This talk will highlight FEM modeling for Plasticity in Solid Mechanics, focusing on key updates in material behavior simulation, including plastic deformation and hardening . Attendees will gain insight into current FEM capabilities for structural analysis and be equipped with the tools to start their own simulations.
​CITATION: Alessandro Mastrofini, Solid Mechanics: Plasticity, Wolfram Virtual Technology Conference 2024. https://www.wolfram.com/events/technology-conference/2024/presentations/

Agenda

◼
  • Introduction to plasticity
    • ◼
  • Computational plasticity
    • ◼
  • Implementation details
    • ◼
  • Cook’s membrane application example

    • Still in development

    Disclaimer: this code is still under development

    Material properties

    In materials science plasticity is the ability of a solid material to undergo permanent deformation.

    Theoretical background

    Tresca’s memoir in 1864
    In[]:=

    Theoretical background

    Continuum approach

    Modeling elasto-plastic materials

    ◼
  • Yield criteria
    • ◼
  • Flow rules
    • ◼
  • Hardening models
    • ◼
  • Yield criteria
    • ◼
  • von Mises yielding:
    Fσ,
    σ
    y0
    =
    1
    2
    
    2
    (
    σ
    1
    -
    σ
    2
    )
    +
    2
    (
    σ
    2
    -
    σ
    3
    )
    +
    2
    (
    σ
    3
    -
    σ
    1
    )
    -
    σ
    y0
    • ◼
  • Tresca yielding:
    Fσ,
    σ
    y0
    =
    1
    2
    max[|
    σ
    1
    -
    σ
    2
    |,|
    σ
    2
    -
    σ
    3
    |,|
    σ
    3
    -
    σ
    1
    |]-
    σ
    y0
    • ◼
  • Flow rules
    • 
    ϵ
    p
    =λ
    ∂
    Q
    p
    ∂σ
    (
    1
    )
    ◼
  • Hardening models

    • Small strain plasticity

    Flow rule in term of the yielding function:
    Discrete time derivative:

    Predictor - corrector algorithm

    Implementation

    1
    .
    Solve elastic - problem
    2
    .
    Evaluate Yielding state
    3
    .
    Plastic flow equations
    4
    .
    Iterative solution (come back to 1)

    1. Solve elastic problem

    2. Evaluate Yielding state

    3. Plastic flow equations

    Stress evaluation

    4. Iterative solution

    Implementation

    1
    .
    Solve elastic - problem
    2
    .
    Evaluate Yielding state
    3
    .
    Plastic flow equations
    4
    .
    Iterative solution (come back to 1)

    1. Solve elastic - problem

    2. Evaluate Yielding state

    4. Iterative solution

    Discretized solution

    Discretization

    Iterative solution

    Plastic flow step

    Examples

    ◼
  • Cook’s membrane

    • Cook’s membrane

    Scale factor = 3
    Scale factor = 3

    Current implementation

    Future implementation

    ◼
  • Integrating several Yielding criteria
    • ◼
  • Implementation of Kinematic Hardening
    • ◼
  • Implementation of different hardening laws

    • CITE THIS NOTEBOOK

    FEM modeling for plasticity in solid mechanics​
    by Alessandro Mastrofini​
    Wolfram Community, STAFF PICKS, November 4, 2024
    ​https://community.wolfram.com/groups/-/m/t/3314703