Modeling Stresses in Polyimide Films
- PDF / 468,978 Bytes
- 8 Pages / 414.72 x 648 pts Page_size
- 63 Downloads / 250 Views
MODELING STRESSES IN POLYIMIDE FILMS MICHAEL T. POTTIGER* AND JOHN C. COBURN** *DuPont Polymers, Experimental Station, P. 0. Box 80101, Wilmington, DE 19880-0101 **DuPont Electronics, Experimental Station, P. 0. Box 80336, Wilmington, DE 19880-0336 INTRODUCTION The trend towards higher density and smaller feature sizes in today's devices, and the increasing costs associated with designing and manufacturing these devices, has placed a greater emphasis on obtaining an a priori understanding of how various materials will perform in a device. A number of manufacturers have turned to computer modeling, utilizing finite element analysis to aid in the design of new devices and reduce the costs associated with preparing prototypes. The use of computer modeling requires a constitutive equation relating the response of a material to an applied load. Polymer behavior is complex and writing an equation or a series of equations that describe the behavior of the polymer over the entire range of possible temperatures and deformations is nontrivial. Instead, series of equations that describe ideal material behavior are used in an attempt to describe the behavior of real materials over a narrow range of temperatures and deformations [1]. For solids, the ideal material response that is generally used to describe real polymer behavior is linear elasticity. The simplest linear elastic constitutive equation, which relates the stress, a, to the strain, E, in a uniaxial deformation of an ideal elastic isotropic solid, is Hooke's law
S=
Ee
(1)
where E is the Young's modulus. The most general form of equation (1) is the generalized Hooke's law Eij = SijrsTrs
(2)
where the subscripts i, j, r and s can take the values 1, 2 and 3, Eij are the components of the small strain tensor (not to be confused with Young's modulus), Sijrs are the components of the compliance matrix and Tij are the components of the Cauchy stress tensor. Assuming orthotropic symmetry, equation (2) reduces to a series of nine equations of the form
Ci = -i [0i - X iv. i.j ]
i
iij
(3)
where the Ei and 7i are the tensile strains and stresses, respectively, yij and tij are the shear strains and stresses, respectively, vij are Poisson's ratios, Ei are the tensile moduli and Gij are the shear moduli. The coefficients of the compliance matrix are in general functions of temperature, but the dependence on temperature is usually assumed to be slight or the variation is usually considered to be sufficiently small that the coefficients may be treated as constants. Thermal expansion, which can produce dimensional changes that are often as large as those induced from the applied loads, is generally included in the following manner [1]
AT=_1-a t i
Vij]
Ej1 I
1--
7 =Yii~t
(4)
where the ai are the linear coefficients of thermal expansion (CTE). For the case of isotropic materials, equations (4) reduce to
Mat. Res. Soc. Symp. Proc. Vol. 308. @1993 Materials Research Society
528
ei--AT
=
(Ti-i-vjaj1
= -ij 1ij
(5)
Consider a linear elastic coating applied to a sub
Data Loading...
