cylinder) controls the stress singularity. If they are equal at the center no stress singularity develops but if they are not equal then stress always develops a logarithmic singularity. In both cases, the energy density and strains are everywhere finite. As a related problem, we consider annular inclusions for which the eigenstrains vanish in a core around the center. We show that even for an anisotropic distribution of eigenstrains, the stress inside the core is always hydrostatic. We show how these general results are connected to recent claims on similar problems in the limit of small eigenstrains.     

Modeling segmentation cracking of a brittle coating due to underneath periodic eigenstrains

The formation mechanism for a segmented Cr coating by the hybrid technique of laser pre-quenching steel substrate plus post-electroplating was investigated. The discrete laser quenched zones (LQZs) were modeled as multiple inclusions with prescribed eigenstrains. The stress field was determined to account for the onset of segmentation cracking. Also addressed was segmentation crack growth through the evaluation of the stress intensity factor (SIF). The computations were implemented by using the finite element (FE) method. The dependence of a wide range of dimensionless variables of interest on both the stress field and SIF was assessed through a detailed parametric study.     

Design of patterned multilayer films with eigenstrains by topology optimization

We develop a formal approach to design shaped microstructures from multilayer films with eigenstrains in the layers. The eigenstrains are inelastic strains that vary from layer to layer resulting in elastic misfit between the layers. Examples include thermal expansion mismatch between the layers, piezoelectric strains, and strains in shape memory alloys. In our approach, the eigenstrains are manipulated by spatially patterning the films to generate structures that, although fabricated by a conventional, planar thin film technology, deform into desired three-dimensional shaped surfaces. The material patterns in the individual layers are determined by topology optimization allowing the creation of arbitrarily complex, geometric layouts. In contrast to existing topology optimization methods for patterning plate structures, the goal of the proposed approach is to generate large deformations via eigenstrains, rather than to increase the stiffness of plate via reinforcement patterns. The optimization methodology is demonstrated by the design of two- and three-layer thin film structures. The performance of the optimized designs is verified by experiments showing the importance of accounting for a nonlinear kinematics in order to obtain the desired shape in the deformed configuration. While our approach is demonstrated in the context of the design of three-dimensional microstructures, it can be easily applied to a variety of problems where it is desired to control the complex shape of plate-like structures by spatial actuation—the spatial actuators are represented by eigenstrains.     

Non-uniform eigenstrain induced stress field in an elliptic inhomogeneity embedded in orthotropic media with complex roots

This paper deals with an elastic orthotropic inhomogeneity problem due to non-uniform eigenstrains. The specific form of the distribution of eigenstrains is assumed to be a linear function in Cartesian coordinates of the points of the inhomogeneity. Based on the polynomial conservation theorem, the induced stress field inside the inhomogeneity which is also linear, is determined by the evaluation of 10 unknown real coefficients. These coefficients are derived analytically based on the principle of minimum potential energy of the elastic inhomogeneity/matrix system together with the complex function method and conformal transformation. The resulting stress field in the inhomogeneity is verified using the continuity conditions for the normal and shear stresses on the boundary. In addition, the present analytic solution can be reduced to known results for the case of uniform eigenstrain.     

纤维复合材料损伤过程的数值模拟

利用界面断裂力学和有限元法数值模拟纤维增强复合材料的细观损伤过程，研究各种主要破坏模式之间的相互转变和影响，指出以断裂能和混合度表示的界面性能是控制复合材料损伤过程的主要细观参数。分析了界面韧度对破坏性能的影响，探讨了基于破坏模式控制的复合材料韧度设计的新途径。     

Field fluctuations in a heterogeneous elastic material—an information theory approach

Internal stress and strain fields in disordered elastic solids such as multiphase materials or polycrystals are considered. In order to derive a probability distribution for those random internal fields, the information theory entropy is maximized subject to constraints representing the basic equations of elasticity and certain experimental data. Thus one can find the probability distribution which agrees with all known facts but makes no assertions about the internal fields which cannot be supported by the available information. This approach is in accordance with the formal exact solution of the statistical problem if one has complete microstructural information. In case of incomplete microstructural data, useful approximate solutions can easily be obtained. In particular, the following set of data is sufficiently detailed for the prediction of internal field fluctuations: the average strain, the one-point probability density of the random elastic constants, and the effective (overall) elastic constants. Especially the information supplied by the effective elastic constants plays a major role since it reflects the microstructural topology of the heterogeneous material. One obtains Gaussian probability distributions for stress and strain, which are applied to calculate mean values and fluctuations of stresses in a cemented metal carbide and a zinc polycrystal.     

