A dislocation-based constitutive model for
viscoplastic deformation of fcc metals at
very high strain rates
Ryan A. Austin, David L. McDowell , International Journal of Plasticity (2010)
Objective
The purpose of this paper is to present a physically-based
constitutive model for the viscoplastic response of polycrystalline fcc
metals with micron-scale grains in the weak shock loading regime
(25GPa).
Why physically-based ?
 Existing constitutive models are often idealized and lack fidelity to the physical
mechanisms at high rates.
 A physically based model because macroscopic properties are depends on microstructure
(i.e., crystal structure, the configuration of dislocations, grain size, secondary
phases, etc.).
Model Provide
(i) A microstructure-sensitive description of the kinematics and kinetics of viscoplastic
flow, and
(ii) A framework for rate- and path-dependent microstructure evolution.
Mechanism
1) Generation of Mobile dislocation segments
2) Dynamics of damped glide dislocations
Kinematics
Macroscopic plasticity
For isochoric plastic deformation
Velocity Gradient 𝐿 = 𝐷 + 𝑊
D : Rate of deformation tensor Capture strain Part
W : Vorticity tensor capture the rigid rotation
J2
plasticity
formulation
under
associated flow rule and isotropic
hardening
• DP : Plastic part of the rate of deformation tensor
• ψ : Equivalent plastic strain rate
•𝜎: Equivalent (von Mises) stress
• s: Deviatoric stress tensor
Clifton (1970)
: is the rate of change of the angle between twoline segments in the current configuration
Orowan equation
•b: Burgers vector
•𝑁𝑚 ​: Mobile dislocation density
•𝑣: Mean dislocation velocity
Finally We get
Kinematics (Cont.)
Crystal plasticity ( For single crystal )
Decomposition of the deformation gradient
plastic part of the velocity gradient in the
intermediate configuration
•𝛾 𝑎 : Shearing rate on slip system 𝑎
•𝑠 𝑎 : Slip direction (unit vector)
•𝑚𝑎 : Slip plane normal (unit vector)
•L𝑃 : Plastic part of the velocity gradient
By Orowan Relation , The shearing rate is
The mean dislocation velocity ( 𝑣 𝑎 ) and the evolution of
are driven by the shear stress resolved on the 𝑎𝑡ℎ slip
system.
Resolved shear stress
•𝑠 𝑎 : Slip direction (unit vector) in current configuration
•𝑚𝑎 : Slip plane normal (unit vector) in current configuration
•𝜎 : is applied stress
Kinematics (Cont.)
Key Considerations in Shock Wave Modeling
W is Shock front width , d is Grain size
𝑊
•If d ≫1
→ Use macroscopic (polycrystal) model
𝑊
•If d << 1
→ Use single crystal model to resolve slip
→ But: Computationally expensive for thousands of grains
Our Approach : Macroscopic Model
The macroscopic model serves as an approximation of shock-wave-induced plastic
deformation when w/d<1.
Kinetics
Dislocation Mobility
Mechanical Threshold Stress
The mechanical threshold stress 𝜏0 is defined as the shear stress
required to propagate dislocations on their respective slip
planes at the temperature T =0K (Kocks 1975).
Thermal fluctuations is sufficient to overcome short range
resistance but not long range .
𝜏𝜇 : Called athermal threshold , Below this no
plastic flow
𝜏 ∗ 𝑜 : Thermal threshold
: Dislocations wait for
thermal
assistance
: Glide dislocations are driven continuously
on the slip planes
Kinetics (Cont.)
Dislocation Mobility
Threshold For Pure FCC
Metals
Short Range Obstacles – Dislocation forest
Obstacles to dislocation glide --- lattice friction,
the dislocation network, and the grain
boundaries
Threshold stresses for fcc-based
alloys
Additional Obstacles --- Precipitates and
solid solutions strengthening (weak )
Kinetics (Cont.)
Dislocation Mobility
According to the model of Clifton (1970)
Mean velocity of the glide dislocations
Waiting time
The waiting time
where 𝑣𝐺 is the attempt frequency of a
dislocation waiting at an obstacle,
L : is the average glide distance between successive
obstacles on the slip planes
t 𝑤 : is the time a dislocation spends waiting at an
obstacle for thermal assistance.
t 𝑟 : is the time a dislocation spends ‘‘running” in-between
obstacles.
Finally Rate of deformation
instantaneous velocity
Kinetics (Cont.)
Comparison
Model without any resistance
Model with resistance/obstacle
Substructure evolution
Substructure : Arrangement and interaction of dislocations
Total Dislocation segments
•Nm ​: mobile dislocation segment
•N𝑖m ​: immobile dislocation density
• Nucleation: formation of new dislocations
• Multiplication: Frank-Read sources ( generation of
dislocation by existing dislocations)
• Annihilation: mutual annihilation
• Trapping: immobilization by obstacles
• Recovery: thermally driven reduction
Mobile and immobile Dislocation Evolution
The rise in shear stress behind the leading edge of the
shock front is the main reason for nucleation and
multiplication, recovery is somehow related with this.
Substructure evolution (Cont.)
Nucleation
Homogenous Nucleation
In defect free lattice when , shear stress approaches the
ideal shear strength of the crystal.
( Armstrong 2007-09)
Probability
Nucleation
model
for
Heterogeneous
The probability that 𝜏𝑐𝑟 falls in the range
(0, 𝜏] is given as
 Mostly Happens in strong shock condition.
Heterogeneous Nucleation
 nominal shear stresses con siderably lower than the ideal
shear strength of the lattice.
 Source : grain boundary or precipitate-matrix interface
 Cause: stress concentrations
Happens when
𝜏 = 𝜏𝑐𝑟
Total Nucleation
n: is the number of sources that have emitted a
dislocation.
N: is the total number of potential sources
(nucleation sites)
Differential length of dislocation line segment
Substructure evolution (Cont.)
Dislocation Multiplication
where 𝛿 is the multiplication coefficient
The relation given above expresses the idea that the
dislocation segment length generated by
multiplication is proportional to the area swept out
on the slip plane by the glide dislocations.
Annihilation and recovery
For high stress
where 𝛼𝑎𝑛𝑛 is a material constant
Trapping
where is the effective mean free path of statistical
trapping
Reasons for trapping – Grain boundary ,
principates, dislocation network.
Steady plastic wave analysis
Shock front is planar, steady. one-dimensional Problem and uniaxial strain, and load is
compression, plastic wave velocity is constant.
Decompose of deformation gradient
The rate of plastic stretching
Transformation to get plastic front
Conservation of mass and momentum
( Rayleigh line equations )
Where lambdas are principal value of F
Where shock wave velocity D given by
EOS
The temperature in the wave front
Finally, after
Transformation
Alloy Composition and Microstructure
6061-T6: Polycrystalline Al age-hardening alloy
Threshold stress for precipitates
is the mean spacing (surface-to-surface) of precipitates
Threshold stress for solid solution
𝑐 is concentration and 𝑘 is material constants
Fine-scale β″ (MgSi) precipitates
dominate strengthening
Material Properties
Damping coefficient
Shear modules
( Steinberg 2009 )
( Leibfried 1950 )
Results
Results (Cont.)
Effective plastic strain
Results (Cont.)
Conclusion
 Developed a macroscale dislocation-based model for FCC metals and alloys under:
 Model Accounts for:
 Thermally-activated dislocation glide and nucleation
 Microstructure-sensitive glide resistance
 Evolving mobile/immobile dislocation densities
 Incorporation of heterogeneous dislocation nucleation
 Accurate Shock Wave Predictions with experiments
Thank you