非均质材料力学中的界面效应

作者姓名：段慧玲

论文题目：非均质材料力学中的界面效应

作者简介：段慧玲，女，1970年5月出生，2001年9月师从于北京大学王建祥教授，于2005年6月获博士学位。

中文摘要

基于科学发展的内在动力和技术进步的现实需求，人们迫切需要了解和表征小尺度下物质的力学行为，给经典连续介质力学框架提出了挑战，也给其发展和突破提供了难得的机遇。因此，跨物质层次的固体变形和强度理论被认为是力学领域最前沿的基础问题之一。由于纳米尺度下表面/界面原子数占很大的比例，表面/界面原子处于与体内原子不同的环境，因而固体的表面/界面应力对纳米结构材料的力学性能以及相关物理性能具有重要影响。纳米尺度下表面/界面应力的研究不仅对于促进连续介质力学和细观力学的发展具有基础理论意义，而且在量子点的生长和性能分析、纳米器件的自组装、微型传感器、纳/微米电子器件、生物工程、材料工程等领域具有重要的应用价值。特别是，界面性能是影响甚至控制复合材料性能的重要因素之一，对各种界面条件的分析可为复合材料性能的预测和材料设计提供理论基础。本文系统地研究了表面/界面应力以及界面相对非均质材料力学行为的影响。具体完成的主要工作如下：

(1) 根据虚功原理得到了界面应力模型的界面条件，在此基础上给出了计及界面应力效应的球形夹杂Eshelby 体系的解(Mechanics of Materials, 2005; Proceedings of the Royal Society A, 2006)。经典的Eshelby体系(Eshelby formalism， 1957)是20世纪固体力学领域一项奠基性的工作，被认为是如今蓬勃发展的细观力学的基石之一。经典的Eshelby体系没有包括界面效应，本文首次给出了计及界面应力效应的Eshelby张量、应力集中张量、Eshelby等效夹杂方法和Eshelby能量公式。发现，与经典结果不同的是，在均匀本征应变和远场应力作用下，内场Eshelby张量和应力集中张量是位置相关和尺度相关的。

(2) 首次建立了具有界面应力效应的细观力学框架，利用复合球模型、Mori－Tanaka方法和广义自洽方法预测了具有界面应力效应的含有球形夹杂和圆柱形夹杂的非均质材料和纳米介孔材料的等效模量，与经典结果不同的是，它们是材料界面特性和夹杂尺寸的函数 (Journal

of the Mechanics and Physics of Solids, 2005; Acta Materialia, 2006)。从理论上发现，通过调整表面特性可以使纳米介孔材料的弹性模量高于基体的弹性模量，并且建议了两种提高介孔材料刚度的途径。这一发现为研制在航空航天领域具有重要用途的轻质高模多孔材料提供了理论基础。

(3) 根据表面、界面效应对纳米结构材料性能影响的物理本质，首次系统地总结和提出了表征由于表面/界面效应导致的纳米结构材料多种性能尺度相关性的两类标度律(Proceedings of the Royal Society A, 2006)。发现，对于各向同性表面/界面， Eshelby 张量、应力集中张量、应力集中系数、等效弹性常数的尺度相关性皆可精确地用一个简单的标度律L l l H L H /)(1)(/)(βαβα++=∞来描述，这一简单的标度律揭示了跨越纳观尺度-细观尺度-宏观尺度的耦合关联。

(4) 利用 P －N 位移势函数和旋转椭球调和函数展开公式求解了轴对称载荷作用下具有界面相的任意取向的旋转椭球夹杂的局部弹性场，首次给出了旋转椭球夹杂在远场纵向剪切载荷作用下的位移势函数，并且给出了迄今为止最简单的旋转椭球夹杂Eshelby 问题的外场解(Proceedings of the Royal Society A, 2005)。这项结果可用于分析多种形状夹杂的弹性场。

