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

1. Aurenhammer F. Voronoi diagrams – A survey of a fundamental geometric data

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.