基于非局部LSM优化的近场动力学及脆性材料变形模拟  

PERIDYNAMIC METHOD BASED ON NONLOCAL LSM OPTIMIZATION AND DEFORMATION SIMULATION OF BRITTLE MATERIALS

作　　者：马鹏飞 李树忱[1] 王修伟 王曼灵 MA Peng-fei;LI Shu-chen;WANG Xiu-wei;WANG Man-ling(Geotechnical and Structural Engineering Research Center,Shandong University,Jinan,Shandong 250061,China)

机构地区：[1]山东大学岩土与结构工程研究中心,山东济南250061

出　　处：《工程力学》2022年第6期1-10,共10页Engineering Mechanics

基　　金：国家自然科学基金项目(51879150,41831278)。

摘　　要：在经典近场动力学模型的基础上,通过小变形假定将近场动力学中的微模量与经典理论中的弹性常数建立联系,引入可以反映非局部作用特性的核函数提高计算精度,利用刚度等效的方式建立有关微模量的线性方程组,并通过寻求不定线性方程组最小二乘(LSM)最小范数解的方式对近场动力学中微模量进行优化,根据二次规划得到最优非负模量。利用优化后的方法对二维平板在单轴和双轴荷载作用下的变形及含预制裂纹脆性材料在荷载下的裂纹扩展进行了模拟并将结果与理论经典近场动力学方法结果对比。结果表明:优化后的方法可以较好的反映结构在荷载条件下的变形与破坏特性,与经典方法相比材料变形模拟在最大误差及误差范围具有良好的改善,并且模拟裂纹扩展过程在同等计算成本下具有更优的收敛速度及收敛结果,进一步验证了所提出方法的有效性,有着较为广泛的应用前景。Based on the peridynamic model, the relationship between the micro modulus in peridynamics and the elastic constants in classical theory is established by assuming small deformation. The kernel function which can reflect the nonlocal action characteristics is introduced to improve the calculation accuracy, and the linear equations related to the micro modulus are established by means of stiffness equivalence. The least square minimum(LSM) norm solution of uncertain linear equations is used to optimize the micro modulus in peridynamics, and the optimal nonnegative modulus is obtained through quadratic programming. The deformation of two-dimensional plate under uniaxial and biaxial loads and the crack growth of brittle material with prefabricated cracks under loads are simulated by the optimized method, and the results are compared with those by classical peridynamics method. The results show that the optimized method can better reflect the deformation and failure characteristics of the structure under load conditions, and the maximum error and error range of material deformation simulation are better than the classical method;and moreover, the simulation of crack growth process has better convergence speed and convergence results under the same calculation cost, which verifies the effectiveness and the wide application prospect of the proposed method.

关 键 词：数值模拟 非局部方法 近场动力学 最小二乘法 材料变形 

分 类 号：O346.11[理学—固体力学]