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LayerSlayer_files/LayerSlayerBlanket User`s Manual

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1. y B fh E v a To g k Tb Tt y Temperate mn drop Multilayer A B C D E E ae associated so profile with finite shown if for interface is Cease conductance top location Ey cae kzy Deformation analysis bottom loca layer 2 location 0 Thot a 1 location 1 Ttop Le t gt Pz t 1 K t Interface n Temperature lies between layer n e 0 plane strain and layer n 1 or z biaxial Global view y Energy release rate G S E S E 5 S E S E Interface n lies between layer n and layer n 1 N 0 plane strain X or z biaxial S E Nx Mz is an option for plane strain analysis FIG 1 Schematic illustrations of the geometry variables and assumptions that underlie LayerSlayerBlanket Temperatures are assumed to be uniform in the plane of the blanket i e in the x z plane with gradients occurring only in the through thickness direction i e in the y direction The thermal analysis solves for the steady state distribution through the stack which yields a piece wise linear temperature profile if finite conductance is specified at the interfaces the temperature is not continuous across the interface but rather experiences jumps proportional to the inverse of the interface conductance hence the temperatures on either side of the interface i e bottom of one layer and top of layer beneath it are re
2. 0 Trot h Top h Th hy h Toph a 9 That is a list of absolute global positions in the stack and temperatures at those positions is returned positions are repeated at interfaces such that there are discrete points for the top and bottom of the interface The command ListPlot creates a temperature position plot illustrating any temperature jumps at the interfaces that are a result of finite interface conductance Again when the interface conductance is large the temperatures are continuous across the interfaces NOTE running the thermal analysis command overwrites the layer definition replacing the previous temperatures at the top and bottom of each layer with those resulting from the thermal analysis Hence there is no need to redefine the layer definitions to account for the steady state temperature distribution as the new definitions have the proper values inserted If you would like to calculate the deformation ERR for a temperature distribution that is not steady state simply insert the proper temperatures at the top and bottom but do not run the thermal analysis B Deformation There are two subroutines to calculate multilayer deformation one for biaxial deformation with no loading and one for plane deformation with an applied moment and axial force In this calculation one can specify a uniform strain parallel to the crack front i e constant The biaxial deformation is found with BiaxialDeformation Multilaye
3. location T Q 6 where location 0 for temperature specified at the bottom of the layer and location 1 for temperature specified at the top of the layer Q is the heat flux in the layer defined as positive from bottom to top If the top temperature is higher than the bottom temperature then Q lt 0 Again when interface conductance is essentially infinite it doesn t matter if you specify the temperature at the top of one layer i e the bottom of the interface or the temperature at the bottom of the adjacent layer i e at the top of the interface That is for infinite interface conductance specifying either 1 1 71 or 2 0 71 yields the same result When interface conductance is finite such that there is a temperature drop across the interface you must specify the temperature at the proper side of the interface V SUBROUTINES FUNCTIONS A Temperature Analysis There are two function calls for steady state thermal analysis depending on whether one prescribes two temperatures or one temperature with the heat flux These commands are TempSol Multilayer Inter faceProps BCs 7 TempSo1Q Multilayer Inter faceProps BCs 8 where Multilayer Inter faceProps and BCs are the names of the lists defined earlier E g according to the above example BCs is replaced with TempBCs These commands return a list of position temperature pairs corresponding from bottom to top of the stack That is they return the following
4. interface i lies between the i and i 1 layers In the ClassicERR command the deformation ahead of the crack is assumed to be biaxial curvature Behind the crack the layers are assumed to experience plane deformation with the strain in the z direction set to the value determined from the fully intact biaxial curvature That is it is assumed that debonding relieves the net resultants acting on the two sections behind the crack front but does not relieve the stresses in the z direction This recovers the usual semi infinite substrate result for a blanket film In the BiaxialERR command the deformation is assumed to be biaxial both before and after debonding It is assumed that the intact multilayer experiences biaxial curvature and that the multilayers behind the crack also experience biaxial curvature In the PlaneStrainERR command zero strain is imposed in the out of plane direction both before and after debonding The quantity M refers to the applied moment applied to both ends of the multilayer this moment is applied to the whole stack ahead of the crack and only to the bottom stack i e the bottom stack formed by the debond behind the crack front Plane strain conditions are enforced i e 0 for all sections VI ILLUSTRATION A Layer and stack definition Consider a three layer system with the following layer properties Layer Thickness Modulus Poisson s Ratio CTE Ref Temp Growth strain Th
5. where the top surface temperature is known to be 1300 and the heat flux from top to bottom with is known to be Q 169244 then we would specify TQbcs 3 1 1300 169244 layer number O bot 1 top temp 25 ifc 104 107 26 where a negative heat flux is prescribed between the top temperature is known to be higher than the bottom temperature i e heat flows from top to bottom which is negative by convention C Results from Temperature Analysis The two temperature boundary condition thermal analysis is conducted by executing Tout TempSol Multilayer ifc TwoTbcs 27 which returns the following Heat flux 169294 28 0 800 0 003 820 315 0 003 876 747 0 005 1215 34 0 005 1215 35 0 006 1300 29 Since the heat flux is defined as positive from bottom to top and the top temperature is highest the heat flows in this case from top to bottom which is a negative heat flux according to the adopted convention Note the 56 temperature drop between Layer and Layer2 Here is a ListPlot of the result Top and bottom temp given Temperature 0 000 0 001 0 002 0 003 0 004 0 005 0 006 Position FIG 2 Temperature distribution from the two temperature boundary condition Note thay if the temperature is known at the bottom of Layer2 we could get the same results by running the following boundary conditions twoTbcs 2 0 876 8 3 1 1300 layer number O bot 1 to
6. 0 0 36 elb kib BiaxialDeformation Multilayer 37 where the first command gets the elongation and curvature for the plane strain case while the second command gets the elon gation and curvature for the biaxial case These results are then used to compute the associated stresses in this case in the x direction scaled by MPa Pstresses GetStress eip k1p 0 0 Multilayer 1 10 38 Bstresses GetStress e1b k1b e1b kib Multilayer 1 10 39 0 28 7655 0 003 18 7446 0 003 162 367 0 005 11 5865 0 005 28 9895 0 006 130 041 0 99 3313 0 003 84 399 0 003 133 796 0 005 20 8355 0 005 52 112 0 006 129 011 where the first data set are the stresses assuming plane strain deformation and the second data set are those assuming biaxial deformation ListPlot is then used to plot the stresses in each layer as shown below E Energy Release Rate Results The calculation of energy release rates are simply function calls to the ERR commands outlined above NOTE If a thermal analysis has been run the calculation will correspond to the ERR at the steady state temperature distribution If you want the ERR at a different temperature distribution you have to simply directly specify the top bottom temperatures for each layer by re writing the layer definitions The following command makes a list of ERRs at all the interfaces Gc Table i Classic
7. ERR Multilayer i i 1 Length Multilayer 1 Here is a ListPlot of the ERRs corresponding to three different deformation assumptions Stress MPa 0 0000 0010 0020 0030 0040 0050 006 Position FIG 3 Stress distribution at elevated temperatures from steady state thermal analysis in the x direction The dashed lines are the stresses assuming biaxial curvature is the deformation state while the solid lines are the results from a plane strain analysis 1 0 1 2 1 4 1 6 1 8 2 0 Interface number FIG 4 ERRs at the two interfaces for the example problem according to different deformation assumptions blue ClassicERR where the stress state is biaxial ahead of the crack and only the x direction stresses are relieved by cracking purple BiaxialERR where the deformation state is biaxial both ahead of the crack and behind the crack gold PlaneStrainERR where the out of plane strain is zero ahead of the crack and behind the crack
8. Ei Ey Ky 13 using the appropriate values for the elongation and curvature for the given direction The command that computes stress is GetStress K amp K z Layer dir scale 14 where K define the total strain in the x direction while define the total strain in the z direction Layer is the property list dir 1 for Oy and dir 3 for z the variable scale is simply a scale factor that allows you to recover appropriate 6 units E g scale 10 yields stresses in MPa if all the units quantities are in base SI units The command returns an ordered list with each entry being a list of the position and stress at the bottom and top of the layers as in Aneel Thon Sbor Diop Stopt ath 15 where the positions e g Vis are measured from the bottom of the stack Thus using ListPlot with the output of GetStress produces a plot of the stresses in the layers with a different linear curve representing the stress in each layer See examples D Energy Release Rates There are three commands to calculate the energy release rate for debonding ClassicERR Multilayer 16 BiaxialERR Multilayer 17 PlaneStrainERR Multilayer Mg 18 In all commands Multilayer is the name of the list that has the properties of all layers in the stack while the entry denotes the interface number where debonding occurs The interfaces are numbered from the bottom of the stack such that
9. LayerSlayer Transient LayerSlayerBlanket User s Manual Matthew R Begley Mechanical Engineering University of California Santa Barbara Dated 24 December 2012 This document provides user instructions for the multilayer analysis code LayerSlayerBlanket The framework is general for an arbitrary number of layers which are assumed to be elastic isotropic layers described by a modulus coefficient of thermal expansion thermal conductivity a reference strain free temperature growth strain and the temperatures at the top and bottom of each layer which can be solved for using steady state thermal analysis The code predicts the following e steady state temperature distribution through the layers for either two prescribed temperatures or a prescribed flux with one temperature e deformation of the multilayer assuming plane strain or biaxial deformation stresses at the top and bottom of each layer steady state solution is a linear stress distribution through each stack energy release rate for debonding assuming biaxial curvature ahead of and behind the crack e energy release rate for debonding assuming biaxial curvature ahead of the crack and zero stress in the direction of the crack but non zero stress in the out of plane direction dictated by the curvature ahead of the crack e energy release rate for debonding under applied moment assuming plane strain deformation 0 everywhere GETTING STARTED Laye
10. a thermal analysis step is conducted the layer definitions are overwritten such that T T correspond to the results of the steady state analysis the other properties are not changed If desired one can re define the reference temperatures to be those associated with the steady state temperatures found via the thermal analysis and re set the current temperature to be the temperature of interest The default is that the reference temperature remain unchanged after thermal analysis while the current temperatures are set to those solved for in the thermal analysis B Stack multilayer definition The multilayer is defined as a list of layers as in Multilayer Substrate BondCoat Coating 3 where Substrate BondCoat and Coating are previously defined layers The order must be from bottom to top i e the order in the Multilayer list indicates the relative position of the layers Note that you can define the multilayer with any name involving layers with any name e g CaseA Layer1 Layer2 For the thermal analysis the analysis incorporates interface elements with zero thickness to allow for temperature drops between layers associated with finite values of interface conductance For the thermal analysis commands one must provide a list of interface conductances with the numbering ordering being from bottom to top e g Interfacesconductance k1 k 4 where k is the interface conductance between layers 1 and 2 k is the in
11. ated to applied moments forces This fundamentally determines the elongation and curvature of the stack the stresses and strain energy densities are then computed using those deformation variables and the above strain definition A variety of scenarios can be addressed with regards to the behavior parallel to the crack front including plane strain deforma tion 0 or biaxial deformation In the classic ERR computation it is assumed that cracking alleviates curvature about the z axis whereas the curvature about the x axis is fixed to be the same ahead of the crack Il LAYER AND STACK DEFINITION A Individual layer definition Each is layer given a name and the list of its properties defined according to Layer h E v a T T k Tp Ti 2 where Layer is the name of the layer can be anything A is the layer thickness F is the layer modulus v is the layer s Poisson ratio T2 T are the bottom and top reference temperatures for that layer i e there is a linear distribution of reference temperature which defines the zero thermal strain temperature for that location is the growth strain in the layer k is the thermal conductance of the layer and 7 T are the current temperature of the layer at the top and bottom of the layer The order must be as prescribed above The temperature distribution in each layer is linear as would result from a steady state thermal analysis If
12. ermal conductance Bot Top Temp Layer1 0 003 200x10 0 3 15x10 70 70 0 25 0 0 Layer2 0 002 40x10 0 2 11x10 70 70 0 1 0 0 Layer3 0 001 100x10 0 2 11x10 70 70 0 2 0 0 The temperatures at the top and bottom of the layer are set to zero since they will be calculated by the thermal analysis In the above the reference temperatures are uniform and equal to room temperature The multilayer is defined as follows Layer1 0 003 200 x10 0 3 15x10 70 70 0 25 0 0 19 Layer2 0 002 40 x10 0 2 11x10 70 70 0 1 0 0 20 Layer3 0 001 100 x10 0 2 11x10 70 70 0 2 0 0 21 Multilayer Layer1 Layer2 Layer3 22 Thus the layers are named appropriately from bottom to top i e Layerl is on the bottom and Layer3 is on the top B Thermal boundary conditions First consider the case where the temperature at the bottom of the stack is fixed to be 800 while the temperature at the top of the stack is fixed to be 1300 Let s assume that the interface between Layer1 and Layer2 has finite conductance such that there is a temperature drop across that interface the interface between Layer2 and Layer3 has a large conductance nearly perfectly conductive such that there will be no temperature drop This is stated as TwoTbcs 1 0 800 3 1 1300 layer number O bot 1 top temp 23 ifc 10 107 24 Second consider the case
13. mperatures at the top and bottom but do not run the thermal analysis A Prescribing two temperatures The function TempSol1 requires two temperatures to be specified within the entire multilayer stack One can specify tem peratures either at the bottom or the top of any given layer The boundary conditions are specified as TempBCs Layer location T Layer location T 5 where location 0 for temperature specified at the bottom of the layer and location 1 for temperature specified at the top of the layer Again note that you can call this list according to any name you choose I e the names used for lists here are just examples When interface conductance is essentially infinite it doesn t matter if you specify the temperature at the top of one layer i e the bottom of the interface or the temperature at the bottom of the adjacent layer i e at the top of the interface That is for infinite interface conductance specifying either 1 1 7 or 2 0 7 yields the same result When interface conductance is finite such that there is a temperature drop across the interface you must specify the temperature at the proper side of the interface B Prescribing the heat flux and one temperature The function TempSo1Q requires one temperature and the heat flux to be specified One can specify the temperature either at the bottom or the top of a given layer The boundary conditions are specified as TempBCs Layer
14. p temp 30 ifc 10 107 31 That is one can prescribe two temperatures at any locations within the stack including those on either side of an interface with finite conductance Conversely if we knew the top surface temperature was 1300 and the heat flux was from top to bottom with magnitude Q 169 294 we could run the one temp and heat flux be analysis as in Tout2 TempSolQ Multilayerex ifc 3 1 1300 169294 32 0 800 001 0 003 820 317 0 003 876 748 0 005 1215 34 0 005 1215 35 0 006 1300 33 This result is identical to that obtained for the other boundary condition as expected NOTE Now that a temperature analysis has been conducted the layer definitions are redefined with the results of the temperature analysis Typing the following returns the layer definitions in matrix form with each row representing the properties in one of the layers MatrixForm Multilayer 34 0 003 2 x 10 0 3 0 000015 70 70 0 25 800 820 315 0 002 4 x 10 0 2 0 000011 70 70 0 1 876 747 1215 34 35 0 001 1 x 10 0 2 0 000011 70 70 0 2 1215 35 1300 That is the current temperatures at the top and bottom of each layer have been re defined to match the results of the thermal analysis D Stress Results To get stresses in the layer we first run the analysis that gives us the total elongation and curvature of the stack Examples are elp kip PlaneDeformation Multilayer 0
15. r 10 where Multilayer is the name of the stack that was defined previously by creating a list of layers This function returns a pair defined as a Mathematica list that contains the elongation of the bottom of the stack and the curvature of the stack K as in K For the biaxial deformation case these deformation quantities are calculated by setting the net resultant moment and the net resultant normal force equal to zero One can substitute an explicit list of the layers desired for analysis as in BiaxialDeformation Substrate Coating 11 That is you can define the multilayer in the function call itself and avoid having to name define additional multilayers if you want to chop out a layer Similarly there is a separate function call for generalized plane strain deformation in which one prescribes the applied moment the applied axial resultant and the value of the out of plane strain This type of deformation is found with PlaneDeformation Multilayer Ma Na z 12 where Multilayer is the name of the stack currently being analyzed M is the value of the applied moment N is the value of the applied axial force and is the constant value of the strain in the z direction C Stresses Stresses are found by providing the elongation and curvature of the stack in the x and z directions That is one provides and where the direct total strain in these directions is given by
