微尺度晶体塑性的离散位错和非局部理论研究.doc

附件2论文中英文摘要作者姓名：柳占立论文题目：微尺度晶体塑性的离散位错和非局部理论研究作者简介：柳占立，男， 1981年11月出生，2004年9月师从于清华大学庄茁教授，于2009年7月获博士学位。中 文 摘 要随着微纳米器件的广泛应用，微电子微机械设备加工中的塑性成型控制及其工作过程中力学性能的可靠性预测成为近年来力学和材料学研究的热点问题。尺度位于几百纳米到几十微米的微尺度晶体材料的塑性变形具有很多新的特性：在均匀载荷下，屈服应力仍具有明显的尺寸效应；塑性变形呈现很强的时空不连续性，在时间上，塑性应变发生突跳，以间歇的方式进行；在空间上，发生局部化，晶体表面形成滑移线与滑移带；由于空间限制，位错存储及增殖机制发生变化，出现新的应力应变关系。这些新的特性给微纳米尺度晶体塑性的研究带来很多挑战。分子动力学模拟在晶体塑性研究中得到了广泛的应用，但巨大的计算规模使其目前受限于纳秒时间尺度和纳米空间尺度，不能满足工程中的应用要求。连续介质方法被运用到微尺度晶体塑性的研究中来解脱分子模拟所受的制约，但是，现有的基于连续介质的晶体塑性理论在解决该问题上仍面临诸多困难，如随位错存储增殖机制的转变，传统的Taylor硬化律可能不再适用、难以在连续介质框架下对离散位错进行时空描述等。本文将位错动力学方法引入到传统晶体塑性理论中，基于连续介质力学框架建立离散位错塑性模型和非局部塑性模型对微尺度下晶体材料的塑性流动问题进行深入系统研究，取得了如下几方面的成果： 1.离散位错塑性研究（1）在离散位错研究中，通过耦合三维离散位错动力学（3D DDD）和连续介质有限元（FE）首次建立统一的完全基于连续介质力学框架的离散位错塑性计算模型。在该模型中，微尺度晶体中的离散位错塑性完全在连续介质力学框架下进行求解：1）引入初始内应力场来表示晶体中预先存在的静态位错；2）外部边界条件通过有限元自然求解；3）本构关系基于传统晶体塑性的有限变形理论，但是通过3D DDD方法计算由位错滑移产生的离散塑性应变，然后通过引入Burgers矢量分布函数，把离散塑性应变较好地局部化到连续介质材料点上。应用该统一离散位错塑性模型，我们模拟了位错线、位错环及位错锁等微结构演化过程，并把与之相对应的应力场与解析结果进行比较，验证了该计算模型的可靠性。相关工作发表于塑性力学的权威杂志International Journal of Plasticity（2009，影响因子4.791, paper no.1）。文章发出不久，ASME Journal of Engineering Materials And Technology 主编HM Zbib教授在Materials Division of ASME的综述报告“Advances in Discrete Dislocations Dynamics and Multiscale Modeling” 中对该工作在离散位错塑性研究中的贡献给予了肯定“Liu et al. extended the work of Lemarchand et al., based on the finite deformation theory of crystal plasticity. The major improvement proposed by Liu et al. is concerning the pre-existing stationary dislocations. To be visible in the continuum, dislocations have to be introduced from the free surfaces in the formulation of Lemarchand et al., while Liu et al. represented pre-existing stationary dislocations by an internal stress field. With the new formulation, discrete dislocation plasticity is completely handled under a continuum mechanics framework. ”（Journal Of Engineering Materials And Technology, 131:041209，2009）。该模型解决了传统晶体塑性理论中唯象本构演化方程难以描述离散塑性应变的问题，并能用于大规模数值模拟，在微纳米器件的塑性成型研究中具有实际应用价值，同时对金属材料的研究也具有重要意义。法国National Centre for Scientific Research 和Laboratory of Microstructures的材料科学家B. Devincre教授应用该方法研究了镍基超级合金塑性变形的晶向依赖性（Acta Materialia, 58:1938，2010），清华大学高原博士生应用该方法研究了微纳米柱压缩实验中压头和晶体柱的接触及晶体柱的椎角对塑性变形的影响（Computational material science,49：672，2010）。（2）应用该统一计算模型对单晶铜柱的单轴拉伸和压缩实验进行了模拟，研究了均匀载荷下应变率及尺寸对塑性变形的影响。在单轴拉伸中研究了微观尺度下不同加载速率对单晶铜力学性能及变形模式的影响。对一个给定尺寸及初始微结构的单晶铜块，存在一个临界应变率：低于该临界应变率，屈服应力对应变率相对不太敏感；一旦高于该应变率，屈服应力将随应变率的增加而增加，并和应变率的对数近似成线性关系。该临界应变率的大小由初始位错密度，位错平均自由程及位错运动障碍的强度决定。随着应变率的增加，位错微结构图案从均匀形态变成单一形态，位错滑移都集中到某一个特定滑移系内，并随之在晶体内部出现变形带。变形带的宽度随着应变率的增加而增加，其中的剪应力明显高于相邻区域，这些区域往往是材料中进一步发生失效的位置。该研究为高应变率下微纳米器件的变形及失效模式预测提供依据。相关工作发表于固体力学主要杂志International Journal of Solids and Structures（2008, paper no.2）。在微单晶铜柱压缩研究中首次模拟了亚微米尺度单晶铜柱中的反常三阶段力学行为，并解释了其产生机理。在微尺度单晶柱中形成稳定位错连接(如三重连接)的可能性很小，相对不太稳定的二重位错连接是位错网的主要组成部分。由二重位错连接构成的L形旋转位错源为主要增殖源，其快速、高效的操作来维持微尺度晶体中的塑性变形。在压缩过程中，一旦位错网中的位错连接被破坏或者从该有限空间中逃逸，位错增殖将停止，正常的塑性应变硬化也将结束。随着位错逃逸的进行，控制晶体材料强度和塑性的过程从位错扩展及相互作用转变为位错形核，随之出现反常的应力应变行为。相关工作发表在金属材料领域著名刊物Scripta Materialia （2009，影响因子2.949, paper no.3）及多尺度工程计算杂志International Journal for Multiscale Computational Engineering（2009, paper no.4）。微柱塑性是当前力学和材料学研究的一个热点问题，其塑性变形导致软化、退火导致硬化的行为与块体材料截然相反，美国艺术与科学院、国家工程院、国家科学院院士（全世界仅有6位材料科学家获得美国国家科学院院士荣誉），Stanford University 的WD Nix教授提出“位错源控制塑性”理论对该现象进行解释，而我们研究中得到的位错匮乏机制对该理论具有重要验证意义。该工作发表后， WD Nix教授来信对该工作进行高度评价（“I have read with interest your nice paper on a dislocation dynamics study of dislocation starvation in copper, recently published in Scripta Materialia. I would like to mention your work in up-coming talks”），并多次引用该工作（Acta Materialia, 57:4404, 2009; Materials Science And Engineering A, 27:1903, 2010）作为“位错源控制塑性”理论的有力证据。