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This 21 page Class Notes was uploaded by Bethany Lindgren on Wednesday September 23, 2015. The Class Notes belongs to MATE460 at Drexel University taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/212240/mate460-drexel-university in Materials Engineering at Drexel University.
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Date Created: 09/23/15
Tom Juliana and Ilan Penrose InslruclorAlavalianlos MATE 460580 class Proiecl June 12002 Introduction amp Motivation Indentation is a technique used to determine material properties such as modulus of elasticity hardness and stiffness In Fig 1 there is a schematic of many of the important values needed to determine these properties There are a variety of standard microhardness indentation tests available but as material feature size decreases on materials such as silicon it is important to be able to characterize these materials and their features at a nanoscale For this reason machines that can create extremely small indentations have been created and the term nanoindentation comes into play It is so named because the resolution of the indent depth can be measured on the order of nanometers however the residual indentation diameter may be either on the order of nanometers or microns depending on the material s hardness P Fig 1 Parameters involved in sunaammru nnman a standard indentation test where P is load hf is residual PMquot 3 1351 Mama depth h is elastic recovery depth hC is the di erence between max depth and hi and J a is the distance from the center of indenter tip to E mmmm projected outermost contact Law I area of indent There are many different shaped tools used for nanoindentation testing The most common is the Berkovich tool which is a threesided pyramid Other shapes used include the conical tool cone with a specified included angle and tip radius and the spherical tool We are currently interested in examining the case where ductile onset of silicon occurs Singlecrystal silicon a normally brittle material may turn ductile and behave in a metallic manner in ambient conditions if sufficient contact pressure is reached above 12 GPa1 and SiII is formed This is especially useful in the machining of silicon since this ductile phase allows for minimum or no surface or micro cracking Therefore sensitive components found on integrated chip devices may not be harmed during machining processes However there currently is no way to measure these contact pressures in situ that is we can only measure the residual stresses in the indentation after it has occurred This is a weakness in understanding how this ductile mechanism behaves For this work we are going to be exploring a simulation of nanoindentation using ABAQUS We will make use ofa 2D mesh of elements and nodes and will look at the cases of indentation with spherical and conical tools This has already been done for general case of spherical and conical contact but the literature found did not address the critical values of pressure in silicon that we are looking for One may ask why we are simulating indentation if the interest is machining and this is because the condition of ductility is found in both machining and indentation and the two may be correlated A 2D mesh is used because of the axisymmetric condition imposed from orthogonal loading of the surface and tool geometry The measure of material pressure with included angle for the conical tool case 60 90 and 120 degrees will be looked at and the critical 12 GPa will be paid the most attention Also the case for an indentation done with a spherical radius tool will be seen The role that mesh size plays on convergence of resulm will be examined as well as critical pressure areas existent at the point of contact under the indenter tip Modeling Considerations For all of the models created there were different considerations that needed to be addressed First we decided that due to computation time the maximum mesh size should be 24x24 or 625 nodes unlike meshes made previously by other authors Fig 223 At a 29x29 mesh the computer starts to take excessively long to complete analysis Second each model needed to include a bias value such that the elemenm closest to the point of indentation were the most spatially dense and tapered out from the point of contact A schematic of this is shown in Fig 3 Third we assumed that the silicon being indented was isotropic Silicon in actuality is not but values in each direction are not far from each other So the modulus of elasticity was set to be 180 GPa Poisson s ratio was taken to be 02 and yield stress was taken as 7 GPa from literature and also experimental data Fourth we assumed that the indenter tip is rigid