Finite Element Method analysis of carrier processing ... · Finite Element Method analysis of...

Finite Element Method analysis of carrier processing flexible substrates

MT 05.06

Bas Adams

Second internship Coach (Philips CFT): dr.ir. H. Zuidema Coach (TU/e) : dr.ir. R.H.J. Peerlings Philips CFT Process Technology Group Eindhoven University of Technology Faculty of Mechanical Engineering Materials Technology Group Eindhoven, December 2004

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Finite Element Method analysis of carrier processing flexible substrates, 14-3-2005 - 1 -

Table of contents 1. Introduction.......................................................................................................................... 2 2. Problem definition ............................................................................................................... 3 3. Model system....................................................................................................................... 4

3.1 Introduction ............................................................................................................... 4 3.2 Geometry ................................................................................................................... 4 3.3 Boundary conditions ................................................................................................. 5 3.4 Computational aspects .............................................................................................. 5

4. Material properties............................................................................................................... 6 4.1. Introduction .............................................................................................................. 6 4.2. Silicon wafer............................................................................................................. 6 4.3. Pressure sensitive adhesive...................................................................................... 6 4.4. Polyimide foil ........................................................................................................... 8 4.5. Indium Tin Oxide (ITO) .......................................................................................... 9

5. Reference model ................................................................................................................10 5.1 Introduction .............................................................................................................10 5.2 Abaqus versus Marc-Mentat...................................................................................11 5.3 Overlay.....................................................................................................................12 5.4 Stresses.....................................................................................................................14 5.5 Typical results (verification) ..................................................................................15

6. Overlay sensitivity.............................................................................................................16 6.1. Introduction ............................................................................................................16 6.2. Reduction of variations..........................................................................................16 6.3. Sensitivity results ...................................................................................................17 6.4. Response Surface Model (Kriging).......................................................................18

7. Stress sensitivity ................................................................................................................20 7.1 Introduction .............................................................................................................20 7.2 Critical values ..........................................................................................................20 7.3 Modelling and reduction of variations...................................................................21 7.4 Typical results (verification) ..................................................................................22 7.5 Sensitivity results ....................................................................................................22

8.Removal of flexible substrate from carrier .......................................................................24 8.1 Introduction .............................................................................................................24 8.2 Methods to numerically remove the flexible substrate .........................................24 8.3 Delamination and failure behaviour.......................................................................25

9. Conclusion and recommendation .....................................................................................27 9.1 Conclusions .............................................................................................................27 9.2 Recommendations ...................................................................................................27

References ..............................................................................................................................28 Appendix A. Determination of the parameters of a Maxwell model .................................30 Appendix B. Microsoft Excel layout for overlay sensitivity ..............................................33 Appendix C. User Manual to add layer on deformed mesh................................................34 Appendix D. Microsoft Excel layout for stress sensitivity .................................................42

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1. Introduction Display technology has experienced a major growth during the last five decades. It all started with the black and white CRT (Cathode Ray Tube). Rapidly new developments such as color CRT, wide screen, LCD- (Liquid Crystal Display) and plasma screens were introduced. The latter two are nowadays most popular due to their thin design. Currently a trend can be observed of sizing the display to thinner ones. To comply with this market request a switch to flexible displays is needed. More design freedom becomes available (roll up of digital newspapers, futuristic advertisement); they are less space consuming and might be cost saving in mass production (plastic substrates versus high-quality glass of LCD). Startup companies, universities and the existing companies developed various concepts to add the parameter flexibility to the display technology. Alien Technology [11] created a Fluidic Self Assembly process, which takes tiny trapezoidal integrated circuits (IC’s) in a liquid and pours the mixture over a substrate with holes (connected by screen-printed electrode) in the desired positions. FlexICs [11] has developed a process that forms a polysilicon layer at temperatures below 100 degrees centigrade, which can be used to build semiconductor components including transistors for active-matrix displays. E-ink [4] uses tiny, charged, colored particles floating in cells. Depending on the electric charge applied to each cell, the particles are attracted to the surface or to the bottom. A different route to bistable displays comes from the French company Nemoptic [11] that has developed an approach using standard liquid crystal material. This material normally loses its image when the electrical charge is removed. The company has developed a way to break the alignment bond of the bottom layer of the liquid crystal substrate, so that it can come to rest in its alternative state. Hitachi, IBM, Mitsubishi and also Philips [11] are experimenting with plastic semiconductor materials. These materials are flexible and can be screen-printed directly onto a plastic sheet to create an active matrix. With all these possibilities of producing a flexible display it will be a competitive game between the startup and the existing companies. Philips [16] developed flexible PolyLED (polymer light emitting diodes) as their standard. On an ITO-coated (Indium Tin Oxide) substrate a matrix of ISO (Indium Sulfide Oxide) is applied using lithography. A cathode is deposited on the ribs of the matrix and enables to transport electrons from anode to cathode. The matrix cells are filled with a hole injection polymer and emissive polymer (serves as the light emitting source when an electron is recombined with a hole). In the end a foil is laminated on top of the stack to shield the flexible display from the environment. During processing brittle layers like ITO have to resist repetitive loading and straining. In this internship report some of these problems are discussed and an onset towards a solution can be found.

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2. Problem definition In theory, during processing steps different aligned layers are applied to the foil. These layers have to be aligned properly to create a functional display. However, the thermal expansion coefficients of all these layers are not equal. Consequently, during processing, non-aligned layers will appear which could cause non-functionality of the flexible screen. Moreover, stresses will be the result of the different deposition temperatures. Therefore, the following problem definition is formulated. The alignment problems of the different layers due to expansion coefficient differences, so called overlay, will be discussed in chapter 5. This is preceded by defining the model system and the material properties in chapters 3 and 4. The foil is glued onto the silicon carrier to have a stable carrier during processing. This alignment problem is a source of inaccuracies with respect to the different layers and connections. Therefore a so-called Design of Experiments (by varying parameters finding an optimum combination) within a Finite Element Method (FEM) package is done in chapter 6 to minimize the overlay. In chapter 7, a functional layer (here Indium Tin Oxide) is applied onto the foil to investigate the failure properties of these extremely thin and brittle materials. FEM-analyses are used to check whether or not the stresses stay below the critical values. Otherwise one can conclude where and why failure occurs. Again a Design of Experiments is done to define the optimum process parameters. A choice for a particular carrier-glue-foil system can now be made based on the sensitivity analysis performed. For this configuration a method to numerically remove the foil from the carrier-glue system has been studied and is reported in chapter 8. After removing the foil from the carrier the strain / stress distribution in a (typical) functional layer can be studied. High-risk areas for failure can be determined and a conclusion of the functionality of the display can be drawn.

Qualify phenomena that exist during processing flexible substrates using a carrier and define routes to minimize / solve these by means of variations of the different parameters of influence.