WD Nix教授在“Exploiting New Opportunities in Materials Research by Remembering and Applying Old Lessons” （2009年7月在中国大连举行的中美自然科学基金会微纳米力学学术报告会和在中国清华大学），和“Prestraining and Annealing of Metallic Microcrystals: Strengthening and Weakening Turned Upside Down”（2009年在美国哈佛大学，2010年在美国西北大学）等多个国际学术会议报告中展示了该工作。2.非局部塑性理论研究（1）发展了Gurtin提出的基于缺陷能量的非局部塑性理论，提出了微尺度塑性变形中存储缺陷能量的物理机制，认为缺陷能是相邻滑移面上的非协调位错在弹性相互作用中存储的势能，并据此推导出了缺陷能的具体形式：它是几何必需位错密度的二次函数，并包含一个弹性系数和一个大小为几个位错滑移面间距的长度参数，该尺度参数反映了位错近程相互作用距离。著名的基于位错机制的应变梯度（MSG）理论忽略了位错近程相互作用，而其在亚微米尺度对晶体材料内应力有重要影响，本研究弥补了MSG理论的不足。相关工作发表于Acta Mechanica Sinica。该工作在投稿期间得到美国Carnegie Mellon University的ME Gurtin教授（缺陷能理论的提出者，2004年获得有国际力学最高成就奖之誉的Timoshenko奖，每年全世界仅评出1人）的关注“I recently saw a preprint of a very, very nice paper by you with Liu, Liu, Zhao, and Gao“Bauschinger and size effects in thin-film plasticity due to defect-energy of geometrical necessary dislocations” and would very much like to receive a pdf file and to know the current status of the paper and any related work”。（2）基于位错动力学的连续介质描述，建立适合于微尺度塑性流动研究的非局部晶体塑性模型。该模型包含三个耦合方程：1）关于晶体滑移的标准扩散方程，它在连续介质层次上描述了位错运动；2）位错密度演化方程，该方程根据位错增殖和湮灭速率的平衡导出；3）传统的宏观力学平衡方程。在亚微米尺度位错和界面（如晶界）的相互作用对金属材料力学性能有重要影响，而现有的高阶晶体塑性模型仅能处理位错完全穿过或完全堆积在晶体界面，这是两种极端条件。而本模型的优点在于它可以把“位错流”类比成热扩散中的“热流”来处理其在晶体界面处的堆积、扩散、反射等行为。因此该模型弥补了现有晶体塑性模型的不足，拓宽了晶体塑性理论的应用范围。应用该模型我们研究了带钝化层的单晶及多晶薄膜在平面拉伸中的塑性变形问题，结果表明亚微米尺度下背应力硬化明显主导了薄膜约束塑性中流动应力的强化。相关工作发表于塑性力学的权威杂志International journal of plasticity（2010，影响因子4.791，paper no.5）。关键词： 离散位错塑性；非局部塑性；应变率效应；尺寸效应；薄膜塑性； 界面模型The Investigation of Crystal Plasticity at Microscale by Discrete Dislocation and Nonlocal TheoryZhanli LiuABSTRACTAs the wide application of nanodevice, the control of plastic forming during micromachining of micro-electro-mechanical systems (MEMS) and the prediction of mechanical reliability in service have become hot topics in the study of material science and mechanics. The plastic deformation of crystalline material at microscale ranging from hundreds of nanometers to tens of micrometers exhibits many new characters: strong size effect of yield stress appears even under the uniform loading; the plastic flow proceeds in a strongly temporal intermittency and spatial localization manner, slip lines or slip bands are observed at the surface of deformed crystal; because of the space limitation, the mechanisms which dominate dislocation storage and multiplication change, leading to new stress-strain relations. These new characters bring challenge to the research in this area.There are extensive atomistic studies of crystal plasticity. Atomistic simulations, in general, are limited in time scale (up to 1 ns) and size scale (up to 1 mm), which fall short to meet the requirement for engineering applications. The continuum models of crystal plasticity are applied to overcome the limitations. However, the conventional continuum based crystal plasticity theory still has some difficulties in dealing with the plasticity at microscale, e.g. the conventional Taylor hardening law may not apply, the spatial and temporal description of discrete dislocation in the continuum framework, etc.In this thesis, by introducing the discrete dislocation method into the conventional crystal plasticity theory, discrete dislocation plasticity model and non-local plasticity model are established, respectively, based on the continuum mechanics framework to systematically study the plastic flow of crystal at microscale. The detailed work includes following parts.1. The study of discrete dislocation plasticity(1) A computational model for discrete dislocation plasticity completely based on continuum mechanics is developed for the first time by directly combining 3D discrete dislocation dynamics (DDD) and finite element method (FEM). In this model, the discrete dislocation plasticity in the microscale crystal is solved completely under continuum mechanics