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POLITECNICO DI MILANO (BOVISA CAMPUS)
School of Industrial and Information Engineering Master of Science in Mechanical Engineering
DESIGN & ANALYSIS OF DIFFERENT HOLE SHAPES IN 2D
PERIODIC STRUCTURES
Supervisor: Prof. MARIO GUAGLIANO
Co-Supervisor: Dr. SARA BAGHERIFARD
Author: GOPI SUBRAMANIAN SANTHOSH KUMAR (853323)
2 Table of contents Table of Contents ………. 2 List of figures ……….. 4 List of Graphs ……….. 7 Abstract ………... 8 1. Introduction ………... 9
1.1 Idea behind the study ………. 9
1.2 State of the art ………. 10
1.3 Specific considerations ……….. 15
1.4 Manufacturing Technology ………. 16
1.4.1 Laser Drilling ………. 16
1.4.2 Advantages of laser drilling process ………. 17
2. Model Validation and Considerations ………. 18
2.1 Auxetic property ………. 21
2.2 Periodic boundary condition ……….. 23
2.2.1 Constraint equations ……….. 23
2.2.2 Dummy Node ……… 23
2.2.3 Definition of equation ………... 24
2.2.4 Input file for a 2D structure with a circle ……….. 25
2.3 Validation of S Shape ……….. 28
2.4 Validation of Ellipse shape ……….. 32
2.5 Calculation of stress intensity factor ……… 36
3. Study and presentation of Optimized shapes ………. 37
3.1 Material Properties ……….. 37
3.2 Load Conditions ……….. 37
3.3 Software for analysis ……….. 37
3.4 Effects of different shapes and arrangements ………. 38
3.4.1 Different shapes under consideration ……….. 38
3.5 Analysis and Results ……… 39
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3.5.1.1 Shapes arranged in horizontal configuration ……… 39 3.5.1.2 Shapes arranged in vertical configuration ……….. 44 3.5.1.3 Shapes arranged in alternate configuration ……….. 49 3.5.2 Shapes arranged in alternate pattern without maintaining porosity ………….. 54 4. Conclusions ………. 61 5. References ……….. 63
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List of figures
Figure 1.1 showing circular shape (Left) & S shape (Right) ……….. 11 Figure 1.2 showing U1 & U2 for circle and ellipse shape [2] ……… 13 Figure 1.3 (a,b,c) showing different arrangements of diamond shapes and figure (d) showing the star shaped arrangement ……… 14 Figure 1.4 showing the minimum cutting distance and minimum distance between two elements ……… 16 Figure 1.5 shows the process of laser drilling ………. 17 Figure 2.1 showing structure with circular hole 1) Journal (Left)
& 2) Validated Result (Right) ……….. 18 Figure 2.2 showing structure with elliptical hole 1) Journal (Left)
& 2) Validated Result (Right) ………. 18 Figure 2.3 showing structure with circular hole 50X50mm 1) Journal (Left)
& 2) Validated Result (Right) ………. 19 Figure 2.4showing structure with elliptical hole 50X50mm 1) Journal (Left)
& 2) Validated Result (Right) ………. 19 Figure 2.5 showing Journal (Top) Vs Validated results (Bottom). Effect of the hole aspect ratio a/b on the deformation of the structure. Distribution of Von-Mises (Normalized by the bulk material elastic modulus E) for elliptical holes with different values of aspect ratio subjected to monotonic uniaxial tensile loading
(in vertical direction) at strain = 1*10-3 ………. 21 Figure 2.6 shows arrangement of shapes leading to Negative Poisson’s ratio ………. 22 Figure 2.7 showing Definition of * equation ………. 24 Figure 2.8 showing Von-Mises (Left) and Von-Mises within limits 0 to 205MPa (Right) for S shape in the horizontal configuration ……… 29 Figure 2.9 showing displacement U1 & U2 for S shape in horizontal configuration ………. 29 Figure 2.10 showing Von-Mises (Left) and Von-Mises within limits 0 to 205MPa (Right) for S shape in the vertical configuration ……….. 30
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Figure 2.11 showing displacement U1 & U2 for S Shape in vertical configuration ………….. 30 Figure 2.12 showing Von-Mises (Left) and Von-Mises within limits 0 to 205MPa (Right) for S shape in the alternate configuration ………. 31 Figure 2.13 showing displacement U1 & U2 for S Shape in vertical configuration …... 31 Figure 2.14 showing Von-Mises (Left) and Von-Mises within limits 0 to 205MPa (Right) for ellipse shape in the horizontal configuration ……….. 32 Figure 2.15 showing displacement U1 & U2 for ellipse shape in
horizontal configuration ………. 32 Figure 2.16 showing Von-Mises (Left) and Von-Mises within limits 0 to 205MPa (Right) for ellipse shape in the vertical configuration ………... 33 Figure 2.17 showing displacement U1 & U2 for ellipse shape in
vertical configuration ……… 33 Figure 2.18 showing Von-Mises for Ellipse shape in alternate configuration ……… 34 Figure 2.19 showing displacements U1 & U2 for ellipse shape in
alternate configuration ……… 34 Figure 2.20 showing Von-Mises (Limits 0 To 205MPa) for ellipse shape in alternate configuration ……….. 35 Figure 2.21 showing normalized stress for Journal (Top) vs Validated result (Bottom) ……. 35 Figure 2.22 showing Plain Sheet Vs Plate with a circle ……….. 36 Figure 3.1 shows the different shapes under study ………. 38 Figure 3.2 showing Von Mises data for all the shapes under consideration in horizontal configuration ……….. 40 Figure 3.3 showing Von Mises (Limit 0 to 205MPa) data for all the shapes under consideration in horizontal configuration ……… 41
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Figure 3.4 showing minimum cutting distance and minimum distance between two elements ……… 42 Figure 3.5 showing displacements U1 & U2 for all the shapes under consideration in horizontal direction ………. 43 Figure 3.6 showing Von Mises data for all the shapes under consideration in vertical configuration ……….. 45 Figure 3.7 showing Von Mises (Limit 0 to 205MPa) data for all the shapes under consideration in vertical configuration ………... 46 Figure 3.8 showing displacements U1 & U2 for all the shapes under consideration in vertical configuration ……….. 47 Figure 3.9 showing Von-Mises stress for all the shapes in alternate configuration …………. 50 Figure 3.10 showing Von-Mises stress within limits 0 to 205MPa for all the shapes in alternate configuration ……….. 51 Figure 3.11 Showing Displacements U1 & U2 for all the shapes
in alternate configuration ……….. 51 Figure 3.12 showing Von-Mises for all the shapes in alternate configuration (Porosity not maintained) ………. 55 Figure 3.13 showing Von-Mises within Limits 0 to 205MPa for all the shapes in alternate configuration (Porosity not maintained) ……… 56 Figure 3.14 showing displacements U1 & U2 for all the shapes in alternate configuration (Porosity not maintained) ………. 57
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List of Graphs
Graph 3.1 showing the values of Smax, Kt, Poisson’s, Yield area (%) for all the shapes in Horizontal configuration ………. 43 Graph 3.2 showing the values of Smax, Kt, Poisson’s, Yield area (%) for all the shapes in Vertical configuration ……….. 48 Graph 3.3 showing the values of Smax, Kt, Poisson’s, Yield area (%) for all the shapes in alternate configuration ……….. 52 Graph 3.4 comparing the values of Smax, Kt, Poisson’s, Yield area (%) for ellipse shape, C shape modified, Double C shape and S shape and for all the configurations ……….. 53 Graph 3.5 showing the values of Smax, Kt, Poisson’s, Yield area (%) for all the shapes in alternate configuration (Porosity not maintained) ………. 58 Graph 3.6 comparing the values of Smax, Kt, Poisson’s, Yield area (%) for ellipse shape, Double C shape and S shape and for all the configurations ……… 59
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ABSTRACT
The main objective of our study is to analyse the distribution of stresses affected by the presence of different shaped holes in a 2D periodic structure. The perforated sheets or sheets with holes are used in a variety of engineering applications. In most of the applications the sheets are subjected to loads and these loads tend to induce stress concentration factor on the hole edges and so the material gets failed in less number of cycles compared to a not perforated ones. However, in some applications the presence of holes is inevitable, and thus a major study is required to analyze their effect of stress distribution. It has been observed that when we induce different shapes other than a circular one we can reduce the stress concentration factor; thus the main objective of our study is to identify the most promising hole shapes in this regard. In the current study we have considered a 2D periodic structure subjected to a uniaxial tension. We have also maintained porosity of the material as 5%, to minimize the number of variables. In this regard we have considered three parameters 1) Maximum stress and stress concentration factor, 2) Yielded area (%), 3) Poisson’s ratio. We have introduced different hole shapes in three different configurations. We have analysed the static behaviour of the structure and calculated the stress concentration factor for each case. Apart from this we have also considered auxetic property, which is the negative Poisson’s ratio exhibited by the material upon inducement of different shapes for the holes. So bringing auxetic property with such low value of porosity is achieved in some shapes when placed in alternate manner. These configurations have also exhibited low value of stress. As a trade-off among the parameters discussed above Double C shape and S shape perforations could be chosen as optimized shapes for having low value of stress and good auxetic property. In the second part of our study we have relaxed the porosity constraint and placed the elements as close as 0.2mm; even here the minimum cutting distance of 0.1mm is considered, as laser manufacturing technique implies. All the elements are placed in alternate manner. We have also calculated the porosity for each shape. When calculating porosity we could have a direct relationship among different shapes, since we have maintained same number of hole elements for each shape. The results show that the Double C shape has quite low value of stress and exhibits good auxetic property. Thus the optimized shapes can be chosen as per the configuration and the final application.
