Post on 13-Apr-2017
transcript
Ben Menke
Finite Element Model Theory and Formulation
Abstract
Copper possesses many desirable qualities that have made it popular in many applications
for several millennia. Copper is ductile and malleable, qualities that allow it to be used in
applications requiring a soft material that can be stretched or compressed. In addition, copper is
characterized by high thermal and electrical conductivity. This allows it to be used in many high
value industries such as electrical connectors and heat exchangers. Many of these desirable
attributes are compromised when copper becomes embrittled and cracks. In a temperature range
of approximately half its melting point, copper displays a severe reduction in ductility. This
behavior is well documented among ductile metals and is referred to as an intermediate
temperature embrittlement mechanism. The ductility loss is co-modulated by several
mechanisms, including void nucleation and diffusion in the solid state. Liquid metal
embrittlement along with intermediate temperature embrittlement promotes structural
degradation of copper, ultimately leading to a catastrophic failure at unusually low stress levels.
The failure is known to originate at the grain boundaries, where the embrittler preferentially
transports. Bismuth is known to induce faceting of copper grain boundaries. Completely faceted
boundaries exhibit brittle behavior and suggest that this structural transition is necessary for
grain boundary embrittlement.
Introduction
This paper will describe how a bismuth-copper finite element model is calibrated using
compact tension experiments. Oxygen-free, 99.999% copper specimens were doped with
bismuth in two autonomous experiments. These samples were then placed in tension and heat
treated in an inert atmosphere at temperatures from 300-650°C. Finally, the compact tension
specimens were pulled to fracture to obtain stress-strain data. Mechanical property information
collected from tension tests along with MATLAB-generated microstructures is then used to
create ABAQUS finite element models for fracture response evaluation.
Stress-strain data for compact tension specimens – 400C, 12h
(Red are control samples, green are affected by bismuth)
When considering the intermediate temperature region associated with the loss of
ductility for liquid metal embrittlement, it is important to note that cracks tend to initiate and
propagate along grain boundaries. This behavior is commonly referred to as intergranular
fracture. As fracture initiates and propagates along the grain boundaries, it is essential to have an
algorithm that is capable of generating a microstructure which follows a similar statistical
distribution to that observed by examining micrographs for the materials which are of interest.
Experiments have shown that metals commonly follow a lognormal or Weibull distribution,
which is not captured through the use of classical Voronoi algorithms. Therefore the modified
Voronoi algorithm is employed here and lognormal distribution has been assumed. The created
MATLAB code can be easily modified to consider distributions which are not lognormal as
desired.
MATLAB Data Point Creation
The first step in the creation of a model to predict crack growth along copper grain
boundaries is to characterize the statistical distribution that the microstructure follows. This is
done by modifying a MATLAB file designed to generate a hypothetical copper grain structure
using primarily the mean of the desired statistical distribution and the standard deviation of the
desired statistical distribution, both collected from experimental data. The number of grains
which are generated by the algorithm is based on the size of the specified domain and the
random sampling of the lognormal distribution. Once the vector of grain sizes has been created,
the grains are then placed inside the domain using random placement. In order of decreasing
grain size, circles are placed within the prescribed domain. Random values are generated for the
coordinates of the center of the circle, and then a check is made to ensure that the circle does not
overlap a previously placed circle or lie outside of the allowable domain. If the circle placement
is not sufficient, new coordinates for the center of the circle is generated and the placement is
verified. The current code allows for 100,000 attempts to place a circle before giving up. Based
on the chosen domain size and the random sampling, there may be some cases where the
placement of circles without any overlapping is not possible. If circles cannot be placed without
overlapping, a warning message is printed to the MATLAB screen.
Example of circles corresponding to sampled grain size distribution placed in domain
Once the circles have been placed into the domain without overlapping, the vertices of the
resulting Voronoi diagram are calculated. First a Delaunay Triangularization is applied to all of
the centers of the circles in the domain. A modified Voronoi vertex is found for each triangle
identified by the Delaunay algorithm.
