Total Lagrangian - Smoothed Particle Hydrodynamics (TL-SPH) is a classical algorithm to improve the tensile instability encountered in the conventional Eulerian kernel-based SPH method for solid mechanics. Meanwhile, the gradient correction usually is employed in TL-SPH to enhance the consistency of approximation, which would generate unavoidable time costs. Therefore, a simplified gradient correction under the initial configuration is derived. Aiming at different particle configurations, the corresponding selection modes are provided and verified. Then the simplified gradient correction is introduced into TL-SPH, which is the Improved TL-SPH (ITL-SPH) method. Firstly, a classical example – elastic rubber collision is simulated to verify ITL-SPH has the same ability as TL-SPH to improve tensile instability. Then, the convergence and high efficiency of ITL-SPH is demonstrated via the uniaxial tension of a plate. Moreover, the plate with square and triangle holes also is simulated to show the adaptation and stability of ITL-SPH for the extreme boundary. Under the Total Lagrangian frame, the treatment to model the crack is necessary. So, this paper proposes a damage model and combines it with the ITL-SPH method for crack initiation and propagation, which is verified by the plate with holes and the notched beam examples.

总拉格朗日 - 平滑粒子流体动力学 (TL-SPH) 是一种经典算法，用于改善固体力学中基于欧拉核的常规 SPH 方法中遇到的拉伸不稳定性。同时，TL-SPH中通常采用梯度校正来增强近似的一致性，这会产生不可避免的时间成本。因此，推导出初始配置下的简化梯度校正。针对不同的粒子配置，提供并验证了相应的选择模式。然后将简化的梯度校正引入TL-SPH，即改进的TL-SPH（ITL-SPH）方法。首先，通过一个经典的例子——模拟弹性橡胶碰撞来验证ITL-SPH具有与TL-SPH相同的改善拉伸不稳定性的能力。然后，通过板的单轴张力证明了 ITL-SPH 的收敛性和高效率。此外，还模拟了带有方形和三角形孔的板，以显示 ITL-SPH 对极端边界的适应性和稳定性。在总拉格朗日框架下，需要对裂纹进行建模处理。因此，本文提出了一种损伤模型，并将其与裂纹萌生和扩展的 ITL-SPH 方法相结合，并通过带孔板和缺口梁实例进行验证。