Menk, Alexander (2011) Simulation of complex microstructural geometries using X-FEM and the application to solder joint lifetime prediction. PhD thesis http://theses.gla.ac.uk/2519/ Copyright and moral rights for this thesis are retained by the author A copy can be downloaded for personal non-commercial research or study, without prior permission or charge This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given. Glasgow Theses Service http://theses.gla.ac.uk/ [email protected] Simulation of Complex Microstructural Geometries using X-FEM and the Application to Solder Joint Lifetime Prediction Alexander Menk (Dipl.-Ing.) Submitted in fulfillment of the requirements for the Degree of PhD Department of Civil Engineering School of Engineering University of Glasgow April 18, 2011 Declaration I declare that this thesis is a record of the original work carried out by myself under the supervision of Prof. S. P. A. Bordas and Professor Chris Pearce in the Department of Civil Engineering at the University of Glas- gow, United Kingdom, and at the research department of the Robert Bosch GmbH, Germany, during the period of April 2008 to March 2011. The copyright of this thesis belongs to the author under the terms of the United Kingdom Copyright acts. Due acknowledgement must always be made of the use of any material contained in, or derived from, this thesis. The thesis has not been presented elsewhere in consideration for a higher degree. Alexander Menk Abstract In electronic devices solder joints form a mechanical as well as an electrical connection between the circuit board and the component (e.g. a chip or a resistor). Temperature variations occurring during field use cause crack initiation and crack growth inside the joints. Accurate prediction of the lifetime requires a method to simulate the damage process based on mi- crostructural properties. Numerical simulation of developing cracks and microstructural entities such as grain boundaries and grain junctions gives rise to several problems. The solution contains strong and weak discontinuities as well as weak sin- gularities. To obtain reasonable solutions with the finite element method (FEM) theelement edgeshave to alignwiththecracks andthegrainbound- aries, which imposes geometrical restrictions on the mesh choice. Addition- ally, a large number of elements has to be used in the vicinity of the sin- gularities which increases the computational effort. Both problems can be circumvented with the extended finite element method (X-FEM) by using appropriate enrichment functions. In this thesis the X-FEM will be developed for the simulation of complex microstructural geometries. Due to the anisotropy of the different grains forming a joint and the variety of different microstructural configurations it is not always possible to write the enrichment functions in a closed form. A procedure to determine enrichment functions numerically is explained and tested. As a result, a very simple meshing scheme, which will be introduced here, can be used to simulate developing cracks in solder joint microstruc- tures. Due to the simplicity of the meshing algorithm the simulation can be automated completely. A large number of enrichment functions must be used to realize this. Well-conditioned equation systems, however, cannot be guaranteed for such an approach. To improve the condition number of the X-FEM stiffness matrix and thus the robustness of the solution process a preconditioning technique is derived and applied. This approach makes it possible to develop a new and fully automated procedure for addressing the reliability of solder joints numerically. The procedure relies on the random generation of microstructures. Performing crack growth calculations for a series of these structures makes it possible to address the influence of varying microstructures on the damage process. Material parameters describing the microstructure are determined in an inverse procedure. It will be shown that the numerical results correspond well with experimental observations. Contents 1 Introduction 20 1.1 Numerical simulation in industrial applications and the ex- tended finite element method . . . . . . . . . . . . . . . . 20 1.2 Lifetime prediction for solder joints . . . . . . . . . . . . . 22 1.3 Development of X-FEM towards the simulation of complex polycrystalline structures . . . . . . . . . . . . . . . . . . . 25 2 Theoretical background 28 2.1 Material description and constitutive relations . . . . . . . 28 2.2 Functional analysis . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Finite element method . . . . . . . . . . . . . . . . . . . . 36 2.4 Extended finite element method . . . . . . . . . . . . . . . 41 2.5 Iterative solvers . . . . . . . . . . . . . . . . . . . . . . . . 53 3 X-FEM for polycrystalline structures 59 3.1 Numerical integration . . . . . . . . . . . . . . . . . . . . . 60 3.2 Numerically determined enrichment functions . . . . . . . 63 3.2.1 Determination of the enrichment functions . . . . . 64 3.2.2 Numerical experiments . . . . . . . . . . . . . . . . 68 3.3 Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . 83 3.3.1 Problems in current enrichment schemes . . . . . . 86 3.3.2 Calculation of the preconditioner . . . . . . . . . . 93 3.3.3 Numerical experiments . . . . . . . . . . . . . . . . 101 3.4 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.4.1 Refinement . . . . . . . . . . . . . . . . . . . . . . 119 3.4.2 Element clustering . . . . . . . . . . . . . . . . . . 122 3.5 Contact Modeling . . . . . . . . . . . . . . . . . . . . . . . 126 3.6 Transient simulations and creep with the X-FEM . . . . . 131 4 Application of X-FEM to solder joints in electronic devices 135 4.1 Solder joints in electronic devices . . . . . . . . . . . . . . 136 4.2 Standard methods for predicting the lifetime of solder joints 140 4.3 New methodology . . . . . . . . . . . . . . . . . . . . . . . 143 4.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . 144 4.3.2 Grain structure generation . . . . . . . . . . . . . . 146 4.3.3 Crack growth criterion . . . . . . . . . . . . . . . . 150 4.3.4 Crack growth algorithm . . . . . . . . . . . . . . . 151 4.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 154 4.4.1 Determination of material parameters . . . . . . . . 155 4.4.2 Crack growth calculations . . . . . . . . . . . . . . 160 5 Conclusions 172 A Stress-strain relation in cylindrical coordinates 176 B Woodbury matrix identity 181 List of tables 3.1 Integrationorderforthesubdomainsoftheenriched elements 62 3.2 Asymptotic behavior of the numerical enrichments for differ- ent discretizations . . . . . . . . . . . . . . . . . . . . . . . 70 3.3 Angular component error of the numerical enrichments . . 72 3.4 Elastic constants for materials in a bi-material structure . 77 3.5 Elastic constants for copper grains . . . . . . . . . . . . . . 79 3.6 Euler angles for polycrystalline structure formed by three grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.7 Material constants for structure with slanting material inter- face . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.8 Characteristics ofthestiffness matricesfordifferentprecondi- tioned versions of the X-FEM applied to a bi-material struc- ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.9 Estimates for the overall computational effort to calculate the preconditioner for a discretized bi-material structure . 107 3.10 Characteristics ofthestiffness matricesfordifferentprecondi- tioned versions of the X-FEM applied to a cracked structure 112 3.11 Estimates for the overall computational effort to calculate the preconditioner for the cracked structure . . . . . . . . 112 3.12 Time measurements . . . . . . . . . . . . . . . . . . . . . . 118 3.13 Number of geometric operations needed to perform mesh re- finement with and without element clustering . . . . . . . 126 4.1 Creep constants taken from the literature . . . . . . . . . . 155 4.2 Deformation of the homogeneous solder model for different loading conditions . . . . . . . . . . . . . . . . . . . . . . . 157 4.3 Tin elastic constants . . . . . . . . . . . . . . . . . . . . . 158 4.4 Tin thermal expansion . . . . . . . . . . . . . . . . . . . . 159 4.5 Elastic deformation of different polycrystalline structures ob- tained by simulation . . . . . . . . . . . . . . . . . . . . . 159 4.6 Thermal expansion of different polycrystalline structures ob- tained by simulation . . . . . . . . . . . . . . . . . . . . . 160 4.7 Adjusted creep constants for SnAgCu grains . . . . . . . . 160 4.8 Displacement rates of different polycrystalline structures ob- tained by simulation . . . . . . . . . . . . . . . . . . . . . 161 4.9 Material properties of the PBGA 676 package assembly . . 163 4.10 Temperature profile used in the crack growth simulation . 167 4.11 Accumulatedcreepstrainevaluatedindifferentpolycrystalline joint models . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.12 Comparison of mean crack length and standard deviation in experiment and simulation . . . . . . . . . . . . . . . . . . 170