Analysis of heterogeneous deformation and dislocation dynamics in single crystal micropillars under compression

The size dependent deformation of Cu single crystal micropillars with thickness ranging from 0.2 to 2.5 μm subjected to uniaxial compression is investigated using a Multi-scale Dislocation Dynamics Plasticity (MDDP) approach. MDDP is a hybrid elasto-viscoplastic simulation model which couples discrete dislocation dynamics at the micro-scale (software micro3d) with the macroscopic plastic deformation. Our results show that the deformation field in these micropillars is heterogeneous from the onset of plastic flow and is confined to few deformation bands, leading to the formation of ledges and stress concentrations at the surface of the specimen. Furthermore, the simulation yields a serrated stress–strain behavior consisting of discrete strain bursts that correlates well with experimental observations. The intermittent operation and stagnation of discrete dislocation arms is identified as the prominent mechanism that causes heterogeneous deformation and results in the observed macroscopic strain bursts. We show that the critical stress to bow an average maximum dislocation arm, whose length changes during deformation due to pinning events, is responsible for the observed size dependent response of the single crystals. We also reveal that hardening rates, similar to that shown experimentally, occur under relatively constant dislocation densities and are linked to dislocation stagnation due to the formation of entangled dislocation configuration and pinning sites.     

Statistical aspects of the identification of material parameters for elasto-plastic models

Summary The presented method to identify material parameters for inelastic deformation laws is based on the numerical analysis of inhomogeneous stress and strain fields received from suitable experiments. Tensile and bending tests were carried out to obtain elastic and hardening parameters. The deformation law for small elasto-plastic strains is presented as a system of nonlinear differential and algebraic equations (DAE) consisting of the stress–strain relation, evolution equations for the internal variables and the yield condition. Different rules for the evolution equations of isotropic, kinematic and distorsional hardening are proposed. The DAE are discretized using an implicit Euler method, and the resulting system of nonlinear algebraic equations is solved using the Newton method. Deterministic optimization procedures are preferred to identify material parameters from a least-squares functional of numerical and measured comparative quantities. The gradient of the objective function was calculated using a semianalytical sensitivity analysis. Due to measurement errors, the optimal sets of material parameters are non unique. The approximate estimation of confidence regions and the calculation of correlation coefficients is presented. The results of several optimization processes for material parameters of elasto-plastic deformation laws show a good agreement between measured and calculated values, but they show also problems which may occur if systematic errors will not be recognized and deleted. Received 30 September 1999; accepted for publication 8 March 2000     

基于中间构形的大变形弹塑性模型

在大变形弹塑性本构理论中,一个基本的问题是弹性变形和塑性变形的分解.通常采用两种分解方式,一是将变形率(或应变率)加法分解为弹性和塑性两部分,其中,弹性变形率与Kirchhoff应力的客观率通过弹性张量联系起来构成所谓的次弹性模型,而塑性变形率与Kirchhoff应力使用流动法则建立联系;另一种是基于中间构形将变形梯度进行乘法分解,它假定通过虚拟的卸载过程得到一个无应力的中间构形,建立所谓超弹性-塑性模型.研究了基于变形梯度乘法分解并且基于中间构形的大变形弹塑性模型所具有的若干性质,包括:在不同的构形上,塑性旋率的存在性、背应力的对称性、塑性变形率与屈服面的正交性以及它们之间的关系.首先,使用张量函数表示理论,建立了各向同性函数的若干特殊性质,并导出了张量的张量值函数在中间构形到当前构形之间进行前推后拉的简单关系式.然后,基于这些特殊性质和关系式,从热力学定律出发,建立模型在不同构形上的数学表达,包括客观率表示的率形式和连续切向刚度等,从而获得模型所具有的若干性质.最后,将模型与4种其他模型进行了比较分析.      