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1. INTRODUCTION
1.1 IDEA BEHIND THE STUDY
In many engineering applications the usage of perforated sheets is of huge necessity. Perforated materials are used in several applications for a variety of reasons. The different applications include automotive, acoustics, manufacturing etc. For example they are used as liners, casings, screens, filter which can control passage of air, liquid, heat, electromagnetic waves and so on. In many of its application these perforated sheets are subjected to stress which may be due to cyclic load, temperature or combination of both. When the structures are induced with perforations the elements are prone to fail earlier than its life due to the increment of stress concentration factor in the edges of the pore elements. But the pores are unavoidable as many applications need them and so there are many studies performed regarding the optimization of pores. One of the methods will be shape optimization and this is the base of our current study. In case of shape optimization we introduced different shaped pores to the structure under consideration and study the effect of stress concentration. In parallel there are studies performed to analyse the auxetic property of the material. The auxetic property increase plain strain fracture toughness and shear stiffness in a given structure. The auxetic property is analysed by calculating the Poisson’s ratio for each shaped structure. In this study we have considered 2D structures with periodic boundary condition implemented and analysed the stress concentration and auxetic property for different shaped pores. We have another constraint, which is the porosity of the material. In the first part of our study we have analysed the results of all the shapes with constant porosity (5%), the percentage of porosity is taken from the previous work [1]. We have maintained the same porosity in our study since we needed to compare the results of shapes from the journal [1], [2] and the shapes introduced in our current study. In the second part of our study we have relaxed this constraint and analysed the effects of different shapes. Since the porosity is quiet low, we must choose shapes with minimum area, thanks to laser drilling which helped us to choose a minimum cutting distance of 0.1mm and minimum distance between two elements as 0.2mm. The design of 2D systems capable of retaining a negative Poisson’s ratio with very low porosity is very difficult. In the first case it is a real challenge to bring in the desired properties with such low porosity. So we need to try many shape optimizations. We have tried two different configurations for each shape and analysed all the parameters under consideration. From the graphs we could infer that some shapes gave results close to the
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desired values and so we have continued our study by trying alternate configurations to analyse the change in the properties using those features. With this approach we were able to choose optimized shapes as a trade-off among different parameters. In the second part of our study we have extended this analysis by placing them very close to each other. Even in this case a minimum cutting distance of 0.1mm and minimum distance between two elements of 0.2mm are considered. The study showed some interesting changes in the stress values and the optimized shape is chosen as a trade-off between among different parameters.
1.2 STATE OF THE ART
The main idea behind the research work is from the study of the porous structures in 2D structures and its usage in different engineering applications. With this objective we have made a literature review of the study which is concurrent to the present topic and we found a journal [1] where they have worked on the shapes for a specific application. In that journal they have considered the perforated sheets used as combustion liners, ducts, casings and sealing surfaces and they are used as cooling holes for the passage of cold air to the turbine blades. Gas turbines are used in a variety of power needs from driving tanks, jets, helicopters, to power generation and industrial power uses. They operate at temperatures that exceed the melting points of the materials, so there is a need to install cooling functionality to the system. Film cooling is an efficient method to reduce the component temperatures and is currently used in many aircraft turbines and many power generation turbine engines. These cooling holes are highly prone to the fatigue failure due to the stress induced by temperature variations. Several studies have considered the shape optimization of the cooling holes, but however most of them focussed on the cooling functionality. The cooling holes used in these types of structures were typically circular holes. But the usage of circular holes had limitations with increased stress concentration factors on the edges of the holes and this lead to the failure of cooling holes before 100k cycles. So, in their study they have replaced circular holes with S-shaped holes and analysed the results. In case of S-shaped holes there was a remarkable change in the deformation mechanism and the way crack propagates. From their analysis they have determined that the S shapes experience low value of stress since the deformation mechanism is found to induce rotation of domains between neighbouring holes. So the cracks that are initiated at the stress concentrations along the s shaped pores get trapped in low stress regions and propagate at a significantly lower rate.
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In their study they have considered two stainless steel dog bone samples patterned with 5 X 5 square array of circular holes and S shaped holes. All the parameters have been explained in chapter 2 where we have validated their models. In case of s shaped holes, the deformation mechanism is due to the rotation of the domains unlike in circular shape where the deformation mechanism is stretching. Also, S shapes arranged in alternative manner resulted in negative Poisson’s ratio i.e the material when stretched becomes thicker perpendicular to the direction of the applied force.
