Self-consistent Modeling of Polycrystal Plasticity
B. Clausen
†, C.N Tomé
‡, P. Maudlin
*and Mark Bourke
‡LANL,
†LANSCE-12,
‡MST-8,
*T-3
TMS 2000, Nashville, Tennessee March 12-16, 2000
Polycrystal versus Continuum Constitutive Description
Single Crystal Properties and Deformation Mechanisms
Polycrystal Texture
Polycrystal Model Experimental Data
Constitutive Response
Continuum mechanics
Simulation of Forming Operations using Finite Element Codes
Outline
z
Self-consistent model (SCM)
z Input, assumptions, output
z
Internal and residual strain development (EPSC)
z Tensile testing of stainless steel
z
Influence of deformation modes on texture development (VPSC)
z Clock rolled Zircaloy
z
Incorporating SCM into FEM formulation
z Bending of highly textured zirconium bars z
Conclusions
Self-consistent model (SCM) - EPSC
z
Material parameters
z Single crystal stiffnesses and coefficients of thermal expansion
z Description of texture with discrete set of grain orientations
z Crystal structure, slip (and twinning) systems
z CRSS and hardening law z
Model Assumptions
z Eshelby inclusion theory
z HEM properties equal to
weighted average of the grains z
Output
z Direct comparison with neutron diffraction measurements
z Averages over grains sets representing reflections
σ σ σc σc
HEM
Self-consistent model (SCM) - basic equations
¼
¼
¼
EPSC (small strain) VPSC (large deformations) z Basic equations for the EPSC and VPSC formulations
z Solve last two sets of equations iteratively
Comparison to neutron diffraction data - EPSC
z Uniaxial tensile loading of austenitic stainless steel
z ND measurements made at load levels marked by the symbols
z Schematic set-up of the NPD at LANSCE. Measurement time is about 2-3 hours
z Measure elastic strains in two directions simultaneously
0 50 100 150 200 250 300 350
0 0.5 1 1.5 2 2.5
Measured Calculated
Stress [MPa]
Macroscopic strain [%]
+ 90°
Detector Bank Incident Neutron Beam
- 90°
Detector Bank Tensile Axis
Q⊥ Q||
Comparison to neutron diffraction data - EPSC
0 500 1000 1500 2000 2500 3000 0
50 100 150 200 250 300 350
111 200 531
Elastic strain [µε]
Stress [MPa]
σ0.2
Active slip systems after
10µε plastic strain Active slip systems after 100µε plastic strain
z The reflections carry different amount of the load
z Plasticity starts around the <531> orientation
z => the <531> reflection deflects to the left
z The <100> orientation stays elastic the longest
z => the <200> reflection deflects to the right
z Applied stress versus measured elastic lattice strain
z ND measures lattice spacing changes and thereby only elastic strains
z Symbols are measurements, lines are model predictions
Self-consistent model (SCM) - VPSC
pr ttw ctw
pr ttw pyr
Predicted <0002> and
<1010> pole figures
Predicted and measured intensities along ND-TD and ND-RD
Predicted relative activity of the
deformation modes
z Prediction of rolling textures for Zircaloy, rolled to a true strain of 1.0
z Two different model assumptions of plastic deformation modes
z prismatic, tensile twins and compressive twins versus prismatic, tensile twins and pyramidal
z Comparison to measured textures for Zircaloy with two different grain sizes
SCM and FEM
(0001) 1
2
levels 2 4 6 8 10 12 max=13.7
0.0 0.1 0.2
0 500
1000 293 K
IPT in-plane tension IPC in-plane compression TTC through-thickness comp.
IPT IPC TTC
stress [MPa]
strain
^ τ (Γ)[MPa]
0.00 0.25 0.50
0 100 200 300 400 500
tens twin 293 K
pyram slip
prism slip
shear strain Γ
z Highly textured “clock” rolled Zirconium
z One set of deformation modes, CRSS and hardening parameters
z Possible to reproduce measured material behavior for all three deformation tests
z Through thickness compression (TTC), in-plane compression (IPC) and in-plane tension (IPT)
SCM and FEM
C0
C90
Measured cross-sections
(Red dots are
outlining the predicted cross-section)
C0 C90
C0 C90
Predicted cross-sections
z 4-point bending of heavily textured Zirconium bars
z Difference in deformed cross-section depending on orientation of preferred c-axis orientation (C0 or C90)
z 377 grain orientations for each element, 4000 elements
z Good agreement between predicted cross-sections and measured data
Conclusions
z
Self-consistent modeling (polycrystal constitutive model)
z Very useful tool for interpreting neutron diffraction data
z Pinpoint active deformation mechanisms
z
Prediction of internal and residual stresses and strains (EPSC)
z
Prediction of texture development for large strains (VPSC)
z
Incorporated SCM into FEM formulation
Self-consistent model (SCM) - EPSC
z Known elastic properties
z Calculate HEM stiffness using Voigt, Reuss or Hill average
z Calculate stress and strain increments in all grains for given macroscopic stress or strain increment
• Requires knowledge of stiffness of HEM and all the grains
• Determines a new self-consistent average stiffness
z Compare the new and old average stiffness/compliance
• Iterate until sufficient convergence is obtained z Further deformation
z Yielding in grains (stress exceeds yield criteria)
• Determine the elastic-plastic stiffness z “Bookkeeping”
z Calculate elastic strains for grain sets representing reflections
z Statistics on; # of slip systems, Taylor factor, plastic strain, etc.
Talk
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Somewhat different length scale - “meso scale”
z
Self-consistent polycrystal models
z
Tool to predict the macroscopic behavior of a material using microscopic material behavior, such as single X-tal elastic
stiffness, inelastic deformation modes like slip and twining, and texture
z
Show how we can validate this tool with neutron diffraction measurements of elastic lattice strains and texture
measurements
z
