CPFEH

Crystal Plasticity Finite Element Homogenization model for Dyalog APL

Example

The script eg.apls runs several example simulations using a randomly generated microstructure with Fe-gamma and Fe-alpha grains applying boundary conditions of uniaxial tension.

Expected output:

    grains      steps       ∆time       iter        iterb       seconds
        50          50   2.50E¯1           292           6        0.54
        50         100   2.50E¯1           414          14        0.87
       100         100   2.50E¯1           647          15        1.41
       250         100   2.50E¯1           646          13        2.37
       500         100   2.50E¯1           604          12        5.05
        50         250   2.50E¯1           741          39        2.11
        50         500   2.50E¯1          1246          75        3.68
      1000        1000   2.50E¯1          3612         132       88.54   r.csv  tex.csv  f.csv

Timings (in the last column) may vary. Plotting the strain and stress in the generated file r.csv should result in the following tensile diagram:

CPFEH Model

The model uses a gradient descent method to solve the self-consistent polycrystal formalism. Grain interaction is determined such that each grain is in a relaxed-constraints relationship with respect to its direction dependent environment.

Grain Behaviour

Each grain is assumed to have elasto-viscoplastic behaviour. Plastic strain is determined using the strain rate sensitivity approach.

Slip Systems

Plastic strain happens by shear along crystallographic directions on crystallographic planes. Each slip system is thus defined by a direction (Burges vector), and the normal to the plane.

Strain Hardening

Accumulated shear causes the increment of the resolved shear stress according to the Voce law.

Grain-Environment Interaction

The environment of each grain along each direction is determined depending on the distribution measured experimentally. It is assumed that the neighbour grains will have relaxed the components not associated with the common face.

Self-Consistent Formalism

Localisation tensors allow to correlate macrospic and grain stresses. These magnitudes are also related according to the interaction equation. Using an iterative method, a solution is found that satisfies all these conditions.

Microstructural Update

After a solution has been found for each simulation step, the microstructure of the material is updated. The strain hardening law is used to increase the resolved shear stress, grains are rotated depending on plastic deformation and angular velocity, and deformation tensors are modified.

Input and Output

The CPFEH dyadic operator takes as right operand convergence parameters and as left operand the target error and maximum number of steps. As right argument it takes the microstructure, including materials (which can be read from json files with the MATERIAL function), orientations, topology (given as common areas in each direction) and volume. It returns a table with results by time (strain, stress and number of iterations to solve stress and to reach self-consistency), the deformed texture, and the deformation of each grain.

See example.

Experimental microstructures

The MICRO function is provided to process experimental data. It can generate the input needed by CPFEH either from EBSD ang or ctf files or from (discrete) distributions of crystallographic orientations and disorientation angles on different planes.

Usage:

p e x y      ← aci MICRO  f[crop]          ⍝ grid and cell size from ang file
p e v[q m n] ← [d] MICRO  a c i f[crop]    ⍝ volumes and disorientations from ang
p e v[q m n] ← [d] MICRO  p e s            ⍝ volumes and disorientations from ang
p e v x y z  ←  d  MICRO  p e v q m[n o]   ⍝ CPFEH parameters from distributions
p e v        ← [v] MICRO,⊂p e v            ⍝ merge volumes
q m          ← [v] MICRO,⊂q m              ⍝ merge disorientations
d            ←  d  MICRO  e                ⍝ disorientations namespace

Parameters:

• f[crop] EBSD file and optional crop region (four additional x0 y0 x1 y1 parameters)
• a c i angle increment (zero to not round) and minimum image quality and confidence index
• d disorientation increment or disorientation namespace
• p e v phases, euler angles and volumes
• q m n z pairs of phases and istributions of disorientations in x y z directions

The script dc.apls shows an example of how to load an EBSD file, partition the microstructure according to grain axis length in the horizontal and vertical direction, and use the obtained texture and distribution of disorientations (in horizontal and vertical directions) to generate input data and run a simulation.

References

• A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys. RA Lebensohn, CN Tomé. Acta metallurgica et materialia, 1993

• Material modeling with the visco-plastic self-consistent (VPSC) approach: theory and practical applications. CN Tome, RA Lebensohn. 2023

• An improved algorithm for the polycrystal viscoplastic self-consistent model and its integration with implicit finite element schemes. J Galán, P Verleysen, RA Lebensohn. Modelling and Simulation in Materials Science and Engineering, 2014

• A multivariate grain size and orientation distribution function: derivation from electron backscatter diffraction data and applications. J Galán López, LAI Kestens. Applied Crystallography. 2021