Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

M20100376

138 views

Published on

  • Be the first to comment

  • Be the first to like this

M20100376

  1. 1. 1 M20100376 Development of an Analytical tool for Multilayer Stack Assemblies Rammohan B , Sanjay Singh Chauhan, Arvind Krishna Delphi Automotive Systems Copyright © 2011 SAE INDIA ABSTRACT The development of an Analytical model for multilayer stack subjected to temperature change is demonstrated here. Thin continuous layers of materials bonded together deform as a plate due to their differing coefficients of thermal expansion on subjecting the bonded materials to the change in temperature. Applications of such structures can be found in the electronic industry for the study of warpage issues in printed circuit boards or in the aerospace industry as laminated thin sheets used as skin structures for load bearing members such as wings and fuselage. In automotive electronics, critical high-power packages (IGBT, Power FETs) include several layers of widely differing materials (Aluminum, Solder, Copper, Ceramics) subjected to wide temperature cyclic ranges. Modeling of such structures by the application of three dimensional finite element methods is usually time consuming and may not exactly predict the inter-laminar strains. An attempt has been made here to obtain the closed form solution for such a multilayered stack using a set of recursive polynomial equations on subjecting the stack to temperature change under steady state conditions. Based on the closed form solution technique, a simple excel based tool has been developed to predict the radius of curvature, bending strains at both the bottom and the top layer of stack and compared with the Numerical standard codes. Results show a close comparison for both the analytical and the finite element simulation. Analytical solutions are further extended to predict the interlayer displacements. INTRODUCTION Co-efficient of Thermal expansion (CTE) and Fatigue induced due to thermal loads on electronic devices are a major concern for the failure of the electronic devices. Inter-laminar strains that are developed due to the bending of the stack results in de-bonding of such multilayered assemblies. In this study, a relatively simple analytical excel based tool is developed based on linear first order polynomial equations to obtain the closed form solution to predict the radius of curvature, warpage, bending strains at both top and bottom layers of the stack. Commercially available software - Ansys has been used to verify the VBA- Excel analytical tool. ANALYTICAL MODEL An analytical model based on conventional thin plate theory has been used here. The assumptions are such that the dimensions of the stack in the thickness direction are less than the other two directions. The stack tends to deform symmetrically on the application of temperature gradient. Bending moment, radius of curvature, in-plane forces, and the strains at both top and bottom layer of the stack are deduced using the set of equations in [1].
  2. 2. 2 Fig 1a: Illustrative example: undeformed stack Fig 1b: Illustrative example: Deformed stack The stack shown in the Fig 1a & 1b above has the same dimensions even in the out of plane direction. Radius of curvature of the stack in the in-plane direction is larger than thickness of the stack. r tE M i ii i 1 )1(12 3   = 2 2 dx wd From the basic static equilibrium at the neutral axis, the summation of all forces and moments about the neutral axis is zero. (In the Equation 3 above the index k ranges from 1 to j). Since the interlayer displacements (u) between the two consecutive layers are the same, and then re-writing equations we get, Equations 1 to 4 are a set of recursive polynomial equations which are solved numerically using FORTRAN code[1] . In this paper a closed form solution is developed using the following algorithm with F1 and 1/r being the primary unknowns.  iF and in the flow chart above leads to a set of recursive equations which can be reduced to 2x2 matrix of the form [A]{B}={C}. The coefficients A11,A12,A21 and A22 obtained as a function of Young’s Modulus, thickness and Poisson’s Ratio is shown below. 0 iF 0) 2 (   j kji t tFM     r tt T tE F tE F ii ii ii ii ii ii 2 )1()1( 1 1 11 11            0) 2 (   j kji t tFM                                                                                                                                                     i j i j i ii N i i N k k kk n i i j i j i ii i i j j i i j j N i i ii i ii N i i i ii N i i i j j N i i ii t tT tE T tE t t tE ttt A t t tE tE t A T tE T tE ttt A tE tE A T T r F AA AA 12 1 1 3 1 1 1 1 22 1211 11 21 2 1 2 1 1 12 11 1 2 11 1 2221 1211 2)1( )1(122 21 2 ) 2 ( 1 1 2 1 12 )(2 1 ) 1 (1 1              (1) (2) (3) (4) (5) r
  3. 3. 3 Thus a closed form equation is obtained as a function of only two variables which can be solved using simple matrix inversions. By knowing the force in the first layer and the radius of curvature of the stack , bending moments and forces in all other layers can be computed. This can be further extended to obtain the forces and moments in all layers of stack, using the recursive functions explained above. COMPARISON USING ANSYS A 3-D Finite Element model has been created in Ansys to exactly simulate the Multilayered stack when subjected to a temperature change under steady state condition. 2-D and 3-D structural solid elements [2,3] have been used for creating the three dimensional FE Model. Each layer has a different co-efficient of thermal expansion, thickness and also has different Modulus of Elasticity. A close correlation is observed between VBA excel tool outputs and FE results shown in Table-1 and 2. Similarly analytical tool gives a very close comparison for prediction of warpage shown in Fig (4) in the out of plane direction by providing the inputs of the stack assembly assumed in Pan et al[1] . Fig 2: Displacements for three layered stack Table 1: Comparison of results for a 3 layered stack Comparison between the Multi-stack tool and the Ansys results are tabulated in the Example 1. Similar comparison for a five layered stack is shown below in Table 2. Fig 3: Displacements for five layered stack Table 2: Comparison of results for a 5 layered stack Fig 4: Warpage from the excel tool for a 5 layered stack – A close correlation with Pan et al [1] . Fig 5: Intermediate displacements from the excel tool for a 5 layered stack –A close correlation with Pan et al [1].
  4. 4. 4 CONCLUSION First order polynomial linear equations have been used for developing closed form solutions using thin plate theory. This correlates well with 3-D finite Element method (FEM) and can be used for quicker study for linear models without actually modeling the 3-D structure in FEM. The results obtained from the tool are the warpage, bending strains both at the top and bottom layer of stack, forces and moments on each layer of the stack. REFERENCES 1. Tsung-Yu Pan and Yi-Hsin Pao, “Deformation in Multilayer stack assemblies”, ASME Journal of Electronic Packaging, Vol.30, pp. 30-34, 1980. 2. Cook, R. D., Concepts and Applications of Finite Element Analysis, 2nd ed. Wiley, New York, 1981 3. Theory Reference for ANSYS and ANSYS Workbench: - Release 11.0 Documentation for ANSYS Symbols used D = flexural rigidity, Nmm2 E = Young's modulus, N/mm2 F= Force, N L= length and width of each layer in the stack, mm M = Bending moment, N mm R = Radius of curvature, mm ΔT= change in temperature t = thickness of each layer u= Displacement in x-direction, mm V = Displacement in y-direction, mm w= Displacement in z-direction, mm Subscripts 1,2,..., j ,... α = thermal expansion coefficient, mm/mm/°C v = Poisson's ratio j= layer indicator

×