姜汉卿
论文题目：应变梯度塑性理论断裂和大变形的研究
作者简介：姜汉卿
，男,1975年03月出生，1996年09月师从于清华大学黄克智教授，于2001年01月获博士学位。
摘
要
形形色色的宏观材料都是由微观世界的原子、分子构成的，而这些原子分子的不同排列组合导致了材料的宏观性能各异。所以从微观尺寸研究材料的性能一直是材料学家、力学家及物理学家所关心的问题。尤其是随着二十一世纪以微电子技术为核心的微制造业、微观材料科学的兴起，材料在微观尺寸的行为就更加引起科技界和工业界的共同重视。这主要在于材料的宏观性能与微观性能相差甚远，已有的宏观理论在微观情况下不再适用。
例如按照经典的宏观理论，材料的硬度是一个材料属性，与材料的尺寸无关，可是越来越多的微观试验发现对微观材料，硬度不再是一个材料属性，而依赖于材料的尺寸。最能体现这种尺寸效应的一类试验就是微压痕试验，Stanford大学的Nix，Cambridge大学的Brown，Fleck和Ashby等很多人都在试验中发现随着压痕深度减小到微米和亚微米尺寸，材料的硬度增加二到三倍。这种尺寸效应也在其它试验中反复出现，例如Cambridge大学的Fleck等人的细铜丝扭转试验、加州大学Santa
Barbara分校的Evans等人的薄梁弯曲试验、加拿大铝公司Lloyd的颗粒增强复合材料和德克萨斯仪器公司Douglass的微电力系统（MEMS）都发现了这种尺寸效应。这些尺寸效应的共同点是尺寸越小，强度越高。
然而这些尺寸效应却成为困扰力学家的一个难题。因为在传统力学理论的本构模型中不包含任何尺寸，所以它不能预测材料在微米尺度下发生的这种尺寸效应。而目前的设计工具，如有限元分析（FEM）、计算机辅助设计（CAD）都是基于传统力学理论的，在微米和亚微米尺度下不再适用。所以建立新的理论来架设宏观理论和微观变形的联系成为近期力学家研究的热点。
另一个促使建立新连续理论的动机来自于人们试图在韧性材料宏观和微观断裂之间建立联系。因为微观材料另一个明显不同于宏观材料的性质是它的断裂行为，德国Max
Planck研究所的Elssner和
Ruhle等人在试验中发现在微观尺度下韧性材料会出现脆性断裂。这种新的破坏模式和人们以往的宏观材料破坏模式完全不同。按照经典塑性理论，韧性材料只能出现由于位错的累积导致的韧性破坏模式，在破坏时裂纹尖端已经由于累积的塑性变形钝化了。而Elssner发现当材料发生脆性破坏时，依然在原子尺度上存在尖裂纹，虽然当时已经有大量的位错出现，却并没有出现按照经典塑性理论预测的钝化裂纹。要触发这种脆性断裂，裂纹尖端的应力水平达到材料的理论强度（剪切模量/10），大约为8到10倍的材料屈服应力（
），而Harvard大学Hutchinson指出按照经典塑性理论，裂纹尖端的应力水平只能达到4到5倍的
。它们之间存在明显的差别。著名材料学家Evans称这此问题为断裂力学中的“佯谬”（paradox）。但是这种破坏模式在微电子元器件中是致命的，它会瞬间导致器件的失效。出于理论及实际应用的出发点，这种新的破坏模式吸引了大批力学家投身其中，其中包括Harvard大学的Rice，Hutchinson，Brown大学的Shih，及Princeton大学的Suo等人。在他们这些研究中，裂纹尖端被一个无位错区包围，以提高裂纹尖端的应力水平。但是实际上裂纹尖端附近存在大量的位错，连续塑性理论（并不是经典塑性理论）应该可以解释这种破坏现象。
基于这两个动机，Fleck和Hutchinson提出了一个唯象的应变梯度理论。他们在本构模型中除了包含经典的应力、应变外，还引入了应变的梯度和与之功共轭的高阶应力。由于应变梯度项的出现，与它相连的内禀材料常数li自然被引进入到本构关系中，而这些li需要拟合微观试验得到。这种唯象的应变梯度理论是传统理论的直接推广，但缺少与材料塑性变形机制的紧密结合，所以在材料界没有被广泛接受，在实际应用中效果也不很理想。
与研究宏观材料的经典塑性理论对应，位错理论是用来研究位错随着变形的增加而产生、运动和储存，这就产生了材料的塑性变形。所以从本质上来说，位错是材料塑性变形的根源。然而，这两种有密切关联的学科（塑性力学理论、位错理论）却没有结合起来。突破点来自Gao（Stanford大学），Huang（伊利诺伊大学），Nix（Stanford大学）和Hutchinson（Harvard大学）联合在位错模型的基础上提出了“基于细观机制的应变梯度（MSG）塑性理论”，这种全新的理论是基于材料微观变形机制而提出的，是由位错理论和连续力学相结合而建立的。在位错理论中广泛应用的联系微观位错密度和宏观流动应力的Taylor律，做为一个基本原则进入MSG塑性理论的本构关系。为了在微观尺度上的统计储存和几何必须位错密度与宏观尺度上的力学表征（应变和应变梯度）之间建立联系，
MSG理论中还引入了多尺寸框架。这种旨在应用于微米和亚微米的全新的理论发表于1999和2000年后，引起力学研究者的注意，期待着这种理论与试验的成功比较。但是由于MSG理论刚建立时的一些假设（比如只能适用于小变形和不可压缩材料），限制了它的应用范围，难于与试验直接进行比较，所以它还不能直接应用到实际工程材料中。所以迫切需要对MSG理论进行改进使之可以直接应用于实际工程材料及与试验进行比较来验证该理论的适用性。
在本论文中，作者对MSG塑性理论进行了系统研究：
（1）
在重新分解应变梯度张量的基础上，发展了适用于可压缩材料的MSG塑性理论；
（2）
结合位错力学和经典塑性理论，发展了MSG大变形理论；
（3）
应用更新的MSG理论，研究了微压痕问题，与微压痕试验进行了比较；
（4）
提出了韧性材料多尺寸断裂观点，解决了韧性材料的脆性断裂问题。
针对原来的MSG理论中的材料不可压缩的假设，而实际工程材料都是可压缩的这种情况，作者发展了适用于可压缩材料的MSG理论。作者研究发现Hutchinson对应变梯度张量的分解只局限于不可压缩材料，这限制了MSG理论的适用，使之不能直接应用于大部分工程材料。为了解决这一问题，作者提出了新的对三阶应变梯度张量静水和偏斜部分的分解，以适用于可压缩材料，得到了适用于一般工程材料的MSG理论。
针对原有的MSG理论中小变形的假设，但是很多问题，尤其是材料的破坏问题，如断裂、剪切带等问题，都涉及到大变形；另外在微电子器件中，例如微尺度的传感器，在应用中都涉及到大变形，为了使MSG理论能应用于以上提到的种种实际工程情况，非常有必要发展大变形理论。在结合传统大变形理论和位错力学的基础上，作者发展了大变形MSG理论。作者推导出了即时构型和初始构型中应力、高阶应力的转换关系，并且建立了任意曲线坐标下这两个构型的平衡方程和边界条件。得到了一整套对应于高阶理论的控制方程。使MSG理论可以直接应用于实际工程材料。为了数值实现的需要，作者发展了高阶理论的全Lagrange格式大变形有限元方法。
为了检验推广的MSG理论的适用性，作者应用发展的高阶有限单元方法，研究了微压痕问题。微压痕中发现的尺寸效应不仅得到了众多的试验验证，研究微压痕问题本身也是对应变梯度塑性理论的检验。作者的研究给出了在试验中重复出现的压痕硬度平方和压痕深度的线性关系，而且还显示在从毫米到微米尺寸上，计算结果从数值上也和实验结果符合得很好。并且与伊利诺伊大学的Huang的计算结果比较后发现，可压缩性对微压痕的影响为20％左右，说明可压缩性在此问题中不能忽略。
推广的MSG理论的另一个重要应用就是成功地解释了德国Max
Planck研究所Elssner和Ruhle等人著名的试验，即在微尺寸下，韧性材料出现脆性断裂。这种新现象吸引了大批著名力学家对微观断裂的兴趣。例如Harvard大学的Rice，Hutchinson，Princeton大学的Suo和Brown大学的Shih提出的无位错区模型等。虽然这些理论可以提高裂纹尖端的应力水平，以触发脆性断裂。但是实际上，即使在微尺度下，韧性材料材料破坏时已经有大量的位错产生，无位错区的假设失去了存在的前提，所以应该应用基于位错的连续统力学来解决这个问题。在本论文中，作者利用MSG理论和高阶有限元方法研究了裂纹尖端的应力场。研究表明，由于应变梯度的贡献，在裂纹尖端，新的MSG塑性理论预测的应力水平比经典HRR场预测的应力水平高出2至3倍以上，即裂纹尖端应力场由4倍的屈服应力
到达了10倍的
，这足以触发脆性断裂，并且裂纹尖端应力奇异性等于弹性场的平方根奇异性。在此基础上，结合前人的无位错区模型，作者提出了一个从Princeton的Suo的无位错区，新提出的MSG塑性区，到Harvard大学的Rice和Hutchinson的经典塑性区（HRR）的韧性材料跨尺寸断裂模型。这是第一次成功地用连续统观点解释在韧性材料中出现的脆性断裂现象。
作者通过坚实的证据表明，
MSG塑性理论可以直接应用于微米和亚微米尺度。同时还表明，通过力学与材料科学的跨学科推动，力学可以有助于发展新的先进材料，而同时近代材料科学也将导致新的力学理论。
关键词:
尺度效应,
应变梯度塑形理论,
细观机制,
断裂,
大变形
Abstract
Recent experiments have repeated shown that the microscale material behavior is very different from that on the macroscale, and the classical continuum theories become inapplicable on microscale. For example, the classical continuum theories suggest that the hardness is a material property. But the recent micro-indentation hardness experiments of Nix (Stanford University), Brown, Fleck, and Ashby (Cambridge University) shew that the hardness of material may increase by a factor of two to three as the indentation depth decreases to microns and sub-microns. The size effect has also been repeatedly observed in other experiments, such as twisting of thin copper wires (Fleck, Cambridge University), bending of ultra thin beams (Evans, University of California at Santa Barbara), particle reinforced composites (Llyod, Alcan), microelectromechanical system-MEMS (Douglass, Texas Instrument). These experiments all suggest one common feature: the smaller, the harder.
The classical continuum theories fail to capture this size effect since their constitutive models do not possess any intrinsic materials lengths. However, the current design tools, such as finite element analysis and computer aided design, are based on the classical continuum theories and are not applicable at this micron and sub-micron scales. There is a critical need to develop a new continuum theory for the rational design and analysis of materials and structures at the microscale.
Another
objective that warrants the development of a micron scale continuum theory is to
link macroscopic fracture behavior to atomistic fracture processes in ductile
materials. Recent experiments of Elssner and Ruhle (Max Planck Institute,
Germany) observed that ductile materials exhibited cleavage fracture at the