Calculation of modified Voronoi vertices
The centroid of the sub-triangle gives the modified Voronoi vertex. All circles are assigned
tracking numbers and the circles which were used to create each vertex is also stored. Once all
modified vertices are identified, a loop is made through each circle. The circle center and
modified Voronoi points created with the current circle are used as inputs into a second
Delaunay Triangularization. The edge of each triangle in this resulting tessellation is checked to
see whether or not one of the end points corresponds to the center of the current circle. All edges
with an end point corresponding to the center of the current circle in the loop are removed from
the set of linear segments corresponding to grain boundaries. The resulting edges are used to
define the grain boundaries in the microstructure. Finally, due to some numerical noise along the
edges, the domain is trimmed to a smaller size, which would be used for actual simulations.
Thus, a larger domain must be considered for the creation of the grain boundaries than will be
used in the resulting simulations.
Full domain microstructure Trimmed domain to remove noise along domain edges
Running this trimming file will output only the coordinates of the vertices that are within the
window bounded by the coordinates of “x” and “y” input by the user. It is important to note that
the bounds may need to be slightly larger than the desired modeling area. For edges which are
cut along the boundary, some vertices that are outside of the modeling area are needed to ensure
that the model is accurately reproduced.
Microstructure Modeling in Abaqus
The remainder of the finite element model is created within ABAQUS. To make the
transition between programs, all of the information produced by the MATLAB files must be
converted. To start, the microstructure vertices created in MATLAB are formatted in ABAQUS
compatible language and imported into the program as vectors between pairs of data points.
Vertices to be imported into ABAQUS Vectors received by ABAQUS
Using a tool to create isolated points, the endpoints of the vectors received from MATLAB are
converted to data points and the vectors are erased.
Creating the Model Geometry
The geometry will be created using a different “part” to represent each grain in the
problem. Using the sketch of data points created previously, and microstructure figure generated
by MATLAB, the user will pick a grain and connect the points using a tool which will connect
the points with a line. If a grain exists along the edge of the modeling domain, then it may be
necessary to trim the geometry so that it fits correctly in the window.
Data point connection Completed grain
Once the grain is successfully modeled by the part, the user will need to define a mesh for the
part. The density of the initial mesh for each part is automatically generated by ABAQUS
according to the size of each part. The points at which the mesh lines intersect each other and
the grain boundaries serve as nodes which will report displacement information in the tests. The
creation of part geometry and mesh is repeated for all grains in the microstructure.
Defining Material Properties
Each model of the copper-bismuth system is created for a constant temperature that falls
in the intermediate temperature range. Previously accepted orthotropic data for copper is entered
into ABAQUS to accurately simulate the behavior of the microstructure at the desired
temperature. Once the copper material property is defined, it is applied to each grain in the
microstructure. Next, each part is assigned a local coordinate system which defines the
orthotropic material orientation for that part.
Creating an Assembly of the Grains
Up until this point, the grains have existed individually in the program. When the
assembly is created, all of the parts are fit together as they appear in the MATLAB
microstructure, and they are now one step closer to being tested as a group.
Defining the Cohesive Contact between Grains
Though the assembly serves as a completed puzzle made up of the individual parts, there
is not yet anything defining the contact between them. Beginning with the first part, the user
defines each surface of the part. Each edge is assigned a letter, and the corresponding edge of
the neighboring part should have the same assigned letter. After every pair of touching edges is
given a unique name, an interaction must be defined. The copper-copper interaction property is
defined using actual copper grain boundary interaction data for the appropriate temperature.
Image of assembly with defined grain interactions and local coordinate systems
Creating Loading and Boundary Conditions
The incrementation is set simply as an initial set of conditions for the time stepping used
to solve the problem. Depending upon the problem these numbers may be either conservative or
unconservative and should be adjusted on a problem by problem basis. Adjustments are most
typically made if there are convergence problems after running the simulation. The simulation
represents stress-strain tests by “pulling” on one edge of the microstructure while the three
remaining edges are held still. To achieve this, a negative pressure is applied to the top edge, the
sides are set for no displacement in the x-direction, and the bottom edge is set for no
displacement in the y-direction.
