Peridynamic Modeling of Process-Dependent Material Strength in Additive Manufacturing

Presentation link
Prof. Dr.-Ing. Christian WillbergORCID Symbol, M.Sc. Jan-Timo HesseORCID Symbol

96th Annual Meeting of the International Association of Applied Mathematics and Mechanics, GAMM 2026
16-20 March, 2026 - Stuttgart

Presentation URL: https://perihub.github.io/Presentations/GAMM_2026

Introduction

  • Additive extrusion processes enables manufacturing of complex structures

  • Many process parameters influence the final properties and individual process parameter-property relations are often unclear

  • Process simulations can help to predict the properties and evaluate the process parameters

Material properties

  • created during process
  • not measurable in advance, because process and path influences them

Polymer crystallization

  • Crystallization influences the mechanical and technical properties of the material

  • Degree of crystallization depends on material properties and cooling conditions

  • Complex processes during cooling in deposition processes

Figure Source: Yang et al., Influence of thermal processing conditions in 3D printing ...
transparent

What is Peridynamics?

  • Alternative to classical continuum mechanics:
  • PD integral equation:
  • Focus material modeling and crack propagation; no continuity for the displacement

PD Solving the integral - Material point method

Advantages

  • Fast to implement
  • Failure propagation
  • Discretization

Disadvantages

  • Convergence is lower
  • Surfaces are not known

Peridynamics correspondence formulation

Derivation

force density stress states

with First Piola-Kirchhoff-stresses

forces at each point

deformation gradient in discrete form

with shape tensor as

and linearized strain

Including thermal effects

  • Convection
  • Heat transfer
  • Thermo-mechanical coupling

Thermo-mechanics

computing thermal stresses to determine the internal forces

Thermal flux (PD)

M.A. Zeleke & M. B. Ageze (2021). A Review of Peridynamics (PD) Theory of Diffusion Based Problems

Single Bond

Time integration

Determine the surface (2D / 3D)

Figure Source: Willberg et al., Peridynamic Framework to Model Add ...

Stable time step

  • thermal time step multiple orders above the mechanical one

Peridynamic Framework

  • No pre-processing required, mesh will be generated based on the gcode
  • Material Models:
    • Elastic
  • Thermal Models:
    • Thermal Flow
    • Heat Transfer
    • HETVAL subroutine
  • Damage Models:
    • Critical Stretch

Subroutine

  • Calculation of crystallization, dual kinetic model (by Velisaris & Seferis)

  • Implementation in Fortran HETVAL Subroutine for usage in Abaqus

    • Calculates crystallization kinetics through process simulation
    • Degree of crystallization at every time step

  • Temperature and time from the process simulation are inputs for the subroutine

  • Stiffness value of each node will be adapted based on the degree of crystallization

  • Fitting function:

Dogbone Specimen

  • Three step simulation process:
    • Printing specimen
    • Cooling step
    • Tensile test
  • Layer height = 0.2mm (20 Layers)
Specimen Geometry: ASTM D638

Simulation Properties

Material: PEEK (Polyetheretherketon)

Parameter Value

Thermal Properties

Parameter Value

Simulation Results


Simulation Results

  • Crack initiation and propagation similar, only initiation time slightly differs

Load-Displacement

Stiffness matrix assembly

Zero energy mode compensation





Analysis comparison

Fully Dynamic Static Mechanical and Thermal 3D Printing

Fully dynamic simulation

  • 10 hours total runtime
  • Time increment: s
  • per step (average)

Quasi-static

  • 2 seconds total simulation time
  • 1 second per increment
  • 90.8 ms per step (average)

Discussion and further work

  • Basic influence of different process parameters can be captured
  • PeriLab allows efficient and statistical analysis of the AM process
  • implementation of UMATs in stiffness matrix approach

Thank you!

Christian Willberg (h2)
Jan-Timo Hesse (DLR)

References

  1. J. Shah, B. Snider, T. Clarke, S. Kozutsky, M. Lacki & A. Hosseini (2019). Large-scale 3D printers for additive manufacturing: design considerations and challenges.

  2. C. Yang, X. Tian, D. Li, Y. Cao, F. Zhao & C. Shi (2017). Influence of thermal processing conditions in 3D printing on the crystallinity and mechanical properties of PEEK material.

  3. C. Willberg, J.-T. Hesse, R. Hein & F. Winkelmann (2024). Peridynamic Framework to Model Additive Manufacturing Processes.

  4. C. Willberg, J.-T. Hesse & A. Pernatii (2024). PeriLab - Peridynamic Laboratory.

  5. M.A. Zeleke & M. B. Ageze (2021). A Review of Peridynamics (PD) Theory of Diffusion Based Problems.

Funding

Name Logo Grant number
German Research Foundation WI 4835/5-1
Saxon State Parliament 3028223
Federal Ministry for Economic Affairs and Climate Action 20W2214G