Analytical solutions of peridynamic equations. Part II: Elastic wave propagation
Analytical solutions of peridynamic equations. Part II: Elastic wave propagation
复制标题
近场动力学方程的解析解。
DOI：
10.1016/j.ijengsci.2023.103866
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发表时间：
2023
影响因子：
6.6
通讯作者：
Bobaru, Florin
中科院分区：
文献类型：
--
作者：
Chen, Ziguang;Peng, Xuhao;Jafarzadeh, Siavash;Bobaru, Florin
本文采用分离变量的方法建立了动力弹性动力学模型的解析解。我们证明，与瞬态扩散的周期动力学模型类似，通过在解的时间指数部分插入“周期动力学/非局部因子”，可以直接从相应经典模型的解中得到周期动力学弹性的无穷级数非局部解。解析解表明，非定域性引起的波色散包含在视界相关的非定域性因子中。用三种常用的周动力核，得到了一维和二维弹性波传播的波频散和群速度公式。我们观察到由非局部波色散产生的非局部解的有趣复杂性，并且与非局部相互作用区域的大小“成正比”。与扩散情况不同的是，当时间趋于无穷大时，弹性的非局部解不收敛于固定视界尺寸下的经典解，这意味着非局部效应在时间上持续存在。我们求解了几个具有Dirichlet边界条件和光滑或不连续初始条件的波传播的例子，并将这些解析解与经典模型对应的解析解进行了比较，经典模型被视为视界为零的PD模型的特殊情况。有趣的是，我们发现在PD解中，空间上的初始不连续在同一位置持续存在。虽然我们在这里给出的大多数解析解都是形式化的，但对于某些情况，我们能够证明级数解的一致收敛。这项工作是在二维有限域的周期动力学问题的系统分析处理的第一次提出。
We use the separation of variables technique to construct analytical solutions for peridynamic models of dynamic elasticity. We show that, similar to the case of peridynamic models for transient diffusion, infinite series nonlocal solutions for peridynamic elasticity can be obtained directly from the solutions of the corresponding classical model by inserting “peridynamic/nonlocal factors” in the time-exponential part of the solution. The analytical solutions show that wave dispersion, caused by nonlocality, is contained in the horizon-dependent nonlocal factor. We obtain formulas for wave dispersion and group velocities for 1D and 2D peridynamic elastic wave propagation with three commonly-used peridynamic kernels. We observe interesting complexity in nonlocal solutions, generated by nonlocal wave dispersion, and “proportional” to the size of the nonlocal interaction region. Different from the diffusion case, as time goes to infinity, the nonlocal solution for elasticity does not converge to the classical one for a fixed horizon size, meaning that nonlocal effects persist in time. We solve several examples of wave propagation with Dirichlet boundary conditions and smooth or discontinuous initial conditions and compare these analytical solutions with those corresponding to the classical model, which is seen as a particular case of the PD model for horizon equal to zero. Interestingly, we find that in PD solutions, initial discontinuities in space persist at the same location, in time. While most of the analytical solutions we present here are formal, for some of the cases, we are able to prove uniform convergence of the series solutions. This work is the first presentation of a systematic analytical treatment of peridynamic problems in 2D finite domains.
DOI：
10.1007/s42102-022-00080-7
发表时间：
2022-03
期刊：
Journal of Peridynamics and Nonlocal Modeling
影响因子：
--
作者：
Ziguang Chen;Xuhao Peng;S. Jafarzadeh;F. Bobaru
通讯作者：
Ziguang Chen;Xuhao Peng;S. Jafarzadeh;F. Bobaru