DYNAMIC FRACTURE ANALYSIS WITH THE FRAGILE POINTS METHOD
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摘要: 工程中的冲击防护结构在撞击、爆炸等冲击载荷下可能发生动态断裂并最终破坏, 抑制结构的动态断裂是提升结构防护能力的重要手段, 为此需要准确预测结构在动态载荷下的断裂行为. 数值仿真是预测动态断裂的重要手段, 然而当前工程中常用的有限元法在模拟断裂方面仍存在网格畸变和难以显式引入裂纹等问题. 碎点法是近年来提出的一种不连续型伽辽金弱形式无网格方法, 适合模拟断裂问题, 本文提出一种显式动力学格式的碎点法并将该方法应用于动态断裂分析. 一方面, 碎点法参考弱形式无网格类方法, 将求解域离散为空间中的节点和子域, 并基于支持域内的节点群构造子域的位移试函数, 因此该方法的子域具有抵抗畸变的能力. 另一方面, 碎点法参考间断伽辽金有限元法, 使用分片连续的位移试函数, 并引入内部界面数值通量修正保证方法的一致性和稳定性, 因此该方法易于在结构中显式引入裂纹. 本文首先介绍碎点法的核心思想和离散形式, 接着推导了动力学碎点法弱形式动量方程, 然后建立了碎点法的显式动力学求解格式, 最后通过算例验证动力学碎点法预测应力波传播和动态断裂行为的能力.Abstract: Impact resistant structures in engineering are likely to undergo dynamic fracture when they are subjected to impact or explosion. Restricting the dynamic fracture has been a key method to reinforce structures’ impact resistance. Thus, an accurate prediction on structures’ fracture behavior under dynamic loads is needed. Numerical simulation has been an important tool for the prediction of dynamic fracture. But, finite element method, commonly used in engineering practices, has some difficulties in fracture simulations, such as mesh distortion and inserting crack explicitly. The recently proposed fragile points method (FPM) is a discontinuous Galerkin meshless method which is suitable for fracture simulations. This paper aims on extending the FPM to analyze dynamic fracture problems. On the one hand, taken the weak form meshless methods as references, the FPM uses points and subdomains to discretize the problem domains. The shape function of an FPM subdomain is determined based on the point cloud in its supporting domain, and thus the FPM is not sensitive to mesh distortion. On the other hand, taken the discontinuous Galerkin finite element method as a reference, piece-wise continuous trial functions are used in the FPM, and the interior interface numerical flux correction is introduced in the weak formulations to guarantee the consistency and stabilization of the FPM. Thus, explicit cracks can be easily introduced in the FPM models. This paper starts with the introduction of the core idea and discretization method of FPM. Then the derivation of the equation of motion in weak form for the dynamic FPM is presented. After that the explicit dynamic solution scheme of the FPM is established. Finally some examples are employed to verify the dynamic FPM’s capability regarding the prediction of stress wave propagation and dynamic fracture.
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图 1 (a) 碎点法模型的离散和(b) 支持域及其节点群的定义
Figure 1. (a) Discretization of FPM model and (b) definition of support range and its point cloud
图 2 (a) 碎点法形函数示意图 (b) 位移场为
${{\rm{e}}^{ - 10{{\left( {{x_1} - 0.5} \right)}^2} - 10{{\left( {{x_2} - 0.5} \right)}^2}}}$ 的碎点法位移试函数[34]Figure 2. (a) Shape function of FPM (b) FPM’s trial function of a field of displacement :
${{\rm{e}}^{ - 10{{\left( {{x_1} - 0.5} \right)}^2} - 10{{\left( {{x_2} - 0.5} \right)}^2}}}$ 
图 7 拉力作用下平板应力波传播问题精确解及FPM结果
Figure 7. Exact solutions and FPM’s results of the displacement and stress at different points
7 拉力作用下平板应力波传播问题精确解及FPM结果(续)
7. Exact solutions and FPM’s results of the displacement and stress at different points (continued)
图 8 (a)悬臂梁示意图和(b)计算模型节点分布
Figure 8. (a) Configuration of the beam and (b) the scatter of subdomains
图 9 自由端中点竖直方向位移响应曲线
Figure 9. Vertical displacement curve of the mid-point at the free end of the beam
图 10 固定端中点应力响应曲线
Figure 10. Curves of stress components of the mid-point at the fixed end of the beam
图 13 计算模型: (a) 6516个子域节点和(b) 11995个子域节点
Figure 13. Domain discretization with (a) 6516 FPM subdomains and (b)11995 FPM subdomains
表 1 有限元、碎点法的位移峰值与理论解的比较
Table 1. Relative error of maximum displacement obtained from FEM and FPM
Maximum ${u_1}$ at A Maximum ${u_1}$ at B Exact FEM FPM Exact FEM FPM results 0.0182 0.0182 0.0181 0.0100 0.0100 0.0100 error − −0.12% −0.58% − −0.01% 0.18% 
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表 2 有限元、碎点法的应力峰值与理论解的比较
Table 2. Relative error of maximum stress obtained from FEM and FPM
Peak ${\sigma _{11}}$ at B Peak ${\sigma _{11}}$ at C Exact FEM FPM Exact FEM FPM results 399.9910 399.5224 399.1767 399.9936 400.2177 399.8739 error − −0.12% −0.20% − −0.10% 0.13%
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