On a formulation for anisotropic elastoplasticity at finite strains invariant with respect to the intermediate configuration

The paper is concerned with a formulation of anisotropic finite strain inelasticity based on the multiplicative decomposition of the deformation gradient F=F_eF_p. A major feature of the theory is its invariance with respect to rotations superimposed on the inelastic part of the deformation gradient. The paper motivates and shows how such an invariance can be achieved. At the heart of the formulation is the mixed-variant transformation of the structural tensor, defined as the tensor product of the privileged directions of the material as given in a reference configuration, under the action of F_p. Issues related to the plastic material spin are discussed in detail. It is shown that, in contrast to the isotropic case, any flow function formulated purely in terms of stress quantities, necessarily exhibits a non-vanishing plastic material spin. The possible construction of spin-free rates is discussed as well, where it is shown that the flow rule must then depend not only on the stress but on the strain as well.     

Scale-dependent constitutive relations and the role of scale on nominal properties

Size effects in strength and fracture energy of heterogeneous materials is considered within a context of scale-dependent constitutive relations. Using tools of wavelet analysis, and considering the failure state of a one-dimensional solid, constitutive relations which include scale as a parameter are derived from a ‘background’ gradient formulation. In the resulting theory, scale is not a fixed quantity independent of deformation, but rather directly dependent on the global deformation field. It is shown that strength or peak nominal stress (maximum point at the engineering stress–strain diagram) decreases with specimen size while toughness or total work to fracture per nominal area (area under the curve in the engineering stress–strain diagram integrated along the length of the considered one-dimensional specimen) increases. This behavior is in agreement with relevant experimental findings on heterogeneous materials where the overall mechanical response is determined by variations in local material properties. The scale-dependent constitutive relations are calibrated from experimental data on concrete specimens.     

Mo-ZrC梯度金属陶瓷的冲击响应行为

分层梯度材料特定的梯度变化能有效增强材料性能。为研究梯度结构、冲击方向对分层梯度材料冲击响应的影响,利用分离式霍普金森压杆结合高速摄影技术对Mo-ZrC分层梯度金属陶瓷进行了动态压缩实验,基于数字图像相关技术讨论了梯度结构、冲击方向对金属陶瓷材料破坏模式的影响,利用Mori-Tanaka理论计算得到金属陶瓷等效性质,结合应力波理论研究波在分层梯度复合材料中的传播规律。结果表明：(1)相同加载条件下,梯度结构对材料的强度、韧性和破坏产物的完整性具有重要影响,在冲击过程中,样品响应可以分为压紧阶段、裂纹成核发展阶段和贯穿阶段,对于不同梯度结构和冲击方向,样品在加载过程中呈现出不同的破坏时序和失效模式;(2)利用数字图像相关方法跟踪分层梯度陶瓷的局部变形发展,分析发现局部增量达到临界状态后,局部变形发展转化为微裂纹的形成和累积,最终导致整体性破碎失效;(3)通过分层梯度材料一维应力波传播理论推导得到,改变冲击梯度方向对应力波透反射系数存在一定影响,不同梯度结构设计对改变冲击梯度方向敏感性不同,且存在极值情况。     

Finding minimum energy configurations for constrained beam buckling problems using the Viterbi algorithm

In this work, we present a novel technique to find approximate minimum energy configurations for thin elastic bodies using an instance of dynamic programming called the Viterbi algorithm. This method can be used to find approximate solutions for large deformation constrained buckling problems as well as problems where the strain energy function is non-convex. The approach does not require any gradient computations and could be considered a direct search method. The key idea is to consider a discretized version of the set of all possible configurations and use a computationally efficient search technique to find the minimum energy configuration. We illustrate the application of this method to a laterally constrained beam buckling problem where the presence of unilateral constraints together with the non-convexity of the energy function poses challenges for conventional schemes. The method can also be used as a means for generating “very good” starting points for other conventional gradient search algorithms. These uses, along with comparisons with a direct application of a gradient search and simulated annealing, are demonstrated in this work.     

