International Conference on Advanced Material Technologies (ICAMT)-2016
27th and 28th December 2016
Dadi Institute of Engineering and Technology (DIET), Visakhapatnam, Andhra Pradesh, India.
Software for Mathematical Modeling of Plastic Deformation in
FСС Metals
A.E. Petelin, A.S. Eliseev
Tomsk State University, 36 Lenin Avenue, Tomsk, 634050, Russia
Abstract: The question on the necessity of software implementation in the study of plastic
deformation in FCC metals with the use of mathematical modeling methods is investigated. This
article describes the implementation features and the possibility of using the software
Dislocation Dynamics of Crystallographic Slip (DDCS). The software has an advanced user
interface and is designed for users without an extensive experience in IT-technologies.
Parameter values of the mathematical model, obtained from field experiments and accumulated
in a special database, are used in DDCS to carry out computational experiments. Moreover, the
software is capable of accumulating bibliographic information used in research.
Keywords: mathematical modeling, FCC-metals, crystallographic slip, research automation,
dislocation dynamics, database, software.
1. INTRODUCTION
More and more subtle combinations of physical, chemical, and mechanical properties are used in the
development of advanced materials for various purposes and in the technology of their production [1].
There is an increasing tendency in the role of plastic deformation in technologies of materials
development and processing. When studying the patterns of processes occurring at different structural
and scale levels that determine the behavior of materials under different conditions, it is impossible to
omit a set of experimental and theoretical methods, as a rule, including mathematical or simulation
modeling [2], [3].
At the present level of computer technology, mathematical models allow describing finer details of
the material behavior, but they become more difficult [4]. For this reason, the development of software
support for the mathematical model makes the process of model development complete and allows, at
its implementation in the problem-oriented software with an interface oriented on a high level of user
support, to make the model available to outside researchers.
2. BASIC SOFTWARE REQUIREMENTS
Mathematical models that describe physical, chemical, mechanical, and other processes in materials
contain a sufficiently large number of parameters which are characteristics of the material or the
exposure on the material (e.g. [5][7]). To carry out computational experiments the researcher must
somehow estimate the parameter values of the mathematical model. Typically, the results of
independent experimental and theoretical research or reference books are used. The search and the
selection of parameter values is a challenge because, as a rule, they are obtained by different authors
using different methods and conditions. It seems reasonable to develop an information support in
problem-oriented application software, enabling both to accumulate and to use the information on
values of material characteristics under chosen conditions (e.g., shear modulus, Young's modulus,
material density at different temperatures, and so forth). To select parameter values that correspond to
given conditions of the computational experiment it is desirable to have information on the conditions
and methods used in the evaluation of these parameters. Moreover, the possibility to visualize research
International Conference on Advanced Material Technologies (ICAMT)-2016
results in real time, as well as the presence of at least simple statistical processing, may greatly
facilitate the initial processing of research results.
3. SOFTWARE “DISLOCATION DYNAMICS OF CRYSTALLOGRAPHIC SLIP”
One example of the implementation of such problem-oriented software is Dislocation Dynamics of
Crystallographic Slip (DDCS), designed to study the dislocation dynamics of the crystallographic slip.
3.1 Mathematical models
Four mathematical models [8][11] of the dislocation loop dynamics, in which the obstacle field of
the dislocation or another nature is replaced with a homogeneous medium rendering the same
resistance to the moving dislocation as the original obstacle field (using the continuum theory of
dislocations [12]) are currently implemented in the software DDCS to carry out the study of the
crystallographic slip. All mathematical models [8][11] are presented as a system of ordinary
differential equations (ODE).
The dynamics model of a prismatic near-surface dislocation loop [8] accounts for viscous drag,
lattice, impurity, and dislocation friction. The mathematical model [8] is the simplest in terms of the
implementation, since it has small dimensionality (two equations in the system) and complexity.
