一种新型稳定可结合三维本构的近场动力学方法

A novel stable peridynamics applicable to use three-dimensionalmaterial constitutive models

[目的]针对目前两种近场动力学(PD)理论,键型PD(BPD)无法应用三维材料本构模型,而态型PD在均匀化过程中出现零能模式常导致强非线性问题不稳定的挑战,研究一种新型混合PD(HPD),以增强PD理论在求解复杂强非线性问题时的稳定性和实用性.[方法] HPD方法结合BPD和SPD的优点参照BPD,将PD点与影响域内其他PD物质点相连形成键,键上的力通过键中点截面上的应力及其法向向量的乘积计算得到,应力可由三维本构关系计算,而应变则使用与SPD类似的方法拟合得到.在HPD中,键力方向与键的方向并不一致.最后通过两个数值算例验证了HPD方法的正确性和稳定性.[结果]在第一个算例中,可以观察到两种影响域半径的HPD和有限元方法(FEM)的结果高度相似.力加载行(M-M)中的最大相对误差仅为4.58%,模型顶行(N-N)中的最大相对误差仅为0.51%.相比之下,由于零能模式的存在,SPD的最大相对误差为382.68%和5.61%.在第二个算例中,选择A点作为代表点,比较了3种方法计算的位移时程.在靠近底座的A点,SPD在位移计算的初始阶段表现出较高的精度.然而,当模型进入强非线性时,误差便会显著增加,达到约65.65%.相比之下,HPD有效地解决了这个问题,在最大位移点实现了仅0.2%的相对误差.[结论] HPD可以消除零能量模式的影响,而且在各种影响域半径下都能产生更准确的结果;即使在强非线性情况下HPD也能解决零能模式问题,同时可确保应力和应变的精确预测,可有效解决强非线性和大变形问题.此外,HPD可以与各种材料模型相结合,是一种实用且有效的计算方法,可用于解决各种复杂问题.

[Objective]Presently,as a non-local continuum theory effective for diverse discontinuous problems,Peridynamics(PD)is classified into two types,namely bond-based peridynamics(BPD)and state-based peridynamics(SPD).However,the former cannot be applied to three-dimensional constitutive material models,and the latter encounters zero-energy modes in the homogenization process,leading to the instability in strongly nonlinear problems.To face these challenges,here we introduce a hybrid peridynamics method(HPD)that combines merits of both BPD and SPD to surpass the existing constraints.This proposed approach aims to enhance the stability and the applicability in complicated nonlinear problems.[Methods]In HPD,BPD methods are extended,PD points are connected within an influence domain to form bonds,while the force is computed on these bonds through the stress on their midpoint sections and corresponding normal vectors.Herein,this stress is derived with the three-dimensional constitutive relationship.Additionally in HPD,midpoint’s strains are computed similarly to state-based PD.Notably,the force direction on the bond does not align with the direction of bonds all the time.[Results]Two numerical examples are presented to validate the correctness and the stability of the HPD method.In the first example,the static analysis is utilized to simulate the deformation of a two-dimensional elastic column under the influence of concentrated force loads.The second example investigates the dynamic time history response of a two-dimensional soil column in incorporation of multi-yield surface materials subjected to earthquake base excitations.For these computations,implicit solutions and static equilibrium PD equations are applied,and solutions are computed using the Newton-Raphson method.In the first example,it is observed that HPD and finite element method(FEM)yield close results for two horizon sizes.The maximum relative error in the force loading row(M-M)lies in a mere 4.58%,and in the top row of the model(N-N),it lies in a mere 0.51%.In contrast,due to the presence of zero-energy mode,SPD exhibits maximum relative errors of 382.68%and 5.61%in the same comparisons.This outcome suggests that HPD not only eliminates the impact of zero-energy mode but also produces more accurate results under various horizon sizes.In the second example,point A is chosen as the representative point,and the displacement time histories computed by the three methods are compared.Notably,at point A closer to the base,SPD demonstrates greater accuracy in initial stages of displacement computations.However,as the model enters strong-nonlinearity zones,substantial errors become apparent,and the maximal relative error of SPD reaches 65.65%.In contrast,HPD effectively tackles this issue,achieving a mere 0.2%relative error at the point of maximal displacement.This comparison emphasizes that HPD can eliminate oscillations caused by zero-energy mode,thereby proficiently addressing challenges related to strong nonlinearity and large deformation while ensuring precise predictions of stresses and strains.[Conclusions]For severely nonlinear problems,HPD that combines merits of BPD and SPD is proposed.Two examples,which are involved,respectively,with statics and dynamics,are presented to demonstrate the ability of HPD in tackling computational stability and accuracy.Finally,it can be combined with various material models,thus hopefully in the future serving as a practical and effective tool that can be used for a wide range of severely-nonlinear problems.