A novel stable peridynamics applicable to use three-dimensional material constitutive models
[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.
peridynamicszero-energy mode3D material constitutive modelhigh nonlinearity
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维普
