Strain measurement by optical correlation

The application of optical correlation measurements to evaluating elastic strain and plastic strain in metals is discussed in this paper. A selected area of the metal surface is illuminated by coherent light. The optical correlation intensity is then measured by transmitting light scattered from the surface through a holographic filter, in which information about the topography of the surface at an earlier time is stored. Changes in surface topography arising from rigid-body displacement, elastic strain, or plastic strain, respectively, cause corresponding changes in optical correlation intensity. Correlation changes arising from surface translation or rotation can be compensated for. An analysis of the process of holographic reconstruction from an elastically strained surface gives good agreement with the experimental results. The correlation technique is sensitive to elastic strains of the order of 10^–5 and to monotonic plastic strains of the order of 10^–4; the change in correlation intensity is essentially linear with increasing plastic strain, up to a maximum strain of about 10^–3.     

Coherent elastic energy calculation of ω particles in β matrix for Zr–Nb alloys

The misfit and coherent elastic energy caused by ω particles in β matrix is quantitatively calculated in this study. First, the coherent strain matrixes for four ω variants are established including the misfit parameters based on Khachaturyan’s theory. Then, the misfit and coherent elastic energy in athermal β → ω transition, and isothermal β → ω transition are calculated, respectively. The calculation results indicate that the coherent elastic energy gets maximum value when x _Nb = 0.08 (Nb content) and gets minimum value when x _Nb = 0.1518 in quenching Zr–Nb alloys, which are in fair agreement with experimental results. For isothermal β → ω transition, the misfit and coherent elastic energy depend on composition and aging temperature. The misfit caused by isothermal ω phase is much larger than the one caused by athermal ω phase. This results in larger coherent elastic energy in isothermal β → ω transition. In addition, the misfit is found as an approximate linear function of temperature and composition for Zr–Nb alloys, and the coherent elastic energy is revealed as an increasing function of |v _F –v _S | for the two kinds of transition.     

Equilibrium surface roughness of a strained epitaxial film due to surface diffusion induced by interface misfit dislocations

In recent years, developments in the microelectronics industry have led to extensive studies of the growth and characterization of thin solid films and their implementation in electronic and opto-electronic devices. A goal is to produce thin films with minimal bulk and surface defects. For those systems produced by epitaxial growth of a film on a substrate that has a slightly different lattice parameter, the stress associated with the elastic mismatch strain needed to satisfy the constraint of epitaxy provides a driving force for nucleation and growth of undesirable defects in the film material or on its surface. Among the most common defects are interface misfit dislocations, arranged more or less periodically on the film-substrate interface, which partially relax the elastic mismatch strain in the film. It has been observed that, for some material systems, surface roughness or waviness arises which correlates spatially with the positions of interface misfit dislocations. It is suggested here that the waviness along the surface may be a result of surface diffusion which is driven by a gradient in the chemical potential of the material along the surface. The chemical potential gradient arises from the nonuniform strain field of the interface misfit dislocations, as well as from the unrelaxed elastic mismatch strain. The focus here is on the development of a relatively simple model of this system which leads to an estimate of the magnitude and profile of surface waviness under conditions of thermodynamic equilibrium, i.e., after the material responds to the chemical potential gradient by seeking out a new configuration for which stresses are redistributed and the chemical potential is again uniform. The condition of uniform chemical potential for the final shape leads to an integro-differential equation for the equilibrium surface shape which is solved numerically. For representative values of system parameters, estimates of equilibrium surface roughness are obtained which can vary from less than one percent of film thickness to a significant fraction of film thickness. Although transient aspects of the process are not studied here, the characteristic time for achieving an equilibrium configuration is estimated.     

Interface structure at the large misfit, still common epitaxy β-FeSi2(101) or (110)//Si(111)

In this paper we compare the elastic energies obtained by tight-binding molecular dynamics simulations for several strained structures of β-FeSi₂, corresponding to the most frequent epitaxial relationships ‘on’ and ‘in’ silicon. Our results confirm that, for coherent interfaces, the very common β-FeSi₂(101) or (110)//Si(111) orientation generates a very large contribution to the elastic energy, due to the large misfit. Therefore, we suggest that the frequent nucleation of such epitaxial relationships in precipitates is provided by the correspondence of the two-dimensional crystal structure for the Si sites between Si(111) and β-FeSi₂(101) or (110). We show it to be maintained even after misfit relaxation in the silicon matrix, as simulated by a large-scale molecular dynamics run.     

