Micromechanical modeling and simulation of rate-dependent effects in ferroelectric polycrystals

In this contribution, rate-dependent switching effects of ferroelectric materials are studied by means of a micromechanically motivated approach. The onset of domain switching is thereby initiated based on the reduction in Gibbs free energy by means of energy-based criterion. The subsequent nucleation and propagation of domain walls during switching process are incorporated via a volume fraction concept combined with a simple linear kinetics theory. The key aspect in modeling of the interaction between the individual grains (intergranular effects) are incorporated in this model by making use of a probabilistic ansatz; to be specific, a phenomenologically motivated Weibull distribution function is adopted. The developed framework is incorporated into a finite element formulation whereby each domain is represented by a single finite element and initial dipole directions are randomly oriented so that the virgin state of the particular bulk ceramics of interest reflects an un-poled material. Based on a staggered iteration technique and straightforward volume averaging concept, the model is simulated to capture the non-linear behavior for different loading, for various loading amplitudes and frequencies. Attributes of the model, both symmetric major loops and biased minor loops are illustrated through examples. Simulation results for the rate-independent case are in good agreement with experimentally measured data reported in the literature and, moreover, are extended to rate-dependent computations which captures some important insights.     

微电子材料在微塑成形中的尺度效应

微电子材料在微机电系统（MEMS）的发展中越来越受到青睐，但是其工艺加工的不足限制了实际应用的步伐。微塑性成形可以成形微电子器件，由于其尺寸微型化，在微塑性成形中存在一个不可避免的“尺度效应”问题，尺度效应表现在材料的流动行为、成形中摩擦效应和实验结果的分散性上。在介绍尺度效应的基础上对其进行了分类，给出了判断标准，并从流动应力、晶粒尺度、摩擦效应和温度效应等方面综述了尺度效应对微塑性成形的影响。由于基于连续介质的传统塑性力学理论无法解释微塑性成形过程中的尺度效应，因此引入了非均匀介质的塑性应变梯度理论并进行了探讨，最后指出了尺度效应的研究发展方向，从而促进微电子材料的开发应用。     

Micromechanical modeling of short fatigue cracks

This paper gives an overview over the micromechanical modeling approaches of short fatigue cracks. Until now many approaches have been presented in the literature, which differ significantly in their degree of complexity ranging from simple empirical or analytical models to complex models based on numerical solutions. In recent years different trends were observed: On the one hand detailed models are presented, which describe the propagation of a microstructurally short fatigue crack in a physically sound way based on discrete dislocations. However, their application is somewhat limited due to their complexity regarding the application in real microstructures. Thus, another trend is to develop models closely related to experimental research work to identify and focus on the main aspects of short fatigue crack growth.     

Influences of temperature and grain size on the material deformability in microforming process

This paper investigated the influences of temperature and grain size on the deformability of pure copper in micro compression process. Based on the dislocation theory, a constitutive model was proposed taking into account the influences of forming temperature, Hall-Petch relationship and surface layer model. Vacuum heat treatment was employed to obtain various grain sizes of cylindrical workpieces, and then laser heating method was applied to heat workpieces during microforming process. Finite element (FE) simulation was also performed, with simulated values agreed well with the experimental results in terms of metal flow stress. Both the FE simulated and experimental results indicate that forming temperature and grain size have a significant influence on the accuracy of the produced product shape and metal flow behaviour in microforming due to the inhomogeneity within the deformed material. The mechanical behaviour of the material is found to be more sensitive to forming temperature when the workpieces are constituted of fine grains.     

Micromechanical modeling of imperfect textile composites

After fabrication the carbon-carbon (C/C) plain weave textile composites often show a certain degree of geometrical or material disorder including yarn waviness and misalignment or nesting of individual fiber tows together with intrinsic material porosity observed at all relevant scales. A brief survey of recently developed approaches for estimating overall elastic stiffnesses or thermal conductivities of such composite systems is presented in this paper. Depending on the source and type of available geometrical data the homogenization scheme usually relies either on finite element (FEM) simulations performed for a suitable Periodic Unit Cell (PUC) or employs one of the popular averaging techniques such as the Mori-Tanaka (MT) method. While existing applications of both methodologies are encouraging, there still exists a number of steps to be completed in the course of the future research.     