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3. Model system

3.1 Introduction Within the scope of this internship it is not possible to develop a full model of the processing of a flexible display. Therefore three particular processing steps will be studied:

• Drying and baking of the planarisation layer; • Deposition of the first functional layer (ITO); • Removal of the foil with ITO from the carrier.

In this way the flowchart and the number of layers is reduced. However, knowledge of the process can be gained.

3.2 Geometry The geometry of flexible substrates during processing can be seen in figure 1.

Figure 1. Geometry and a flexible screen on foil

Here one can conclude that the flexible substrate does not have a regular shape and therefore in fact has to be modelled 3D. For this a complicated mesh and a lot of calculation time is needed. Therefore, a circle is assumed and axisymmetric modelling becomes feasible.

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3.3 Boundary conditions The assumptions made in 3.2 provide two mechanical boundary conditions:

• Fixed point to hold carrier on a predefined place; • Displacement restriction in radial direction on the axisymmetric axis.

During processing different materials are deposed and (pre-) baked onto the substrate. After deposition they are cooled to a reference temperature and the surface is cleaned and dried. Thermal-mechanical loading will be present in all these steps. A thermal boundary condition is therefore defined.

• Edge (thermal) film, outside surface is exposed to heat and by means of conduction the material is heated. A convection coefficient and an environmental temperature are defined in each specific step

An overview of all boundary conditions is given in figure 2.

Figure 2. Boundary conditions

3.4 Computational aspects The specifications for the finite element model are enumerated below:

• Coupled thermal-mechanical analysis • Thermo-mechanical axisymmetric solid element types • 4-node elements with full integration • 1500 elements with bias factor in positive radial and axial direction of 0.5 • Large strain formulation • Updated Lagrange • Material properties are isotropic

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4. Material properties

4.1. Introduction In this section all materials used in the report are discussed and the typical values for the thermal and mechanical properties are given. These materials do not vary much with changes in chemical composition, heat treatment and mechanical working [5]. For the present purposes, it is therefore reasonable to work with single values of the properties, which are typical of those properties. This assumption makes theoretical verification of the significant results easier, which enables to understand typical phenomena more easily.

4.2. Silicon wafer The melting temperature of silicon lies far above the temperature range considered so it is assumed to be isotropic and elastic during processing. The typical values of silicon [1] are presented in table 1.

Table 1. Material properties of silicon

Thermal properties Typical value Thermal expansion coefficient 5.2·10-6 [K-1] Conductivity 84 [W·m-1·K-1] Heat capacity 760 [J·kg-1·K-1] Mechanical properties Young’s modulus 11·1010 [N/m2] Poisson’s ratio 0.26 [-] Mass density @ 293 [K] 2300 [kg/m3]

4.3. Pressure sensitive adhesive Since the glass transition temperature [15] of the glue (PSA610) is within the temperature range considered, a complete isotropic visco-elastic description is needed. Visco-elasticity cannot be modeled with a constant Young’s modulus since dependencies of time and temperature are present. Therefore some more attention is paid to this topic. The term visco-elasticity relates two kinds of behavior, namely: Elasticity: material that regains its original shape and size after removing the load due to the stored elastic energy. Viscosity: material that does not recover to its original shape and size after removing the load due to the dissipated energy. These two materials can be described by a spring and a damper (the latter is time- dependent). Combining the two will result in an example of a simplified description of visco-elastic behavior, see figure 3.

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Figure 3. Material models

To describe the visco-elastic glue in a proper way the generalized Maxwell model [12] is often used.

Figure 4. Generalized Maxwell model

The generalized Maxwell model ( ),E t T is easily determined and has a number of time constants to characterize the viscoelastic material response [9].

( ) ( ), i

tT

ii

E t T E E eτ∞= + ⋅∑ , ( ) ( )i

ii

TT

Eη

τ = and ( ) ( )( )ν+⋅

=12

tEtG (1)

Table 2 gives the parameters of PSA610 which have been determined experimentally (see Appendix A)

Table 2. Generalized Maxwell model parameters [15]

Figure 5 shows measurements that are valid for visco-elastic materials. During heating the glue becomes more viscous, see figure 5 bottom, and energy will be dissipated. It will behave like a rubber or at even higher temperatures as a viscous fluid. During these high temperatures the glue acts for example like it has the shear stiffness of water.

Mode i

Relaxation time

iθ [s] Relaxation

modulus Gi [Pa] 8 4.3122·101

1 7.6312·101 5.9976·103

2 4.0733·100 1.2519·104

3 2.3836·10-1 2.5252·104

4 9.9720·10-3 1.0383·105

5 5.6848·10-4 4.2931·105

6 1.4835·10-5 3.2022·106

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Figure 5. Results of a mechanical and dynamical analysis of a dynamical excitation in general [6]

Other mechanical and thermal properties of the adhesive are summarized in table 3.

Table 3. Material properties of the adhesive.

Thermal properties Typical value Thermal expansion coefficient 4.2·10-6 [K-1] Conductivity 0.15 [W·m-1·K-1] Heat capacity 1100 [J·kg-1·K-1] Mechanical properties Poisson’s ratio 0.49 [-] Mass density @ 293 [K] 1200 [kg/m3]

4.4. Polyimide foil The transition from the glass phase to the rubber phase occurs in Kapton between 360°C and 410°C [3]. Since the temperature range considered is below 360°C, Kapton can be modeled as an elastic solid. Again the typical values [3] are given, see table 4.

Table 4. Material properties of Kapton

Thermal properties Typical value Thermal expansion coefficient 20·10-6 [K-1] Conductivity 0.12 [W·m-1·K-1] Heat capacity 1090 [J·kg-1·K-1] Mechanical properties Young’s modulus 2.5·109 [N/m2] Poisson’s ratio 0.34 [-] Mass density @ 293 [K] 1420 [kg/m3]

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4.5. Indium Tin Oxide (ITO) Indium Tin Oxide is a transparent conducting material that is usually used as a thin coating. The material properties of ITO have been experimentally [9] determined and are listed in table 5. Also here the material can be assumed to be an elastic solid because of a very high tensile strength.

Table 5. Material properties of ITO

Thermal properties Typical value Thermal expansion coefficient 7.6·10-6 [K-1] Conductivity - [W·m-1·K-1] Heat capacity - [J·kg-1·K-1] Mechanical properties Young’s modulus 1.0·1011 [N/m2] Poisson’s ratio 0.30 [-] Mass density @ 293 [K] 7.14·103 [kg/m3]

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5. Reference model

5.1 Introduction The processing of the flexible substrates occurs on a stable underground, namely a silicon wafer. A glue (PSA610) is spin coated on the wafer and a polyimide foil (Kapton) is laminated on top. These three layers are defined as the reference model. The carrier-glue-foil model is shown in figure 6. It is repeated here from chapter 3 that the problem can be assumed to be an axi-symmetric one.

