MODELLING OF FUNCTIONALLY GRADED MATERIALS USING …

Journal of Engineering Science and Technology Vol. 15, No. 3 (2020) 1719 - 1730 © School of Engineering, Taylor’s University

1719

MODELLING OF FUNCTIONALLY GRADED MATERIALS USING THERMAL LOADS

ASSETBEK ASHIRBEKOB1, ANUAR ABILGAZIYEV1, SYUHEI KUROKAWA2, MD. HAZRAT ALI1,*

1Department of Mechanical Eng., School of Engineering and Digital Sciences, Nazarbayev University, Kazakhstan

2Department of Mechanical Eng., Kyushu University, Japan *Corresponding Author: [email protected]

Abstract

Functionally Graded Materials (FGMs) are used in specialized industries, notably, in the aerospace industry, due to their unique response to thermal loads which are very prominent during spacecraft flights. The use of FGMs in the rapidly growing 3D printing area is also increasing interest in them, and proper modelling of FGMs is becoming a prominent development topic. This paper suggests a method of modelling FGMs using the ANSYS Workbench environment for dummy thermal loads in ANSYS Mechanical, which excludes the need for extensive FORTRAN programming typical of a significant part of FGM modelling methods. The method can be applied for complex geometries, imported from CAD software, works with shell elements bodies, and any distribution of materials throughout the FGM. Two case studies: one for 2D analysis and another for 3D shell elements analysis are developed. Two solutions are developed for each case: numerical solution, obtainable through the proposed method, and analytical solution. Solutions are compared, and the results yielded by the proposed method matched the analytical solution with high accuracy.

Keywords: ANSYS, Functionally graded materials, Modelling, Thermal loads.

1720 A. Ashirbekov et al.

Journal of Engineering Science and Technology June 2020, Vol. 15(3)

1. Introduction Functionally Graded Materials (FGMs) in the scope of the paper are a special type of composite materials, where the boundary between two material phases is not a clear line but a gradient, from one phase to the other. They have numerous advantages compared to conventional composites, including the absence of stress concentration areas in the boundary of phases, leading to higher bond strength, more smooth wear, dynamic response, and higher tolerance to the difference in thermal expansion modes of the composite components [1]. Due to these benefits, FGMs are used in specialized industries, notably, in the aerospace sector, as they have a unique response to significant thermal loads, which are very common during spacecraft flights. FGMs are also used in applications in 3D printing, with numerous studies dedicated to analysing the impact of print parameters on the performance of an FGM [2, 3]. In 3D printing, the main limitation of FGMs - complex manufacturing process [4], is mostly mitigated, making it suitable media for FGMs technology development. An extensive state-of-the-art review of 3D printing methods suggests additive manufacturing as a fitting means of FGM production [5]. A recent study [6] developed technology for manufacturing highly complex FGM structures using 3D printing with high precision, showing the topic is relevant. Modelling of FGMs is an extensive topic with multiple models existing. One of the later reviews on the subject categorizes them and suggests three theoretical formulations and seven models developed under them [7]. While those models are powerful and are used for complex formulations, for appropriate nonlinear and post-buckling responses, they require complex setups for specific simulation creation and are not suited for general design purposes.

Practical modelling of FGMs for design purposes is complicated, as materials vary throughout the boundary gradient smoothly, and some widely used modelling software solutions, notably, ANSYS, does not support such material boundaries assignment. Existing methods to overcome this include extensive scripting/programming in software unique syntax, making the learning curve for modelling steep. A more straightforward approach is to model the FGM completely with their microstructure, but this is undesirable as the microstructure study has to be conducted, and so involves high computational resources. In the spatially varying material properties method, the FGM is simulated by setting the properties of the material as a function of coordinates, i.e., 𝐸𝐸 = 𝑓𝑓(𝑥𝑥, 𝑦𝑦, 𝑧𝑧), so allowing for an exact FGM definition. However, the only implementations have been done using ABAQUS and not ANSYS [8], which, while a very powerful tool for modelling, is not intuitive and straightforward for non-advanced users. Other methods rely on the discretising of the FGM’s continuum into layers with small steps in property change [9]. Most of the continuum method requires computer program coding for each assignment. Hassan and Keleş [10] developed a method utilizing ANSYS APDL (ANSYS Parametric Design Language), which is, although powerful, faces the same limitation of a steep learning curve. It does not support geometry import from CAD software. APDL has limited CAD capabilities of its own and usually requires more effort to develop a model.

