**6. Conclusions**

We were able to establish a simplified and easy-to-use tool for estimating the warpage in printed circuit boards based on the multilayered plate theory combined with the effects from the temperature-dependent thermal expansion coefficient and the temperaturedependent viscoelastic characteristics of the resin material. The results derived from this method are confirmed to be in agreement with the FE analysis results.

Application of Thermo-Viscoelastic Laminated Plate Theory to Predict Warpage of Printed Circuit Boards 317

domain of 20 digits, the number of terms in the generalized Maxwell model may be selected

The next task is to organize the model so that the contribution from each term is approximately smoothed. In accordance with the knowledge derived by Emri et al. (1993), keeping the smoothness of discrete relaxation spectra is effective in securing the desirable accuracy of approximation results. The Kronecker's delta in the expression is denoted by δ.

1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 Time t [sec]

> 1 () ( )

An example of these relaxation spectra is shown in Fig. 17. An attempt was made in this example such that the envelope for these discrete spectra is approximated to be piecewise quadratic so that the smoothness can be maintained subject to the curvature change along this envelope being not too large. Through the testing of such provisions, followed by an approximate calculation, it becomes possible to perform a curve fit operation for a master curve even though data is missing. For viscoelastic materials with sharp temperature dependency, it becomes very difficult for the temperature control in the measurement device to catch up with the actual material response, and as a result, critical defects are bound to occur (Fig. 18(b)); therefore, smoothing manipulation for those relaxation spectra is

In the curve fit program developed by the author's company, the generalized Maxwell model is identified based on the master curve shown in Fig. 7. This program is designed to completely fulfil the constraint conditions discussed in the preceding section. A sample output from this program is shown in Fig. 19. The user is only required to enter "Input data," "Number of Prony terms," and "Poisson's Ratio" in the specified input field, and then press the "Optimization" button. The program automatically performs an approximate

(27)

Viscoelastic(2elements)

Elastic

*N n nn n n H τ E τδτ τ* 

**Figure 16.** Relaxation behavior of single two-element Maxwell model

**Smoothness of relaxation spectra** 

0

100

Relaxation Modulus Er [Pa]

200

a highly effective measure.

to include 20 or more.