Figure 1.1 showing circular shape (Left) & S shape (Right)
During the review study we had a mention regarding negative Poisson’s ratio. So, to have a deeper study we have referred to journal [2] which dealt with the Low porosity metallic periodic structures with negative Poisson’s ratio. The Poisson’s ratio is the ratio between transverse and axial strain. For isotropic linear material the values of Poisson’s ratio are -1 to 5. The discovery of materials with negative Poisson’s ratio which are called as auxetic materials counterintuitively expand in the transverse direction under tensile load. Major studies of research are being done to find an optimum structure which could produce negative Poisson’s ratio for a trade-off between very low porosity and very high porosity in
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the structure. However, in many applications the target value of porosity is set by the process requirements. The design of 2D systems capable of retaining a negative Poisson’s ratio with very low porosity is a real challenge. Although there were study that diamond or star shaped perforations introduced in 2D sheets can lead to auxetic behaviour the porosity is again a constraint. So, a real study on the shape is required to determine the shapes that could have negative Poisson’s ratio while retaining low porosity. They have chosen a 2D metallic sheet with square array of mutually orthogonal ellipse shape. They have demonstrated that the Poisson’s ratio can be effectively controlled by changing the aspect ratio of voids. For low aspect ratios, the structure is characterized by positive values of Poisson’s ratio. However, as the aspect ratio increases the value of Poisson’s ratio decrease monotonically and becomes negative. In their study they report evaluation of Poisson’s ratio as a function of pore aspect ratio. They have maintained the porosity and modified the shape as a function of aspect ratio. At aspect ratios near 1 the effective Poisson’s ratio is nearly the same as the bulk material. The results clearly show that the aspect ratio a/b of the holes strongly affect the lateral contraction/ expansion of the structure. Significant auxetic behaviour could be produced in metals with low porosity, provided the void aspect ratio a/b is large enough. In their study they have considered a circle (a/b = 1) and ellipse with different aspect ratios (with a/b~30). They have calculated the Poisson’s ratio with the displacements obtained from U1 & U2. The results confirm that hole aspect ratio can be effectively used to design structures with low porosity and negative value of Poisson’s ratio. The high aspect ratio ellipses lead to the material characterized by a large negative value of Poisson’s ratio. There is a limitation in this aspect ratios technique because with aspect ratios approaching 30 leads to the edges becoming so pointed and the stress concentration rises abruptly. We need some shapes which could have negative Poisson’s ratio as well as less stress concentration factor. So, a compromise is required and alternate shapes which could reduce this effect is needed. For the circular shape the Poisson’s ratio is still close to the bulk modulus Poisson’s ratio but for the ellipse it is entirely different where the value has approached to -0.65 for an infinite structure. From this study we could take ellipse shape has a reference to calculate the Poisson’s ratio for the shapes we have considered under our study. In the below figure they have shown the displacements U1 & U2 for a circular shape and ellipse shape.
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Figure 1.2 showing U1 & U2 for circle and ellipse shape [2]
As a next part of our study it is necessary to know deeper about the auxetic property in 2D porous structures. In this regard we have referred Journal [3] & [6] which deals with the perforated sheets exhibiting negative Poisson’s ratio and a general review of auxetic structures. Their study has a started with a limitation that the availability of materials which exhibit negative Poisson’s ratio Is very limited because they are expensive or difficult to manufacture on a large scale. Also, there is another limitation that some of the materials exhibit negative Poisson’s ratio only during stretching or some only during compression. So in their study they have tried to analyse auxetic systems which can operate in both tension & compression and can be readily manufactured. They have induced star shape and diamond shape holes in the structure and tried to analyse the Poisson’s ratio. They have tried changing
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the arrangement of the elements and the minimum distance between two elements. When the minimum distance between two elements was 0.1mm they have achieved a Poisson’s ratio of -1. In some arrangements they have also obtained Poisson’s ratio close to 0. In this they conclude that when these shapes are placed in various conformations we can obtain a negative Poisson’s to a value closer to zero. In this study they have not maintained any porosity also they have not discussed regarding the stress concentration the experienced by the holes.
Figure 1.3 (a,b,c) showing different arrangements of diamond shapes and figure (d) showing the star shaped arrangement
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1.3 SPECIFIC CONSIDERATIONS
From the contents discussed above we have had a basic idea on how to proceed with our study. In our study we tried to bring in all the parameters discussed in [1], [2], [3], [6]. From [3] we have an idea that the rotating rigid units with proper arrangements can lead to negative Poisson’s ratio. From [2] we have an idea that the ellipse shape with high aspect ratio induced in 2D structures has a lead to negative Poisson’s ratio owing to low porosity. From [1] it is evident that they have chosen S shape based on low stress concentration and good auxetic properties. Now combining the thoughts from these journals, we have picked three important parameters under consideration 1) Maximum Stress and stress concentration factor, 2) Yield area in %, 3) Poisson’s ratio. Also, we need to optimize these ideas like from [3] we have an idea by proper arrangements and less minimum distance between elements we could achieve negative Poisson’s ratio, but the part to be refined are that they don’t consider porosity and stress concentration which are two main parameters of our concern. From [2] they have introduced that the ellipse shape with high aspect ratio leads to high values of negative Poisson’s ratio, but the part to be refined was the stress concentration factor which is to be reduced by inducing optimized shapes. From [1] they have tried to optimize both the stress concentration factor and negative Poisson’s ratio, but we don’t have much comparison since they have considered only a circle and ellipse as an alternate shape. Also, the application of their study is very much limited to the turbine. These are taken as a reference for our study and should be enhanced. In our study we have introduced different shapes with different configurations and analysed the results with the three parameters mentioned above. We have used 2D structures with periodic boundary conditions implemented and performed all the analyses. We have performed static analysis in every part of our study. We have performed our study in two parts. In the first part we have maintained the porosity for all the configurations and in the second part we have relaxed this constraint and analysed the structures. We have considered a minimum cutting distance of 0.1mm and minimum distance between two elements as 0.2mm. This could be achieved by the advancement in manufacturing technology process. The results are calculated for each shape and each configuration.
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1.4 MANUFACTURING TECHNOLOGY
The manufacturing technology considered in our study is laser drilling. This technology has been chosen based on its capabilities of precise machining. The minimum cut distance considered in our study is 0.1mm and the minimum distance between two elements is considered to be 0.2mm. We have confirmed laser drilling based on the information from [9], [10], [11].
Figure 1.4 showing the minimum cutting distance and minimum distance between two elements
1.4.1 LASER DRILLING
Drilling with pulsed lasers is a thermal processing technique. As a result of the high energy depth, the strongly focused beam melts and evaporates the material. There are two techniques involved in pumping laser in laser drilling. They are vaporization and melting. For vaporization Q-switched ND-YAG and for melting flashtube pumped NG-YAG is used. The melt expulsion arises as a result of the rapid buildup of gas pressure. As an outcome of the resulting steam pressure, particles are flung from the drill hole. Only pulsed lasers achieve the performance and energy depths required for this. Depending on the hole diameter, the material strength, the quality requirements, as well as processing speed, there is a choice of four laser drilling methods 1) Single shot, 2) Percussion, 3) Trepanning, 4) Helical. Laser drilling has made it possible to drill very small, precisely tapered, unusually shaped and blind holes in a variety of objects. The minimum cutting distance achievable by laser drilling is 1 μm and the minimum distance between two elements is 80 μm.
0.2mm
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Figure 1.5 shows the process of laser drilling
1.4.2 ADVANTAGES OF LASER DRILLING PROCESS
• It is a non-contact process, which means that a minimal amount of distortion is caused to the material
• Laser drilling offers a high level of accuracy and control over beam intensity and duration and heat output
• Laser drilling can produce holes with small diameters and high aspect ratios • Multi-tasking is another advantage of laser drilling
Laser drilling is flexible and could be used on a number of materials including various metals, plastic and even diamond
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2 MODEL VALIDATION AND CONSIDERATIONS
Based on the requirement the models from Journal [1] & [2] mentioned in the reference has been validated. The models are validated using finite element package ABAQUS. At first the models from Journal [2] are validated. 3D structure with dimension 300mm X 50mm and thickness of 0.4046mm are modelled. The material is modelled as linearly elastic and perfectly plastic with E = 70000 MPa and Poisson’s ratio 0.35. The yield stress is taken as 275 MPa. The mesh used was C3D10. The applied strain is 0.34%. The loading is done by fixing the translation at one edge and specifying a static displacement at the opposite edge. The remaining boundaries are traction free. In the first sample circular holes with radius a=b=3.154mm.
Figure 2.1 showing structure with circular hole 1) Journal (Left) & 2) Validated Result (Right)
Then the validation was made by changing the ratio as a/b=30 where a=33.27mm and b=1.16mm. In this case the applied strain is 0.07%. All the other procedures are maintained the same and the results are validated with the journal
Figure 2.2 showing structure with elliptical hole 1) Journal (Left) & 2) Validated Result (Right)
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Then the validation is done on a 300mm X 150mm with the same properties and same strain values as mentioned in the earlier validation. After the validation an area of 50mm X 50mm are selected as shown below and the results are shown below
Figure 2.3 showing structure with circular hole 50X50mm 1) Journal (Left) & 2) Validated Result (Right)
Figure 2.4 showing structure with elliptical hole 50X50mm 1) Journal (Left) & 2) Validated Result (Right)
The Poisson’s ratio is found by sampling displacement at 8 points along each of the four boundaries of the central regions. Each set of 8 points are averaged (arithmetic mean) to compute the average displacements at the boundaries: (ux)L, (ux)R, (uy)T, (uy)B
(ϵxx) = (ux)R - (ux)L (ϵyy) = (ux)R - (ux)L L0 L0
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where L0 is denoting the distance between the top/ bottom and left/ right boundaries in the underformed configuration. The local strain averages are then used to calculate an effective Poisson’s ratio
ν = - (ϵxx)
(ϵyy)
The corresponding Poisson’s ratio for a/b=1 and a/b=30 are 0.34 and -0.65 respectively. For the case of circular holes, the pores are found only to locally perturb the displacement field, so that the displacement field typical of the bulk material can be easily recognized. In contrast the array of elliptical holes is found to significantly affect the displacement field, so the linear distribution is disturbed.
Thus, the auxetic properties are studied for different a/b ratios and the results show that the auxetic property could be produced with even low porosity, provided the void aspect ratio a/b is large enough. High aspect ratio ellipses lead to a material characterized by a large negative Poisson’s ratio.
Further to this we have also validated the Journal 1 mentioned in the reference. ABAQUS was used to model and evaluate the results. In this case 2D structure with dimension 10mm X 10mm from a dog bone sample of 90mm X 25mm are considered for modelling. The material is modelled as linearly elastic with E = 193000 MPa and Poisson’s ratio 0.33. The mesh used was CPS8. The applied strain is 0.001%. The length is 10mm and so the corresponding displacement is 0.01. Monotonic tensile load was applied. To ensure the computational cost and to ensure the response is not affected by boundary effects we have considered a unit cell with periodic boundary condition as described in chapter 2 and performed a linear elastic response for the material. The contour maps for the Von-Mises are shown in the below figure. The models are validated for circular shape and S shape. It is also extended to elliptical shapes with ratios a/b=9, a/b=18, a/b=27. The porosity is maintained at 0.05% of the total area. The results were compared for all the shapes. The normalized stress is calculated by the formula Von Mises/ E. In both circle and S shape the stress concentrates along the boundaries of the holes, in the structure with the S shaped pores most of the material experiences low value of stress, while the one with circular pores there are crosses that are highly stretched. The results are shown in the below figure.
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Figure 2.5 showing Journal (Top) Vs Validated results (Bottom). Effect of the hole aspect ratio a/b on the deformation of the structure. Distribution of Von mises (Normalized by the bulk material elastic modulus E) for elliptical holes with different values of aspect ratio subjected to monotonic uniaxial tensile loading (in vertical direction) at strain = 1*10-3
2.1 AUXETIC PROPERTY
Poisson’s ratio defines the ratio between the transverse strain and axial strain in a loaded material. The extent of deformation a material undergoes when it is uniaxially stretched or compressed are quantified through Poisson’s ratio. For most of the conventional materials the value of Poisson’s ratio is positive. But the Poisson’s ratio need not be always positive. The classical theory of elasticity suggests that the Poisson’s ratio of isotropic materials can have the values within -1 to 0.5. and the range is wider for orthotropic and anisotropic materials. Negative Poisson’s ratios have now been discoveredfor naturally occurring or man-made metals including foams, polymers, cubic lattice, natural layered ceramics, ferroelectric polycrystalline ceramics, metals, zeolites etc. Various model structures and mechanisms which exhibit negative Poisson’s ratio include re-entrant units, rotating rigid units. The materials which exhibit negative Poisson’s ratio are auxetic materials. The auxetic materials exhibit the very unusual property of becoming wider when stretched and narrower when compressed. Auxetic materials are of interest due to their counter intuitive behaviour under
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deformation and enhanced properties. It has been found that auxeticity can be described in terms of geometry of the material system and deformation mechanism when loaded. The negative Poisson’s ratio is a scale independent property i.e the auxetic behaviour could be achieved at a macroscopic or microscopic level, or even at the mesoscopic and molecular level. These materials demonstrate unique and enhanced mechanical properties compared to the conventional materials. It has been shown experimentally that the indentation resistance of auxetic materials has been enhanced up to four times when compared with their conventional equivalent. Other enhanced properties are mechanical hardness, fracture toughness, and stiffness. In terms of the dynamic performance, auxetic materials also show an overall superiority regarding damping and acoustic properties compared to the conventional materials. Owing to their lower price, easy availability, and desirable mechanical properties, auxetic materials find a broad range of applications in many industrial sectors, especially in automotive, aerospace, and marine industries. More and more auxetic materials and structures with different microstructures are used to replace the conventional counterparts and have achieved satisfactory results.
To date, a variety of auxetic materials and structures have been discovered, fabricated or synthesized ranging from the macroscopic down to the molecular levels. The design of 2D systems capable of retaining a negative Poisson’s ratio at low values of porosity remains a challenge. Geometrical optimization of 2D structures could lead to the exhibition of auxetic properties which is also included in the current study. The combination of various geometry optimization lead to the increase in negative Poisson’s ratio. With some combinations we have also obtained Poisson’s ratio values which are close to 0. This is also an enhanced property of the system that these systems neither get farther nor thinner when stretched or compressed.
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2.2 PERIODIC BOUNDARY CONDITION
Periodic boundary conditions are a set of boundary conditions which are used to simulate an infinite structure by simply modelling a finite representative volume. A finite representative volume is often called a unit cell. In our study we use PBC to approximate a 10mm X 10mm to infinite structures. PBC is a great option to reduce computational time and computational cost in case of modelling large structures. Periodic boundary conditions are commonly applied in molecular dynamics, dislocation dynamics and material modelling to eliminate the existence of surface and avoid huge number of molecules or large size of simulation. We have followed [5] to define PBC equations in our study. Also we have been some modifications to suit our requirement. A strain controlled PBC may be specified by following equations. u(x+L) = u(x)+
x ∀ x ∈ B2𝛿 (1) u(x+L) = u(x) (2) t(x+L) = -t(x) ∀ x ∈ B1
𝛿 (3) where u is the displacement at x,
is the strain applied to the RVE, t is the traction force andB1
𝛿 represents the boundary B whose normal is along “1” direction.
2.2.1 Constraint Equations
We apply PBC through linear constraint equation in ABAQUS. Multi points may be constrained by a general linear combination of nodal variables. The summation of the product of a coefficient and the corresponding nodal variable is equal is equal to zero. We define a general linear homogenous equation
A1uPi + A2uQj + ………. + ANuRk = 0 (4) Where r is node, k is degree of freedom and AN is a constant coefficient that define the relative motion of nodes.
2.2.2 Dummy Node
To apply PBC using constraint equations described above, one abstract concept of “dummy node” is introduced in Abaqus. We rewrite equation (4) by replacing zero by a non zero value u.
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A1uPi + A2uQj + ………. + ANuRk = u (5)
2.2.3 Definition of equation
Figure 2.7 showing Definition of * equation
*Equation
3 ** equation has 3 terms
Left, 1, 1 ** left surface node set, dof = 1, coeff = 1 Right, 1, -1 ** right surface node set, dof = 1, coeff = -1 *Equation
3 ** equation has 3 terms
Left, 2, 1 ** left surface node set, dof = 2, coeff = 1 Right, 2, -1 ** right surface node set, dof = 2, coeff = -1 10000, 2, 1 ** dummy node z=10000, dof = 2, coeff = 1 RVE description as per our study requirement
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u2top – u2bottom = -0.01 (7)
Sample input file used in our study on different shapes
2.2.4 Input file for a 2D structure with circle
*Heading
** Job name: circle Model name: Model-1 ** Generated by: Abaqus/CAE 6.14-1
*Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=Part-1 *End Part ** *Part, name=dummy-LR *End Part ** *Part, name=dummy-TB *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly **
*Instance, name=Part-1-1, part=Part-1 *Node 1, -18.6296806, 4.62969542 2, -18.4903049, 4.49031925 3, -18.1200008, 4.36999989 4, -17.7496948, 4.49031925 5, -17.6103191, 4.62969542 6, -17.4899998, 5. 7, -17.6103191, 5.37030458 8, -17.7496948, 5.50968075 9, -18.1200008, 5.63000011 10, -18.4903049, 5.50968075 ……… 9194, -19.6230736, 7.55718994 9195, -20.9239616, 3.52641296 9196, -21.3657303, 7.19272757 9197, -21.3561554, 7.08833408 9198, -19.5791016, 7.65781736 9199, -21.4266281, 7.22704697 9200, -21.4724998, 7.2937727 9201, -17.336031, 7.77302265 9202, -17.2476501, 7.71204567 9203, -17.1652679, 7.70722485 9204, -20.5425739, 4.57812023 9205, -20.4600601, 4.62432575 9206, -21.3876438, 7.30040216 9207, -19.687706, 2.51486325
26 9208, -20.1431427, 2.47530007 *Element, type=CPS8 1, 607, 953, 755, 1, 3109, 3110, 3111, 3112 2, 953, 132, 1008, 1294, 3113, 3114, 3115, 3116 3, 1383, 1987, 1889, 1374, 3117, 3118, 3119, 3120 4, 1374, 1889, 2788, 207, 3119, 3121, 3122, 3123 5, 2542, 2407, 2894, 2893, 3124, 3125, 3126, 3127 6, 2535, 2857, 2577, 128, 3128, 3129, 3130, 3131 7, 2806, 2271, 1896, 2778, 3132, 3133, 3134, 3135 8, 1672, 125, 1717, 126, 3136, 3137, 3138, 3139 9, 1011, 124, 951, 1087, 3140, 3141, 3142, 3143 10, 825, 951, 627, 23, 3144, 3145, 3146, 3147 11, 727, 22, 728, 926, 3148, 3149, 3150, 3151 12, 728, 22, 626, 824, 3149, 3152, 3153, 3154 13, 183, 728, 824, 827, 3155, 3154, 3156, 3157 14, 403, 931, 729, 828, 3158, 3159, 3160, 3161 15, 426, 1219, 2000, 829, 3162, 3163, 3164, 3165 16, 1022, 830, 731, 449, 3166, 3167, 3168, 3169 17, 31, 636, 732, 832, 3170, 3171, 3172, 3173 18, 1127, 2657, 1053, 1192, 3174, 3175, 3176, 3177 19, 1506, 1786, 1763, 516, 3178, 3179, 3180, 3181 20, 1670, 1536, 540, 1684, 3182, 3183, 3184, 3185 21, 806, 846, 1030, 1363, 3186, 3187, 3188, 3189 22, 847, 1235, 1030, 846, 3190, 3191, 3187, 3192 23, 587, 1305, 1209, 1180, 3193, 3194, 3195, 3196 24, 849, 733, 940, 588, 3197, 3198, 3199, 3200 25, 850, 589, 940, 733, 3201, 3202, 3198, 3203 ……… 2986, 3088, 1977, 3108, 487, 8162, 8782, 9011, 8189 2987, 3091, 2519, 2437, 3090, 9110, 9076, 9202, 9201 2988, 2899, 208, 3092, 3093, 9166, 5171, 9205, 8551 2989, 3093, 209, 2998, 2315, 9204, 9144, 4990, 8552 2990, 3094, 2170, 3108, 2268, 6041, 9012, 8781, 8650 2991, 3098, 3097, 3107, 508, 6459, 9073, 5990, 8965 2992, 3102, 2527, 2470, 3101, 9123, 5217, 8699, 9075 *Nset, nset=Set-1, generate
1, 9208, 1
*Elset, elset=Set-1, generate 1, 2992, 1
** Section: Section-1
*Solid Section, elset=Set-1, material=Material-1 *End Instance
*Instance, name=dummy-LR-1, part=dummy-LR *Node
40000, -70., 10., 0.
*Nset, nset=dummy-LR-1-RefPt_, internal 40000,
*End Instance
*Instance, name=dummy-TB-1, part=dummy-TB *Node
50000, 70., 10., 0.
**This dummy node can be arbitrary *Nset, nset=dummy-TB-1-RefPt_, internal 50000,
*End Instance
*Nset, nset=Set-dummy-LR, instance=dummy-LR-1 40000,
27
*Nset, nset=Set-dummy-TB, instance=dummy-TB-1 50000,
**
*Nset, nset=Bottom, instance=Part-1-1
38, 39, 40, 41, 42, 43, 44, 45, 46, 52, 53, 54, 55, 56, 57, 58
59, 60, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 658, 659, 660 661, 662, 663, 664, 665, 666, 667, 668, 4080, 4082, 4119, 4121, 4187, 4193, 4194, 6073 6438, 6445, 6454, 6528, 6534, 6537, 6539, 6541, 6548, 6552, 6553, 6558, 6560, 6564, 6568, 6571 6575, 6579, 6689, 6759, 6761, 6762, 6767, 6768, 6820, 6821, 6822, 6838, 6841, 6846
*Nset, nset=Left, instance=Part-1-1
13, 14, 15, 16, 17, 18, 19, 20, 21, 27, 28, 29, 30, 31, 32, 33
34, 35, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 630, 631, 632 633, 634, 635, 636, 637, 638, 639, 640, 3170, 3985, 3992, 3993, 3996, 4000, 4091, 4126 4512, 5524, 5807, 6489, 6493, 6496, 6498, 6502, 6503, 6507, 6509, 6513, 6517, 6524, 6527, 6674 6684, 6686, 6688, 6734, 6740, 6742, 6746, 6815, 6824, 6825, 6826, 6831, 6833, 6835
*Nset, nset=Right, instance=Part-1-1
63, 64, 65, 66, 67, 68, 69, 70, 71, 77, 78, 79, 80, 81, 82, 83
84, 85, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 686, 687, 688 689, 690, 691, 692, 693, 694, 695, 696, 3361, 3680, 3688, 4221, 4223, 4259, 4290, 4353 4356, 4374, 4437, 5804, 6580, 6586, 6590, 6594, 6596, 6599, 6603, 6606, 6608, 6612, 6616, 6618 6624, 6627, 6698, 6700, 6701, 6712, 6772, 6780, 6781, 6787, 6855, 6857, 6862, 6864
*Nset, nset=Set-5, instance=Part-1-1 38,
*Nset, nset=Set-7, instance=Part-1-1
13, 14, 15, 16, 17, 18, 19, 20, 21, 27, 28, 29, 30, 31, 32, 33 34, 35, 63, 64, 65, 66, 67, 68, 69, 70, 71, 77, 78, 79, 80, 81 82, 83, 84, 85, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 630 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 671, 672, 673, 674, 675, 676 677, 678, 679, 680, 681, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696 3170, 3361, 3680, 3688, 3985, 3992, 3993, 3996, 4000, 4091, 4126, 4221, 4223, 4259, 4290, 4353 4356, 4374, 4437, 4512, 5524, 5804, 5807, 6489, 6493, 6496, 6498, 6502, 6503, 6507, 6509, 6513 6517, 6524, 6527, 6580, 6586, 6590, 6594, 6596, 6599, 6603, 6606, 6608, 6612, 6616, 6618, 6624 6627, 6674, 6684, 6686, 6688, 6698, 6700, 6701, 6712, 6734, 6740, 6742, 6746, 6772, 6780, 6781 6787, 6815, 6824, 6825, 6826, 6831, 6833, 6835, 6855, 6857, 6862, 6864
*Elset, elset=Set-7, instance=Part-1-1
17, 68, 154, 156, 240, 242, 243, 244, 245, 269, 278, 307, 308, 317, 324, 343 344, 351, 367, 391, 700, 788, 789, 1002, 1003, 1004, 1005, 1006, 1007, 1009, 1010, 1011 1012, 1014, 1015, 1035, 1037, 1039, 1040, 1041, 1042, 1044, 1045, 1046, 1047, 1048, 1049, 1050 1051, 1071, 1076, 1077, 1078, 1085, 1086, 1087, 1096, 1109, 1112, 1113, 1115, 1132, 1137, 1138 1141, 1160, 1166, 1167, 1169, 1174, 1176, 1178, 1196, 1198, 1203, 1205
*Nset, nset=Set-8, instance=Part-1-1 38,
*Nset, nset=Top, instance=Part-1-1
88, 89, 90, 91, 92, 93, 94, 95, 96, 102, 103, 104, 105, 106, 107, 108 109, 110, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 714, 715, 716 717, 718, 719, 720, 721, 722, 723, 724, 3465, 3467, 4394, 4415, 4430, 4432, 4454, 4461 4468, 5235, 5248, 5271, 6463, 6467, 6470, 6628, 6634, 6637, 6638, 6639, 6642, 6647, 6648, 6653 6657, 6662, 6667, 6670, 6673, 6702, 6722, 6799, 6809, 6814, 6868, 6869, 6874, 6875 *Equation 3
Left, 1, 1.,Right, 1, 1.,Set-dummy-LR, 1, 1. *Equation
3
Top, 2, 1.,Bottom, 2, -1.,Set-dummy-TB, 2, 1. *End Assembly
**
28 ** *Material, name=Material-1 *Elastic 193000., 0.33 ** --- ** ** STEP: Step-1 **
*Step, name=Step-1, nlgeom=NO *Static
1., 1., 1e-05, 1. **
** BOUNDARY CONDITIONS **
** Name: BC-1 Type: Displacement/Rotation *Boundary
Set-dummy-LR, 1,1,0
** Name: BC-2 Type: Displacement/Rotation *Boundary
Set-8, 1, 1
** Name: BC-3 Type: Displacement/Rotation *Boundary
Set-dummy-TB, 2, 2, -0.010 **
** OUTPUT REQUESTS **
*Restart, write, frequency=0 **
** FIELD OUTPUT: F-Output-1 **
*Output, field, variable=PRESELECT **
** HISTORY OUTPUT: H-Output-1 **
*Output, history, variable=PRESELECT *End Step
Using the concepts and procedures mentioned above we have validated S shape and Ellipse shape in all the three configurations
2.3 Validation of S Shape
Using the procedures mentioned previously the results have been evaluated for S shape and the values of Von-Mises and normalized stress are plotted below. Also, we have established other properties like Poisson’s ratio to study the auxetic property. We have established 3 configurations for S shape as the general procedure to evaluate the behaviour of structure in three different configurations horizontal, vertical and alternate. First, we tried the shapes with horizontal configuration to evaluate four main parameters the Maximum Von-Mises,
29
stress concentration factor, Yield area and Poisson’s ratio. The values are as Maximum Von-Mises stress – 990MPa, Kt – 5.12, yield area – 4.8% and Poisson’s ratio – 0.12
Figure 2.8 showing Von-Mises (Left) and Von-Mises within limits 0 to 205MPa (Right) for S shape in the horizontal configuration
Figure 2.9 showing displacement U1 & U2 for S shape in horizontal configuration
In the horizontal configuration we could notice that the stress concentration is high because
the shapes are placed at a minimum distance of 0.2mm from each other. Also, the variation is the Poisson’s ratio also accounts to the same minimum distance between two elements In the next step the s shapes are tried with vertical configuration and the values of Maximum Von-Mises stress – 600MPa, Kt – 3.1, yield area – 5% and Poisson’s ratio – 0.2
30
Figure 2.10 showing Von-Mises (Left) and Von-Mises within limits 0 to 205MPa (Right) for S shape in the vertical configuration
Next the displacements U1 & U2 are calculated for the vertical configuration and the results are as shown in th below figure
Figure 2.11 showing displacement U1 & U2 for S Shape in vertical configuration
When the s shapes are placed in vertical configuration the Von-Mises decreases compared to the horizontal configuration and so the stress concentration factor, but in contrast the % of yield area has also increased compared to the horizontal configuration which is very different and an interesting point to be observed. Also, the Poisson’s ratio has decreased from the normal values and is seen at 0.20.
Now as the next step we are intersted in placing these S shapes in alternate configuration and try the results for the same.
31
Figure 2.12 showing Von-Mises (Left) and Von-Mises within limits 0 to 205MPa (Right) for
S shape in the alternate configuration
Figure 2.13 showing displacement U1 & U2 for S Shape in vertical configuration We have obtained the displacements U1 and U2 from the model as shown in the below figure. From the displacement values we have calculated the Poisson’s ratio by the procedure mentioned in Journal 2 and the value was found to be -0.44. We have also established the maximum stress and the yield area of spread 290 MPa and 1.5% respectively. The value of stress intensity factor is 1.5. Here we could observe a drastic change in the values of Von-Mises when placed in the alternate configuration and these will be compared to the shapes under consideration in the following chapters
32
2.4 Validation of Ellipse Shape
The ellipse shape is also tried in three configurations. The first type is by placing horizontally, the second one is by placing vertically and the third type is alternate one. Using the procedures followed for S Shape we have evaluated all the results for the ellipse shape. Here we have considered a/b ratio of ~ 30 and now the results for all the configurations are presented below. The below figure on the left shows the Von-Mises distribution for the horizontal configuration and on the right Von-Mises within limits 0 to 205MPa is shown. From this we have calculated the maximum value of Von-Mises as 2000MPa, Stress concentration factor kt as 10.36, Yield area as 5.8%, Poisson’s ratio as 0.12.
Figure 2.14 showing Von-Mises (Left) and Von-Mises within limits 0 to 205MPa (Right) for ellipse shape in the horizontal configuration
33
The similar procedure is followed for ellipse with vertical configuration and the results are obtained as maximum value of Von-Mises as 211MPa, stress concentration factor as 1.09MPa, Yield area is 0.1% and the Poisson’s ratio as 0.32. While placing the ellipse in vertical configuration we could obtain the best results in case of maximum value of stress and stress concentration factor.
Figure 2.16 showing Von-Mises (Left) and Von-Mises within limits 0 to 205MPa (Right) for ellipse shape in the vertical configuration
34
The Maximum Von-Mises, Yield area and Poisson’s are analysed from the ABAQUS results. The Von-Mises stress for an ellipse with a/b~ 30 in a 10mm X 10mm 2D plate with periodic boundary conditions implemented are as shown below
Figure 2.18 showing Von-Mises for Ellipse shape in alternate configuration
We have obtained the displacements U1 and U2 from the model as shown in the below figure. From the displacement values we have calculated the Poisson’s ratio by the procedure mentioned in Journal 2 and the value was found to be -0.65. We have also established the maximum stress and the yield area of spread 2000 MPa and 2.8% respectively. The value of stress intensity factor is 10.36.
35
Figure 2.20 showing Von-Mises (Limits 0 To 205MPa) for ellipse shape in alternate configuration
From the journal we need to analyse and compare the normalized stress for different shapes and are plotted in the below figure
Figure 2.21 showing normalized stress for Journal (Top) vs Validated result (Bottom)
The contour maps of the normalized stress shown in the above figure indicate that the two structures carry the applied deformation differently. In both systems the stress concentrates
36
along the boundaries of the holes and in the structure with S-shaped pores experienced low value of stress, while in the circular pores there are crosses that are highly stretched. In the sample with circular holes the deformation mechanism is strectching, while in the S-shape it is mostly rotation of the domains defined by the pores. The stress concentrations on the boundary of the pores significantly reduce when compared with the circular and elliptical pores.
2.5 CALCULATION OF STRESS INTENSITY FACTOR
To calculate the stress concentration factor, we have applied the periodic boundary condition on a 10mm X 10mm plain 2D structure as shown in the below figure and calculated the Von-Mises stress of the plain sheet. The value was 193MPa. Then to verify the same we have applied the same on a plate with a circle in the centre and we have obtained the value of stress intensity factor as 2.9 which is very close to 3. Thus, the procedure for calculating the stress intensity factor is validated.
Figure 2.22 showing Plain Sheet Vs Plate with a circle
37
3. STUDY AND PRESENTATION OF OPTIMIZED SHAPES
The main work involved inducing different shapes in the given structure and get the results for the same. The parameters of concern were Von Mises, Stress concentration factor, Poisson’s ratio, Area of extent of stress distribution around the maximum stress concentrated area. The properties of the material and parameters and techniques used are listed below For all our study we have considered a 2D structure
We have considered a 10mm X 10mm 2D sheet in which we have induced the different shapes and tried the results
3.1 MATERIAL PROPERTIES
Material: Stainless steel SS316L Modulus of elasticity = 193000MPa Poisson’ ratio = 0.33
Yield strength = 205MPa
3.2 LOAD CONDITIONS
Length of the specimen = 10mm Breadth of the specimen = 10mm Given,
Applied strain = 1 x 10-3
And the strain is applied in Y direction and in elongation So,
Displacement, u = 0.01mm
The displacement is applied in Y direction.
3.3 SOFTWARE FOR ANALYSIS
We have used ABAQUS for all our analysis. We have performed static analysis in all of our study.
38
Periodic boundary condition is implemented in all the specimen of our study which is presented hereon. In all part of our study we have used static analysis.
MESH: CPS8
3.4 EFFECTS OF DIFFERENT SHAPES AND ARRANGEMENTS
We have tried a number of shapes with different arrangements for each shape and found some interesting behaviour of the different shapes.
For every shape under study we are interested in calculating the Maximum Von Mises, Stress concentration factor, Extended area of stress distribution from the critical point when it is considered under the limits 0 to 205MPa. We have made 205MPa as a standard for comparison because this tends to be the yield strength of the material.
3.4.1 DIFFERENT SHAPES UNDER CONSIDERATION
The main idea behind the selection of different shapes comes from the parameters as listed in the previous page. The different shapes considered were sown in the below figure
Figure 3.1 shows the different shapes under study
(1) Circle Shape (2) Ellipse (3) U shape
(4) Inverse ellipse shape (5) Square ellipse shape (6) Double D shape (7) Tube shape
39 (8) C shape
(9) C Shape modified (10) Double C shape (11) S Shape
3.5 ANALYSIS AND RESULTS
The main aim our study is to implement these shapes with in two conditions 1) Maintain porosity 2) Without maintaining porosity. The porosity is the amount of void area in the available 10mm X 10mm structure. We have used periodic boundary condition in all of our study to reduce computational time. In the first part the porosity is maintained constant at 5% of the given area. So, it makes to 5mm. In the second part of our study we have removed this constraint to check the effects of the shapes when the elements are placed closer to each other. Thanks to the laser drilling technology which allowed us to use the minimum distance between two elements as 0.2mm.
3.5.1 ANALYSIS OF DIFFERENT SHAPES WITH CONSTANT POROSITY 3.5.1.1 SHAPES ARRANGED IN HORIZONTAL CONFIGURATION
At first, we have studied the effects of shapes by placing them in horizontal manner. Shapes tried in horizontal arrangement
(1) U shape
(2) Inverse ellipse shape (3) Square ellipse shape (4) Double D shape (5) Tube shape (6) C shape
(7) C shape modified (8) Double C shape
For the horizontal arrangement the results obtained from ABAQUS are shown in the below figure. The model with C shape had the highest value of Von Mises compared to the other shapes. The value of Von Mises for C shape is around 1700MPa. All the other shapes had Von Mises stress as C shape modified – 950MPa, Inverse Square Ellipse – 900MPa, U shape –
40
800MPa, Double C Shape – 700MPa, Double D Shape – 675 MPa, Tube Shape – 675 MPa, Square Ellipse – 650 MPa and the values of stress concentration factor are calculated as per the procedure explained in the chapter model validation. The corresponding values of stress concentration factor are C Shape – 8.8, C shape modified – 4.9, Inverse Square Ellipse – 4.6, U shape – 4.1, Double C Shape – 3.6, Double D Shape – 3.4, Tube Shape – 3.4, Square Ellipse – 3.3. The figures for Von Mises stress are shown below.
Figure 3.2 showing Von Mises data for all the shapes under consideration in horizontal configuration
We have also calculated the area of stress distribution by calculating the percentage of grey area (yield area) that appear when the Von Mises stress is limited between 0 to 205MPa. The 205MPa limit has been chosen based on the value of yield strength of the material. Since we have considered SS316L as the material the corresponding value of yield strength for the material is considered in our analysis. We have used Image J software as explained in the literature section to calculate the area and the values are expressed in percentage as a ratio of grey area to the total area. The corresponding values of percentage of grey area (Yield area) are C Shape – 9.7%, C shape modified – 3.1%, Inverse Square Ellipse – 8.1%, U shape – 7.3%, Double C Shape – 2%, Double D Shape – 6%, Tube Shape – 3.4%, Square Ellipse – 3.3%. The figures for Von Mises stress limited between 0 to 205MPa are shown below.
41
Figure 3.3 showing Von Mises (Limit 0 to 205MPa) data for all the shapes under consideration in horizontal configuration
In the third part we have analysed the displacements U1 & U2 for all the shapes and calculated the Poisson’s ratio by following the procedures as described in [2] and the results as listed. The corresponding values of stress concentration factor are C Shape – 0.33, C shape modified – 0.094, Inverse Square Ellipse – 0.25, U shape – 0.098, Double C Shape – 0.096, Double D Shape – 0.25, Tube Shape – 0.25, Square Ellipse – 0.3. The C shape modified and Double C shape had the least minimum distance between two elements and the least minimum distance in perpendicular to the direction of application of load. With the given porosity it was a challenge to obtain reduced minimum distance between two elements. Thanks to the C shape modified and double C shape which helped in achieving the least minimum distance between two elements. Because of this property we could see a drastic change in the Poisson’s ratio of the material. The Poisson’s ratio has almost reached 0 for both shapes. The figures for Displacements U1 & U2 are shown below.