Influence of external stress and plastic strain on morphological evolution of precipitates in Ni-based superalloys

An analytical method is proposed to investigate the morphological evolution of γ^′ precipitates in Ni-based superalloys. Two types of superalloys are considered with different ratio between the Young’s moduli of the precipitates and the matrix. Based on Eshelby’s equivalent inclusion theory and Mori–Tanaka’s mean field method the elastic energy is calculated as a function of the particle shape. The plastic strain induced by the formation of dislocation networks at the phase interface is taken into account. The results show that the external stress and the plastic matrix strain play a significant role for shape stability and the rafting mechanism of the precipitates. The elastic–plastic analysis provides a satisfactory explanation for all available experimental observations superalloys undergoing plastic creep.     

Transient creep behavior of a metal matrix composite with a dilute concentration of randomly oriented spheroidal inclusions

The transient creep behavior of a metal matrix composite containing a dilute concentration of randomly oriented spheroidal inclusions is derived explicitly from the constitutive equation of the matrix. This theory can account for the influence of inclusion shape, elastic inhomogeneity between both phases, and the volume fraction of inclusions. The micro-macro transition is carried out by considering the mechanics of incremental creep, which discloses the nature of stress relaxation in the ductile matrix and the connection between the micro and macro creep strains. The transient creep curves of the composite are displayed with several inclusion shapes. Consistent with the known elastic behavior, spherical inclusions are found to provide the weakest reinforcing effect, whereas thin, circular discs possess the most effective strengthening shape. According to this theory and in line with the experimental data, the creep resistance of cobalt at 500°C can improve by more than 80% after adding a mere 5% of rutile particles into it.     

On evolution equations for elastic deformation and the notion of hyperelasticity

For elastic-plastic and elastic-viscoplastic materials it is possible to introduce an evolution equation for an elastic deformation measure. Also, it is possible to develop constitutive equations for which the stress and strain energy are functions of elastic deformation only, the stress is determined by a derivative of the strain energy function and the associated material response is rate-independent and non-dissipative in the absence of the rate of inelasticity. Yet, these equations do not necessarily exhibit hyperelastic response in the elastic range. The objective of this paper is to emphasize the importance of satisfying an additional condition that requires the work done between two configurations to be insensitive to the history and rate of total deformation. This work condition places restrictions on the evolution equation which ensure that the integrated elastic deformation measure is a function of total deformation only. Also, it is argued that there is no need to complicate the evolution equation for elastic deformation to accommodate alternative strain measures since the nonlinearities of these strain measures can be absorbed into the form of the strain energy function.     

A self-consistent approach to dynamic effective properties of a composite reinforced by distributed spherical particles

Summary A self-consistent approach to dynamic effective properties of a composite reinforced by randomly distributed spherical inclusions is studied. The coherent plane waves propagating through the particle-reinforced composite are of attenuation nature. It implies that there is an analogy between the particle-reinforced composite and the effective medium with complex-valued elastic constants from the viewpoints of wave propagation. A composite sphere consisting of the inclusion, the matrix and the interphase between them is assumed embedded in the effective medium. The effective wavenumbers of the coherent plane waves propagating through the particle-reinforced composite are obtained by the dynamic self-consistent conditions which require that the forward scattering amplitudes of such a composite sphere embedded in the effective medium are equal to zero. The dynamic effective properties (effective phase velocity, effective attenuation and effective elastic constants) obtained by the present dynamic self-consistent approach for SiC-Al composites are compared numerically with that obtained by the effective field approach at various volume concentrations. It is found that there is a good agreement between the two approaches at a relatively low frequency and low volume concentration but the numerical results deviate from each other at a relatively high frequency and high volume concentration.     

Analysis of strain error sources in micro-beam Laue diffraction

Micro-beam Laue diffraction is an experimental method that allows the measurement of local lattice orientation and elastic strain within individual grains of engineering alloys, ceramics, and other polycrystalline materials. Unlike other analytical techniques, e.g. based on electron microscopy, it is not limited to surface characterisation or thin sections, but rather allows non-destructive measurements in the material bulk. This is of particular importance for in situ loading experiments where the mechanical response of a material volume (rather than just surface) is studied and it is vital that no perturbation/disturbance is introduced by the measurement technique. Whilst the technique allows lattice orientation to be determined to a high level of precision, accurate measurement of elastic strains and estimating the errors involved is a significant challenge. We propose a simulation-based approach to assess the elastic strain errors that arise from geometrical perturbations of the experimental setup. Using an empirical combination rule, the contributions of different geometrical uncertainties to the overall experimental strain error are estimated. This approach was applied to the micro-beam Laue diffraction setup at beamline BM32 at the European Synchrotron Radiation Facility (ESRF). Using a highly perfect germanium single crystal, the mechanical stability of the instrument was determined and hence the expected strain errors predicted. Comparison with the actual strain errors found in a silicon four-point beam bending test showed good agreement. The simulation-based error analysis approach makes it possible to understand the origins of the experimental strain errors and thus allows a directed improvement of the experimental geometry to maximise the benefit in terms of strain accuracy.     