Figure 6. Shape of the carrier-glue-foil system and a cross section

The accompanying dimensions are presented in table 6.

Table 6. Thickness of the different layer of the carrier-glue-foil system

Material Thickness [m] Radius [m] Foil Kapton 50·10-6 75·10-3 Glue PSA610 4·10-6 75·10-3 Carrier Silicon 700·10-6 75·10-3

This reference model will be subjected to the production steps as stated in section 3.1. Consequently it will undergo a temperature sweep in the range of 20°C to 180°C. So a typical heating and cooling step is chosen to gain insight in the phenomena like overlay, see table 7 and figure 7.

Table 7. Thermal load to the edges of the carrier-glue-foil system

Time Description Overall convection Ambient temperature

0 – 600 s The stack is placed in a convection oven with a

mild breeze

30 W/m2K at the outside of the

product 180°C

600 – 1200 s The product is naturally cooled in a room

13 W/m2K at the outside of the

product 20°C

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Figure 7. Temperature versus time in the carrier-glue-foil system

5.2 Abaqus versus Marc-Mentat So far a FEM-model of the reference model is available within the Abaqus package. During this simulation hourglass elements arose, which made the results unreliable. In this section the reference model is built within the FEM-package Marc-Mentat and compared with the Abaqus model. Figure 8 shows the result.

Figure 8. Difference in displacement and temperature of the foil in time between the FEM-packages

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The difference in displacement of the foil between the two FEM packages is approximately 12 %. Possible causes for these differences are:

• During the simulation in Abaqus so called hourglass elements arose which influence the accuracy of the results

• The number of elements in the Marc-mentat model is lower. Bias factors to the glue and end of the carrier-glue-foil system are used for compensation. The influence of the number of elements on the accuracy is small and does not justify the increased CPU-time. A little less accuracy (0.05%) results in a 30 times faster FEM-analysis.

• The Abaqus model uses fewer time-steps during heating and cooling. Steps are concentrated at 180°C, this makes the simulation during the cooling and heating stages not very accurate.

The influence of different parameters on the dimensional stability and the accompanying displacement of the foil were studied within Abaqus [8]. These parameters were:

• Polymer foil thickness • Foil material • Carrier material • Processing temperatures • Processing speeds

Also for comparison the influence of these parameters were calculated with Marc-Mentat. Results are given in table 8.

Table 8. Results of the parameter variation on the displacement of the foil (in µm) in the end as a result of the temperature sweep

Foil

thickness (100 µm)

Foil material (Teonex)

Carrier material (Glass)

Process temperature

(125 °C)

Heat flux (20

W/m2K) Abaqus 58.6 46.8 55.4 67 66.9 Marc-Mentat 54.2 31.4 48.2 58.4 58.4

Difference 7.5 % 32.9% 13 % 12.8 % 12.7 % Since the Abaqus model shows hourglass elements and the results are not quite similar the Marc-Mentat model gives more confidence. Therefore the Marc-mentat model will be used.

5.3 Overlay Overlay can be defined as the result of alignment problems of the different layers due to expansion coefficient differences. The definition of overlay is clarified in figure 9 and calculated using equation (2).

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Figure 9. Definition of the overlay problem

41 nodenode xxoverlay ∆−∆= (2) The displacement of nodes 1 to 4 is plotted in figure 10. Using equation (2) the overlay versus time is calculated. The figures clearly show the heating (0 – 600s.) and cooling (600 -1200s.) stages.

Figure 10. Displacement of different nodes and the overlay versus time

The overlay phenomenon probably arises due to the fact that the glue gains stiffness in a very short time and prevents the foil from shrinking further to its equilibrium. Because energy was dissipated in the glue it will not come back to its initial state. So during heating (more viscous) one can expect low stresses and during cooling the stress has to build up to its final value. Figure 11 shows this aspect clearly.

Figure 11. Stresses in the glue at r = 0.07m. during processing

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In the first 50 s. the glue is has not yet reached its transition temperature, which results in higher stiffness and higher stresses. This explains the transient behaviour during initial heating.

5.4 Stresses In the end, next to the overlay as an important parameter, the stresses present in the functional layers are important as well. Too high stresses will cause failure phenomena like crack growth and delamination. Therefore stresses at some important locations are investigated. To check the reliability of the stress calculation some figures are created to see if they comply with the expectation. During cooling the glue gains stiffness and consequently shear stresses will be built up in time. At the edge the glue has a direct contact with the environmental temperature and will cool more rapidly. Consequently, it gains more and faster stiffness than closer to the symmetry axis. A peak of shear stress at the end (r = 0.075) of the substrate in the neighborhood of the glue can be seen in figure 12 (red square). The other validation for this peak stress is the presence of an edge effect. There is less shear stress closer to the symmetry axis (yellow, red and green line). The shear stress block clearly explains the sign (negative / positive) of the stress.

Figure 12. Stress on the white line of the carrier-glue-foil system at increment 100

Due to the symmetry axis the radial stresses will decrease radially. This is verified with figure 13. At the end the stresses have to become zero because of the free surface. This can be observed in the figure during the processing steps.

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Figure 13. Evolution of the radial stress in the foil

5.5 Typical results (verification) The model was verified by giving the glue a very small so that the carrier and foil could expand freely. The displacement could then be calculated theoretically using equation (3) and compared with the value of the FEM-analyses. Theory and FEM matched clearly so the glue is causing the overlay.

0lTl ⋅∆⋅=∆ α (3) with: l∆ = elongation of carrier and foil [m]

a = thermal expansion coefficient [1/K] T∆ = temperature difference [K]

0l = initial length [m] Within Philips CFT stresses in multilayer components are often verified with a laminate tool [13]. Material properties are input and bi-axial stresses are the result. In table 9 these values are compared with the axial stresses which resulted from the FEM-program.

Table 9. Verification of FEM-results with laminate tool, reference model is heated from 20°C to 180°C

s axial laminate tool [MPa]

s axial FEM [MPa]

Kapton Bottom -1 -1.16963 Top -1 -1.16963 PSA610 Bottom 0 -0.0186837 Top 0 0.016109 Silicon Bottom -0.15 -0.130268

Top 0.29 0.276070 Both verification methods validated the model.

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6. Overlay sensitivity

6.1. Introduction Since overlay is one of the problems that arise during processing the flexible substrate, it should be minimised (see problem description). Therefore a number of parameters are varied to find an optimum configuration. For all these variations a FEM simulation has been performed in section 6.3. Before that, the sensitivity of each parameter (influence of a parameter on the failure mechanisms) is determined to reduce the number of simulations (see section 6.2.) This sensitivity analysis contains the following parameters:

• Thickness of the carrier, glue and foil • Radius of the carrier-glue-foil system • Material properties of the carrier and foil

Hereafter a process area (range in which parameters are varied) can be defined in which FEM simulations have been performed. As a consequence a cloud with overlay points will arise. A Response Surface Model (in this case normal Kriging) is used to process and interpret this data in section 6.4.