Micromechanical modeling of fracture initiation in 7050 aluminum

Mechanical testing and finite element calculations have been carried out to characterize the fracture initiation behavior of the high-strength aluminum alloy 7050-T7451. Results show that fracture initiation is well predicted for two specimen types of differing constraint using the stress-modified, critical plastic strain micromechanical model. The relation between stress triaxiality and critical plastic strain was found from a series of notched tensile specimens. Data from these tests are interpreted using both companion finite element modeling and common, semi-empirical relations, and these two approaches are compared. Multiple, interrupted tests of standard, highly constrained single edge notched bend specimens are used to obtain the J–R curve in 7050 for small amounts of tearing to experimentally identify initiation. Companion modeling and the stress-modified, critical plastic strain relation are used to find the length scale for fracture, l^*, needed for initiation predictions. The calibrated stress-modified, critical plastic strain relation and length scale are then used to predict fracture initiation of a low-constraint specimen. The prediction is within 5% of the experimental measurements. Finally, various aspects of the procedure followed in the present work are compared to previous efforts using similar approaches.     

Micromechanical modeling of load transfer in fibrous composites

A unified analytical treatment is presented for the study of micromechanical stress distribution in unidirectional fibrous composites loaded with various thermal and mechanical loads. Two models are considered to represent the composite. Both use a concentric cylindrical system with the difference that one requires laterally free while the other requires laterally constrained outer boundaries, broadly describing situations of plane stress and plane strain, respectively. The present work has been motivated by the recent work of McCartney (McCartney, Proc. Roy. Soc. London, Ser. A 425 (1989) 215–244) who analyzed the laterally free system, and by our previous work (Nayfeh, Fibre Sci. Technol. 10 (1977)) in which we analyzed the laterally constrained one. For axisymmetric loading, and upon adopting some appropriate restrictions on the radial behavior of some field quantities, an elasticity-based procedure reduces the two-dimensional field equations, which hold in both the fiber and matrix components, together with the appropriate interface and boundary conditions, to a quasi-one-dimensional system. The resulting system is capable of identifying the stress distribution in each component as influenced by the other component via the readily identifiable interaction (transfer) terms. The model is general and applicable to a large variety of situations. These include situations of matrix cracking, fiber break and even regions of slip at the fiber–matrix interface. As a by-product, the model was capable of obtaining the classical Lamé solutions (the iso-strain case) as a degenerate case. Confidence in the modeling was gained when it identically reproduced all of the numerical examples presented by McCartney. Numerical results that parallel some of the ones presented by McCartney are included in the form of comparisons between results obtained based upon the laterally constrained and the laterally free systems.     

R-curve modeling of rate and size effects in quasibrittle fracture

The equivalent linear elastic fracture model based on an R-curve (a curve characterizing the variation of the critical energy release rate with the crack propagation length) is generalized to describe both the rate effect and size effect observed in concrete, rock or other quasibrittle materials. It is assumed that the crack propagation velocity depends on the ratio of the stress intensity factor to its critical value based on the R-curve and that this dependence has the form of a power function with an exponent much larger than 1. The shape of the R-curve is determined as the envelope of the fracture equilibrium curves corresponding to the maximum load values for geometrically similar specimens of different sizes. The creep in the bulk of a concrete specimen must be taken into account, which is done by replacing the elastic constants in the linear elastic fracture mechanics (LEFM) formulas with a linear viscoelastic operator in time (for rocks, which do not creep, this is omitted). The experimental observation that the brittleness of concrete increases as the loading rate decreases (i.e. the response shifts in the size effect plot closer to LEFM) can be approximately described by assuming that stress relaxation causes the effective process zone lenght in the R-curve expression to decrease with a decreasing loading rate. Another power function is used to describe this. Good fits of test data for which the times to peak range from 1 sec to 250000 sec are demonstrated. Furthermore, the theory also describes the recently conducted relaxation tests, as well as the recently observed response to a sudden change of loading rate (both increase and decrease), and particularly the fact that a sufficient rate increase in the post-peak range can produce a load-displacement response of positive slope leading to a second peak.     

