Module
1: MATLAB Project - mathematical modeling of plasticity
Project Brief
In this project you will write MATLAB code defining several functions
from the plastic behavior of materials, and use curve-fitting approaches
to extract parameters from experimental measurements.
Successful completion will demonstrate competence in MATLAB data
manipulation, analysis, and visualization. These skills are invaluable
in analyzing the output of scientific computing software.
Deliverables
You should submit your scripts by creating a subdirectory
called
/class/mse404pla/sp22/<your_net_id>/Project1
and copying your code into that directory by 11:59pm on 4 April
2022. Late submissions will not be accepted; let me know in advance if
you will have difficulty with completion..
I will give you feedback on the expectations listed below and for the
overall script. Plots possessing unlabeled axes (variable and
units!), illegibly small text, or inappropriately scaled axes will be
penalized.
Specific Expectations
Multiaxial loading. Two of the equations to
convert a uniaxial yield stress into the yield “surface”
for a general stress state are the Tresca criterion, and the von Mises criterion, where yield occurs when the criterion is
true. Make two three dimensional plots showing the yield
surface (where the inequality is a strict equality) in , , : one plot for the Tresca
criterion (use =100
MPa) and one plot for the von Mises criterion (use = 100 MPa). Comment on
the differences between the two surfaces.
Solute/dislocation interaction. The hydrostatic
pressure of an edge dislocation is given by the equation for bulk
modulus , Burgers vector , distance from slip plane and angle from slip plane . If a solute is introduced at some
location, that changes the volume by , the interaction energy is given by ; this interaction depends on
the solute position, because the pressure is position dependent.
Consider a solute of magnesium in aluminum ( = 29 GPa, = 0.33, = 0.405 nm, and for Mg in Al, = 0.2). If the solute is placed
at above the slip
system, (1) make a plot of the interaction energy as the dislocation
glides in its slip plane, (2) plot the interaction force in the glide
plane as a function of position, and (3) determine the maximum
interaction force. Comment on the size of the interaction energies and
forces.
Work-hardening model. Kocks (http://dx.doi.org/doi:10.1115/1.3443340) suggested a
physically-motivated model for the evolution of dislocation density with
two contributions: dislocation production due to jogs, that is
proportional to the inverse distance between dislocations divided by a
mean-free path of
dislocation motion, and dislocation annihilation that assumes a constant
density of “recovery sites,” so that the rate of annihilation is
proportional to the dislocation density,
for true strain , and where (a unitless strain)
is an empirical parameter governing the recovery of dislocations.
Combined with the equation for work-hardening, with (FCC-averaged Schmid
factor) and for the
dislocation hardening, we can determine a true-stress / true-strain
relationship for work-hardening that includes two empirical
parameters.
- First, consider nickel ( = 125
GPa, = 0.25 nm) which has a yield
stress of 15 MPa at an initial (well-annealed) dislocation density of
. Using = 30 nm and = 0.7, use MATLAB to
numerically integrate your dislocation density vs. strain from 0 to a
true strain of 1, and plot it.
- Next, with your values of dislocation density, make a plot of true
stress vs. true strain over the same range.
- Convert your expressions into a parameterized model of stress and
strain, and determine the interaction parameters corresponding to the
data in
/class/mse404pla
.