6.2. Reduction of variations The process area is dependent on the boundaries of the commercially available thicknesses and accompanying material properties. This results in the process area defined in table 10.

Table 10. Sensitivity parameters

Range Number of simulations Unit

Foil thickness 10 < d <100 10 [µm] Glue thickness 2 < d < 10 9 [µm] Carrier thickness 4·10-4 < d < 10·10-4 7 [m] Radius carrier-glue-foil system 0.05 < r < 0.1 6 [m]

Young’s modulus foil 6.2·108 and 2.5·109 2 [Pa] CTE carrier 4·10-6 and 7·10-6 2 [-]

The total number of simulations will be 15120 for this particular process area. To reduce this number, the parameters are individually tested for their influence on the results. The overlay response upon their variation of these parameter (constant, linear, quadratic) is investigated. An example is given in figure 14. All simulations with various thicknesses of the carrier gave approximately the same overlay so that a constant value can be chosen. A reduction of 70 to 10 simulations is the result (red dots).

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. 3 4 5 6 7 8 9 10 11

x 10-4

0

10

20

30

40

50

60

70

80

90

100

110

δcarrier [m]

δfoil

[µm ]

Figure 14. Sensitivity of the thickness of the carrier and reduction of the number of simulations

After this reduction, a reliable conclusion can still be drawn and it is saving computing time. All parameters are tested on their influence in this way and results are shown in table 11. In this way the number of simulations is reduced to 140. These simulations are automatically run with Python [7]. Such a procedure is often called a Design of Experiments (DOE).

Table 11. Reduced sensitivity parameters

range relation number of simulations unit

Foil thickness d = 10, 20, 30, 70, 100 Non-linear 5 [µm] Glue thickness d = 2, 3, 4, 7, 8, 9, 10 Non-linear 7 [µm] Carrier thickness d = 9·10-4 Constant 1 [m] Radius r = 0.075 Constant 1 [m] Young’s modulus foil E = 6.2·108, 2.5·109 Linear 2 [Pa] CTE carrier a = 4·10-6, 7·10-6 Linear 2 [-]

6.3. Sensitivity results In the previous section a design of experiments resulted in a configuration with the minimum overlay, see table 12.

Table 12. Minimum overlay parameter values

dfoil [m] dglue [m] Young’s modulus of foil [Pa]

Coefficient of Thermal Expansion [-

] Optimum

value 1·10-5 2·10-6 6.2·108 7·10-6

These optimum values give an overlay of 20.5 µm instead of 58.4 µm in the reference model and at the same time the maximum value of overlay of the 140 simulations. The optimum values followed from the DOE are the lowest in the range for dfoil, dglue and Efoil. Decreasing both thicknessess means that the layers can exert less force on the substrate

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and therefore the overlay will be less. Since the glue and foil layer are mainly responsible for the overlay, these have to be minimised to minimize the overlay. The same is valid for the Young’s modulus of the foil. A lower Young’s modulus gives a lower stress at the same strain. This stress gives less resistance against shrinking so the overlay will be less. The value of the CTE of the carrier has a minor influence on the overlay. Since this layer is much thicker and isotropic elastic it will come back in its initial state. Only during processing, a higher CTE provides more expansion. As a consequence node 1 and 4 come closer together and overlay during processing is minimized.

6.4. Response Surface Model (Kriging) A process can be optimised by changing the material properties and dimensions. Before a new configuration / experiment is established, simulation can give a first guess of the result. Again this is time and cost saving. The ‘x’-dimensional cloud, which followed from the DOE of the previous section, will be reduced to a single equation to create overview and manageability. Such a tool is created with Response Surface Modelling (in this specific case a Kriging model), which is processed in a Microsoft Excel layout. Kriging’s method is referred to as Design and Analysis of Computer Experiments (DACE) models. Georges Matheron named this process Kriging after Danie Krige, who has done a tremendous amount of empirical work on weighted averages. The Kriging mathematics [14] is described by equation (4).

)()()( xxx Zfy += (4) in which:

=)(xy unknown function of interest =)(xf known function of x (“globally” approximates the design space)

=)(xZ realization of a stochastic process with mean zero, variance 2σ and non-zero covariance (creates “localized” deviations)

)(xf is a known polynomial function of x and in this specific case it is taken as the mean

of ( )xiy . From here this constant value is called β . The basic idea behind ( )xZ is to estimate the value of y at a location x where the true value is not known.

Figure 15. Unknown point (red dot) in cloud of data points

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Each point ix has a specific contribution to the unknown point x̂ . This contribution is determined by means of a weight factor, see equation (5)

( )ii Z xR ⋅=λ (5) in which =R correlation matrix composed of correlation functions (‘weighed distance’) in the form ( )jiR xx , . These weighed distances can be seen in figure 16 and calculated with equation (6).

Figure 16 Weighed distances to compose the correlation matrix

( )

−−= ∑ =

dvn

k

pjk

ikk

ji xxR1

exp, θxx (6)

in which

jikx , = value at points i and j for a specific design variable xk.

dvn = number of design variables pk ,θ = unknown parameters to fit the model

These fit parameters are determined with a Maximum Likelihood Estimate (MLE) [10]. The correlation matrix R can now be composed. Since the function of interest ( )xy for the data points is known, the variance ( )iZ x can be calculated from equation (4). The weight factors can than be calculated using equation (5). For an unknown point x̂ all ‘distances’ between x̂ and ix are calculated using equation (6) and multiplied by their specific weight factor. With equation (7) every function of interest for an unknown point x̂ can now be determined.

( ) ( ) ( )∑=

⋅+=+=n

i

ikki RxZxy

1

,ˆˆˆ xxλββ (7)

For customer use this formula is reworked into an Excel cockpit which can be found in Appendix B.

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7. Stress sensitivity

7.1 Introduction The maximum achievable bending radius and reliability of the display is dictated by the fracture properties of extremely thin and brittle films in the stack of the display. This is controlled by a complex interplay between process-induced defects, residual film stresses, as well as cohesive and adhesive properties of the individual layers. In sections 7.3 and 7.4 a thin film of ITO will be deposited on a carrier-glue-foil system as the first functional layer. FEM-analyses have to prove whether or not the stresses stay below the critical values defined in section 7.2 or else one can conclude where and why failure occurs. The temperature sensitivity mainly causes these stresses. Therefore a temperature has to be found in section 7.5 for which the stresses in the ITO stay below the critical value.