framework: 1) an initial internal stress field is introduced to represent the preexisting stationary dislocations in the crystal; 2) the external boundary condition is handled by finite element method spontaneously; 3) the constitutive relationship is based on the finite deformation theory of crystal plasticity, but the discrete plastic strains are calculated by dislocation dynamics methodology. These discrete plastic strains are then localized to the continuum material points by a Burgers vector density function proposed by us. Various microstructure evolution processes, such as dislocation loop evolution, dislocation junction formation etc., are simulated to verify the reliability of this computational model. The related work has been published in the authoritative journal International Journal of Plasticity (2009; impact factor 4.791, paper no.1)In a short time after the publication of this work, Prof. HM Zib of Washington State University, who is the editor-in-chief of ASME Journal Of Engineering Materials And Technology, cited it in the review report “Advances in Discrete Dislocations Dynamics and Multiscale Modeling” for the Materials Division of ASME and fully affirmed the contribution of our work to the discrete dislocation plasticity, “Liu et al. extended the work of Lemarchand et al., based on the finite deformation theory of crystal plasticity. The major improvement proposed by Liu et al. is concerning the pre-existing stationary dislocations. To be visible in the continuum, dislocations have to be introduced from the free surfaces in the formulation of Lemarchand et al., while Liu et al. represented pre-existing stationary dislocations by an internal stress field. With the new formulation, discrete dislocation plasticity is completely handled under a continuum mechanics framework” (Journal Of Engineering Materials And Technology, 131:041209, 2009). This model solves the problem existing in the conventional crystal plasticity theory in which it is difficult to describe the discrete plastic strains by phenomenological constitutive evolution equations, and is suitable for large-scale simulation of plastic forming during the micromachining of MEMS. It is also meaningful for the material science research. Prof. B Devincre, famous material scientist in French National Centre for Scientific Research and Laboratory of Microstructures, used this model to study the orientation dependence of plastic deformation in nickel-based single crystal superalloys (Acta Materialia, 58:1938，2010). Yuan Gao of Tsinghua University used it to study the effects of the contact and the taper angle on the plastic deformation in micro-pillar compression test. (Computational material science, 49：672，2010)(2) By applying this computational model, uniaxial tension and compression simulations for Cu single-crystal micro-pillar are carried out to investigate the strain rate and size effect on plastic deformation under uniform loading condition. The loading rate effects on the yield stress and the deformation patterning of single crystal copper are investigated in the tension simulation. A critical strain rate exists in each single crystal copper block for the given size and dislocation sources, below which the yield stress is relatively insensitive to the strain rate. However, above this critical value, the yield stress of single crystal copper increases rapidly with the increasing of strain rate, and approximately has a linear relationship with . At the same time the dislocation microstructure patterning changes from non-uniform to uniform and deformation bands are also observed in crystal which often are the places where the damage further occurs. These results are helpful for understanding