microscale. This is totally different from the well known macroscopic fracture
behavior, i.e., the blunting of crack tip due to significant dislocation
activities around a crack tip. The classical plasticity theory predicts the
stress level of no more than 4 times the tensile yield stress (
) around a crack tip. However, Elssner and Ruhle observed the atomistically
sharp crack tip. In order to trigger this cleavage fracture, the stress level
near a crack tip needs to reach the theoretical strength of the material (shear
modulus/10), which is about 8~10 times 
. The classical plasticity theory clearly falls short to trigger this cleavage
fracture in ductile materials (---the so-called “paradox” by Evans, the
well-known material scientist), even though this fracture mode is important to
the microelectronics industry since it can cause instantaneous failure of the
micro components. This new fracture mechanism has attracted many top
mechanicians, such as Rice, Hutchinson (Harvard University), Shih (Brown
University), Suo (Princeton University). They have all used a dislocation-free
zone to explain this gap between the experiments and the classical plasticity
theory. However, there are always a large numbers of dislocations around a crack
tip such that there should exist a continuum plasticity theory (but not the
classical plasticity theory) that can explain this new fracture phenomenon.
The macroscopic plastic deformation results from the dislocation motion and storage on the microscale. But the macroscopic continuum plasticity theory is never quantitatively linked to dislocation models. Motivated to overcome the limitation of classical plasticity theory on the microscale, Gao (Stanford University), Huang (University of Illinois), Nix (Stanford University) and Hutchinson (Harvard University) proposed a “Mechanism-based Strain Gradient (MSG) Plasticity Theory” based on the widely accepted Taylor dislocation model. A multi-scale hierarchical framework is established to link the microscale notion of statistically stored and geometrically necessary dislocations to the macroscopic quantities of plastic strain and strain gradient. This dislocation-based continuum plasticity theory has drawn significant attention since its publication in 1999 and 2000, and is anticipated to be successfully applied to the micron and sub-micron scale problems. However, the original MSG plasticity theory involves some critical assumptions, such as material incompressibility, and infinitesimal deformation. These assumptions may not hold for engineering materials, which limit the application of MSG plasticity theory, as well as the validation of MSG plasticity theory by comparing with microscale experiments. It is impendent to improve the MSG plasticity theory in order to be directly applied to engineering materials.