Q235钢缺口试样拉压低循环疲劳的实验研究

以Q235钢制U型缺口板试样为研究对象,用有限元方法计算其缺口根部等效应变幅对应的试样标距段位移,以此控制试验机进行拉压循环疲劳试验。然后用局部应力应变法对试验测得的寿命结果进行分析。结果表明：无论用有限元还是修正Neuber公式计算缺口根部的应力应变,局部应力应变法的疲劳寿命评估只适用于缺口半径较大的试样;对缺口半径较小试样的估计寿命明显低于实测值,且有限元法比修正Neuber法更保守。进而又对试样缺口区域应变梯度的影响进行了探讨：参照有限元计算的应变梯度,利用Taylor模型估算了缺口根部的屈服应力和流动应力;在此基础上重新计算应变分布并估计试样的疲劳寿命,结果证实考虑应变梯度影响可改善缺口试样的疲劳寿命估计。     

Full-field characterization of mechanical behavior of polyurethane foams

The foam material of interest in this investigation is a rigid closed-cell polyurethane foam PMDI with a nominal density of 20 pcf (320 kg/m³). Three separate types of compression experiments were conducted on foam specimens. The heterogeneous deformation of foam specimens and strain concentration at the foam–steel interface were obtained using the 3-dimensional digital image correlation (3D-DIC) technique. These experiments demonstrated that the 3D-DIC technique is able to obtain accurate and full-field large deformation of foam specimens, including strain concentrations. The experiments also showed the effects of loading configurations on deformation and strain concentration in foam specimens. These DIC results provided experimental data to validate the previously developed viscoplastic foam model (VFM). In the first experiment, cubic foam specimens were compressed uniaxially up to 60%. The full-field surface displacement and strain distributions obtained using the 3D-DIC technique provided detailed information about the inhomogeneous deformation over the area of interest during compression. In the second experiment, compression tests were conducted for cubic foam specimens with a steel cylinder inclusion, which imitate the deformation of foam components in a package under crush conditions. The strain concentration at the interface between the steel cylinder and the foam specimen was studied in detail. In the third experiment, the foam specimens were loaded by a steel cylinder passing through the center of the specimens rather than from its end surface, which created a loading condition of the foam components similar to a package that has been dropped. To study the effects of confinement, the strain concentration and displacement distribution over the defined sections were compared for cases with and without a confinement fixture.     

花岗岩单轴压缩侧向变形及脆性破坏机制实验研究

为研究花岗岩侧向变形及脆性破坏机制,对花岗岩试件进行单轴压缩实验。利用动态应变采集系统、数字散斑相关方法(DSCM)和显微观测手段,记录并分析花岗岩试件在单轴压缩过程中的宏观侧向应变、局部侧向应变以及破裂面形貌,并与水泥砂浆试件的破坏过程对比,讨论了花岗岩脆性破坏机制。实验与分析结果表明:(1)花岗岩试件在加载初期发生侧向收缩变形,产生并发展于压密阶段,消失于线弹性阶段初期,这主要由于试件内部裂纹闭合造成的;此后,宏观侧向应变持续增长,当侧向应变与轴向应变之比接近0.5时试件破坏;(2)在峰值载荷前很长一段时间内,局部侧向应变在一定范围内波动,临近试件破坏时局部侧向应变最大值和最小值均出现较大幅度的波动,二者差值迅速增大,试件不均匀程度增大,最终导致试件破坏;(3)在峰值载荷前有无塑性屈服阶段是峰值载荷后脆性破坏程度的重要影响因素,而宏观裂纹的贯通程度是峰值载荷后应力降大小的决定因素。     

一种从离散模拟到连续介质弹性模拟的过渡方法

提出了一种从离散分子动力学模拟(MD)到连续介质弹性有限元计算分析(FEA)的过渡方法, 简称MD-FEA方法. 首先通过MD计算获得晶体材料原子的移动位置, 然后根据晶体结构的周期性特征构造连续介质假设下的有限单元变形模型, 进一步结合材料的力学行为本构关系获得应变和应力场. 为了检验MD-FEA方法的有效性, 将该方法应用于详细分析Al-Ni软硬组合两相材料纳米柱体的拉伸变形问题和基底材料为Al球形压头材料为金刚石的纳米压痕问题. 采用MD-FEA方法获得了上述两种问题的应力?应变场, 并将计算结果分别与传统MD方法中通过变形梯度计算的原子应变以及原子的位力应力进行了比较, 详细讨论了用MD-FEA方法计算的应力?应变场与传统MD原子应变和位力应力的区别, 并对MD-FEA方法的有效性及其相较于传统MD方法所具有的优势进行了探讨. 结论显示, MD-FEA方法与传统MD方法在应力?应变变化平缓的区域得到的结果接近, 但在变化剧烈的区域以及材料的表/界面区域, MD-FEA方法能够得到更加精确的结果. 同时, MD-FEA方法避免了传统MD方法中, 需要人为选取截断半径以及加权函数所导致的误差. 另外, 当应变较大时, MD-FEA方法计算的小应变与传统MD方法计算的格林应变存在一定差异, 因此, MD-FEA方法更适合应变较小的情形.      

