Abstract Applying parameter mapping theory, this paper establishes a new three dimension lattice-spring model with rotational degree of freedom (3D-LSMR). This 3D-LSMR contains nearest neighbor, next nearest neighbor and third nearest neighbor which highly increases the accuracy of system. Compared with the general lattice spring model, 3D-LSMR adds the rotational and tangential degrees of freedom, whose spring parameters are obtained by finite element parameter mapping. The traditional lattice spring model only considers the normal interaction between two points (the Poisson’s ratio is fixed), which limits the application of the lattice spring model in a wider range of materials. Some scholars proposed to add shear spring into the traditional model to solve the problem of fixed Poisson’s ratio, but the rotation invariance could not be maintained. The main reason is that the shear spring can’t distinguish the difference of tangential velocity (or displacement) between the two particles due to the common rotation or shear. Therefore, the rotation of the whole rigid body may cause incorrect generation of additional strain energy inside the shear spring. 3D-LSMR model is able to maintain rotation invariance and reproduces different Poisson’s ratios because of the spring parameters with rotational degrees of freedom and the rotation effect of lattice points. In this paper, the finite element stiffness matrix with rotational freedom is derived by using 3D-LSMR as the basic mechanical model, and the parameter mapping theory of spring stiffness coefficient is introduced. By means of numerical simulation, the large deflection of slender cantilever beam, elastic symmetrical collision and the simulation of dynamic fracture for concrete L-specimen are checked and calculated respectively, and then the correctness of the model is verified.

中文翻译：

---

具有旋转自由度的三维点阵弹簧模型及其在弹性脆性材料断裂模拟中的应用

摘要 应用参数映射理论，建立了一种新的具有旋转自由度的三维晶格弹簧模型（3D-LSMR）。这个 3D-LSMR 包含最近邻、次近邻和第三近邻，这大大提高了系统的准确性。与一般晶格弹簧模型相比，3D-LSMR增加了旋转和切向自由度，其弹簧参数通过有限元参数映射获得。传统的晶格弹簧模型只考虑两点之间的法向相互作用（泊松比是固定的），这限制了晶格弹簧模型在更广泛的材料中的应用。有学者提出在传统模型中加入剪切弹簧来解决固定泊松比的问题，但不能保持旋转不变性。主要原因是剪切弹簧由于共同的旋转或剪切而无法区分两个粒子之间切向速度（或位移）的差异。因此，整个刚体的旋转可能会导致剪切弹簧内部不正确地产生额外的应变能。由于具有旋转自由度的弹簧参数和格点的旋转效应，3D-LSMR模型能够保持旋转不变性并再现不同的泊松比。本文以3D-LSMR为基本力学模型，推导出具有旋转自由度的有限元刚度矩阵，并介绍了弹簧刚度系数的参数映射理论。通过数值模拟，细长悬臂梁的大挠度，