The author has conducted a systemic study on the MSG plasticity theory in the following aspects:
(1) The author has proposed a new decomposition of the strain gradient tensor so as to remove the constraint on material incompressibility. An MSG plasticity theory for compressible material has been developed.
(2) The author has developed a finite deformation theory of MSG plasticity via a multiscale, hierarchical framework linking the continuum plasticity theory to the Taylor dislocation model.
(3) The author has used the generalized MSG plasticity theory to study the micro-indentation experiments, and provided the important validation of the MSG plasticity theory as well as the linkage between continuum plasticity theory and dislocation models.
(4) The author has used the MSG plasticity theory to successfully explain the experimental observed cleavage fracture in ductile materials.
In order to remove the assumption of material incompressibility, the author has developed a generalized MSG plasticity theory for compressible material. A new decomposition of the strain gradient is developed for compressible materials, and it replaces Hutchinson’s original decomposition for incompressible solids. This generalization significantly broadens the application range of MSG plasticity theory, and is suitable for engineering materials.
Another limitation in the original MSG plasticity theory is the assumption of infinitesimal deformation. The finite deformation analysis is particularly important for materials and structures at the microscale, such as failure (e.g., fracture and shear bands) and instability analysis. The micro-electronic components may also undergo large deflection and rotation. It is therefore impendent to develop a finite-deformation MSG plasticity theory. The author has combined the dislocation model with the continuum plasticity theory and established a finite deformation MSG theory via a multiscale, hierarchical framework linking microscale dislocation activities to the macroscopic strain gradient plasticity theories. The author has derived the expressions of stress and higher-order stress for both current and reference configurations from the work conjugate relations. The finite deformation MSG plasticity theory can be straightforwardly applied to engineering materials.
In order to validate the proposed MSG plasticity theory, the author has developed the finite element method for this higher-order continuum theory, and has used the finite element method to study the micro-indentation hardness experiments. The micro-indentation experiments have repeatedly shown the indentation size effect, i.e., the smaller the indentation depth, the larger the hardness of the material. The indentation experiments have been considered as a benchmark test for the developments of strain gradient plasticity theories. The indentation experiments on various metals have all shown the linear dependence between the square of indentation hardness and the reciprocal of indentation depth. The results based on the generalized MSG plasticity theory agree very well with the micro-indentation experiments over a wide range of indentation depths, therefore provide the validation of the MSG plasticity theory. The author has also found that the influence of compressibility is significant. If not accounted for, the error is around 20% as compared with Huang’s (University of Illinois) incompressibility analysis.
Another important application of the generalized MSG
plasticity theory is to explain Elssner and Ruhle’s (Max Planck Institute,
Germany) well-known experiments, i.e., cleavage fracture in ductile material.
This new failure phenomenon has attracted many mechanicians, such as Rice,
Hutchinson (Harvard), Suo (Princeton), and Shih (Brown). They have all used a
model of dislocation-free zone. Although their analyses can increase the stress
level around a crack tip to trigger the cleavage fracture, there are some
inconsistencies with the experiments since ductile materials always contain
numerous dislocations. The assumption of dislocation-free zone may not always
hold. It is therefore desirable to use the dislocation-based continuum theory
for studying this new fracture mechanism. The author has studied the crack tip
stress field using the generalized MSG plasticity theory and the finite element
method. It is shown that, due to the strain gradient effect, the stress level
around a crack tip predicted by the MSG plasticity theory is more than twice or
even three times higher than that in classical plasticity, i.e., the stress
level near a crack tip increases from 4 to 5 times the tensile yield stress 
as predicted by the classical plasticity to 8 to 10 times 
, which is high enough to trigger cleavage fracture. These observations, in
conjunction with a dislocation-free zone model, provide a multi-scale view of
cleavage fracture in ductile materials. The author has shown the transition from
a dislocation-free zone (Suo, Princeton University), through a new MSG
plasticity zone, to classical plasticity zone (i.e., HRR field). This provides,
for the first time, a continuum plasticity model that successfully explains
cleavage fracture in ductile materials.
The author’s study provides the solid evidence that the MSG plasticity theory works very well at the micron and sub-micron scales. It also demonstrates that, through the inter-disciplinary interaction between mechanics and materials, mechanics can help the development of new advanced materials, while the modern material science may lead to the new advances of mechanics theories.
Key words: size effect, strain gradient plasticity theory, mesoscopic mechanism, fracture, finite deformation
