纤维增强复合材料的弹力理论及其在几何中的应用

This programme is motivated by the theory of linear elasticity in composite materials and the theory on harmonic maps in geometry. In 1999, Professor Babuska, a member of National Academy of Engineering of United States of America, prosed an open problem on the uniform boundedness of the stress of fiber-reinforced composite materials, which caused more attentions of many mathematicians, including Professor Nirenber, a member of National Academy of Science of United States of America. This problem is concerning with a class of elliptic systems in divergence form with piecewise constant (continuous or smooth) coefficients. The study on such systems is more challenging and of significance in both pratice and mathematics, especially in the classical theory on partial differential equations. . This programme is mainly concened with Babuska problem on perfect elastic composite material. We will establish the gradient estimate of solutions to the corresponding elliptic systems, investigate the relationship between the gradient and the distance of subdomians, and improve the damage analysis on composite material.. The study on the above problem in composite material allow us innovate to extend these result to harmonic maps, which from a Riemmanian manifold with piecewise smooth metric. We will investigate the optimal regularity of these generalized minimizing harmonic maps, weak harmonic maps and stationary harmonic maps, respectively, and attempt to consider their heat flows.

本项目的研究动机来源于材料学中的线性弹力理论和几何学中的调和映照理论。1999年，美国工程院院士Babuska 提出的关于纤维增强复合材料应力一致有界性的公开问题引起了包括Nirenberg 院士在内的众多数学家的兴趣。该问题对应的一类散度型椭圆方程组的系数不再具有整体连续性，而是分片常数（连续或光滑）的。对这类方程的研究不仅具有现实的应用价值，而且对经典的偏微分方程正则性理论也具有挑战性。.本项目将通过建立椭圆方程组解的梯度一致有界性估计，刻画应力大小与纤维距离之间的精确依赖关系，解决超弹性复合材料的Babuska问题，完善复合材料的变形与破坏机理理论。.受复合材料问题的启发，本项目的另一个创新点就是将欧氏区域上分片常数系数的方程问题推广到流形上的调和映照。分别建立带有分片光滑（连续）度量底流形上的极小（弱、稳态）调和映照的最优正则性，并尝试考虑对应的热流问题。

本项目主要研究了一类来自复合材料的偏微分方程组, 对于高对比度的复合材料的破坏机制是材料工业界和应用数学界普遍关注的一个问题. 本项目从基础的数学理论出发, 研究这一类具有实际背景的线性偏微分方程组, 利用能量估计的办法, 建立了纤维增强复合材料经典模型其纤维之间应力大小与纤维距离的依赖关系, 明确了其爆破的速度的阶数, 回答了Babuska问题的极限情形, 部分结果在《Arch. Ration. Mech. Anal.》等国际主流应用数学期刊发表..另外项目组还研究复合材料中的Eshelby猜想的相关问题, Riemann几何中调和映照的部分正则性, 以及 Hessian等完全非线性方程的外Dirichlet问题解的存在性, 部分结果发表在《Calc. Var. Partial Differential Equations.》、《J. Differential Equations》等国际主流数学期刊发表.