准晶非线性弹性与断裂力学及其相关问题研究

Quasicrystal was discovered in 1982 as a new solid structure, this discovery was awarded the 2011 Nobel Prize in Chemistry. Du to incommensurate phase appears in the quasi-periodic symmetric structure, leading to the characterization of the quasi-crystal elastic not only need the phonon field to describe the lattice vibrations, and also need the phason field to portray the atomic quasi-periodic arrangement, and the two are coupled. This determines the diversity of the quasicrystalline structure and the complexity of the quasicrystal elasticity. The studying on quasicrystal linear elastic fracture mechanics has been proposed a number of successful and effective methods, such as the Green's function method, the Fourier transform method, perturbation method and the complex variable function method et al, and many important results has been achieved. Due to nonlinear problems closer to reality, with the deepening of the study of quasicrystal, researching on the nonlinear elasticity of quasicrystals has attracted the attention of many researchers. The purpose of this application is to make use of the specialty of our group in the study of nonlinear mathematical physics methods and quasicrystal fracture mechanics to study the nonlinear elasticity and fracture mechanics of quasicrystal, rather than the linear elastic fracture mechanics of quasicrystal which is our past studying. In this application, we will mainly consider nonlinear constitutive relation of quasicrystals, and their geometric non-linear fracture mechanics problems and nonlinear wave propagation form, further reveal the unique mechanical properties of quasicrystals.

准晶是1982年发现的一种新的固体结构，这一发现荣获了2011年诺贝尔化学奖。由于准周期对称结构中无公度相的出现，导致准晶弹性问题的刻画不仅需要描写晶格振动的声子场，还需要刻画原子准周期排列的相位子场，且二者是相互耦合的。这就决定了准晶结构的多样化和弹性的复杂性。关于准晶线弹性断裂力学问题的研究，已经提出了若干成功和有效的求解方法，如Green函数法、Fourier变换法、摄动法和复变函数法等，并取得了许多重要成果。由于非线性问题更加接近于现实问题，随着准晶研究的不断深入，关于准晶的非线性弹性的研究吸引了不少学者的注意。本项申请的目的在于将本小组在非线性数学物理方法和准晶断裂力学方面的研究有机结合，从项目组过去重点研究准晶的线弹性断裂力学问题深入到开展准晶非线性弹性与断裂力学的研究，重点考虑准晶的非线性本构关系、几何非线性断裂力学问题和非线性波的传播形式等，揭示准晶材料特有的力学性质。

由于准周期对称结构中无公度相的出现，导致准晶弹性问题的刻画不仅需要描写晶格振动的声子场，还需要刻画原子准周期排列的相位子场，且二者是相互耦合的。这就决定了准晶结构的多样化和弹性的复杂性。关于准晶线弹性断裂力学问题的研究，已经提出了若干成功和有效的求解方法，但关于准晶非线性弹性断裂力学问题的研究相对较少。由于非线性问题更接近于现实问题，因此开展准晶非线性断裂力学问题的研究对提高准晶材料的可靠性和利用率具有重要的理论意义。. 项目组按计划开展了准晶非线性弹性与断裂力学的研究，重点研究了准晶压电材料中的椭圆孔口带多裂纹、唇形裂纹、翼形裂纹等具有非线性边界缺陷的断裂力学问题、多缺陷的相互作用和局部边界受力问题、塑性变形和热载荷等非线性问题，给出了场强度因子、能量释放率和相互作用力的解析解。在国内外学术期刊上已发表学术论文43篇，录用2篇，其中SCI、EI检索19篇。在本项研究资助期间，2位骨干成员晋升为教授；3位青年学术骨干晋升为副教授；项目在研期间，共培养硕士研究生10人，博士研究生1人。. 由于准晶独立弹性常数测定的实验难以进行，故将准晶材料非线性的断裂力学研究调整为准晶压电材料复杂缺陷断裂力学的理论研究，并增加了准晶热载荷下的断裂力学问题研究。