In addition to forces accounted for in [8], the mathematical model of the expansion dynamics of the
dislocation loop [9] also takes into account the Peach-Keller force conditioned upon the applied force,
as well as the resistance force to expansion of the dislocation loop resulting from viscous drag, line
tension, and generation of point defects. Similar to the model [8], the mathematical model of the
expansion dynamics of the dislocation loop consists of two equations, but the equations are much more
difficult to implement because there is a number of additional terms describing the unaccounted for in
[8] resistance forces to dislocation motion. The assumption that a dislocation loop in the initial
configuration and during the expansion has the shape of a circle was used in writing of the
mathematical model [9]. However, this assumption is quite rough. Therefore, this model is used only
for an approximate evaluation of the dislocation loop dynamics.
A consideration of changes in the shape of the dislocation loop has been implemented in the
mathematical model [10]. The ODE system [10] has the dimensionality equal to the number of the
considered expansion directions of the dislocation loop. Therefore, the more accurately the shape of
the dislocation loop has to be described, the greater will be the dimensionality of the ODE system [10].
It has been shown that the shape of the loop is essentially independent of the ODE system [10] if its
dimensionality is more than 720 equations. The authors have also shown that the model [10] has a
pathological stiffness. Increased stiffness and large dimensionality of the ODE system imposes
significant limitations on the applied numerical method.
A mathematical model [11] which, in addition to forces considered in [9] takes into account the force
of elastic interaction among all dislocation loops formed by one Frank-Reid dislocation source, is
implemented in the software DDCS. In actual practice, the dislocation source can emit from few tens to
few tens of thousands of dislocations. Therefore, the dimensionality of the mathematical model [11]
can be extremely large. This imposes severe limitations on the selection of the numerical method
implemented in the applied software.
3.2 Computational module
The analysis of mathematical models [8][11] of the dislocation loop dynamics has shown that they
are stiff [13], with a variable stiffness on the integration interval. ODE systems [8], [9], as a rule, tend
to be moderately or highly stiff, and models [10], [11] are pathologically stiff [14].
The Gear numerical method [15] of the variable integration step in the representation of the
Nordsieck vector [16] has been selected and implemented in the program complex DDCS as a result of
the analysis of numerical methods for solution of problems [8], [9]. A number of performed
computational experiments have shown high efficiency (in terms of the calculation speed and
International Conference on Advanced Material Technologies (ICAMT)-2016
accuracy) of applying the given numerical method for studying the dynamics of dislocation loops using
models [8], [9]. However, application of the Gear numerical method to problems [10], [11] turned out
to be impractical due to a long computation time (up to few weeks). Therefore, the analysis of
numerical methods and solvers intended for problems of large dimensionality and pathological
stiffness is currently carried out.
3.3 Databases
Three databases are implemented in the program complex DDCS in support of the implementation
of computational experiments: database of materials characteristics, designed to accumulate and
efficiently process the information on characteristics of different materials; bibliographic database – to
store bibliographic information, brief annotation, and reference lists; and database of computational
experiment results, which stores the obtained results and values of material characteristics and the
effect.
The following is stored for each material in the database of material characteristics: values of
characteristics; conditions under which these values are obtained (for example, the temperature of the
experiment); and a link to the information source. The reference to the information source can be: a
reference to a literary source (output data of articles, monographs), a link to a network resource (page
address on the Internet), or a full path to the file on the computer with the program complex DDCS. It
is possible to store a set of several above listed types of links to the source (for example, it is possible to
specify the output data of the article and the address of the network resource where the article is
located). Each record in the database of material characteristics contains a comment box which can
contain additional information, such as reviews on the performed study or peculiarities of its
implementation, and so on.
Frequently, a number of material characteristics and experimental conditions are not specified in the
literature. Therefore, some fields in the database can be unfilled, including comment fields.
To select the value of a certain material characteristic from the database the researcher must use a
special subprogram, whose interface is presented in the tab Material characteristics in the window
Settings of the computational experiment (Fig 1, 2); select the item Selection of characteristic values;
in the drop-down list of material characteristics select the characteristic whose value must be set; in the
tree Literature sources select the material and the literature source which contains essential results for
the implementation of the study. In the tab Graph the researcher can view the results graphically.