Assembly with boudary conditions applied Scaled view of boundary condition symbology
Performing a Displacement Convergence Study to Assess the Quality of the Mesh
The finite element mesh is of paramount importance to a finite element analysis. With a
mesh which is too coarse, the solutions of the finite element analysis may have insufficient
accuracy. If a mesh is needlessly dense then very small if any gain in the resulting solution will
be achieved at a higher computational cost. In general, the relationship between the mesh
density and the solution time can be considered as quadratic. If is important to note that a finite
element analysis solves for the displacement caused by loading and boundary conditions for a
solid mechanics analysis and that the stress is then calculated based on that displacement. For
geometries with stress concentrations increasing the mesh density will decrease the distance from
nodes to the locations of higher stress, resulting in a prediction of higher stress values. Thus, the
convergence of the stress with respect to the mesh occurs more slowly than displacement. the
general approach is to ensure that the mesh is sufficiently refined for the displacement to
converge to some value.
To assess the quality of the mesh, and analysis can be performed that assumes that all of
the grains are perfectly bonded and then considers the effect of the mesh on the displacement.
As an unknown solution exists, the following procedure can be used to assess the quality of the
mesh. First, the user must perform and initial analysis on a coarse mesh and record the values of
the maximum and minimum displacements in the “x” and “y” directions. Second, the user must
increase the mesh density by a factor of two, repeat the analysis and record the displacements.
Finally, the user must continue this procedure until the change between the predicted solutions
from one mesh to another is acceptably small. When the procedure is complete, the user will
take the final mesh density value and apply if to each part individually, overwriting the automatic
value for mesh density set by ABAQUS.
Scaled view of part mesh
Data Acquisition
An initial “job” can now be run with the new value for mesh density. This will serve as
the simulation of the pure copper microstructure. All of the following simulations will represent
copper grain boundaries “doped” with different levels of bismuth. To simulate a bismuth doped
grain boundary, a new interaction property must first be defined according to accepted data for
bismuth doped copper grain boundaries. Afterward, a spreadsheet will be created to serve as a
guide for the systematic doping of the copper grain boundaries.
Image of Excel file for systematic doping of the copper grain boundaries
Using the spreadsheet will ensure proper concentration and distribution of bismuth doped grain
boundaries. A total of thirteen simulations will be run, including the concentrations of bismuth
doped grain boundaries shown above in the image, as well as the pure copper test and a 100%
doped test.
Data Analysis
Reports written for each of the simulations provide grain displacement data to be used in
the data analysis. Once all of the values are entered into a spreadsheet, they will be plotted on a
graph of % change in displacement versus the grain designations.
Graph of % change in displacement versus the grain designation
By looking at the graph in the image above, the user can see that the last reliable grain
designation is number 40. From here, the user will cut out the data provided by the grains
designated higher than 40 and plot the % change in displacement versus the % grain boundaries
doped.
Graph of the % change in displacement versus the % grain boundaries doped
Future Work and Conclusions
Many aspects of ABAQUS are not fully understood by the research team. Things such as
the inability to demonstrate necking of the microstructure and diffusion of bismuth in copper
cannot currently be accounted for in the simulation. However, information provided by the finite
element model will be useful in all applications of copper. Bismuth has been found to be a
natural impurity of copper, and it is therefore a point of concern because of its embrittlement
effects. The finite element model provides us with information about the copper-bismuth
relationship at varying concentrations of bismuth and temperatures, and could be used for testing
copper in any of its environments.
References
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structure, ACM Computing Surveys 1991; 23 (3): 345-405.
2. Duscher, Gerd; Chisholm, Matthew; Alber, Uwe; Ruhle, Manfred. Bismuth-induced
embrittlement of copper grain boundaries. Nat Mater 2004, 621-625.
3. Laporte, V; Mortensen, A. Intermediate temperature embrittlement of copper alloys.
International Materials Reviews 2009, 54, 94-116.
4. Luo, J; Cheng, H; Asi, KM; Kiely, CJ; Harmer, MP. The role of a bilayer interfacial
phase on liquid metal embrittlement. Science 2011.
5. Luther T, Könke C. Polycrystal models for the analysis of intergranular crack growth in
metallic materials. Engineering Fracture Mechanics 2009; 76(15): 2332-2343.