7.2 Critical values Two types of failure will be investigated, cohesive and adhesive failure, see figure 17.

Figure 17. Adhesive and cohesive failure

Adhesion = the mutual attraction between unequal molecules without chemical bonding Cohesion = the mutual attraction between equal molecules without chemical bonding The failure behavior of ITO is described in the report ‘Layer mechanics of optimized materials’ [9]. Here a description is given of how a low-strain regime analysis and the saturation stage of tensile cracking lead to the determination of the tensile strength and the interfacial shear strength (IFSS) respectively. The substrate used to measure these values consists of a Arylite foil (ARY), hardcoat (HC) and on top a layer of ITO. Here it is mentioned that delamination occurred at the interface ITO – HC and not directly between the foil and ITO.

Table 13. Cohesive and adhesive properties of ITO layers on polymer substrates [9]

Tensile strength (cohesive criterion) [GPa]

IFSS (adhesive criterion) [MPa]

ARY/HC/ITO 100nm 2.2 >41* (*) Lower bound to IFSS due to HC-layer failure

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With these critical values, one can predict at which process temperature the ITO layer will be damaged and will become non-functional.

7.3 Modelling and reduction of variations Since the substrate (carrier-glue-foil) is heated to the deposition temperature before sputtering on the layer of ITO, overlay will be present. Numerically this means that the layer is placed onto a deformed configuration. In Marc-Mentat an option to add layers on a deformed mesh is not available. Therefore subroutines (originally created by Martijn Panis) and Matlab script files have been created to realize this. These methods are described in the user manual in Appendix C. From section 6.3, a configuration with minimum overlay and accompanying stress followed. To show that the stresses in the minimum overlay configuration are the lowest, it is compared with the maximum overlay configuration. Figure 18 shows that the stresses of the minimum overlay configuration stay below the value of the maximum overlay configuration in shear (s 12) and axial (s 22) direction. Since these two are of main importance one can conclude that the minimum overlay configuration gives lower stresses. A clear explanation is not found for the disturbance at a radial distance of 0.03 m. The process temperature variation will therefore be performed on the configuration described by table 12.

Figure 18. Stresses in ITO for maximum and minimum overlay configuration

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To deposit the functional layer of ITO on the foil an increased temperature is desired to obtain a good material structure. By means of variation it is tried to find the highest process temperature for which the stresses stay below the critical stress values. Since the Kapton foil can resist a maximum temperature of 180°C the range is chosen below this value (20 - 180°C).

7.4 Typical results (verification) The FEM models show a linear increase of stress for increasing temperature. The only load is temperature and it is constant in space. Since equation (3) gives a linear relation between strain and temperature, the relation between stress and temperature will also be linear. So this linear relation is correct. Finally the stresses (following from the laminate tool) of the ITO are compared with the axial stresses resulting from the FEM-program in table 14. Table 14.Verification of FEM-results with laminate tool, reference model is cooled from 90°C to 20°C

s axial laminate tool [MPa]

s axial FEM [MPa]

ITO Bottom 4 5.91 Top 4 5.91

From table 14 one can conclude that the order of magnitude is correct (the data from the laminate tool is relatively inaccurate but created to give an order of magnitude).

7.5 Sensitivity results The results of varying the temperature are again processed in a Microsoft Excel layout, see Appendix D. With this tool a maximum temperature is found for which the stresses stayed below the critical stress values. These are shown in table 15.

Table 15. Stress sensitivity results

Process temperature 95 [°C] Principal stress minimum 1.51·106 [Pa] Principal stress maximum -4.09·107 [Pa]

Figure 20 shows that there is a peak value at the edge of the substrate, which is equal to the critical values (red lines). The numbers 1-5 in figure 20 correspond with the stress points in figure 19.

Figure 19. Position of stress points

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stress on the interface

-6,00E+07

-4,00E+07

-2,00E+07

0,00E+00

2,00E+07

4,00E+07

6,00E+07

1 2 3 4 5

length in axial direction

stre

ss [

Pa] Prin. stress max.

Prin. stress min.

Prin. stress.int

Figure 20. Stress on the interface between Kapton and ITO

A remark has to be made since edge effects in FEM-simulations could cause the high peak of the principal minimum stress.

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8.Removal of flexible substrate from carrier

8.1 Introduction For handling and stability reasons the flexible substrate is temporarily laminated on a silicon wafer by means of a suitable adhesive, PSA610. After processing, this silicon carrier is removed to regain substrate flexibility. Presently, the delamination of the flexible substrate is performed manually, using a metal blade scraper and a simple chuck for holding the wafer in position [2]. In this chapter several methods to numerically remove the flexible substrate from the silicon carrier are examined.

• metal blade scraper; • pulling off the flexible substrate in mode 1; • bombing the glue with UV-light, as a consequence the glue loses its adhesive

property.

Based on the overlay and stress sensitivity analysis, done in chapters 6 and 7, a choice for a particular system is made. In section 8.2 the impact of the above methods on the stresses is examined and a delamination method will be chosen. After removal from the carrier, the strain and the stress distribution will be studied in section 8.3. High-risk areas for cohesive / adhesive failure are determined in section 8.4.

8.2 Methods to numerically remove the flexible substrate In this section the methods to numerically remove the flexible substrate are clarified. The first method separates the adhesive layer that bonds the flexible substrate and the silicon wafer by forcing / pushing a straight knife blade through the adhesive layer, see figure 21. Using the second method, the silicon carrier is clamped and the flexible substrate is simply separated by pulling in axial direction (mode 1), see figure 22.

Figure 21. Numerical removal of flexible substrate with the use of a metal blade scraper

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Figure 22. Numerical removal of flexible substrate with the use of a mode 1 displacement

These methods both influence the initial / residual stresses present inside the flexible substrate and its functional layers. An external force is caused by the cutter as well as due to the mode 1 displacement. Again the resulting stresses are not allowed to be higher than 41 MPa. Since the main interest goes to the effect of the residual stresses after removal, these two models are not further investigated. The last method is based on exposing the glue to ultra violet (UV) light. Due to the UV-light the glue loses its adhesive property and falls apart. In this way the flexible substrate is separated without use of an external force. Since the residual stresses of the different process steps are still present inside the display, their influences can be better estimated. Therefore this method is chosen to study the failure behaviour and high-risk areas.

8.3 Delamination and failure behaviour The last process step before delamination is a temperature sweep from 95°C to 20°C. The consequence of this step is that positive stresses will exist in the flexible substrate. After delamination these stresses will be released by means of elongation and could cause failure. Different types could occur, for example roll up of the flexible substrate or buckling (adhesive failure of the coating (ITO) on the substrate) of the functional layer, see figure 23.

Figure 23. Buckling and roll-up

FEM-analyses showed that the cohesive and adhesive criterion are not exceeded after delamination (so no buckling failure occurs) and therefore roll-up could be present. Since the thickness of the Kapton layer is 500 times larger than the layer of ITO it will release more stress and can practise a larger force on the substrate. The elongation of Kapton will be larger than ITO and the substrate will roll-up in positive direction. FEM-analyses showed the same results.