Fig. 1 Tab Selection of characteristic
values
Fig 2. Tab Graph
Currently, the database contains information on coefficient values of the viscous friction, shear
modulus, material density, Poisson's ratio, and the Burger’s vector for metallic materials with FCC
structure (copper, nickel, aluminum, lead, silver, and gold), depending on the temperature [17][20].
The database of material characteristics can be expanded to store a wider range of materials, as well as
their characteristics and conditions under which they were obtained.
Since the experiments are usually carried out under a certain specified set of conditions (certain
International Conference on Advanced Material Technologies (ICAMT)-2016
material density, temperature, and so forth), the results of the study are represented in the database in a
discrete form. The linear interpolation on the known values (field interpolation) is currently used to
determine the values of material characteristics under intermediate conditions (for example, under
temperatures that have never been described in the literature). The field Interpolation is designed as a
drop-down list, since the implementation of other interpolation methods is assumed.
When choosing the values of material characteristics the researcher must take into account a large
amount of information: who carried out the research, what techniques were used, what kind of material
was studied. To accumulate the information on the results of theoretical and experimental studies of
mechanisms, processes, and patterns of plastic deformation, the software DDCS utilizes a
bibliographic database which stores information on: last name, first name, and middle name of the
researcher; description of r results; description of research techniques; reference list; reference to the
information source; year(s) of research. Since in a publication the information on the study can be
incomplete, some fields in the bibliographic database may be uncertain (empty).
The search for bibliographic data in the software DDCS is implemented by means of a specialized
subprogram Bibliographic database (Fig. 3). Currently, the information search can be carried out by
setting search conditions in one or more columns (the columns tab), but it is also expected to
implement data search using a search tree (the search tree tab) and by user request (tab on request).
To perform data search by column, the researcher must select this column and to enter the selection
criteria in one or more of the search fields (field all words, exact phrase, with any word, without a
word). It is possible to use the selection criteria of table rows containing the following in corresponding
boxes: all listed words (field all words); specified phrase (field exact phrase); at least one listed word
(field any word); none of listed words (field without a word). The search can be carried out by multiple
columns. To do this, it is necessary to specify the selection criteria in corresponding tabs in each
column.
The user interface of the subprogram Bibliographic database is intuitive. The capability of
customizing elements of the user interface in order to display data in the most convenient and visual
way is implemented: data in each column can be aligned to the left, right, or center (dropdown list
Alignment); column titles can be changed (field Title); data can be sorted in the forward or reverse
alphabetical order (dropdown list Sorting); the column width can be changed (field Column width).
Fig. 3. Interface of the subprogram Bibliographic database
The database of computational experiment results is implemented in the software DDCS to store and
accumulate the results of computational experiments. In addition to the results of computational
experiments, this database stores the values of material characteristics used in modeling. Depending on
user settings, the results of computational experiments and the used values of material characteristics
International Conference on Advanced Material Technologies (ICAMT)-2016
are stored in the database in the course of the computational experiment or upon its completion.
It shall be noted that the databases implemented in the software DDCS are universal and can be used
not only in the program complex DDCS but can be integrated with other software products.
4. CONCLUSION
The developed problem-oriented software DDCS is designed for modeling of the dislocation loop
dynamics. It has an advanced user interface and is designed for users without an extensive experience
in IT-technologies. In the near future it is planned to carry out the adaptation of the software DDCS for
arbitrary mathematical models, presented in the form of systems of ordinary differential equations
(ODE). Therefore, such possibility together with the already implemented functionality of the system
will allow the software DDCS to become a popular research tool for scientists engaged in plasticity
modeling. This software is unique in its functionality and purpose and, no doubt, has to generate a great
interest in the community of researchers specializing in the study of metals.
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FirstAuthor Petelin Alexander Evgenevich, PhD, Associate Professor, Faculty of Innovation
Techology, National research Tomsk State University, [email protected]
Second Author Eliseev Andrey Sergeevich, second-year student, Faculty of Innovation Technology,
National research Tomsk State University, [email protected]