since it is typically made of diamond and has minimal deformation as compared to the material Fifth we constrain the indenter tip to the node at the upper left corner of the mesh for the case of only indentation but for the case of indentation and retraction these nodes are free from each other Last we ensure that the total width and depth of the mesh is at least three times in magnitude as the projected indentation diameter This factor is determined from experimental observations 1 x i e 55 w s s 5 V amp w Fig 39 Arrangement of elements N13 000 used for previous indentation simulations Fig 3 Sample 19x19 element mesh Simulation Results 7 Indentation Sixteen different simulations dealing with only indentation and no retraction of the tool were run All meshes were 30x30 pm in dimension and indentation depths are 2 pm In all simulations with the 60 degree included conical angle the tip had to be blunted to a small extent to evade ABAQUS constraint errors A chart that summarizes the simulation logistics and the corresponding simulations are shown below Of particular interest is the reported zone of ductile transformation which is above a pressure of 12 GPa The transformed zones are shown in red and the darkest blue regions are below 300 MPa tion Angle a1 Radius Table I 39 Summary of indentation simulation parameters Sim liTip atis 02um Sim 47Tip atis 04um Sxmuhhm Results r1deth deeuuum Sxmuhhmswue 51m catnedaulfm the use am 145 ags mduded mg mm mama bang mdemed mdxetxacud mm m mat1151 Th2 depth uf mdemum enms cauldmtbe xesalved became 5 lug deg af eedam h maternal mahm m md m Enable 5 mm xetxachm fth indent1 Th2 results m z a m Shawn b 1 M mss xeadual messes because n 1915125 magma messes as We are m hydrang messes mm sleace fth maternal place mam a znndznuhm 5m Drscussmpuhcxusuhs Accaningm mx smuhnms wefmd a cumergehce uf area e enedbypzesme As canbe 52m faxmast cases 39he l xl meshyulds remrvexy paanesul39s whzn cmnparedtathz 25x25 mesh Huwwer ch gmual femxesfamdm39heZSxZS mesh cm alsabeseenmthz 1qu meshes Lhzrefme camergmceheghswhehusru mesh use ee r a where pressure reachst muss Vesue um GPA Hawever farm m x xamanxesul39s n S rhcreese asLhz mcluded mge unhe camcal 1ndm39zrs rhcreese Fxm y Lb rune rs deepestabaml m fax ch use unhe sphech mam39er These maehuuuh resuus ere m fact m er cenem ageemmththwhnt hes heen uhserveu erpemmemmyrur Lb case uf uymmrc scre39ehhg Anhuugr we are we able tameamethz pressuresmum we cm see ewdznce uf use kmsfmnahmsthat ere 39 3 E T E E g 2 E i E E F 39he use afmdenlman and museum we see 39hu Lb reams suesses ha uh Lb Mac are uh39he uruer urzes GPA These uehghznhmwhathasbezn expenmema y uhserveuthrm mess elds azde erkumchmuemueuhs Fxg 5 Hawwar mx made uhes manure mm accmmt Lb Sen ta Sum unusual phese urme sch s manmthzmntenal m an as am Haweverbmhmadzlsmuncampleubecamethzydamnakemtaaccaml ssure andwhxne changes assauaudw hphase kmsfmnahms Fume damn by experimental xesults cm W Pyrimldu mm mm mm th Berkawch mdentatmnx m xtltcan Reference 1 M Ekman K Persson G Grimvall Physical Review B 62 2000 147849 2 S Carlsson and P Larsson Acta Mater 49 2001 217991 3 S Suresh A Giannakopoulos J Alcala Acta Maren 45 1997 130721 4 Y Gogotsi G Zhou S Ku S Cetinkunt Semicond Sci Tech 16 2001 34552 Appendix A Input File This section includes a reproduction of the input le with explanations for the commands All commands are preceded by an asterisk and the arguments of the commands are directly beneath them Any line preceded by a double asterisk is a comment line and is disregarded by the modeling program Additional explanations will be preceded by a dash Certain terms should be de ned before discussing the input le The term node refers to a discrete point in space where the analysis should be performed An element refers to the material between nodes The meaningful de nition of nodes comes from an understanding of finite element analysis In nite element analysis equations are chosen which will best represent the behavior of a system with respect to a variable For example the behavior of a plate under heating with respect to temperature at each coordinate is a nite element analysis The coordinates in that example are the nodes The important part about nite element analysis is that equations are chosen which have no basis in physics but are chosen numerically to reduce residual values between real and modeled experiments HEADING Silicon Indent The heading on the program output NODE 1 0 0 15 30 0 225 3030 21 1 030 300 00530 400 0052995 500 02995 This command de nes the absolute coordinates ofthe nodes 115 225 211 300 400 and 500 The rst four of these nodes form the comers of the mesh while the final three will be used later to form a very small element