Remarks on the energy release rate for an antiplane moving crack in couple stress elasticity

This paper is concerned with the steady-state propagation of an antiplane semi-infinite crack in couple stress elastic materials. A distributed loading applied at the crack faces and moving with the same velocity of the crack tip is considered, and the influence of the loading profile variations and microstructural effects on the dynamic energy release rate is investigated. The behavior of both energy release rate and maximum total shear stress when the crack tip speed approaches the critical speed (either that of the shear waves or that of the localized surface waves) is studied. The limit case corresponding to vanishing characteristic scale lengths is addressed both numerically and analytically by means of a comparison with classical elasticity results.     

弱界面复合材料中的基体裂纹

利用二维弹性力学模型研究了纤维增强复合材料中基体裂纹与弱界面的相互作用机理．文中首先导出各向异性弹性多层介质中刃型位错的基本解，然后运用这些基本解建立了弱界面复合材料中典型的Ｈ型缺陷的奇异积分方程组，通过求解这些方程得到外载荷的大小、弱界面的结合强度、界面的残余压力和摩擦系数、纤维与基体的弹性模量比等微结构参量与基体裂纹附近的应力场的关系     

Microstructural parameters affecting creep induced load shedding in Ti-6242 by a size dependent crystal plasticity FE model

This paper is aimed at identifying critical microstructural parameters that cause local stress concentration due to load shedding between microstructural regions of varying strengths. This stress is viewed as one of the fundamental reasons for crack initiation in Ti-6242. A rate dependent, anisotropic, elasto-crystal plasticity based finite element model (CPFEM) for poly-phase Ti-6242 is used in this study to identify the critical variables responsible for localized stress concentration due to load shedding. The model can account for various microstructural features like grain size, orientation and misorientation distributions. Various microstructural variables, such as crystal orientation, misorientation, grain size, Schmid factor and composition of phases, are considered in a detailed parametric study. Critical combinations of these parameters that result in high stress due to load shedding are identified. Finally, load shedding in a microstructure model of polycrystalline Ti-6242 is discussed from the results of CPFEM simulations. The model is statistically equivalent with respect to features observed in OIM scans.     

循环接触下安定状态问题的研究

基于线性随动强化理论,运用算子分离技术,研究将弹塑性问题转换为弹性问题和残余问题的分析方法,且针对循环载荷接触安定状态,建立了计算机分析程序,该研究能够分析计算弹塑性接触载荷在安定状态下的应力、残余累积应变及残余应力,分析计算了不同载荷的安定状态,并探讨其残余应力场的分析方法。     

An eigenstrain formulation for the prediction of elastic moduli of defective fiber networks

A method for predicting the elastic moduli of a regular network populated by a large number of randomly located defects is presented. The prediction is based exclusively on the stiffness of individual fibers and the location of defects. The method requires a preliminary calibration step in which the eigenstrains associated with “elementary defects” of the regular network are fully characterized. Each type of defect is represented by a superposition of singular point sources in 2D elastostatics producing a field identical to the eigenstrain of the respective defect. The amplitude of the point sources is determined by probing the eigenstrain with a series of path independent integrals. This “spectral decomposition” represents the generalization that allows applying methods developed to account for crack–crack interaction in fracture mechanics to situations in which the interacting sources have eigenstrains obtained by the superposition of multiple types of singularities. Once the representation of each elementary defect is determined, any distribution of defects in the network can be mapped into a distribution of point sources in an equivalent continuum. This allows inferring the elastic behavior of a defective network of any distribution and concentration of defects. The method discussed here provides an efficient way to treat the non-affine deformation of defective regular fiber networks.     

Some basic problems in anisotropic bimaterials with a viscoelastic interface

This research is devoted to the study of anisotropic bimaterials with Kelvin-type viscoelastic interface under antiplane deformations. First we derive the Green’s function for a bimaterial with a Kelvin-type viscoelastic interface subjected to an antiplane force and a screw dislocation by means of the complex variable method. Explicit expressions are derived for the time-dependent stress field induced by the antiplane force and screw dislocation. Also presented is the time-dependent image force acting on the screw dislocation due to its interaction with the Kelvin-type viscoelastic interface. Second we investigate a rectangular inclusion with uniform antiplane eigenstrains embedded in one of the two bonded anisotropic half-planes by virtue of the derived Green’s function for a line force. The explicit expressions for the time-dependent stress field induced by the rectangular inclusion are obtained in terms of the simple logarithmic and exponential integral functions. It is observed that in general the stresses exhibit the logarithmic singularity at the four corners of the rectangular inclusion. Our results also show that when one side of the rectangular inclusion lies on the viscoelastic interface, the interfacial tractions are still regular at the two corners of the inclusion which are located on the interface. Last we address a finite Griffith crack normal to the viscoelastic interface by means of the obtained Green’s function for a screw dislocation. The crack problem is formulated in terms of a resulting singular integral equation which is solved numerically. The time-dependent stress intensity factors at the two crack tips are obtained and some interesting features are discussed.     