The paper introduces a method of modelling the FGM in the ANSYS Workbench environment based on the above-mentioned dummy thermal loads ANSYS APDL method. The proposed method does not require code writing. Compared to ANSYS APDL, ANSYS Workbench is more accessible for new users. It allows import from CAD, so supporting complex geometries; it can work

Modelling of Functionally Graded Materials Using Thermal Loads . . . . 1721


with thermal loads on shell elements, which is not possible in APDL due to lack of thermomechanical shell element; and, it can work with any distribution of materials and their boundaries throughout the FGM. The method is based on obtaining dimension-dependent properties of the FGM, not supported in ANSYS, by defining properties as dependent on the thermal condition (temperature) as proposed by Hassan and Keleş [10]. Subsequent inducing dummy thermal loads corresponding to the FGM distribution allows for effective FGMs modelling using a single material definition.

2. Analytical Conditions ANSYS supports material property variation dependent on temperature, and that is used to recreate property variation based on dimensions. Similar to the APDL method, the main steps can be generalised as:

Defining material properties as a function of temperature. Two arbitrary temperature points are chosen, e.g., 0°C and 99°C. The first temperature point is given properties of material A of the FGM, and the second is given properties of material B. Then, temperature points between (from 0°C to 99°C) are treated as a percentage of dimensional variation, following any distribution rule of interest. The amount of temperature points should consider the size of the mesh elements. Thermal expansion and stresses are set to zero, as no actual thermal behaviour is studied.

The model, with geometry, mesh, initial, and boundary conditions, is developed in a Static structural section of ANSYS. Solid or shell elements are supported, for both 2D and 3D analysis.

Dummy thermal conditions are applied according to the FGM distribution in the model. Any distribution can be used. For simpler cases, the Static structure’s internal “Thermal condition” tool is used, and for more complex cases, the more flexible Static thermal tool can be coupled. Results are obtained, and no solution or post-solution changes are necessary.

3. Modelling Using ANSYS To test the simulation method, two case studies were conducted. The first one used the same model with the same geometry, boundary conditions, and material definition as proposed by Hassan and Keleş [10], with their analytical solution shown in Table 1. The case study is a static mechanical analysis of an FGM long cylinder, long enough to be considered a plane strain problem. As the model is radially symmetrical, both geometrically and load wise, the only a quarter sector needs to be considered. Boundary conditions set were frictionless support at the radii of the sector and pressure at the tangential side, as shown in Fig. 1. The second case is a shell extruded ring, to show the application of the method both in 3D and working with surface shell elements. Both analytical results, which are shown in Table 2 and calculated by using Eq. (3), and numerical results, obtained with the method proposed and shown in Table 3, were obtained and compared. Boundary conditions are fixed support at one end and force at the other end of the cylinder, as shown in Fig. 2. The material definition is the same, as for the first case.



Table 1. Analytical results for 2D.

Table 2. Analytical results for 3D.

Table 3. Comparison between the simulation and analytical results for 3D.

Fig. 1. Geometry and boundary conditions of the first case.



Fig. 2. Geometry and boundary conditions of the second case.

For the material definition, the elasticity modulus was chosen as the variable property of the FGM, and the Poisson’s ratio was considered constant. The exponential distribution model was chosen, as given in Eq. (1).

𝐸𝐸(𝑟𝑟) = 𝐸𝐸0𝑒𝑒−𝑛𝑛𝑅𝑅𝜂𝜂 ,𝑛𝑛 = ln 𝐸𝐸0

𝐸𝐸𝑜𝑜𝑜𝑜𝑜𝑜,𝑅𝑅 = 𝑟𝑟

𝑏𝑏 (1)

Values to be evaluated in the 2D case were radial displacement and the stresses, radial, and hoop (tangential). Radial stress is defined as the stress in the radial direction of the cylinder, and hoop stress is tangential to the radial direction. To compare with the analytical solution, values were non-dimensionalised, as shown in Eq. (2).

𝑈𝑈 = 𝑢𝑢𝐸𝐸0𝑏𝑏𝑃𝑃0

,𝜎𝜎′ = 𝜎𝜎𝑃𝑃0

(2)

In the 3D case, the strain is evaluated, with formulae for normalization given in Eq. (3).

𝑒𝑒(𝑙𝑙) = 𝜎𝜎𝐸𝐸(𝑙𝑙)

,𝜎𝜎 = 𝐹𝐹𝐴𝐴

,𝐴𝐴 = 𝜋𝜋(𝑅𝑅2 − 𝑟𝑟2) (3)

The dimensional values used for both 2D and 3D cases are presented in Table 4. Table 4. Parameter selection.