directly under the indenter NGEN NSETBOTTOM 115 1 NGEN NSETTOP 21 1225 1 NGEN NSETRIGHT 1522515 NGEN NSETLEFT 121 115 NGEN NSETREDLEFT 119615 The preceding 5 commands create and de ne the node sets BOTTOM TOP RIGHT LEFT and REDLEFT The node set TOP is the top surface of the mesh The case is the same with the right left and bottom of the mesh The node set REDLEFT contains all nodes on the left hand side of the mesh but not the node directly under the indenter This is the case because if that element is too constrained then there will be overconstraint errors in the evaluation of the model NFILLNSETALL BIAS1 15 BOTTOMTOP1415 NFILLNSETALI BIAS 8 LEFTRIGHT 141 These two commands ll in the mesh vertically and horizontally to give a full mesh The command bias provides for a larger number of nodes at one end of the mesh as opposed to the other This option was chosen because more information of interest is to be found near the indenter tip instead of far away from the indentation site This provides both a vertical and horizontal bias for the mesh ELEMENT TYPECAX4 1000500400300211 ELSET ELSETCENTER 1000 SOLID SECTION ELSETCENTER MATERIALSILICON ELEMENT TYPECAX4 l121716 ELGENELSETPLATE l14lll415l4 SOLID SECTION ELSETPLATE MATERIALSILICON The preceding commands define the elements and their materials This is done twice once for the small element 1000 directly under the indenter tip and the other for all other elements in the set The material is defined as silicon MATERIALNAMESILICON ELASTIC 180E3 02 PLASTIC 7000 The previous lines define the material properties of silicon First for the elastic properties a modulus of 180 GPa is chosen and a Poisson s ratio of 02 These values have been reported previously and have been confirmed experimentally using nano indentation The PLASTIC command argument is the yield stress of silicon This value ranges largely from article to article but this value appeared often and some list it as a preferred value NODE NSETRIGID 999903 1 This defines an additional node 9999 which will be used to control the motion of the rigid body which is the indenter RIGID BODY ANALYTICAL SURFACEINDENT REF NODE9999 SURFACE TYPESEGMENTS NAMEINDENT START 6 32 LINE 030 LINE 632 The previous two commands define the indenter rigid body The indenter is conical so in this 2D simulation can be modeled as two lines SURFACE NAMESURF TYPEELEMENT PLATE This command de nes the top of the element set as a surface named SURF CONTACT PAIR INTERACTIONSMOOTH SURFINDENT SURFACE INTERACTIONNAMESMOOTH These two commands de ne the surface contact pair SURF and INDENT namely the sample and the indenter It also addresses and names the surface interaction SMOOTH SURFACE BEHAVIOR NO SEPARATION This command even though inoperative defines the behaviour of the contact pair Originally NO SEPARATION was specified such that the surfaces would not separate during the indentation When it was decided to examine indentation and retraction this condition had to be removed in order to allow realistic retraction of the indenter without the surface sticking to the indenter FRICTION 005 This command describes the friction behavior of the contact pair A low value was chosen since the sample is highly polished and the indenters are usually finely ground BOUNDARY REDLEFT XSYMM This command specifies the boundary condition for the left of the mesh excluding the topmost element the one directly under the indenter The argument XSYMM states that there is a line of symmetry in the xdirection along the node set REDLEFT EQUATION gtxlt 2 211ll99991l gtxlt 2 2112l99992l This command constrained the motion of the indenter and the node directly under the indenter to prevent overconstraint errors However this command had to be removed when it was decided to indent and retract RESTART WRITE Command to write the output data to a file STEP NLGEOM UNSYMMYES This command starts the step for indentation allowing nonlinear geometric analysis and an unsymmetrical matrix storage solution STATIC 0001l005 This command is used for the static analysis The three arguments are the initial time increment the total time period and the maximum time increment These arguments are used because the program uses and automatic time step BOUNDARY BOTTOM20 9999 2l5 9999 10 9999 60 The preceding are the boundary conditions for the rst step indentation The rst argument is the nodeset to be constrained the second is the degree of freedom in which the constraint will take place 1 displacement 2y displacement 62 aXis rotation and the third argument is the value OUTPUT HISTORY FREQ1 NODE OUTPUT NSETZRIGID URF OUTPUT FIELD FREQ3 ELEMENT OUTPUT S PEEQ NODE OUTPUT U NODE OUTPUT NSETRIGID URF The remaining lines specify the output to be recorded including