Strain-rate effect on initial crush stress of irregular honeycomb under dynamic loading and its deformation mechanism

The seemingly contradictory understandings of the initial crush stress of cellular materials under dynamic loadings exist in the literature, and a comprehensive analysis of this issue is carried out with using direct information of local stress and strain. Local stress/strain calculation methods are applied to determine the initial crush stresses and the strain rates at initial crush from a cell-based finite element model of irregular honeycomb under dynamic loadings. The initial crush stress under constant-velocity compression is identical to the quasi-static one, but less than the one under direct impact, i.e. the initial crush stresses under different dynamic loadings could be very different even though there is no strain-rate effect of matrix material. A power-law relation between the initial crush stress and the strain rate is explored to describe the strain-rate effect on the initial crush stress of irregular honeycomb when the local strain rate exceeds a critical value, below which there is no strain-rate effect of irregular honeycomb. Deformation mechanisms of the initial crush behavior under dynamic loadings are also explored. The deformation modes of the initial crush region in the front of plastic compaction wave are different under different dynamic loadings.     

刚体碰撞下直杆的塑性屈曲和后屈曲

研究了受刚体轴向碰撞直杆中塑性屈曲变形发展的机理.推导了增量形式的非线性动力方程,并用差分法求解,将特征值分析给出的初始局部屈曲位移作为方程解的初始条件.数值结果说明:随着轴向压缩波的传播,碰撞端附近发生的局部屈曲变形增长和向前传播,由半波形的初始模态发展成后屈曲高阶模态;在后屈曲变形的早期阶段无应变率逆转发生,切线模量理论适用于该阶段弯曲变形的计算.     

Mechanics of hexagonal atomic lattices

A theoretical framework for describing the kinematics and energetics of hexagonal atomic lattices, including planar carbon graphene sheets and cylindrical nanotubes, is proposed. By analogy with the membrane theory of thin shells, the deformation of the particulate lattice in the neighborhood of each atom is described in terms of a uniquely defined deformation gradient and companion local inner displacement. Expressions for the pointwise tensions developing in the plane of the lattice are developed, and a rational procedure for deriving discrete equilibrium equations is discussed. An alternative formulation involving the second-order deformation gradient that parallels the strain gradient theory of bulk media is proposed, and a tentative analogy with a the theory of micropolar elastic media is outlined.     

高速铁路长隧道压缩波波前变形规律分析

高速列车进入隧道时产生的压缩波在长隧道中传播时会产生波前变形,即波前压力梯度发生变化。线路测量和学者的研究表明,隧道出口处微气压波的强度与压缩波波前压力梯度最大值成正比,而微气压波的大小又与隧道出口的爆破音直接相关,因此有必要研究压缩波在长隧道中传播时的波前变形规律。采用计算流体力学三维动态仿真计算方法,对长隧道内压缩波的生成和传播过程进行研究。证明了压缩波的波前变形不仅与初始压缩波生成时的惯性运动和气体摩擦相关,而且与列车在隧道中的运动相关,即列车运动产生的能量输入会影响压缩波波前的变形。通过多工况计算,获得了隧道内压力梯度的最大值及其出现位置与列车速度和隧道阻塞比之间的变化规律。     

Adaptive Rotated Gaussian Weighted Digital Image Correlation (RGW-DIC) for Heterogeneous Deformation Measurement

Background

Digital image correlation (DIC) has advanced to become a flexible, reliable and fast optical method for the measurement of non-contact and full-field surface deformation. However, the accuracy of existing methods in measuring heterogeneous deformation fields—especially for the high gradient strain field – can be improved.