集中载荷作用下弹性平面刚性夹杂的形状优化

当夹杂的弹性模量比基体的弹性模量大得多时,可将其看成刚性夹杂.对于这类硬夹杂与软基体的复合材料,采用复变函数方法中的保角映射技术和推广的Schwarz延拓原理,并结合对复应力函数的奇性主部分析,导出了焊接任意形状刚性夹杂的全场基本奇异解,求解了集中力与集中力偶作用下弹性平面刚性夹杂的形状优化问题,描绘出了夹杂界面应力最大值、界面主应力最大值随夹杂形状的变化规律,精确地定位出了界面应力最大值、界面主应力最大值的作用位置.通过对这些不同形状夹杂的界面主应力最大值的比较,确定了夹杂的最优形状.     

Misfit dislocations and stress in In<Subscript>1 − <Emphasis Type="Italic">x</Emphasis></Subscript>Ga<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>As/GaAs heterostructures

We have estimated the elastic properties of In_{1 − x} Ga _x As/GaAs heterostructures and the characteristics of misfit dislocations in such heterostructures: misfit dislocation spacing, Burgers vector length in various interfaces, surface density of dangling bonds, film/substrate interface energy, critical film thickness below which pseudomorphic growth is possible without misfit dislocations, elastic strain energy of the film-substrate system, average elastic strain of a thin-film island as a function of its radius, thermal stresses induced by the thermal-expansion and lattice mismatches between the layers in contact, and crack length in the film.     

The nonlinear effect of lattice mismatch parameter on morphological and compositional instabilities of epitaxial layers

Summary The misfit deformation in a film-substrate system is mostly concentrated in the film when the stiffness of the system is dominated by that of the substrate. Such is generally the case in microelectronics applications. For many semiconductor materials with electronic properties suitable for device applications, the associated mismatch parameter routinely falls in the range, say, from –5% to +5%. Elastic strains of such magnitude provide a source of free energy for configurational modifications. To examine the configurational stability of such a uniformly strained film, the elastic deformation associated with a perturbation in configuration is first obtained. This elastic deformation and the misfit deformation are then used to calculate the film elastic energy. For a strictly linear analysis, the resulting stability condition is either a function of the mismatch strain energy density, for the case of a morphological perturbation, or completely unrelated to the underlying mismatch, for the case of a compositional perturbation. In either situation, the sign of the mismatch is totally immaterial. In this paer, the elastic deformation brought about by a perturbation in the configuration is taken as a small deformation superimposed on the large mismatch deformation in a nonlinear setting. The role of the mismatch is thus more fully explored and uncovered.     

Large elastic strains and ductile necking of W nanowires embedded in TiNi matrix

The deformation behaviors of W nanowires embedded in a TiNi matrix were investigated by means of in-situ synchrotron high energy X-ray diffraction(HEXRD) and in-situ transmission electron microscopy(TEM) analysis during tensile deformation.The HEXRD measurement indicated that the W nanowires exhibited an average lattice strain of about 1.50 %,whereas the TEM examination revealed a local elastic strain of about 4.59 % in areas adjacent to the TiNi matrix where stress-induced martensitic transfo rmation occurred.This strain corresponds to a stress of ~15 GPa for the W nanowires.In addition,in areas adjacent to the TiNi matrix where plastic deformation and cracking were generated,the W nanowire showed significant ductile necking with ~80 % reduction in cross-section area.The ductile necking of W nanowire is attributed to the lack of protection from the stress-induced martensitic transformation of the TiNi matrix.     

Topological derivatives for fundamental frequencies of elastic bodies

In this article a new method for topological optimization of fundamental frequencies of elastic bodies, which could be considered as an improvement on the bubble method, is introduced. The method is based on generalized topological derivatives. For a body with different types of inclusion the vector genus is introduced. The dimension of the genus is the number of different elastic properties of the inclusions being introduced. The disturbances of stress and strain fields in an elastic matrix due to a newly inserted elastic inhomogeneity are given explicitly in terms of the stresses and strains in the initial body. The iterative positioning of inclusions is carried out by determination of the preferable position of the new inhomogeneity at the extreme points of the characteristic function. The characteristic function was derived using Eshelby's method. The expressions for optimal ratios of the semi-axes of the ellipse and angular orientation of newly inserted infinitesimally small inclusions of elliptical form are derived in closed analytical form.     