stress in the elements and displacement of the nodes END STEP This ends commands for the rst step indentation STEP NLGEOM UNSYMMYES STATIC 0001l005 BOUNDARY BOTTOM20 9999 20 9999 10 9999 60 All is similar except now the indenter controlled by the movement of node 9999 will end up in the initial position of 0 OUTPUT HISTORY FREQ1 NODE OUTPUT NSETZRIGID URF OUTPUT FIELD FREQ3 ELEMENT OUTPUT S PEEQ NODE OUTPUT U NODE OUTPUT NSETRIGID URF END STEP The nal step is completed and the program ends mmonucnow Ralhng empssuuh 1 s wxdzly mined mammal pmcess by wheh s maternal m s pmquot sh xs cmnpacud e um Rnl y 5 m ch pmcessng Fxgu39e 1 Shaws s typical puwueh campamanpmcess Fxgu39e 1 rTypxcal eumpsshuhpmeess paw x hm hg mush awahxmy uques s pmuwwuh s xnfmn shape shes snuws far very puns mxxmre emu Ralhng campamm 5 s ample pmcess but calculahms hf 39he stress musuevexycamplex Fxguel Shaws s lypxcal mumg setup uhur symmeuy Fxglxe 2 e Rallmg empsmuh Sehemsue Huewe ssh see same huh pmeesshg pumems fax mumg eumpsshuh as wen as ch awn aw mews hf 39he maternal ha and he e mushse shes exuheegmsxespemve y 115L112 awn cam e R sthz mans shan shes ammt hg mm mun respectively mhzl rm 39hu camphcntes human 5 Lb use hf can he seen m Feguhez eveemususrm afu ydense maternal m andapmmmntuxal 311 The maternal hehsves ensuesuy as lung 551115 euhzuheu w hn Lb laws but whemhe yield camihms hit Lb em lawsthz maternal awsplashes camihms m 39hu material a m mm was u yuldsby me My rule far that material In am as gum s bmex lad15Lng huh mumg eumpsshuh pmeess sh expheu made hss heen ueu m gm AEAQUS far he smuhuuh paramemc study hf variables sssuuueu wuh mllmg campachm MODEL cowmucnow In order to gain a better understanding of the compaction process a set of ABAQUS models was created with different processing parameters in order to note the effects of these parameters on the nal compact These models are based on the standard two dimensional rolling compaction model for fully dense materials given in the examples in the ABAQUS documentation This model is a 2 Dimensional 2D half symmetry model The model given in the example was one that had a rigid surface acting as a roller with the material moving under the roller and being deformed through this process A schematic of this can be seen in Figure 4 Figure 4 Schematic of the model given in the examples This model has problems that could lead to exaggerated errors in later calculations The greatest problem is the severe deformation that the mesh undergoes as it travels through the rollers In order to limit the effects of these errors the model was converted from a pure Lagrangian model to a Arbitrary Lagrangian Eulerian ALE model This means that instead of the entire mesh moving through the rollers the mesh stays at a constant in ow and out ow points The material then ows through the mesh therefore allowing the mesh to keep a more consistent shape This is also coupled with adaptive meshing to keep the elements as close to the equilibrium shape as possible Figure 5 shows the ALE model schematic free suriaoe 100 INFLOW Figure 5 ALE mode schematic The next modification made was making the material a porous material yielding under the Gurson model for porous materials and with the parameters used for q1 C12 q3 being 1 l and 1 respectively The effect of varying the yield parameters of the material was not examined in this study so all of these parameters are constant throughout the models Table l is a list of the parameters used in this model 785E03 Relative 500E01 Radius 300E01 m Mass 100E05 time 150E00 sec mu 05 Entrance 002 Exit 001 225 001 m Table l Model parameters m m RESULTS MODEL VALIDITY Now that the model is set up it must be tested to assure that it is working properly Mass scaling is the process of arti cially increasing the density of the material ABAQUS uses mass scaling in order to cut down on calculation time This can introduce errors into the calculation if the mass scaling is too high To check for this the inertial energy of the model must be checked Figure 6 shows the energy pro le for the model Figure 6 7 lnertial Energy of the model As can be seen the internal energy is much greater than the kinetic energy meaning that inertial errors should be at a minimum By looking at the velocity at the left edge or in ow area of the mesh the fact that the model has reached steady state can be determined At steadystate the acceleration inside the model should go to zero Figure 7 shows the velocity pro le of a node at left edge of the model x o lt m z u w z 5 a 39o o z a E Figure 77 Velocity pro le of left edge The fact that the velocity is at a constant at D05 is proof that the model has reached steady state after 05 seconds PARAMETRIC STUDY Three