Variable Value used b 200 mm E0 120 GPa P0 100 GPa n [−1,0,1] ν 0.3 η 0.9 F 1000 N Rf 25.5 mm r 24.5 mm L 200



The method applied to the specific case The step-by-step application is shown for the 2D case, as basic steps are the same for any analysis type. Elasticity modulus was calculated in an Excel table using Eq. (1) and parameters’ values from Table 4, for values of R from 0 to 1 with a step of 0.01, resulting in 99 values. Values obtained were copied to ANSYS Workbench Engineering Data as isotropic elasticity property and assigned for values of temperature from 0 to 99, substituting R, shown in Table 5 for the first thirteen points for n=-1. Three materials were defined, one for each of n value: -1, 0, and 1.

Table 5. Values for the initial thirteen points defined in Engineering Data. Temperature (oC) Young's Modulus (Pa) Poisson's Ratio Bulk Modulus (Pa)

0 1.20×1011 0.3 1.00×1011 1 1.20×1011 0.3 1.00×1011 2 1.20×1011 0.3 1.00×1011 3 1.20×1011 0.3 1.00×1011 4 1.20×1011 0.3 1.00×1011 5 1.20×1011 0.3 1.00×1011 6 1.20×1011 0.3 1.00×1011 7 1.20×1011 0.3 1.00×1011 8 1.20×1011 0.3 1.00×1011 9 1.20×1011 0.3 1.00×1011

10 1.20×1011 0.3 1.00×1011 11 1.20×1011 0.3 1.00×1011 12 1.20×1011 0.3 1.00×1011

This resulted in the properties distribution shown in Fig. 3. The material with the n=-1 distribution is essentially a plain, non-composite material, but is shown as well as it can be regarded as a special case of an FGM and method should apply anyway.

Fig. 3. Young’s modulus vs temperature for

n=1 (top left), n=-1 (top right) and n=0 (bottom).



Fig. 4. Dummy thermal loads applied. Pure material A is at the centre,

pure material B at outer circumference, and smooth transition in-between.

Next, the model is defined, with imported geometry, a generated mesh, and boundary loads applied. The geometry was developed using the Workbench build-in SpaceClaim. However, it can also be imported from external CAD software, e.g., SolidWorks or AutoCAD Inventor. Mesh was defined using the quadrilateral dominant method, as shown in the Eq. (3). The element size of 2 mm was chosen, to obtain approximately 100 mesh elements along the radial direction, complementing the 100 temperature points defined, and applied in the following steps. Formal loads and supports were applied, as shown in Fig. 1, and then dummy thermal loads were applied to create the radial distribution of FGM throughout the model, as shown in Fig. 4.

4. Results and Discussion For the 2D case, the analytical solution is referenced from work conducted by Hassan and Keleş [10], and presented in Table 1. Table 6 contains results provided by the method proposed with the percentage error relative to the analytical solution. Figures 5-7 are the normalized displacement, radial stress and hoop stress, plotted versus rational radius R, for both numerical model and analytical solution. As can be seen from Figs. 5-7, the analytical and the model solutions converge for at every radii point and for every n value. This validates the method and its appropriability for FGM modelling. Percentage errors are small, with the values of 0.096177 and - 0.007106 and with the maximum value of 0.388% for displacement obtained, which results in high accuracy, acceptable for most design purposes. Similarly, for the 3D case, the strain was plotted versus rational length for both analytical and modelled solutions in Fig. 8, with Tables 2 and 3 containing analytical and modelled solutions. Stresses were not evaluated, as for cylinder tension, only single directional stress is present – along the axis of length, with uniform distribution.



Again, the percentage error of the modelled solution to analytical is small, with a maximum value of 0.0145% obtained. Based on several trials, the obtained error results were sufficiently small.

Table 6. Comparison between the simulation and analytical results for 2D.

Fig. 4. Radial stress versus radius.



Fig. 5. Radial stress versus radius.

Fig. 6. Hoop stress versus radius.

Fig. 7. Strain versus length.



Although different values are compared for the 2D and 3D cases, all show high accuracy. Tables 1 and 6 present values of normalized displacement and stresses, with modelled values in the case of Table 3, for each rational radius value with a stop of 0.1, as shown in Fig. 8. Models were evaluated for all three n values. Tables 2 and 3 do the same, but with displacement values for the 3D case. Rational length is used for the 3D case, but with the same ten points’ distribution.

Fig. 8. Ten points at which displacement and stresses were evaluated.

In Table 1, analytical solutions predict a value of −1 for normalized stresses for “n = 0” in the 2D case. Table 6 shows that the model predicts it with 100% accuracy, and for stresses, the error does not exceed 0.1%. While normalized displacement for “n = 0” is n-th order of ten thousandths, other cases show a larger error, for nonzero n cases, 0.388% and 0.362% for “n = -1” and “n = 1”, respectively. Those are the largest of evaluated error, but they are still are small for modelling method. In Table 2, analytical solution constant value for all ten points for displacement for “n = 0” in 3D case. Table 3 shows that the model also predicts constant value with 0.006% accuracy, which is small, and even the largest error of 0.014% is negligible, rendering almost exact accuracy.