Objective

In state-of-art local DIC applications, several methods have been put forward to adapt a subset to unknown deformation. Although improvements in performance using these methods are obtained, results are still ungratified for severely heterogeneous deformation such as the Star 2 and Star 5 images from DIC Challenge 2.0.

Methods

In this paper, a rotated Gaussian weighted zero-mean normalized sum of squared difference (RGW-ZNSSD) criterion function is proposed as the basis for RGW-DIC subset size adaptation. RGW-DIC can automatically determine the optimum weight distribution, hence self-adaptivity in subset size and orientation are achieved simultaneously.

Results

The effectiveness of the proposed RGW-DIC is verified using DIC-challenge 2.0 images and simulated sinusoidal deformation images. Results reveal that the adaptively determined subset weight distribution can significantly improve the accuracy of heterogeneous deformation measurement compared with traditional DIC and DIC with isotropic Gaussian weight functions.

Conclusions

The proposed RGW-DIC can be applied to unknown severely heterogeneous deformation measurement.

     

A non-ordinary state-based peridynamic method to model solid material deformation and fracture

In this paper, we develop a new non-ordinary state-based peridynamic method to solve transient dynamic solid mechanics problems. This new peridynamic method has advantages over the previously developed bond-based and ordinary state-based peridynamic methods in that its bonds are not restricted to central forces, nor is it restricted to a Poisson’s ratio of 1/4 as with the bond-based method. First, we obtain non-local nodal deformation gradients that are used to define nodal strain tensors. The deformation gradient tensors are used with the nodal strain tensors to obtain rate of deformation tensors in the deformed configuration. The polar decomposition of the deformation gradient tensors are then used to obtain the nodal rotation tensors which are used to rotate the rate of deformation tensors and previous Cauchy stress tensors into an unrotated configuration. These are then used with conventional Cauchy stress constitutive models in the unrotated state where the unrotated Cauchy stress rate is objective. We then obtain the unrotated Cauchy nodal stress tensors and rotate them back into the deformed configuration where they are used to define the forces in the nodal connecting bonds. As a first example we quasi-statically stretch a bar, hold it, and then rotate it ninety degrees to illustrate the methods finite rotation capabilities. Next, we verify our new method by comparing small strain results from a bar fixed at one end and subjected to an initial velocity gradient with results obtained from the corresponding one-dimensional small strain analytical solution. As a last example, we show the fracture capabilities of the method using both a notched and un-notched bar.     

A gradient smoothing method (GSM) for fluid dynamics problems

A novel gradient smoothing method (GSM) based on irregular cells and strong form of governing equations is presented for fluid dynamics problems with arbitrary geometries. Upon the analyses about the compactness and the positivity of coefficients of influence of their stencils for approximating a derivative, four favorable schemes (II, VI, VII and VIII) with second‐order accuracy are selected among the total eight proposed discretization schemes. These four schemes are successively verified and carefully examined in solving Poisson's equations, subjected to changes in the number of nodes, the shapes of cells and the irregularity of triangular cells, respectively. Numerical results imply us that all the four schemes give very good results: Schemes VI and VIII produce a slightly better accuracy than the other two schemes on irregular cells, but at a higher cost in computation. Schemes VII and VIII that consistently rely on gradient smoothing operations are more accurate than Schemes II and VI in which directional correction is imposed. It is interestingly found that GSM is insensitive to the irregularity of meshes, indicating the robustness of the presented GSM. Among the four schemes of GSM, Scheme VII outperforms the other three schemes, for its outstanding overall performance in terms of numerical accuracy, stability and efficiency. Finally, GSM solutions with Scheme VII to some benchmarked compressible flows including inviscid flow over NACA0012 airfoil, laminar flow over flat plate and turbulent flow over an RAE2822 airfoil are presented, respectively. Copyright © 2008 John Wiley & Sons, Ltd.