16. rSlayer Blanket is written in Mathematica The source code which defines commands used in the analysis is LSBencrypt which is read loaded into any Mathematica notebook The source code is loaded using the Mathematica commands SetDirectory Users Begley Desktop LSB FINAL lt lt LSBencrypt where the pathway in the SetDirectory command should be set to the name of the folder in which the file LSBencrypt is stored NOTE reading in the encrypted files will generate error messages pertaining to function definitions these can be ignored The file LSB Example is a conventional Mathematica notebook which illustrates how various analyses are conducted Il OVERVIEW The analysis is conducted for a stack of blanket films as shown in Figure 1 The films are assumed to be infinite in the x z plane with a semi infinite crack that grows along an interface that lies on a specific x z plane with a crack tip that is aligned with the z direction energy release rates are computed for the steady state growth scenario i e when the crack is long enough so as to not influence the results The problem is essentially one dimensional such that behavior in the x y plane is analyzed in the third dimension out of plane to the analysis one can either assume plane strain deformation i e 0 or biaxial deformation i e x Layer and stack definition Temperature analysis A h E v a To g k Tb Tt
17. tained as possibly independent variables Temperatures on either side of an interface will be identical if the interface conductance is sufficiently high When analyzing cracking it is assumed that the crack has no influence on the temperature distribution i e the intact stack is analyzed While cracking would naturally alter the heat flow in the cracked region it is assumed that cracking events are sufficiently fast so as to not allow for changes associated with debonding That is the scenario being analysis consists of a steady state crack growing at speeds faster than thermal transport The analysis of initiation of crack growth involving a different temperature profile behind the crack due to debonding requires a 2D thermal analysis to account for spatial gradients in the x direction The key assumption is that the strain distribution in bonded layers is given by Ex y y a T y To y 1 where 3 is the elongation in the x direction at the bottom of the stack K is the curvature of the stack about the z axis is the coefficient of thermal expansion in the layer T y is a reference temperature that has a linear distribution in a given layer and T y is the linear temperature distribution dictated by the temperature at the top and bottom of a given layer This strain distribution is used to compute the resultant moment and normal force on any stack including sub stacks created by cracking which are equ
18. terface conductance between layers 2 and 3 and so forth Again note that this list of interface properties can be given any name as in IfaceEx 10 10 etc Setting the interface conductances to a large number in comparison to the effect layer conductances characterized by k h results in nearly perfect conduction with piece wise continuous temperatures throughout the stack For cracking analyses one must provide to the ERR commands see below the interface of interest for which the energy release rate is to be computed these are numbered with interface n falling between layers n and n 1 Thus the limits on possible interface numbers run from 1 to N 1 where N is the number of layers For the purely plane strain ERR you can also specify applied moments and loads IV THERMAL BOUNDARY CONDITIONS STEADY STATE ANALYSIS Two types of steady state analyses are included distinguished by the thermal boundary conditions that are imposed NOTE running the thermal analysis command overwrites the layer definition replacing the previous temperatures at the top and bottom of each layer with those resulting from the thermal analysis Hence there is no need to redefine the layer definitions to account for the steady state temperature distribution as the new definitions have the proper values inserted If you would like to calculate the deformation ERR for a temperature distribution that is not steady state simply insert the proper te

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