Observation of intrinsic size effects in the optical response of individual gold nanoparticles

The photothermal heterodyne imaging method is used to study for the first time the absorption spectra of individual gold nanoparticles with diameters down to 5 nm. Intrinsic size effects that result in a broadening of the surface plasmon resonance are unambiguously observed. Dispersions in the peak energies and homogeneous widths of the single-particle resonances are revealed. The experimental results are analyzed within the frame of Mie theory.     

Micromechanical formulation and 3D finite element modeling of microinstabilities in composites

A 3D micromechanical formulation and a FE-model of fiber micro-buckling in materials with isotropic and transversal isotropic fibers in compression is presented. Three variants of geometrical modeling of the characteristic cell are proposed and compared. An appropriate one is then selected. An eigenvalue analysis of a characteristic cell is performed. The results show that the fiber anisotropy reduces significantly the critical loads and must be taken into account.     

Micromechanical modeling of the behavior of duplex stainless steels

The aim of the present paper is the micromechanical modeling of an aged duplex (austenite/ferrite) stainless steel. While the elastic properties of both phases are almost the same, the plastic mismatch between ferrite and austenite is high due to ferrite aging. Accounting for the complex micro and macrostructure of this material, three scales are investigated. The first one corresponds to each single phase. The second scale consists of a percolated (interwoven) network of both phases. The third scale is an aggregate of percolated bicrystals. For the first scale, each phase is modeled as a single crystal. In the second scale, F.E. simulations are performed on a unit cubic cell representing the percolated networks. A mean field model is then fitted in order to represent that bicrystal. This mean field model is used at the third scale to model the behavior of the aggregate of two-phase grains, using a model for multi-phase materials. At this scale, it becomes possible to represent the distribution of average ferrite stresses in each ferrite-austenite bicrystal. These variations are thought to be responsible for the heterogeneous damage nucleation observed in this material.     

Computational modeling of size effects in concrete specimens under uniaxial tension

The paper presents a follow-up study of numerical modeling of complicated interplay of size effects in concrete structures. The major motivation is to identify and study interplay of several scaling lengths stemming from the material, boundary conditions and geometry. Methods of stochastic nonlinear fracture mechanics are used to model the well published results of direct tensile tests of dog-bone specimens with rotating boundary conditions. Firstly, the specimens are modeled using microplane material and also fracture-plastic material laws to show that a portion of the dependence of nominal strength on structural size can be explained deterministically. However, it is clear that more sources of size effect play a part, and we consider two of them. Namely, we model local material strength using an autocorrelated random field attempting to capture a statistical part of the combined size effect, scatter inclusive. In addition, the strength drop noticeable with small specimens which was obtained in the experiments could be explained either by the presence of a weak surface layer of constant thickness (caused e.g. by drying, surface damage, aggregate size limitation at the boundary, or other irregularities) or three dimensional effects incorporated by out-of-plane flexure of specimens. The latter effect is examined by comparison of 2D and 3D models with the same material laws. All three named sources (deterministic-energetic, statistical size effects and the weak layer effect) are believed to be the sources most contributing to the observed strength size effect; the model combining all of them is capable of reproducing the measured data. The computational approach represents a marriage of advanced computational nonlinear fracture mechanics with simulation techniques for random fields representing spatially varying material properties. Using a numerical example, we document how different sources of size effects detrimental to strength can interact and result in relatively complicated quasibrittle failure processes. The presented study documents the well known fact that the experimental determination of material parameters (needed for the rational and safe design of structures) is very complicated for quasibrittle materials such as concrete.     

Micromechanical modelling of size effects in failure of porous elastic solids using first order plane strain gradient elasticity

In this paper a first order porous strain gradient elasticity model is presented. The constitutive equations have been obtained by higher order homogenization and the model is used with a failure criterion in order to discuss size effects in failure of porous elastic solids. The model contains two microstructural parameters namely the void volume fraction and the half void spacing. After an extended numerical validation of the porous strain gradient elasticity model, the boundary value problem of a plate with a hole under bi- and uniaxial remote tension is investigated. The numerical simulations have been performed varying both microstructural parameters in order to study the influence of different microstructural dimensions on the onset of macroscopic failure. The numerical results show that the presented model is able to predict size effects and that size effects in failure do not only depend on the microstructural properties but also on the macroscopic geometry, loading conditions, and the failure mechanism.     