Linear and elastic–plastic fracture mechanics revisited by use of Fourier transforms – theory and application

This paper investigates whether and how discrete Fourier transforms (DFT) can be used to compute the local stress/strain distribution around holes in externally loaded two-dimensional representative volume elements (RVEs). To this end, the properties of DFT are first reviewed and then applied to the solution of linear elastic and time-dependent elastic plastic material response. The equivalent inclusion method is used to derive a functional equation which allows for the numerical computation of stresses and strains within an RVE with heterogeneities of arbitrary shape and stiffness. This functional equation is then specialized to the case of circular and elliptical holes of different minor axes which eventually degenerate into Griffith cracks. The numerically predicted stresses and strains are compared to the corresponding analytical solutions for a single circular as well as an elliptical hole in an infinitely large plate under tension as well as to finite element calculations (for time-independent elastic/plastic material response).     

The effect of different forms of strain energy functions in hyperelasticity‐based crystal plasticity models on texture evolution and mechanical response of face‐centered cubic crystals

Micro‐mechanical and macro‐mechanical behavior of face‐centered cubic (FCC) crystals is investigated by using different forms of strain energy functions in hyperelastic material models in crystal plasticity finite element framework. A quadratic strain energy function with anisotropic elastic constants, a polyconvex strain energy function with invariants associated with the cubic symmetry, and a strain energy function from an inter‐atomic potential are considered in hyperelastic material models to describe the elastic deformation of FCC crystals. In our numerical experiments, the trajectories of {111} poles in the pole figure and the accumulated plastic slips of FCC coppers under uniaxial tension and simple shear depend on the choice of strain energy functions when the slip resistance of the slip systems is high. The ability of strain energy functions in this study to represent elastic lattice distortions in crystals varies with the amount of elastic deformation and the shape of deformed lattice. However, numerical results show that the change of macroscopic mechanical behavior of FCC coppers is not significant for the choice of strain energy functions, compared with the change of crystallographic texture evolution. Copyright © 2014 John Wiley & Sons, Ltd.     

Misfit dislocation loop nucleation at a quantum dot

An energy criterion for the nucleation of a circular prismatic misfit dislocation loop in a spheroidal inclusion modeling a quantum dot is considered. The critical radius of the inclusion, for which the misfit dislocation loop can nucleate, is studied as a function of the lattice misfit between the inclusion and the matrix.     

岩样单轴压缩峰后泊松比理论研究

研究了单轴压缩岩样应变软化阶段侧向应变与轴向应变的比值(峰后泊松比)的变化规律.岩样的塑性变形假设根源于塑性应变局部化.岩样的轴向及侧向变形被分别分为两部分:弹性变形(由虎克定律描述)及由局部化引起的塑性变形(由梯度塑性理论及几何关系确定).应变软化阶段的轴向应变-侧向应变曲线、轴向应力-轴向应变曲线及轴向应力-侧向应变曲线都得到了实验验证.在峰值强度时,峰后泊松比等于峰前泊松比.当压缩应力降至零时,峰后泊松比达到临界值.该临界值可能比峰前泊松比大,也可能比它小.峰后泊松比还和试件尺寸有关,这与峰前泊松不同.峰后泊松比与轴向压应力之间的关系可能是一条直线,也可能是上凸的,或上凹的.这取决于岩石的本构参数(弹性模量、剪切及软化模量、剪切带宽度及峰前泊松比)、试件的结构尺寸(试件宽度及高度)及剪切带倾角之间的关系.     

Rigid toroidal inhomogeneity in an elastic medium

The contribution of a rigid toroidal inhomogeneity to the overall elastic properties is addressed. The method of asymptotic extension is applied to derive asymptotic approximations for the polarization matrix of a rigid toroidal inclusion embedded in an elastic media.     

Strain Hardening of Initially Isotropic Metals under Loading along Slightly Curved Trajectories

By the example of martensitic steel we study regularities of strain hardening under loading along two-link broken paths corresponding to slightly curved strain paths. It is shown that the loading surface separating the domains of elastic and elastoplastic strains (yield surface) is displaced in the direction of a vector connecting the surface center with the loading path image point, while the shape of its frontal part remains unchanged. The yield surface center displacement versus the intensity of accumulated plastic strains is described by a curve invariant to the loading path.