(5) 首次给出了无限大体中具有非均匀界面相的球形夹杂的Eshelby 张量及应力集中张量，分析了它们的性质；首次给出了自由和固定两种边界条件下有限球域中球形夹杂的 Eshelby 张量。给出了Eshelby 张量和应力集中张量的两个应用实例，其一是分析了纳米芯－壳结构复合颗粒由于晶格失配产生的弹性应变场；其二是基于三相构型的体积平均意义上的 Eshelby 等效夹杂方法提出了预测含球形夹杂的非均质材料等效模量的一种新的细观力学方法，这种细观力学方法的优点是其预测结果和广义自洽方法以及三阶关联近似方法几乎完全相同，但是表达式很简单(Journal of the Mechanics and Physics of Solids, 2006)。

(6) 在统一的泰勒级数展开框架下，建立了数学界面模型和界面相模型之间的联系，首次发现和建立了界面应力模型与薄而硬的界面相模型的等效关系；提出了一个反嵌入界面模型，排除了经典的线弹簧界面模型的缺点 (International Journal of Mechanical Sciences, 2005)。

在实验方面，与他人合作，用原子力显微镜测量了银纳米线的杨氏模量，并且基于表面应力理论分析了实验结果（该项实验工作在攻读博士学位期间与他人合作进行，相关论文2006

年发表于凝聚态物理权威杂志Physical Review B 73, Art. 235409）。

基于力学问题和广义传导（热传导、电传导、磁传导、物质扩散）问题在数学上的类比关系，本文对于表面/界面应力效应的研究方法为研究非均质材料的广义传导问题奠定了理论基础。作者与合作者已经利用这种类比关系建立了具有性能梯度界面相非均匀介质的广义传导性能的界限理论；发展了同时计及夹杂形状、空间分布、界面效应的预测理论。这些研究成果发表于Physical Review B 和 Journal of Applied Physics.

关键词：非均质材料，界面效应，Eshelby 体系，等效模量，标度律

Interface Effect in Mechanics of Heterogeneous Materials

Duan Huiling

ABSTRACT

Due to the inherent impetus of the development of science and the realistic demand of the advance of technology, it is imperative to understand and characterize the mechanical behaviour of materials at the nano- and micro-scale. On one hand, this task challenges the framework of the classical continuum mechanics; it also provides rare favorable chances for the development and breakthrough on the other. Therefore, the trans-scale deformation and strength theories of solids have been regarded as one of the most fundamental frontiers of mechanics. Because nano-structured materials have a large fraction of the atoms at the surfaces/interfaces, and these atoms have a different environment from those in the interior, the effect of the surface/interface stress on their mechanical behaviour can be substantial. The study of the surface/interface stress effect is not only fundamentally important for the development of continuum mechanics and micromechanics, but also essential in the areas of the growth and analyses of quantum dots, the self-assembly of nano-devices, nano- and micro-sensors, microelectronic devices, bioengineering and materials engineering, etc. Particularly, interfacial bonding condition is one of the important factors that control the local elastic fields and the overall properties of composites. The analyses of various interfacial conditions can provide theoretical guidelines for the prediction of the properties and the design of composites. In this Thesis, the effects of the surface/interface stress and interphases on the mechanical behaviour of heterogeneous materials are systematically studied. The primary obtained results are as follows:

(1) The interface condition for the interface stress model is derived using the principle of virtual work. Based on this, the solution of the Eshelby formalism for inclusion/inhomogeneity problems with the interface stress is given (Mechanics of Materials, 2005; Proceedings of the Royal Society A, 2006). The classical Eshelby formalism (Eshelby，1957)，which does not include the interface stress, is landmark work in mechanics of the 20th century, and it is regarded as one of the cornerstones of micromechanics. Taking into account the interface stress, this Thesis first presents

the complete Eshelby formalism, including the Eshelby tensors, the stress concentration tensors, the Eshelby equivalent inclusion method and the Eshelby energy formula. It is found that unlike the classical results, these Eshelby and stress concentration tensors are position-dependent and size-dependent under a uniform eigenstrain and a remote stress.