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In the previous lines it is explained that the flexible substrate will roll up. To prevent this roll-up a part of the substrate is constrained . The axial displacement of the constrained substrate and the stresses which exist due to these constraints can be seen in figure 24 and 25.

Figure 24. Axial displacement with constraints Figure 25. Maximum principal stress and adhesive criterion These constraints can be seen as a force to straighten the flexible substrate. Figure 25 clearly shows that the adhesive criterion is exceeded and failure occurs on the Kapton-ITO interface. Smoothing of the substrate could therefore cause disfunctionality of the display.

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9. Conclusion and recommendation

9.1 Conclusions Based on the research done in this report, the following conclusions are drawn:

• The dimensions and material properties of the foil mainly influence the amount of

overlay. An overlay sensitivity study (with parameters dfoil, Efoil, dglue and acarrier) managed to decrease the overlay from 58,4 um to 20,5 um;

• Visco-elastic properties of the glue PSA610 causes overlay. Glue gains stiffness during cooling and prevents the foil from shrinking;

• The preferred deposition temperature of the functional layer ITO to stay just below the critical failure value is 95°C;

• Delamination by means of exposing the glue to ultra violet light is the recommended method to remove the display from the carrier since no external forces can damage the functional layers on the flexible substrate;

• Due to the smoothing of the substrate, the interface between ITO and Kapton reaches the critical stress failure value;

• The Finite Element Method software of Marc-Mentat is preferred to build a model of the flexible substrate. The Abaqus software was not reliable because hourglass elements arose during simulations.

9.2 Recommendations • The overlay problem could perhaps be solved by placing the foil on a vacuum

chuck (replaces the carrier and glue) during processing. The foil can then expand and shrink freely. Another advantage is that the flexible substrate can be removed by releasing the vacuum;

• After removal of the display the residual stresses are still too large. The different process stages need to be optimised so that the critical failure value will not be reached. Decreasing the deposition temperatures is an option. However, layer quality can be an issue here;

• To obtain comparable results with experiments the simulations of the flexible displays probably have to be modelled in 3-D. The layers deposed after the layer of ITO are not axisymmetric anymore;

• Translating the matlab script-files, which import the deformed mesh of each process stage, to Marc-Mentat subroutines can optimise and accelerate the modelling of the different stages of processing. It is recommended to simulate all process stages to draw conclusion on the residual stresses present inside the flexible display.

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References [1] Brandes, E.A., Brook, G.B., Smithells Metals Reference Book, seventh edition, ISBN 0-7506-1020-4 [2] Doedee, R.T.M., Winters, H.H.A.,2001-11-26, Evaluation of the De-laminator for Polymeric Electronics, CTB594-01-1256 [3] Dupont, material properties Kapton Internet: http://www.dupont.com/kapton [4] E-ink, Principle of E-ink technology Internet: http://www.eink.com [5] Fenner, R.T., Mechanics of Solids, CRC Press, ISBN 0-632-02018-0 [6] Goveart, L.E.,Materiaalkunde, inleiding Polymeer Technologie, lecture notes [7] Lamy, F., Mechanical and Thermo Mechanical analysis of thin layers, April- September 2002, CTB598-02-2215 [8] Leijsen van, C.P., FEM-calculation on FlexLED, internal research [9] Leterrier, Medico, Bouten, Goede, Layer mechanics of optimised materials,

FLEXled Report D17 “Layer mechanics of optimised materials” [10] Montgomery, D.C., George C. Runger, Applied Statistics and Probability for Engineers, Second edition, John Wiley & Sons, Inc., ISBN 0-471-17027-5 [11] PC Magazine, Displaying the future, flexible screens, Internet: http://www.pcmag.com/article2/0,1759,1491664,00.asp [12] Scheurs, P.J.G., Visco-elasticity Internet: http://www.mate.tue.nl/~piet/edu/mcm [13] Silfhout, R., Microsoft Excel laminate tool, server Philips CFT, Process

technology [14] Simpson, T.W., Georgia Institute of Technology, Comparison of Response Surface and Kriging Models in the Multidisciplinary Design of an Aerospike Nozzle, NASA/CR-1998-206935, ICASE Report No. 98-16 [15] Zuidema, H., 2001-09-17, Dimensional Stability of Polymer Electronics, Philips CFT, CTB591-01-7148, Internal Philips report [16] Zuidema, H., 2002-07-23, Design flexible PolyLED, Philips CFT, CTB591-02-

7046, Internal Philips report

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Distribution List

Philips CFT

• Raymond van Agthoven • Peter Bouma • Ric van Doremalen • Marcel van Gils • Arjan v. Leijsen • Olaf van der Sluis • Peter Timmermans

Philips Research

• Giovanni Nisato • Piet Bouten • Kees Mutsaers

EPFL

• Yves Letlerier

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Appendix A. Determination of the parameters of a Maxwell model To determine the visco-elastic parameters of PSA610 a temperature sweep has been performed from 298 until 373 K using the RDA-II of Rheometrics, see research lecture Dimensional Stability of Polymer Electronics by H. Zuidema. The procedure below is used to determine the Maxwell parameters. The phase difference between stress and strain results in a so-called hysteresis loop, when a stress-strain diagram is drawn. In figure A1 one can see that the strain is ahead of the stress, so the material behavior is time dependent. The area enclosed by the hysteresis loop is a measure for the dissipated energy per unit of volume during one cycle.

Figure A.1. Phase difference between s and e Figure A.2. Hysteresis loop

Writing the stress response with two different relations results in relations between E’, E” and d. The amplitude s 0 of the stress response can also be calculated.

( ) ( ) ( ) ( ) ( )ttt ωδσωδσσ cossinsincos 00 += (A.1) ( ) ( ) ( )tEtEt ωεωεσ cossin 00 ′′+′= (A.2)

With equation (A.1) and (A.2) the storage and loss modulus can be calculated.

( )δεσ

cos0

0=′E and ( )δεσ

sin0

0=′′E (A.3)

It is impossible to get through the whole time / frequency range. Experiments have demonstrated that temperature and time / frequency variations gave similar results for shear. So combining these two magnitudes and using ‘Williams-Landel-Ferry’ time-temperature superposition equation gave the opportunity to consider the full time-scale.

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So the single shift curves at single temperatures, ( ) ( )TatTGtTG ,, 21 = , were translated to a master curve in the whole time scale. Equation (A.4) calculates the shift factor aT.

( ) ( )( )ref

refT TTC

TTCa

−+

−−=

2

1log (A.4)

Table A.1. Time-temperature shift constants used in the WLF-equation

Treference 330 [K] C1 7.4508 [-] C2 71.1204 [K]

The result of this equation is a master curve, see figure A.3. Included are the Maxwell fits used to model the material behavior.

