Accompanying dataset for the paper: "One-dimensional elastoplastic models with isotropic hardening as differential inclusions in primal form"

Contributions

Author 1 carried out most of the study, structured and wrote the manuscript, implemented the algorithms, and performed the simulations.

Author 2 contributed substantially to the development of the methodology, assisted throughout the study, and revised and improved several versions of the draft.

Author 3 provided valuable mathematical insights and participated in proofreading.

Author 4 contributed substantially to the construction and mathematical formulation of the model and participated in proofreading.

All authors read and approved the final manuscript.

Licensing

CC BY 4.0 — attribution required

Funding sources

• This work was supported by the WSL Institute for Snow and Avalanche Research SLF, Switzerland.
• This work was supported by the Geobrugg AG, Switzerland

data: folder containing results for case 1

• bouncing_ball - folder containing the data for the bouncing ball example
  + README.md - Description
  + hardening_II_primal_kinematics.csv - t, positions x1,y1,y2 and velocities of ball and table, formulation II primal form; y1,y2,x1 used in figure 18
  + hardening_II_primal_kinetics.csv - t, percussions PN,PF,PP, forces laN,laF,laP, plastic variables p1,p2,gamma1,gamma2, formulation II primal form; PN,PF,PP used in figure 18
• minimal_example - folder containing the data for the minimal example
  + README.md - Description
  + HardeningIIPrimal_force_linear.csv - linear isotropic hardening, box excitation; figures 11, 12 and 15
  + HardeningUnilateralCompressionIIPrimal.csv - unilateral hardening element, compressive only; figure 15
  + HardeningUnilateralTensionIIPrimal.csv - unilateral hardening element, tensile only; figure 15
  + IdealPlasticIIPrimal.csv - ideal plasticity, i.e. no hardening; figure 15
  + NonlinearHardeningBothIIPrimal.csv - nonlinear hardening, exponential and polynomial combined; figure 16
  + NonlinearHardeningExpIIPrimal.csv - nonlinear hardening, exponential law; figure 16
  + NonlinearHardeningPolyIIPrimal.csv - nonlinear hardening, polynomial law; figure 16
  + analytical_sol.csv - t,q,u of the analytical reference solution of the minimal example; figure 12
  + convergence_study_hardening_II_primal_global.csv - global errors of q,u,la_P,q_P,u_P for six time steps dt from 3e-2 to 1e-4, plus the reference slopes dt2…dt6 (only dt is plotted); figure 13
  + convergence_study_hardening_II_primal_iterations_global.csv - fixed point tolerance (1e-3 … 1e-12) versus max and avg number of iterations; figure 14
  + force_box.csv - t,f of the prescribed box excitation force f(t); figures 15 and 16
  + force_box_negative.csv - t,f of the mirrored excitation force -f(t); figure 15
  + solution_diff_abs.csv - absolute difference numerical minus analytical solution in q,u; figure 12 (bottom right)
• rockfall - folder containing the data for the rockfall example
  + README.md - Description
  + net_energy.csv - t, total E, kinetic T and potential V energy [kJ] and rate of dissipation D [kJ/s] of the net; figure 20(b) and (c)
  + rock_energy.csv - t, total E, kinetic T and potential V energy [kJ] of the test block; figure 20(a)

cdr_files.zip: archive containing CorelDraw files of the figures

Software used

• Corel Draw 2024
• Python 3.14
• matlab 2023

Paper Description

To improve the numerical simulation of rockfall protection barriers, this work addresses an extension of a one-dimensional model for elastoplasticity using a friction element and a spring in series, to include isotropic hardening and unilateral behavior. We reformulate the resulting system of differential inclusions in primal form and develop a timestepping scheme via an extended version of Moreau's midpoint rule. The scheme handles nonsmooth dynamics consistently, allowing to include frictional contact and impacts. The results provide a rigorous and implementable framework for simulating coupled elastoplastic frictional contact systems, contributing to the analysis and computation of nonsmooth rate-independent processes.