the deformation and failure modes of nanodevice under high strain rate. This work has been published in famous journal International Journal of Solids and Structures in the field of solid mechanics. (2008, paper no.2)In the study of micro-pillar compression, the atypical three-stage constitutive behavior in Cu single crystal at sub-micrometer scale is observed for the first time and the corresponding physical origin are explained. The simulation results show that the possibility of forming permanent junctions (such as ternary junctions) in micro-pillar is small, the relatively unstable binary junctions are the main components of the dislocation network. These binary junctions form the spiral dislocation sources, which efficiently operate to maintain the plastic deformation in micro-pillar. During the compression, once the dislocation junctions in the network are destroyed or escape from this limited space, the dislocation multiplication will cease, and the normal plastic strain hardening will stop correspondingly. With this dislocation exhaustion, the process controlling strength and plasticity changes from dislocation interaction to dislocation nucleation, leading to atypical constitutive relationship. The related work has been published in famous journal Scripta Materialia (2009; impact factor 2.949, paper no.3) in the field of material science and International Journal for Multiscale Computational Engineering (2009, paper no.4).In micro-pillar plasticity, plastic deformation leads to softening and annealing leads to hardening, just the opposite of what occurs in bulk metals. Prof. WD Nix of Stanford University, who is the member of American for Academy of Arts and Science, National Academy of Engineering, National Academy of Sciences (in the world only 6 material scientists are the member of National Academy of Sciences), proposed “dislocation-source controlled plasticity” theory to explain this phenomenon. The dislocation exhaustion mechanism obtained in our research provides convincing evidence for the “dislocation-source controlled plasticity” theory. Prof. WD Nix cited our work in several papers (Acta Materialia, 57:4404, 2009; Materials Science And Engineering A, 27:1903, 2010), and gave a high appraisal “I have read with interest your nice paper on a dislocation dynamics study of dislocation starvation in copper, recently published in Scripta Materialia. I would like to mention your work in up-coming talks”. Prof. WD Nix showed our work in his several international academic reports, e.g. “Exploiting New Opportunities in Materials Research by Remembering and Applying Old Lessons” (Nanomechanics Report of NSF and NSFC in Dalian, China; Tsinghua University, China in July, 2009), “Prestraining and Annealing of Metallic Microcrystals: Strengthening and Weakening Turned Upside Down” (Harward University, USA, in 2009; Northwestern Universitiy, USA, in 2010).2. The study of non-local crystal plasticity theory(1) A defect energy based non-local plasticity theory established by Gurtin is improved, the mechanism of defect-energy storage during the microscale plastic deformation is proposed by us. The defect-energy is regarded as the potential energy stored during the elastic interactions of incompatible dislocations moving on the closed neighboring slip plane and it can be expressed as a quadratic function of geometrical necessary dislocations, including an elastic coefficient and a length scale parameter about several slip-plane distances. This length scale reflects the scope of short range interactions of dislocation. The famous mechanism based strain gradient (MSG) theory neglects the short range interactions of dislocation, which heavily influence the internal stres