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MODIFICATION IN ADINA SOFTWARE FOR ZAHORSKI MATERIAL
Major Maciej
Major Izabela
Czestochowa University of Technology, Faculty of
Civil Engineering, Address ul. Akademicka 3,
Częstochowa, Poland
e-mail: [email protected]
e-mail: [email protected]
Abstract
This paper presents the modification in
Adina software that allows for the declaration of
hyper-elastic materials such as Zahorski material.
The essence of modifications in Adina software is
the process of multi-stage programming, specified
in this paper. The diagrams comparing stressdeformation for rubbers "A", "B" and "C" for
Mooney-Rivlin and Zahorski materials are
generated from ADINA software.
Key words: rubber, Adina, hyperelastic material,
Zahorski.
INTRODUCTION
Searching for physical relationships that
would describe behavior of materials with large
elastic deformations caused by different external
factors started in the 20th century. In the late 30s,
Murnaghan determined a constitutive relation for a
non-linear compressive elastic material [1]. Further,
in the 40s and 50s, first attempts were made to
determine constitutive relations that describe
behavior of rubber and rubberlike materials [2], [3].
In 1951, a general form of the elastic energy was
obtained by Rivlin and Saunders [4]. Another stage
in development of knowledge on hyperelastic
materials was marked in 1959 by Zahorski's
introduction of the function of deformation energy,
which defined incompressible material with nonlinear correlation to invariants of deformation
tensor. This allowed for describing the elastic
behavior of rubberlike material at larger
deformations.
CONSTITUTIVE RELATIONS FOR
HYPERELASTIC MATERIALS
Constitutive equations describe behavior
of a particular material as affected by a number of
external factors. They are also termed physical
relationships for a particular material medium. The
choice of a material model depends on the factors
that have the most essential importance to the
behavior of the medium. For this reason, the
constitutive equations define a subjectively selected
model of material which describes actual behavior
(better or worse) in a particular area of changes in
certain factors. Therefore, it might be assumed that
constitutive
equations
that
describe
the
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relationships between deformations and stresses or
between deformations and energy for hyperelastic
materials are obtained based on the equations of
mechanical energy balance. In the field of the
theory of elasticity and general problems of
mechanics, elastic materials are used with internal
bonds or without bonds. With regard to elastic
bodies without bonds, properties of such medium
are defined if there is opportunity for definition of
the function W, which, for any deformation d of this
medium, defines the corresponding elastic energy
W=W(d) created in the unit of volume with respect
to configuration BR . Therefore, function W
represents a function of deformation energy.
While meeting the principle of objectivity
and limiting to homogeneous isotropic elastic
bodies, constitutive equations can be written as
W  W ( I 1, I 2 , I 3 )
(1)
where I1, I2, I3 are invariants of deformation tensor.
With respect to the elastic body with
imposed internal bonds, one should bear in mind
that, contrary to the elastic body without bonds, it
cannot be subjected to any deformation. The basic
form of internal bonds is incompressibility. Thus
the incompressible body can be subjected only to
deformations that do not affect its volume
(isochoric deformations). A condition for
acceptable deformations must be met
I3  1
(2)
The condition (2) causes that I3 does not
occur as an argument of the deformation energy
function which, for the incompressible elastic body,
is only a function of two other invariants. This can
be re-written in a form which is analogous to the
conditions (1)
W  W ( I 1, I 2 )
(3)
The equations (2) and (3) define the
constitutive relations for incompressible material.
ZAHORSKI MATERIALS
The constitutive relation that describes
Zahorski material is given by
W  W ( I 1, I 2 )  C1 ( I 13)  C2 ( I 2  3) C 3 ( I12  9) (4)
where C1, C2, C3 are material constants which, for
three types of rubber, were given in the study [5].
Elastic energy for the incompressible
isotropic elastic material is linearly dependent on
invariables of deformation tensor. The constitutive
equation proposed allows for a
more
comprehensive, compared to Mooney-Rivlin
material, analysis of the wave phenomena
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propagating in elastic incompressible materials. A
description that suits the behavior of rubber for the
main elongation was obtained even for λ=3,
whereas for neo–Hookean and Mooney-Rivlin
materials, the acceptable results are observed for
λ=1.4.
A non-linear term in the equation
C 3 ( I12  9)
(5)
allows for a more precise analysis and obtaining
another quality elements for description of wave
processes. The relationship (4) models the effects of
a dynamic behavior of materials and is used for
analysis of wave phenomena that concern
propagation of disturbance (see [6]). Furthermore,
analysis described in the study [7] demonstrated
that the constitutive equation that contains nonlinear relationships with respect to invariants of
deformation tensor defines more precisely the
behavior of the rubber at much higher deformations
than in the case of neo-Hookean or Mooney-Rivlin
materials.
OVERVIEV OF FEM SOFTWARE WITH
LIBRARIES OF HYPERELASTIC
MATERIALS
The relationships between functional
conditions or lack of these relationships with
respect to laboratory tests points to the necessity of
numerical simulations. Analysis of non-linear
hyperelastic materials can be carried out using
numerical software based on the Finite Element
Method. FEM is a method of approximated solving
of partial differential equations. All of the
numerical programs based on FEM contain groups
of selected models of materials in their libraries,
including the models of hyperelastic materials. By
selection of one of the models, one can perform a
numerical analysis of the behavior of the
component designed (see [8]). There are many
pieces of software using the popular elastic
potentials.
The most frequent systems used for such
computations are ANSYS, ABAQUS, MARC,
NASTRAN, ALGOR, ADINA. However, this
group of applications do not support the Zahorski
material presented in this study.
Detailed knowledge on behavior of the
rubber modeled as a hyperelastic incompressible
material (material continuum described with elastic
potential) is a factor that stimulates development of
technologies and design of rubber products in the
industry today, where rubber is a necessary
component in many modern technological
solutions. It might represent a supplementation of
the methods used for evaluation of correctness of
technological processes and evaluation of
parameters for new technologies. Numerical
modeling of rubber and rubberlike materials allows
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for determination of detailed relationships between
the laboratory tests and processes of extrusion
moulding, calendering and preparation of mixtures.
ADINA (Automatic Dynamic Incremental
Nonlinear Analysis) is software based on Finite
Element Method, dedicated to analysis of stress in
solids and static, dynamic, flat and spatial
structures. It helps perform a number of numerical
simulations and model both linear and highly nonlinear behaviors of bodies, allowing for substantial
deformations, non-linearity of material and contact
analysis. The system is composed of pre- and postprocessor and the computation modules:

ADINA-IN (pre-processor): a module for
data preparation,

ADINA: a module for analysis of the
structure's strength,

ADINA-F: a module for analysis of
compressible and incompressible flows,

ADINA-T: a module for analysis of
thermal and electrostatic fields and porous
media,

ADINA-PLOT (post-processor): a module
for visualization of computation results.
In the available and popular FEM software
packages such as ABAQUS, ALGOR, ANSYS,
MARC, NASTRAN and ADINA, there are no
standard option for modeling a hyperelastic
material described with Zahorski potential (eq. (4)).
Such calculations are possible at the moment of
using a procedure discussed in this study, which
involves compilation of scripts for ADINA
software. This allows for obtaining of a modified
library, which represents the basis for numerical
modeling.
The software developer described the
guidelines for compilations in a text file
README.txt, whereas the description of the
author's procedure of modification is presented
below.
MODIFICATION OF ADINA LIBRARIES
FOR ZAHORSKI MATERIAL
Obtaining solutions for the problems
concerning wave phenomena using the finite
elements method in hyperelastic material described
by Zahorski potential became possible with an
author's elaboration dedicated for ADINA software.
The software supports standard modeling of
Mooney-Rivlin material, but there is no option for
computation of less popular materials such as
Zahorski material. In order to perform numerical
modeling of distribution of stress generated with
waves of discontinuities, the authors introduced
some modifications in ADINA software.
The main point of interest of the present
publication is a module responsible for modeling of
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elastic potential using the Mooney-Rivlin method.
Due to its similarity to computations for Zahorski
material, this module represents a perfect basis for
further modifications, thus becoming an object
where all the necessary changes were added.
File /usrdlll32/README.txt contains all
the necessary information to modify software
modules provided by the software developer. This
information was the basis for modifications in the
files, which increased the certainty that they were
performed in a manner consistent with software
developer's instructions and would have an
expected effect, without implying the undesirable
effects or the effects inconsistent with actual status
that should be simulated by the application. The
same folder (usrdll32) contains all the source files
where users can make their own modifications.
The sources contain the codes written in
Fortran language. Modifications concern only
selected files which are connected with the module
that computes elastic potential using the MooneyRivlin method. These files are: ovl30u_moon1.f,
ovl30u_moon2.f and ovl40u_moon.f
The essence of modification introduced to
the software is a multi-stage programming
procedure. The first stage involves localization of
source files that contain coded to be modified.
These files were specified above. Then, blocks of
the source code responsible for computations to be
modified should be identified. After analysis of the
source code, one should modify the selected parts
of the code by replacing them with the code which
is a substitute for the primary code. The
precondition to be met is to preserve, in the new
block, an identical set of input data and returning
output data with the identical format. Only this
approach can guarantee compatibility of the new
version of the code with ADINA. The code input
must also meet all the syntax and semantic
requirements defined in Fortran language.
Introduction of changes in source files does not
have any effect on software operation. It is
necessary to compile them into the DLL library file.
required by ADINA developer. For this purpose, it
is
sufficient
to
input
the
instruction
\DF98\bin\dfvars in the command line at the level
of the main folder in ADINA software. Next, one
should open the above mentioned source files in the
compiler.
After making the modifications, files
should be saved and the compilation can be started.
According to the recommendations of the
developer, compilation is made using the
instruction nmake /f Makefile.<usr> with <usr>
replaced by a string suitable for dll file to be
generated. Full list of possible options is available
in README.txt file.
In order to ensure that the modifications
will be effective in the software, one should replace
all the types of dll files. After starting the
compilation instruction, new dll files are created in
