MODIFICATION IN ADINA SOFTWARE FOR ZAHORSKI MATERIAL

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1 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 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 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 B R. 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, I, ) (1) ( 1 2 I3 where I 1, I 2, I 3 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 I 1 (2) 3 The condition (2) causes that I 3 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 2 W W ( I, I ) C ( I 3) C ( I 3) C ( I 9) (4) where C 1, C 2, C 3 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 38

2 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 ( 2 3 I1 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 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 39

3 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 Mooney- Rivlin 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 C 1, C 2, C 3. 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 C 1, C 2, C 3 for three kinds of rubber, source: [5] Constants [Pa] C 1 C 2 C 3 Rubber A Rubber B Rubber C 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 Figures 1, 2 and 3 presents differences in stress-deformation function between the Mooney- Rivlin 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 40

4 rubber A Zahorski material rubber C Zahorski material Fig. 1 Stress-deformation diagram for rubber A rubber B Mooney - Rivlin material rubber B Zahorski material Fig. 2 Stress-deformation diagram for rubber B rubber C Mooney - Rivlin material Fig. 3 Stress-deformation diagram for rubber C CONCLUSION The study discussed and presented a hyperelastic incompressible material described by Zahorski potential. Mooney-Rivlin material has two constants C 1 and C 2, linearly dependent on invariables of deformation tensor I 1 and I 2, whereas Zahorski material has three constants, of which C 1 and C 2 are linearly dependent on the deformation tensor (such as Mooney-Rivlin material), while the constant C 3 depends non-linearly on the deformation tensor invariable I 1. 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 [1] Murnaghan, F.D.: Finite Deformations of an Elastic Solid, Amer. J. Math., 59, s. [2] Mooney, M.: A theory of large deformations, J. Appl. Phys. 11, s [3] Rivlin, R.S.: Large elastic deformations of isotropic materials, I Fundamental concepts., Phil. Trans. Roy. Soc. Lond. A 240, s

5 [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, s [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 ISSN: [7] Boyce, M.C., Arruda, E.M.: Constitutive models of rubber elasticity: A review, Rubber Chem. Technol., 73, s [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 s ISBN: