MATERIAL MECHANICS, SE2126 COMPUTER LAB 2 PLASTICITY

Transcription

1 MATERIAL MECHANICS, SE2126 COMPUTER LAB 2 PLASTICITY

2 PART A INTEGRATED CIRCUIT An integrated circuit can be thought of as a very complex maze of electronic components and metallic connectors. These connectors are generally made from copper, aluminium, tungsten or some other good conductor. The base material of an integrated circuit is usually some insulating and electrically passive material, for example silicon. Consider a very simplified 2D model of a conductor wire made of copper, which is partially embedded in a silicon matrix, see Figure 1. The matrix is considered to be semiinfinite. This could for instance represent a small part of the central processing unit (CPU) of the computer you are working on right now. Through the copper wire a current is flowing and since the copper is not a perfect conductor, there will be some resistance. This resistance will in turn give rise to an increase in temperature. Luckily computers are fitted with cooling devices. But what if this cooling device fails? Cu Si Figure 1. Copper conductor in a silicon matrix You will in this part of the lab investigate what would happen if the copper was given an increase in temperature ΔT, as a result of a failure of the internal cooling device. This increase in temperature is generally very fast and since the difference in thermal expansion for copper and silicon is a whole order of magnitude greater for copper, the thermal expansion for silicon will be ignored.

3 What if ΔT is large enough to plastically deform the conductor wire? What if the cooling is turned on again, or if the computer is turned off and cools down, could there be any permanent mechanical damage to the integrated circuit as a result of this? Could there be any residual tensile stresses that could cause a fracture? Start the ANSYS Mechanical ADPL Product Launcher from the start menu in Windows. There are a few settings you will NEED to MAKE SURE are correct, see Figure 3 and the table below. When this is done, click on Run to start ANSYS. Figure 3. ANSYS Product Launcher. Make sure that the settings are correct! Simulation environment: ANSYS Licence: ANSYS Academic Teaching Advanced Working Directory: C:\TEMP\[ any subfolder of your choice ]

4 First you will need to get two files from the course web page. There is an ANSYS database file GeometryAndMeshPartA.db and a text file PartA-Solver.txt that you will need for this part. The database file contains the model geometry, the mesh and also the material definitions. The matrix is modelled as a cubic anisotropic elastic material with stiffness components C 11, C 12 and C 44 according to Table 1 below. C 11 C 12 C ,9 GPa 63,5 GPa 79,5 GPa Table 1. Stiffness components for cubic elastic silicon matrix. From Wikström, A. Modeling of stresses and deformation in thin film and interconnected line structures Doctoral thesis, Department of Solid Mechanics, KTH, The conductor wire is modelled as a bilinear isotropic elastic-plastic material with isotropic hardening and a thermal expansion coefficient α, see Figure 3 and Table 2. σ E 117 GPa ν 0,34 σ Y E T α 22e-6 K -1 E ε σ Y E T 100 MPa 1,12 GPa Figure 3. Schematic figure of a bilinear elastic-plastic material. Table 2. Material properties for the copper wire. When ANSYS starts, choose File > Resume from and load the file geometry file from where you saved it on the hard drive.

5 Look at the mesh, it is not beautiful, but it will do. If you can not see the mesh you can plot it from the menu Plot and then choose Elements, or you can type the command EPLOT in the ANSYS Command Prompt. The mesh contains at least one triangular element, which is really a collapsed square element. This would generally not be so good if you where using linear elements, since it would be very stiff in bending. Now issue the APLOT command, or choose Plot > Areas from the menu, to look at how the model is built. The model has a symmetry line at x = 0. The circular outer boundary is supposed to be far enough away from the copper wire to simulate an infinite boundary, at least to a first approximation. The model is meshed with PLANE183 8-noded quadratic elements with a plane strain formulation, which is to say that the out of plane dimension is also considered to be large and that there are no strains in the z-direction. Now open the text file that you downloaded from the course home page. This file contains the information for ANSYS on how to solve the model and what boundary conditions to apply. You will only need to change the value for the temperature load, DeltaT. You can leave it at 10 degrees for now. This file will execute two solutions after each other. First it will apply the temperature change to the copper wire and solve, and secondly it will remove the temperature and solve again. Your first task is to find the temperature load for which you get initiation of plastic deformation. Choose File > Read Input from and make ANSYS read the solver text file to initiate the solution process. It should solve this problem quite fast since 10 degrees is not enough to initiate plastic deformation and therefore no nonlinear iterations are required. Wait until both load steps have solved. Now open up the General Postproc from the ANSYS Main Menu to the left. From here you can access all the results and view them or list them in basically any way you want. (As long as this happens to be the same way as ANSYS wants ) The results you are viewing now are the results after the second load step. Look at the von Mises equivalent stress by Plot Results > Contour Plot > Nodal Solu, and there

6 choose Nodal Solution > Stress > von Mises Stress. There might be some residual stresses, but look at the maximum value! The units of the model are in millimetres and Newton so the stresses are given in MPa. These stresses are, to a very good approximation, zero. Why is that do you think? Now from the General Postproc choose Read Results > By Pick and pick the first load step and click on Read. Close that dialogue box and go back and plot the von Mises equivalent stress again. What you are viewing now is the stress state after heating, but before cooling down again. You will still have stress concentrations at the boundary but look instead at the stress state in the copper. To accurately inquire the stress state in the copper part only you will need to select only those elements. By issuing ESEL,S,MAT,,2 you will select only those elements associated with material number 2, in this case the copper. From these results you can actually figure out at what temperature change you will expect the copper wire to experience yielding. You do not have to iterate a solution. Can you figure out how? Hint: Is the solution linear or nonlinear below the temperature you are seeking? How does the stresses relate to the temperature in this case? You know the yield stress, how is that related to the von Mises stress in the yield criterion? 0, Call this initiation temperature change, and write it down on the answer sheet at the end. Confirm it by doing a new analysis with that temperature as the loading. To do this you can just start over by clearing the current analysis and then solving again. Remember to change the loading and save the updated text file. File > Resume from and then just reload the geometry and solve using the text file. Evaluate the maximum residual tensile stresses in both the copper wire and the silicon substrate for a number of temperature changes ranging from to 10. Remember that it is only tensile stresses that can cause fracture; hence looking at the von Mises

7 stress will not give you a meaningful result. Also, to get accurate results you will have to select and interrogate the results for each part separately using the ESEL,S,MAT,,# command (where # should be 1 for the silicon and 2 for the copper).

8 PART B HARDENING OF NOTCHED SPECIMEN In this part of the lab you will consider an ordinary uniaxial tensile test specimen made of common steel with a circumferential notch, see Figure 4. The specimen will be loaded with cyclic tension and compression and you will investigate the effects of different hardening laws. Specifically, you will look at the difference between isotropic hardening and kinematic hardening. You will need the two files from the course home page for this part. The ANSYS database file GeometryPartB.db and the text file PartB-Solver.txt. 2r t a Figure 4. Test specimen with notch. L First you will need to Clear & Start New to get rid of the geometry and results from Part A. Then you should open up the geometry file in the same way as you did in part A. There are a few things you will need to set before you solve the model. First, there is no nonlinear material law; only the elastic properties of steel are predefined (i.e. a Young s modulus of 209 GPa and a Poisson s ratio of 0.3). Secondly, you will need to mesh the geometry. The model is an axisymmetric 2D model with PLANE183 elements. It is important that you make sure that the element type you are using can handle the type of problem you wish to model. You can look in the Help section for the element formulation if you are unsure about whether or not an element is appropriate.

9 Now you will add a nonlinear material property to the specimen. Click on Preprocessor > Material Props > Material Models to open up the Define Material Model Behavior dialogue box. You will start by defining the hardening law as a rate independent, isotropic hardening, bilinear Mises plasticity formulation. In reality this means that you will need to define two values, the yield stress σ Y and the tangential modulus E T. The figure below shows you how to find the correct material formulation. Set the yield stress to 250 MPa and the tangential modulus to one tenth of the Young s modulus (no, not a tenth of the yield stress!). This is quite a lot, but it will also mean that you get some nice results to look at. Now you will need to mesh the model. Under Preprocessor > Meshing you will find the MeshTool, click on that to open it up. Check the Smart Size box and set the slider to 1. Make sure you are meshing areas, and that you use a free mesh with quad shapes, and then click on the Mesh button. Select the area of the specimen and click OK. If for some reason the MeshTool hangs you can do the same by issuing the commands SMRTSIZE,1 and then AMESH,ALL. Open up the text file and make sure that the load is 120 MPa and that the number of load cycles is set to 7. This means that the solver will solve for 15 load steps, 7 tension loads

10 and 7 compression loads, alternating between tension and compression, and then one unloading, see Figure 5. σ One load step Time One load cycle Figure 5. Load history of the notched specimen. Now you can solve your model. This is done by reading the text file into ANSYS. The bottom edge is constrained in y displacement and the load is applied on the top edge. Wait until it finishes the last load step until you do anything. It might take a little while. You are now to record the values for the maximum equivalent plastic strain for each load step. This is a measure of the accumulated plastic deformation that has occurred during the load history. Start by looking at the first results set and record the maximum equivalent plastic strain on a separate sheet of paper, excel sheet or something similar. Repeat this for each load steps results set until you have a list of 15 values for the equivalent plastic strain, one for each load step. You will have to hang on to these values until you have solved the problem again, but with a different hardening law. The easiest way of doing this is by using the General Postproc > Results Viewer, where a sliding bar can be used to change between load step results.

11 Now you will do the same analysis all over again but this time with a kinematic hardening law instead. Except for that, everything else should be set in the same way as you did before. Which of these hardening laws would you expect to give the highest accumulated plastic strain at the end, and why? Now you can plot your results in the designated graph area of the answer sheet.

12 PART C SPHERICAL INDENTER ON FLAT SURFACE. The last part of this lab concerns the modelling of a spherical steel ball pushed into a flat steel surface, as seen in Figure 6. To make matters a little less complicated and to avoid the contact problem between the ball and the flat surface, the ball will be simulated by application of a Hertz contact pressure instead. What you will investigate in this part is what the critical force is that will initiate yielding in the steel block, and also plot a graph of the residual stresses after unloading as a function of applied force. F r p 0 2a Figure 6. A sphere being pushed into a flat surface and the resulting Hertz pressure distribution. In this part you will use text files to both generate the geometry and to solve the problem. Download the files PartC-Preprocessor.txt and PartC-Solver.txt from the course web page.

13 In the pre-processor text file the Hertz pressure calculations are implemented and the model geometry changes as a function of the applied force F. The Hertz pressure distribution is a half circle with radius p 0 according to the equations below /, /, 2 1 1, Now use the command File > Read Input from and make ANSYS read the preprocessor file to generate the geometry and mesh. Look at the geometry and mesh. It is emphasized that the meshed part represents the flat surface, that the ball is not included but modelled by a contact pressure instead and the curved boundary is supposed to represent infinity (quite well actually...). The geometry is very similar to the geometry in Part A, but there is a distinct difference. Both are 2D but this one is axisymmetric about x = 0 instead of having the plane strain formulation. Since the geometry is more structured compared to Part A the mesh can more easily be controlled. As in Part B, ANSYS understands that the line x = 0 is a symmetry line and no symmetry condition needs to be applied there. Run the solver script and look at the result set at the end of the first load step when the solution is finished. Do this by using the command General Postproc > Read Results > By Pick, which will bring up a list of all the result sets. This time ANSYS will have solved the problem with more sub steps in each load step, and the command OUTRES, ALL, ALL that is issued in the solver file will have made ANSYS write the results for all sub steps to the solution data base. The result set you are to look at is the results that are written for time = 1,00. Look at the von Mises effective stress. What is the maximum value of this? Can you use this to estimate at what F initiation of plastic behaviour will occur? To do this you will

14 need to know that the yield stress is 250 MPa. Call this initiation value F 0 and record it on the answer sheet. Hint: How does yielding relate to the von Mises equivalent stress? What is the relation between the force F, the von Mises equivalent stress and the pressure p 0? What are the values of α and β? Try and solve the model with your value for F 0. Did it give you a small area of residual plastic deformation below the contact point similar to that in Figure 7a below? You can also compare your results to the plot of the first principal residual stress, shown in Figure 7b. Since the model is axisymmetric you can actually bring up a 3D view of the results by doing a symmetry expansion; PlotCtrls > Style > Symmetry Expansion > 2D Axi- Symmetric > ¾ expansion. a) b) Figure 7. a) Residual plastic deformation after loading close to F 0 and then unloading. b) Plot of the first principal stress after unloading. Now you are supposed to solve the model for a few values of F of your own choice and compile a graph of some measure of residual stress as a function of F. What could be a good stress measure to look at, and why? What is it that you might want to know about

15 the stress state after unloading? You are free to choose yourself, but you will have to motivate it. You can also have a look at the remaining deformation. Compared to the other dimensions of the model, this will usually be quite small (unless you give it a real whack, but don t do that). A command that might come in handy then is the Displacement Scaling command accessed via PlotCtrls > Style > Displacement Scaling That s all folks! Until next time, bye!

16 RESULTS FROM PART A INTEGRATED CIRCUIT Threshold temperature for yielding, ΔT 0 : K Maximum residual tensile stress in copper and silicon. (Two curves please!) σ ΔT 0 10ΔT 0 ΔT

17 RESULTS FROM PART B HARDENING OF NOTHCHED SPECIMEN DRAW TWO CURVES, ONE FOR KINEMATIC AND ONE FOR ISOTROPIC HARDENING! ε e pl N steps

18 RESULTS FROM PART C SPHERICAL INDENTER ON FLAT SURFACE Force that will initiate plastic deformation, F 0 : N Residual stress measure, σ res : σ res F F 0