the folder containing the source files. The
compilation instruction should be performed for
each type of file separately. All output files should
be moved to the folder /x32 that contains
executable files and temporary dll files. For safety
reasons, it is recommended to save a backup copy
of the primary dll files before the replacement.
After replacement of files, ADINA
software can be started. After all the modifications
are performed properly, the application should start
without errors. Now all the materials using the
function of software that symbolizes the computing
module using Mooney-Rivlin method will be
actually subject to the method of computation for
Zahorski material, with declaration of constants C1,
C2, C3.
STRESS-DEFORMATION FUNCTION FOR
MOONEY–RIVLIN AND ZAHORSKI
MATERIALS
The table 1 presents elastic constants for
rubbers "A", "B" and "C". The values of the
constants were obtained by computing per SI
system units and are based on the study [5].
tab.1 Constants C1, C2, C3 for three kinds of rubber, source: [5]
Constants [Pa]
Rubber „A”
Rubber „B”
Rubber „C”
C1
6.278·104
2.099·104
3.453·104
Compilation is possible only by means of the
Fortran language compiler (version 6.6a or newer
according to the software developer's declaration).
Compilation yields dll files which should replace
the previous files. After this operation, ADINA
operates based on the modified source code.
After installation of the compiler and
before starting operations on the source code, it is
necessary to define environmental variables
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C2
8.829·103
1.275·104
0
C3
6.867·103
3.924·103
2.256·104
Figures 1, 2 and 3 presents differences in
stress-deformation function between the MooneyRivlin and Zahorski materials for rubbers "A", "B"
and "C" studied (see table 1). The diagrams for
Zahorski material are generated from ADINA
software, after taking into consideration the author's
modifications introduced into material libraries.
rubber „A” Mooney - Rivlin material
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rubber „A” Zahorski material
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rubber „C” Zahorski material
Fig. 1 Stress-deformation diagram for rubber „A”
Fig. 3 Stress-deformation diagram for rubber „C”
rubber „B” Mooney - Rivlin material
CONCLUSION
rubber „B” Zahorski material
The study discussed and presented a
hyperelastic incompressible material described by
Zahorski potential. Mooney-Rivlin material has two
constants C1 and C2, linearly dependent on
invariables of deformation tensor I1 and I2, whereas
Zahorski material has three constants, of which C1
and C2 are linearly dependent on the deformation
tensor (such as Mooney-Rivlin material), while the
constant C3 depends non-linearly on the
deformation tensor invariable I1. This is the reason
of differences in stress-deformation function
between the Mooney-Rivlin and Zahorski materials
for rubbers "A", "B" and "C" (Fig.1). The process
of designing and implementation of rubber products
using Zahorski material and using modifications
discussed in this study also has an economic
significance involving less use of rubber materials
for products used in various industries.
References
Fig. 2 Stress-deformation diagram for rubber „B”
rubber „C” Mooney - Rivlin material
[1] Murnaghan, F.D.: Finite Deformations of an
Elastic Solid, Amer. J. Math., 59, 1937. 140 s.
[2] Mooney, M.: A theory of large deformations, J.
Appl. Phys. 11, 1940. s.582-592.
[3] Rivlin, R.S.: Large elastic deformations of
isotropic materials, I Fundamental concepts.,
Phil. Trans. Roy. Soc. Lond. A 240, 1948.
s.459-490.
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[4] Rivlin, R.S., Saunders, D.W.: Large elastic
deformations of isotropic materials, VII
Experiments of the deformation of rubber, Phil.
Trans. Roy. Soc. Lond. 243, 1951. s.251-288.
[5] Zahorski, S.: Doświadczalne badania niektórych
własności mechanicznych gumy, Rozprawy
inżynierskie, tom 10 (1), 1962
[6] Major, M.: Velocity of Acceleration Wave
Propagating in Hyperelastic Zahorski and
Mooney – Rivlin Materials, J. Theoret. Appl.
Mech.Vol.43 nr 4, s.777-787. 2005. ISSN:
1429-2955
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[7] Boyce, M.C., Arruda, E.M.: Constitutive
models of rubber elasticity: A review, Rubber
Chem. Technol., 73, 2000. s.504-523.
[8] Major, M., Major, I,. Kurzak, L.: Velocity of
Acceleration Wave Propagating in Hyperelastic
Materials, Aktual'nye Problemy Architektury i
Stroitel'stva. Sankt-Peterburg : TU 2013. s.342347. ISBN: 978-5-9227-0423-6