(2) The fundamental micromechanical framework is first generalized to take into account the interface stress effect. The effective moduli of heterogeneous materials containing spherical and cylindrical nano-inhomogeneities are predicted using the composite spheres assemblage model, the Mori －Tanaka method and the generalized self-consistent method (Journal of the Mechanics and Physics of Solids, 2005; Acta Materialia, 2006). The effective moduli are shown to be functions of the interface properties and the size of the inhomogeneities. It is found that the effective elastic constants of nano-channel-array materials can be made greater that those of the matrix materials, and two routes to the stiffening are suggested. This finding may herald a radical breakthrough in sandwich-type structures that have many applications in aerospace engineering.

(3) According to the physical essence of the surface/interface effect on the properties of nano-structured materials, two simple scaling laws to characterize the properties of nano-structural materials are given (Proceedings of the Royal Society A, 2006). It is found that for isotropic surfaces/interfaces, the size-dependency of the Eshelby tensor, the stress concentration tensor, the stress concentration coefficient and the effective elastic constants can all be accurately depicted by a simple scaling law L l l H L H /)(1)(/)(βαβα++=∞. This simple scaling law reveals the trans-scale coupling among the nano-scale, micro-scale and macro-scale.

(4) The elastostatic solution to an arbitrarily-oriented spheroidal inhomogeneity with an interphase embedded in an infinite medium under a remote axisymmetric loading is obtained by using the method of the P －N displacement potentials and expansion formulas for spheroidal harmonics. New displacement potentials for a spheroidal inhomogeneity with an interphase under a remote longitudinal shear loading are first given. The simplest up to now expressions of the stress field outside an arbitrarily-oriented spheroidal inhomogeneity without an interphase in an infinite matrix under a remote axisymmetric loading (i.e. the Eshelby problem) are given in terms of elementary functions (Proceedings of the Royal Society A, 2005). This solution can be used to analyze the

elastic fields of inhomogeneities of various shapes.

(5) The Eshelby tensors and the stress concentration tensors are first obtained for a spherical inhomogeneity with an inhomogeneous interphase embedded in an alien infinite elastic matrix. The solution is then specialised to inhomogeneous inclusions in finite spherical domains with fixed displacement or traction-free boundary conditions. The Eshelby tensor and stress concentration tensor are applied to two examples. The first solves the strain distributions in core-shell nanoparticles with eigenstrains induced by lattice mismatches. In the second example, a micromechanical scheme, based on the equivalent inclusion method of the three-phase configuration in a volume average sense, is proposed to predict the effective moduli of composites containing spherical inhomogeneities. The main advantage of this scheme is that, whilst its prediction of the effective moduli is almost identical to those of the generalized self-consistent method (GSCM) and the third-order approximation (TOA), the resulting expression has a very simple form (Journal of the Mechanics and Physics of Solids, 2006).

(6) Under a unified framework based on the Taylor series expansions, the connections between the thin interphase model and the mathematical interface models are established. The equivalency between the interface stress model and a thin-and-stiff interphase model is first found and established. In addition, an anti-interpenetration interface model is presented, which removes the shortcoming of the classical linear spring interface model (International Journal of Mechanical Sciences, 2005).

In collaboration with other researchers, the Young modulus of silver nano-wires is experimentally measured with the AFM. The size-dependency of the elastic constant is theoretically explained using the theory of surface stress (This work was conducted in collaboration with other researchers during the studies for the PhD degree, but the relevant paper was published in the prestigious journal Physical Review B 73, Art. 235409 in the field of condensed matters in 2006).

Based on the analogy between the prediction of the effective elastic constants and the effective generalized conductivities (heat conductivity, electric conductivity, magnetic permeability, dielectric constant and diffusivity), the methodology in the Thesis for the surface/interface effects also lays the theoretical foundations of the prediction of the effective conductivities. Recently, making use of

the analogy, the author and co-workers have presented the bounds on the effective conductivities of heterogeneous media with a graded interphase, a predictive scheme with consideration of the shape, spatial distribution of the inhomogeneities, and the interface effects. These results are published in Physical Review B and Journal of Applied Physics.

Key words: Heterogeneous materials, interface effects, Eshelby formalism, effective moduli, scaling laws