of the parameters for rolling compaction were varied in order to give a better understanding of the compaction process and the role that these parameters play These three studies are Contact Angle Friction and Gap Size The results were the neutral angle or the angle at which the material is moving at the same speed as the roller the clamping force or vertical force acting on the roller the torque felt on the roller and the void volume action VVF which is a representation of porosity Friction The iction plays an important role in the rolling process as it is the iction between the powder and the roller that caused the material to move through the roller The values of the coef cient of iction that were studied were 01 02 03 05 07 09 01 is omitted om most of this study as it did not yield reasonable results As the material is compacted the porosity goes down and it is the iction that feeds the material through the rollers Figure 8 show the how much of an effect that friction plays on the compaction process WF vs Coef ofFriction 4 SHEEN 4 than 3 SHEEN 3 than u 2 SHEEN i 2 than 1 SHEEN 0 1 qurm f o u u 2 u a 1 Coef of Friction Figure 8 7 VVF dependence on friction There is a moderate improvement at 02 though it takes almost an entire second for the material to reach steadystate and the material only experiences a 15 compaction The neutral angle moves farther away from the exit point from the rollers as the friction coefficient rises This can be seen in Figure 9 Neutral angle vs Coef of Friction Neutral Angle deg n n 6 Coef of Friction Figure 9 7 Neutral angle dependence on friction This is intuitive because as the friction goes up the material can be more easily accelerated to the same speed as the roller The clamping force varies along a similarly predictable path and as reaches its maximum as the material approaches full density Figure 10 shows this note that the force is negative because it points down Norrmlize Clanping Force vs Coef ofFriction Clamping Forcelyeild stresslarea Coef of Friction Figure 10 7 Clamping force dependence on friction The torque exhibits similar behavior as seen in Figure 11 Norrmlize Torque vs Coef of Friction Torquelyeild stresslarea Coef of Friction Figure 11 7 Torque dependence on friction Gap Size Gap size is a key factor in the compaction of the material The values for gap size that were studied are 0005 001 0035 006 011 meters Anything above 001 did not fully compact across the cross section of the green body This is seen in Figure 12 39 c Figure 12 7 Variation in porosity across the cross section for a GS0035m b GS006m and c GS011m As seen in the above gure only the outside edge of the compact reaches its inaccurate Contact Angle The contact angle also plays an important role in the compaction of the powders as the contact angle determines the pressure imparted on the compact Figure 13 shows this relationship between the contact angle and the neutral angle ll eulml Angle vs Cnnlzcl Angle culln Mule lam 51 u 15 2e 25 Mental any new Figure 13 iNeutral angle dependence on contact angle Because there is a greater amount of material owing through the rollers The porosity is expected to go down as the contact angle increases which is shown in Figure 14 wr vs Cnnlz Angle Cullrt Angle new Figure 14 7 Contact angle dependence of VVF The clamping force and torque is also representative of the increasing density of the compac CONCLUSIONS The rolling compaction process depends greatly on the friction between the rollers and the material If the friction between the rollers and the material is not great enough the material cannot be taken into the rollers This results in the extrusion of the material with little to no compaction of the powder This can be seen in Figure 1 Frgure 15 7 VVF ofmu0l But even at lower values of friction mu02 and 03 the material was compacted to less than 50 of its initial dens39ty Conversely the increased friction puts a greater demand on the equipment as the clamping force and torque go up The friction could also limit the motion of the particles in real life The gap sizeroll diameter is an important parameter in rolling compaction The apparent critical value for this would appear to be 13 but without further data between GS001m and GS0035 that conclusion cannot be made An interesting trend is seen in the data in Figure 13 where the neutral angle is increasing rapidly with the contact angle But it is dif cult to see this trend with only 3 data points An expanded look into the in uence of contact angle on the parameters would prove bene cial in providing a trend Overall the model that I have created seems to provide reasonable results for the compaction process
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