Analysis of elastic crack bridging in ceramic matrix composites

A micromechanics analytical model based on the consistent shear lag theory is developed for predicting the failure modes in fiber reinforced unidirectional stiff matrix composites. The model accounts for a relatively large matrix stiffness and hence its load carrying capacity. The fiber and matrix stresses are established as functions of the applied stress, crack geometry, and the microstructural properties of the constituents. From the predicted stresses, the mode of failure is established based on a point stress failure criterion. The role of the microstructural parameters of the constituents on the failure modes such as self-similar continuous cracking, crack bridging and debonding parallel to the fibers is assessed.     

考虑T-stress的非正态随机参数疲劳裂纹扩展寿命的可靠性分析

基于断裂力学的疲劳裂纹扩展寿命问题的研究常常将裂纹尖端应力展开项的高次项忽略,引起了裂纹扩展模拟的误差,本文考虑高次项T-stress对裂纹扩展角的影响,对裂纹扩展过程做了数值模拟,结果显示相同裂纹扩展长度下,考虑T-stress会延长裂纹扩展寿命。文章首先采用修正的Paris-Erdogan 公式计算了两端承受均布拉伸载荷的边缘斜裂纹板的疲劳裂纹扩展寿命,裂纹扩展方向采用两参数修正的最大拉应力准则。由于结构尺寸,材料特性和载荷等因素具有不确定性,导致疲劳裂纹扩展过程带有一定的随机性,本文以材料属性和载荷为随机变量,在随机有限元法的基础上,结合计算可靠度的四阶矩法,Edgeworth级数展开方法,提出随机参数服从任意分布时的结构疲劳裂纹扩展寿命可靠度的计算方法。分析了参数为非正态分布时的平板裂纹扩展寿命可靠度随裂纹扩展的变化过程。本文方法可预测工程中板裂纹的扩展行为,以及预测裂纹板的可靠度。     

New Green’s function for stress field and a note of its application in quantum-wire structures

The elastic field caused by the lattice mismatch between the quantum wires and the host matrix can be modeled by a corresponding two-dimensional hydrostatic inclusion subjected to plane strain conditions. The stresses in such a hydrostatic inclusion can be effectively calculated by employing the Green’s functions developed by Downes and Faux, which tend to be more efficient than the conventional method based on the Green’s function for the displacement field. In this study, Downes and Faux’s paper is extended to plane inclusions subjected to arbitrarily distributed eigenstrains: an explicit Green’s function solution, which evaluates the stress field due to the excitation of a point eigenstrain source in an infinite plane directly, is obtained in a closed-form. Here it is demonstrated that both the interior and exterior stress fields to an inclusion of any shape and with arbitrarily distributed eigenstrains are represented in a unified area integral form by employing the derived Green’s functions. In the case of uniform eigenstrain, the formulae may be simplified to contour integrals by Green’s theorem. However, special care is required when Green’s theorem is applied for the interior field. The proposed Green’s function is particularly advantageous in dealing numerically or analytically with the exterior stress field and the non-uniform eigenstrain. Two examples concerning circular inclusions are investigated. A linearly distributed eigenstrain is attempted in the first example, resulting in a linear interior stress field. The second example solves a circular thermal inclusion, where both the interior and exterior stress fields are obtained simultaneously.     

Deterministic and stochastic optimization of composite cylindrical laminates

The paper is focused on optimization of prestress and placement of fibers in laminated cylindrical composites. It also involves a stochastic study of prestress deviation in particular layers. Optimization (design) parameters considered in control of internal stresses are the eigenstrains. The behavior of a certain functional serving for optimization of the eigenstrains with stochastically perturbed and correlated values in a laminated cylindrical structure is examined. In the first part, a deterministic optimization of composite laminated cylinders is performed by means of the eigenstrains produced in the layers during the fabrication process. Because fabrication of laminates is sensitive to deviation of eigenstrain magnitudes, as shown from stochastic study, an additional minimization of the eigenstrains is introduced.