Both cases represent the simplest load types: tension and compression, and are accurate. However, due to the nature of FGM assignment: as smoothly varying properties along with the same continuous matter, the solution should hold up for more complex cases as well.

5. Conclusions In this paper, the method of FGM modelling was proposed of two different cases based on the existing dummy thermal loads methods, however, with benefits of using a more user-friendly Workbench environment, with the convenience of no manual coding, availability of external CAD importing, more functional Workbench UI and shell elements modelling.



FGMs are applied only in very specialized spheres, for instance, in the aerospace industries. FGMs own unique response features to considerable thermal loads that are very common during aerospace flights. FGMs are also found in smart materials and 3D printing studies.

Compared to the analytical solution in the case study of the mechanical static plane stress model, the proposed method was proved to be accurate down to 0.388%. In the second case, the 3D shell elements study has shown accuracy down to 0.0145% of strain. This proposed method’s result is extremely close to the analytical solutions, so that it can be used for the design of complex geometries with FGMs using CAD software and also gives several advantages.

First, the proposed approach is user-friendly and straightforward. Since FGM is an extraordinarily complicated material, varying throughout the boundary gradient smoothly, it is not supported in ANSYS or other similar products. A simulation by computing at the microstructure level is also inconvenient as it requires high computational resources. Second, complex geometries can be simulated. Other methods are not compatible with 3D CAD files and cannot compute any geometries. The last, it does not require computer coding, whereas other methods need new computer coding for every assignment. Thus, this proposed method is more convenient and more accessible for researchers.

The only limitation of the proposed approach, compared to other discreet analysis methods, is the inability of the thermal and thermomechanical experimental study, as the dummies occupy thermal loads, and preparation of the material is not affordable. Future studies are suggested to work this around, as the thermal behaviour of FGM is prominent.

Acknowledgements

The authors sincerely express their gratitude to Nazarbayev University for this research grant. The award number is 090118FD5327. With the financial support, the research activities are carried out successfully.

Nomenclatures b The radius of the cylinder E0 Elasticity modulus at the cylinder centre Eout Elasticity modulus at the cylinder outer surface E(r) Elasticity modulus at radius r n (also ɳ) FGM distribution parameters P0 Pressure applied R Rational radius r Radius, distance from sector origin U Radial displacement normalized u Radial displacement obtained

Greek Symbols ɳ (also n) FGM distribution parameters σ Stress obtained, applicable for both radial and hoop σ' Stress normalized, applicable for both radial and hoop



References 1. Udupa, G.; Rao, S.; and Gangadharan K. (2014). Functionally graded

composite materials: an overview. Procedia Materials Science, 5, 1291-1299. 2. Zhang, B.; Jaiswal, P.; Rai, R.; and Nelaturi, S. (2018). Additive

manufacturing of functionally graded material objects: a review. Journal of Computing and Information Science in Engineering, 18(4).

3. Li, Q.; and Xu, X. (2015). Self-adaptive slicing algorithm for 3D printing of FGM components. Materials Research Innovations, 19(5).

4. Naebe, M.; and Shirvanimoghaddam K. (2016). Functionally graded materials: a review of fabrication and properties. Applied Materials Today, 5, 223-245.

5. Ngo, T.; Kashani, A.; Imbalzano, G.; Nguyen, K.; and Hui, D. (2018). Additive manufacturing (3D printing): A review of materials, methods, applications and challenges. Composites Part B: Engineering, 143, 172-196.

6. Kuang, X.; Wu, J.; Chen, K.; Zhao, Z.; Ding, Z.; Hu, F.; Fang, D. and Qi, J. (2019). Grayscale digital light processing 3D printing for highly functionally graded materials. Science Advances, 5(5).

7. Thai, H.; and Kim, S. (2015). A review of theories for the modeling and analysis of functionally graded plates and shells. Composite Structures, 128, 70-86.

8. Gangwar, S.; Patnaik, A.; Yadav, P.; Sahu, S.; and Bhat I. (2019). Development and properties evaluation of marble dust reinforced ZA-27 alloy composites for ball bearing application. Materials Research Express, 6(7).

9. Watanabe, Y.; Inaguma, Y.; Sato, H.; and Miura-Fujiwara E. (2009). A novel fabrication method for functionally graded materials under centrifugal force: the centrifugal mixed-powder method. Materials, 2(4), 2510-2525.

10. Hassan, A.; and Keleş, I. (2017). FGM modelling using dummy thermal loads applied with ANSYS APDL. Journal of Selcuk International Science and Technology, 1(10-16).

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