Figure A.3. Master curves for the storage and loss modulus

The master curves for the storage and loss modulus can be modeled using a discrete relaxation time spectrum obtained using the Maxwell model:

( ) ( )( )2

2

1 1 i

in

iiGGG

ωθ

ωθω

++=′ ∑

=∞ (A.5)

( ) ( )( )2

2

1 1 i

in

iiGG

ωθ

ωθω

+=′′ ∑

=

(A.6)

in which n is the number of modes, G8 the equilibrium modulus, Gi the relaxation strength for the ith mode and iθ the relaxation time for the ith mode. The linear visco-elastic parameters for the Maxwell model are shown in the table A.2.

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Table A.2. Linear visco-elastic parameters for the discrete relaxation time spectrum used in the Maxwell model for the glue (PSA610)

Mode i

Relaxation time iθ [s]

Relaxation strength Gi [Pa]

8 4.3122e1

1 7.6312e1 5.9976e3

2 4.0733e0 1.2519e4

3 2.3836e-1 2.5252e4

4 9.9720e-3 1.0383e5

5 5.6848e-4 4.2931e5

6 1.4835e-5 3.2022e6

To model the Maxwell model in Marc Mentat one has to define the input for the Young’s modulus in the following way.

∑=

∞ +=n

iiGGG

1

(A.7)

GE ⋅+⋅= )1(2 ν (A.8) In this way Marc-Mentat determines the initial shear modulus.

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Appendix B. Microsoft Excel layout for overlay sensitivity

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Appendix C. User Manual to add layer on deformed mesh To realize the deposition on the deformed mesh two jobs are needed:

• Heat the carrier-glue-foil system to deposition temperature; • Import the deformed mesh, depose layer of ITO and cool to room temperature.

The following steps are taken: Job 1: -write ‘job_1’-.t19-file; -use subroutine elevar.f which saves all stresses of all elements on last increment. Job 2: -rewrite the ‘job_1’.dat-file with create_dat.m and extract_t19.m (adds the

displacements of the t19-file at last increment to the nodal coordinates); -rewrite the fort.62-file with create_62.m (adds zero stresses for all the new ITO elements); -run add_layer.proc to add the layer of ITO in the correct way; -use subroutine uinstr.f to add the stresses of the last increment of Job 1 as an initial condition in Job 2.

The user subroutine elevar.f and uinstr.f and the matlab script files create_dat.m, extract_t19.m and create_62.m can be seen below. Subroutine elevar.f C====================================================== C This subroutine was originally created by Martijn C Panis and it has been adapted to make it work for C the ITO-layer problem C C Bas Adams, 09-11-2004 C====================================================== subroutine elevar(n,nn,kc,gstran,gstres,stress,pstran, +cstran,vstran,cauchy,eplas,equivc,swell,krtyp,prang,dt, +gsv,ngens,ngen1,nstats,nstass,thmstr) implicit real*8 (a-h,o-z) dimension gstran(ngens),gstres(ngens), +stress(ngen1),pstran(ngen1),cstran(ngen1),vstran(ngen1), +cauchy(ngen1),dt(nstats),gsv(1), +thmstr(1),krtyp(4) C====================================================== C This subroutine reads the stresses of all C elements on the last increment C======================================================

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include 'concom' IF (inc .EQ. 20) THEN WRITE(62, '(I5,2X,I3,2X,4(E11.4,1X))' ) * N, NN, ( STRESS(i), i=1,NGEN1 ) ENDIF return end Subroutine uinstr.f C====================================================== C This subroutine was originally created by Martijn C Panis and it has been adapted to make it work for C the ITO-layer problem. The subroutine reads the C stresses into Marc-mentat as an initial condition C C Bas Adams, 09-11-2004 C====================================================== C====================================================== C Before running this subroutine the matlabroutine C create_62.m has to be runned in order to add the C zero stress value for the new elements C====================================================== subroutine UINSTR (S,NDI,NSHEAR,N,NN,KC,XINTP,NCRD, +INC,TIME,TIMEINC) C====================================================== C M=4*total numbers of elements C (new elements included) C A(M,6) C====================================================== implicit real*8 (a-h, o-z) dimension S(1), XINTP(NCRD), N(2), A(7600,6) OPEN(UNIT=61,FILE = 'step2.marc61') OPEN(UNIT=71,FILE = 'step2.marc71') I = 4 * N(1) + NN - 4 READ(UNIT=62,FMT=*) (A(I,J),J=1,6) S(1) = A(I,3) S(2) = A(I,4) S(3) = A(I,5) S(4) = A(I,6) c S(5) = 00.0d0 c S(6) = 00.0d0 WRITE(UNIT=71,FMT=200) I, N(1), NN, S(1), S(2), S(3), S(4)

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100 FORMAT(I3,2X,I3,2X,E11.4,1X,E11.4,1X,E11.4,1X,E11.4,1X,/) 200 FORMAT(I6,1X,I6,1X,I2,1X,E11.4,1X,E11.4,1X,E11.4,1X,E11.4) M=7600 IF(I .EQ. M) THEN CLOSE(61) ENDIF return end Matlab script file create_dat.m clear all; clc; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %datfile from first marc-mentat loop fr=fopen('model1_job1.dat','r'); %Generate a file in which the coordinates will be written fw=fopen('co.m','w'); %temporary write file fwnew=fopen(strrep('model1_job1.dat','.dat','_new.dat'),'w'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Extract coordinates from datfile of 'old'-file % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %first line in co.m fprintf(fw,'coord=['); flag=1; while(flag); % loop to define the end of the file s=fgets(fr); % read all lines of the old-file if(s==-1); flag=0; else fprintf(fwnew,'%s',s); % write lines from old file to new marc-mentat datfile end if(strcmp(s(1:(length(s)-1)),'coordinates')); % find word 'coordinates' in old datfile (compare strings) s=fgets(fr); fprintf(fwnew,'%s',s); % print line from old datfile in the new datfile nndstr=s(11:29); nnd=str2num(nndstr); for jj=1:nnd;fprintf(fwnew,'%s',s); % change Marc format to Matlab format with string replacement