Micromechanical multiplane finite element modeling of crack growth in fiber reinforced materials

The purpose of this study was to devise and verify a scheme of analysis which can be used to investigate the micromechanical failure mechanisms and determine an effective fracture toughness for a class of fiber reinforced materials. The material of primary interest in this study consists of a linearly elastic matrix material reinforced with rows of parallel, linearly elastic and straight fibers. Micromechanical multiplane finite element and experimental studies of the stress conditions near a crack front in a side cracked fiber reinforced epoxy tensile specimen were conducted. The 2-D multiplane method of analysis, recently developed at Syracuse University for approximate analysis of a class of 3-D problems, was the basis of the micromechanical finite element analytical technique developed in this study. Since failure of a member fabricated from a fiber reinforced material is generally proceeded by local failures, sequential finite element analyses were performed to model the progressive failure mechanism. Local failure modes considered in the analysis are yield in either the matrix material or fibers, crack extension in the matrix material, and failure of the matrix to fiber bond. The agreement between the multiplane analytical and laboratory test results show that the multiplane method provides a useful tool for micromechanical study of fiber reinforced composite materials.     

Micromechanical modeling of damage and failure mechanisms in C/C–SiC

This paper deals with modeling of damage and failure mechanisms observed in 2D C/C–SiC composite samples loaded by tension, shear and compression. In a specimen subjected to tension, the early stage of damage is characterized by transverse cracking. Further load increase induces fiber failure in the longitudinal plies, which leads to fracture. Shear loading gives origin to cracks oriented at 0°/90° and 45° to the fiber axes, the latter causing fracture. Specimens loaded in compression exhibit catastrophic failure due to microbuckling of fiber bundles. Micromechanical models are formulated for the discussed mechanisms. Results of numerical simulations show good agreement with experiments and the ability of the model to describe more complex loadings.     

Numerical simulations of specimen size and mismatch effects in ductile crack growth - Part II: Near-tip stress fields

The effect of ductile crack growth on the near tip stress field in two different specimen geometries has been investigated. For homogeneous specimens it is observed that the peak stress level increases with ductile crack growth. The effect is most pronounced up to about 1 mm of crack growth. For low and intermediate hardening there is a significant effect of specimen size on the stress level. In case of mismatch in yield stress, the simulations show that the increase in stress level in the material with the lower yield stress is of a similar magnitude as is the case for stationary cracks. In case of ductile crack growth deviation from the original crack plane occurs, the highest stresses are still found close to the interface, and not in front of the current crack tip.     

Micromechanical modeling of grain boundary resistance to cleavage crack propagation in ferritic steels

In ferritic steels a propagating cleavage microcrack changes its propagation direction as it advances from grain to grain. This is due to differences in the orientation of the cleavage planes of two neighboring grains. In order to reach a cleavage plane in a new grain, a microcrack must first penetrate the grain boundary. Grain boundaries therefore act as natural barriers in cleavage fracture. The influence of a grain boundary and the associated misorientation in cleavage planes on crack arrest is here examined using a 3D finite element model with axisymmetric periodicity, representing two grains whose cleavage planes are tilted and twisted relative to each other. The temperature dependent mechanical properties of ferrite are modeled using a temperature dependent viscoplastic response. The development of the crack front as the microcrack penetrates through a grain boundary is here presented. The influence of the twist misorientation on the critical grain size, defined as the largest grain size that can arrest a rapidly propagating microcrack, is examined in a temperature range corresponding to the ductile to brittle transition (DBT) region. It is shown that when both tilt and twist misorientation are present, the influence of tilt and twist, respectively, on crack growth resistance can be decoupled.     

Micromechanical modeling of damage and fracture of unidirectional fiber reinforced composites: A review

An overview of methods of the mathematical modeling of deformation, damage and fracture in fiber reinforced composites is presented. The models are classified into five main groups: shear lag-based, analytical models, fiber bundle model and its generalizations, fracture mechanics based and continuum damage mechanics based models and numerical continuum mechanical models. Advantages, limitations and perspectives of different approaches to the simulation of deformation, damage and fracture of fiber reinforced composites are analyzed.