模拟纤维增强复合材料的叠加单元完全光滑积分算法研究

It is obvious that fiber reinforced composite material has the properties of inhomogeneous and anisotropy. Considering the complex compositions of fiber, matrix and joint interface, numerical simulation of fiber reinforced composite material is tedious and challenging works in the field of computational mechanics. In order to develop a high efficient Finite Element Method (FEM) and to improve the computational accuracy of extremely irregular elements, the main contents of this project are: (1) In order to remove the coordinate mapping and calculation of Jacobian matrix, the rebar element is incorporated into the matrix element in the global physical coordinate system, in which the representation of rebar element, shape function and strain matrix of matrix element are required, respectively. (2) A fully smoothed finite element method is developed for smoothed integral of partial derivative terms and non-partial derivative terms of shape function based on overlay element, the calculations of stiffness matrix, mass matrix and damping matrix are transformed into the same framework of smoothed integral of FEM. (3) The overlay element of fiber reinforced composite material is modified to study on the influence of joint interface between fiber and matrix materials. (4) Program of fiber reinforced composite material under impact loads is developed based on overlay elements and fully smoothed integral algorithm, the accurate stress distribution of fiber and matrix materials can be obtained even though extremely irregular elements are used for simulation, i.e., the element distortion is insensitive to the computational accuracy of fully smoothed FEM.

纤维增强复合材料具有明显的非均匀性和各向异性特点，考虑其纤维、基体及连接界面复杂构成的数值模拟是目前计算力学领域极为繁琐且有挑战性的工作。项目为了发展求解效率高且在极度不规则单元情况下仍具有较高模拟精度的有限元法，主要开展以下研究：（1）在整体物理坐标系下对rebar单元进行表征，推导基体材料单元的形函数和应变矩阵，将rebar单元叠加至基体单元中，无需坐标映射和雅可比矩阵计算。（2）基于叠加单元发展完全光滑有限元法，对有限元系统方程中形函数偏导项和非偏导项进行光滑处理，将刚度矩阵、质量矩阵和阻尼矩阵统一到光滑积分的框架下进行求解。（3）考虑纤维和基体连接界面对复合材料力学性能的影响，对叠加单元求解进行修正。（4）基于叠加单元开发冲击载荷下纤维增强复合材料完全光滑积分求解程序，发挥光滑有限元法对单元“畸变”不敏感的优势，即在单元“畸变”的情况下仍可获得纤维和基体材料精确的应力分布。

纤维增强复合材料、生物组织等具有明显的非均匀性和各向异性特点，考虑其纤维、基体及连接界面复杂构成的数值模拟是目前计算力学领域极为繁琐且有挑战性的工作。项目为了模拟纤维增强复合材料和生物组织等而发展了求解效率高且在极度不规则单元情况下仍具有较高模拟精度的有限元法。项目完成的主要研究内容包括：1）拓展光滑积分理论及应用范围，实现有限元系统方程中刚度矩阵、质量矩阵和阻尼矩阵光滑处理，避免了坐标映射和雅可比矩阵计算，使不规则单元亦可获得精确的数值结果，并针对纤维增强复合材料层合板的静/动力学、应力集中等问题开展了研究。2）构建了基于整体物理坐标系的叠加单元，引入应变光滑技术，发展了多种形式的完全光滑叠加单元，大大降低了求解过程中所需的自由度、计算时间及数据存储需求，自主开发了相关程序，并研究了纤维和基体连接界面对复合材料力学的影响。3）基于选择性光滑有限元方法（Selective ES/NS-FEM）研究了考虑剪切效应的人体脊椎纤维环力学响应，并开展了具有不规则空间螺旋结构的生物组织的力学特性和损伤机理研究。4）针对冲击载荷下复合材料力学性能及损伤问题开展了相关研究，获取了不确定性情况下复合材料抗冲击性能，并揭示了其损伤破坏规律。