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s=fgets(fr); % replace the -1 for e-1 so matlab can read it s=strrep(s,' ',' '); s=strrep(s,'-1.',' -1.'); s=strrep(s,'-2.',' -2.'); s=strrep(s,'-3.',' -3.'); s=strrep(s,'-4.',' -4.'); s=strrep(s,'-5.',' -5.'); s=strrep(s,'-6.',' -6.'); s=strrep(s,'-7.',' -7.'); s=strrep(s,'-8.',' -8.'); s=strrep(s,'-9.',' -9.'); s=strrep(s,'+0','e+0 '); s=strrep(s,'-1','e-1 '); s=strrep(s,'-2','e-2 '); s=strrep(s,'-3','e-3 '); s=strrep(s,'-4','e-4 '); s=strrep(s,'-5','e-5 '); s=strrep(s,'-6','e-6 '); s=strrep(s,'-7','e-7 '); s=strrep(s,'-8','e-8 '); s=strrep(s,'-9','e-9 '); s=strrep(s,'+1','e+1 '); fprintf(fw,'%s',s); % write coordinates in matlab format to co.m end fprintf(fw,'];'); % last line in co.m + close file fclose(fw); end end % Define output-file fout=fwnew; % call created matlab file co.m (in extract_coordinate.m) co; extract_t19; load A; % temporary datfile where new coordinates are written % in the correct way with the correct spaces and length newcoord=fopen('newcoord.dat','w') for i=1 write=[-0d0 -0d0 -0d0]; fprintf(newcoord,'%10.1i,%20.13e,%20.13e,%20.13e\n',[i write]); end for i=2:size(A,1) write=[A(i,1) A(i,2) A(i,3)]; fprintf(newcoord,'%10.1i,%20.13e,%20.13e,%20.13e\n',[i write]); end fclose(newcoord); % read from created newcoord.dat

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newcoord1=fopen('newcoord.dat','r'); olddat=fopen('model1_job1.dat','r'); % new datfile newdat=fopen('new.dat','w'); i=0; flag=1; while (flag); read2=fgets(olddat); if (read2==-1); flag=0; end if(strcmp(read2(1:(length(read2)-1)),'coordinates')); % compare strings, find word 'coordinates' fprintf(newdat,'%s',read2); read2=fgets(olddat); fprintf(newdat,'%s',read2); flag1=1; while(flag1); i=i+1; read=fgets(newcoord1); % read new coordinate read1=num2str(read); fgets(olddat); if (read==-1); flag1=0; else % change Matlab format to Marc format read1=strrep(read1,'e',''); read1=strrep(read1,',',' '); end if (read>0); fprintf(newdat,'%s',read1); end end read2=fgets(olddat); end if (read2>0); fprintf(newdat,'%s',read2) ; %as long as we are not in coordinates write end %old data from datfile into new one end fclose(newcoord1) fclose(olddat) fclose(newdat) delete co.m delete matrix.m delete A.mat delete newcoord.dat delete test_job1_new.dat Matlab script file create_t19.m clear all %clc

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fr=fopen('model1_job1.t19','r'); %fw=fopen('matrix.m','w'); %fprintf(fw,'mat=['); flag=1; counter=0; nbinc=21; nnodal=36; mat=[]; i=0; while(flag) s=fgets(fr); %to read through all the file and find the end of the t19-file if(s==-1); flag=0; end if(strcmp(s(1:min(12,length(s))),'Displacement')); %Find the word 'Displacement' counter=counter+1; %Find the last increment end if (counter==nbinc); if not(strcmp(s(1:min(5,length(s))),'=end=')) & ... %statements not write things in not(strcmp(s(1:min(12,length(s))),'Displacement')) & ... %the matrix not(strcmp(s(1:min(5,length(s))),'=beg=')) & ... not(strcmp(s(1:min(4,length(s))),'----')) & ... not(strcmp(s(1:min(4,length(s))),'++++')) & ... not(strcmp(s(1:min(13,length(s))),' 0')) & ... s~=-1 i=i+1; %A=[str2num(s(1:13)) str2num(s(14:26)) str2num(s(27:39)) str2num(s(40:52)) str2num(s(53:65)) str2num(s(66:78))]; if i==541; B=[str2num(s(1:13)) str2num(s(14:26)) str2num(s(27:39)) str2num(s(40:52))]; else C=[str2num(s(1:13)) str2num(s(14:26)) str2num(s(27:39)) str2num(s(40:52)) str2num(s(53:65)) str2num(s(66:78))]; end mat=[mat ; C]; %fprintf(fw,'%s',s); %1.13e,%c,%1.13e,%c,%1.13,%c,%1.13e,%c,%1.13e,%c,%1.13e\n %write the displacement matrix into the new file end end end %fprintf(fw,'];'); %fclose(fw); fclose(fr);

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%call created matlab file co.m (in extract_coordinate.m) co; %call matlab format coordinates of the nodes %matrix %arrange the displacements of node 1 till 36 for x and y disp_x=[]; disp_y=[]; for i=3:540; for j=1:2:5; x=mat(i,j); disp_x=[disp_x x]; end; end; for i=3:540; for j=2:2:6; y=mat(i,j); disp_y=[disp_y y]; end; end; disp_x=[disp_x B(1,1) B(1,3)]; disp_y=[disp_y B(1,2) B(1,4)]; %new coordinates for the deformed mesh A=[coord(:,2)+(disp_x)' coord(:,3)+(disp_y)' coord(:,4)]; save A A; Matlab script file create_62.m ne=1500; % number of elements ne_new=400; % number of new elements % creating temp.txt which writes the lines with zero stress new=fopen('temp.txt','w'); for i=(ne+1):(ne+ne_new) s1=[' ' num2str(i) ' 1' ' ' '0.0000E+00' ' ' '0.0000E+00' ' ' '0.0000E+00' ' ' '0.0000E+00']; s2=[' ' num2str(i) ' 2' ' ' '0.0000E+00' ' ' '0.0000E+00' ' ' '0.0000E+00' ' ' '0.0000E+00']; s3=[' ' num2str(i) ' 3' ' ' '0.0000E+00' ' ' '0.0000E+00' ' ' '0.0000E+00' ' ' '0.0000E+00']; s4=[' ' num2str(i) ' 4' ' ' '0.0000E+00' ' ' '0.0000E+00' ' ' '0.0000E+00' ' ' '0.0000E+00']; fprintf(new,'%s\n',s1); fprintf(new,'%s\n',s2); fprintf(new,'%s\n',s3); fprintf(new,'%s\n',s4); end fclose(new);

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fr=fopen('fort.62','r'); %lezen van de 62-file fr_temp=fopen('temp.txt','r'); %lezen in het textbestand om nullen toe te voegen fw=fopen('fort_new.62','w'); for i=1:4*(ne+ne_new) if i>4*ne s=fgets(fr_temp); fprintf(fw,'%s',s); else s=fgets(fr); %oude regels wegschrijven fprintf(fw,'%s',s); end end fclose(fr); fclose(fr_temp); fclose(fw); delete temp.txt delete fort.62 % 4 x writing fort_new.62 into .62 file (old .62 is overwritten by new one) fr=fopen('fort_new.62','r'); fw=fopen('fort.62','w'); for i=1:4*(ne+ne_new); s=fgets(fr); k(i,:)=char(s); end fclose(fr); for i=1:4; for j=1:4*(ne+ne_new); fprintf(fw,'%s',k(j,:)); end end fclose(fw); delete fort_new.62;

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Appendix D. Microsoft Excel layout for stress sensitivity

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