42
Figure 3.4 showing minimum cutting distance and minimum distance between two elements
• L0: Center to Center distance
• A: Minimum distance between two elements in vertical direction • B: Minimum distance between two elements in horizontal direction
43
Figure 3.5 showing displacements U1 & U2 for all the shapes under consideration in horizontal direction
The values are represented graphically for all the parameters we have considered. The representation is as shown below.
Graph 3.1 showing the values of Smax, Kt, Poisson’s, Yield area (%) for all the shapes in
44
From the graph showing values of Smax and kt we could see that there is a huge variation with the introduction of different shapes. The highest was 1700MPa whereas the lowest that could be achieved are 650MPa. But just with the value of maximum stress we couldn’t conclude and so we have also calculated the yield area which showed the extent of yield for double C shape is the lowest and C shape it is again the highest. We have also calculated the Poisson’s ratio to study about the auxetic property of the material upon introduction of different shapes in the structure. Unfortunately, there is no appreciable difference. But we could also observe that C shape modified and Double C shape had Poisson’s ratio close to 0 and that is due to the fact that the least minimum distance between two elements could be achieved in this configuration maintaining constant porosity.
3.5.1.2 SHAPES ARRANGED IN VERTICAL CONFIGURATION
At first, we have studied the effects of shapes by placing them in vertical manner. Shapes tried in vertical arrangement
(1) Double D shape (2) Tube shape (3) C shape
(4) C shape modified (5) Double C shape
The shapes mentioned above are analysed in vertical configuration. The other shapes which were not analysed are U Shape, Square Shape, Inverse square shape because these shapes will have the same results in both the configurations. The model with C shape had the highest value of Von Mises compared to the other shapes. The value of Von Mises for Double D shape is around 650MPa. All the other shapes had Von Mises stress as Tube Shape – 450MPa, C Shape modified – 450MPa, Double C Shape – 400MPa, C Shape – 285MPa. The corresponding values of stress concentration factor are Double D Shape – 3.3, Tube Shape – 2.3, C Shape modified –2.3, Double C Shape – 2.07, C Shape – 1.4. The figure for Von Mises are shown below
45
Figure 3.6 showing Von Mises data for all the shapes under consideration in vertical configuration
The percentage of grey area (Yield area) for all the shapes are calculated as mentioned in the previous procedure for horizontal manner. The corresponding values for shapes are Double D Shape – 1.4%, Tube Shape – 1.8%, C Shape modified – 1.6%, Double C Shape – 2.5%, C Shape – 1.5%. Here we could see that the area of distribution of stress for this type of configuration are much lesser compared to the ones with the horizontal configuration. The figure for Von Mises (within Limits 0 to 205MPa) are shown below
46
Figure 3.7 showing Von Mises (Limit 0 to 205MPa) data for all the shapes under consideration in vertical configuration
As a next step we have analysed the displacements U1 & U2 for all the shapes and calculated the Poisson’s ratio by following the procedures as described in [2] and the results as listed. The corresponding values of stress concentration factor are Double D Shape – 0.31, Tube Shape – 0.31, C Shape modified – 0.27, Double C Shape – 0.29, C Shape – 0.32. The displacements were very similar to the normal behaviour of the material with a slight variation due to the inducement of the shaped elements. Again, there is no appreciable difference in the value of Poisson’s ratio because of the introduction of different shapes. Also, in case of C shape modified and Double C Shape there is no drastic change because in this configuration the minimum distance is achieved in the direction parallel to the direction of the applied load. So, from this configuration we could infer that there is an appreciable change in the Poisson’s ratio of the material only when they are placed closer in the direction perpendicular to the direction of the applied load. The figures for the displacements U1 & U2 are as shown below.
47
Figure 3.8 showing displacements U1 & U2 for all the shapes under consideration in vertical configuration
For the shapes in vertical configuration we have plotted a graph for the parameters 1) Max Von Mises, MPa
2) Kt
3) Yield Area (%) 4) Poisson’s ratio
48
Graph 3.2 showing the values of Smax, Kt, Poisson’s, Yield area (%) for all the shapes in
Vertical configuration
From the above graph we could infer that the Maximum Von-Mises stress concerning the different shapes has a trend very different from the one we had for the shapes with horizontal configuration. The Maximum value is 650MPa for Double D shape and the minimum is 285MPa for C shape. The extent of yield for Double D shape and C shape was almost the same and it is very low. So, whenever there is a requirement of horizontal arrangement C shape could be a best alternative. Also, the Poisson’s ratio was not so differing from the material properties with small variations because of the element shapes. In case of C shape modified and Double C shape though the minimum distance is maintained the same as for horizontal configuration we couldn’t see a drastic change in the value of Von-Mises and this is due to
49
the arrangement of shapes parallel to the direction of the applied load. After these two configurations we would like to take the best shapes and try combination of both the configuration placing them in alternate manner.
3.5.1.2 SHAPES ARRANGED IN ALTERNATIVE CONFIGURATION
From the previous study we have obtained results for the parameters under study and now we need to have a trade-off between the different parameters to choose the shapes with which the study is to be continued. From the previous results we have chosen C shape modified and Double C Shape are the shapes with which our new trials are to be made. We have also considered Ellipse shape and S shape which were taken from [1,2] to be compared with the shapes which we have introduced in this study. Another reason for choosing these two shapes are least minimum that could be achieved with these shapes. This idea is based is obtained from the previous studies [1,2] where they typically try to induce shapes that have minimum distance while maintaining the porosity.
From the previous study we have also identified that the shapes arranged in alternate manner are preferable and yielded best results in terms of all the parameters we have considered in our study.
We have analysed all the four shapes by placing them in alternate manner. 1) Ellipse Shape
2) C shape Modified 3) Double C Shape 4) S Shape
In this configuration we have used the combination of both horizontal and vertical configuration. We have placed them in alternate manner and the porosity is still maintained. We have got the results of Von Mises for these shapes and are Ellipse shape – 2000MPa, C shape Modified - 950MPa, Double C Shape – 450MPa, S Shape – 290MPa. The corresponding Stress concentration factor are Ellipse shape – 10.36, C shape Modified – 4.92, Double C Shape – 2.33, S Shape – 1.5. The trend of Von-Mises shows that the ellipse shape had the maximum value of stress while S shape had the least and the Double C shape is the second least and a huge difference between C Shape Modified and Double C shape is observed. The Von-Mises for all the shapes are as shown below.
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Figure 3.9 showing Von-Mises stress for all the shapes in alternate configuration
Then we have made the Von-Mises under the limits 0 to 205MPa where the upper limit tends to the yield strength of the material. With this procedure we could obtain the yield area for all the shapes. Then using that we have calculated the % of yield area and the results are Ellipse shape – 2.8%, C shape Modified – 3.2%, Double C Shape – 3.3%, S Shape – 1.5%. The area of extent of yield area were greater for C shape modified and Double C shape compared to the Ellipse Shape and S shape, but the stress concentrations were very less for both C shape modified and Double C shape compare to the ellipse shape. The figure below shows the Von-Mises under Limits 0 to 205 for the above-mentioned shapes.
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Figure 3.10 showing Von-Mises stress within limits 0 to 205MPa for all the shapes in alternate configuration
As a next step the displacements U1 & U2 are calculated for all the shapes and the Poisson’s ratio for each shape is calculated as per the procedure followed in previous arrangements. Thus the Poisson’s ratio for all the shapes are Ellipse shape – -0.65, C shape Modified – -0.23, Double C Shape – -0.45, S Shape – -0.44. The figures for displacements U1 & U2 are as shown below
