**Number of terms in the generalized Maxwell model**

It is common practice to give the abscissas of a master curve, i.e., frequencies using a logarithmic scale covering a range of 20 to 30 digits. To make this master curve approximate a smooth curve, it is said that the number of terms (number of two-element Maxwell models) should be selected so they are equal to or above the number of digits of the frequencies. In order to confirm this, a simple calculation was carried out using a single two-element Maxwell model. The calculation was performed under the following conditions.

Elastic model: E = 100 Pa Viscoelastic model: E1 = 100 Pa, τ1 = 1 sec

The relaxation modulus calculated using this model is shown in Fig. 16. In the figure, the curve with the solid line in the relaxation module is noted to decay one digit over time. It shows that a single Maxwell model is capable of representing relaxation behavior over a time domain of about one digit. Accordingly, when each dashpot is provided with the sequence of τn such that the next one has an order of time greater by one digit than the former, the relaxation behavior over the full range of a time domain can be expressed without a break. If the abscissa of a master curve, for example, is represented with a time domain of 20 digits, the number of terms in the generalized Maxwell model may be selected to include 20 or more.

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

#### **Smoothness of relaxation spectra**

316 Viscoelasticity – From Theory to Biological Applications

**development of a curve fit program)** 

**Ee, En, τn of positive definite** 

to make them positive definite.

conditions.

Elastic model: E = 100 Pa

Viscoelastic model: E1 = 100 Pa, τ1 = 1 sec

**Number of terms in the generalized Maxwell model** 

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

When the generalized Maxwell model for time domain is identified based on the master curve in frequency domain using Eq. (25), the following three points should be noted.

Since the generalized Maxwell model is regarded as a mechanical model, it is preferable that all the values of these coefficients are always positive. However, the input rule for them is different among general purpose FEMs. For example, some codes allow negative value input, while others strictly prohibit negative value input. Hence, there is no unified rule among all the codes. Since it is empirically observed that master curves may oscillate due to the affect of terms with negative values, it is considered reasonable to control the input data

It is common practice to give the abscissas of a master curve, i.e., frequencies using a logarithmic scale covering a range of 20 to 30 digits. To make this master curve approximate a smooth curve, it is said that the number of terms (number of two-element Maxwell models) should be selected so they are equal to or above the number of digits of the frequencies. In order to confirm this, a simple calculation was carried out using a single two-element Maxwell model. The calculation was performed under the following

The relaxation modulus calculated using this model is shown in Fig. 16. In the figure, the curve with the solid line in the relaxation module is noted to decay one digit over time. It shows that a single Maxwell model is capable of representing relaxation behavior over a time domain of about one digit. Accordingly, when each dashpot is provided with the sequence of τn such that the next one has an order of time greater by one digit than the former, the relaxation behavior over the full range of a time domain can be expressed without a break. If the abscissa of a master curve, for example, is represented with a time

method are confirmed to be in agreement with the FE analysis results.

**Appendix (Identification of the generalized Maxwell model and** 

**6. Conclusions** 

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.

$$H\_n(\boldsymbol{\pi}) = \sum\_{n=1}^{N} E\_n \boldsymbol{\pi}\_n \delta(\boldsymbol{\pi}\_n - \boldsymbol{\pi})\tag{27}$$

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 a highly effective measure.

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

calculation. The optimization operation uses the quasi-Newton method. For the quasi-Newton method, it is necessary to set up the initial condition in the vicinity of the optimized value. However, this program incorporates an algorithm that can automatically estimate, from the test results, an initial condition that easily converges.

Application of Thermo-Viscoelastic Laminated Plate Theory to Predict Warpage of Printed Circuit Boards 319

Initial Value

9.95E+08 9.26E+07 1.00E+11 9.95E+08 1.00E-11 1.55E+08 1.00E-11 1.00E-11 1.07E+09

Experimental Data (Fig.1) Proney Series Master Curve (Fig.2)

Auto or Manual Output Output

8.50E+08 1.12E+08 5.72E+09 1.48E+08 1.00E-06 1.51E+08 1.00E-06 6.00E-11 9.52E+08

8.00E+08 1.27E+08 2.65E+09 2.76E+07 1.00E-04 1.40E+07 1.00E-04 8.00E-11 9.34E+08 8.02E+08 1.21E+08 1.28E+09 1.68E+07 1.00E-03 7.25E+06 1.00E-03 9.00E-11 9.26E+08 7.46E+08 1.33E+08 1.23E+09 1.29E+07 1.00E-02 5.16E+06 1.00E-02 1.00E-10 9.19E+08

5.28E+08 1.57E+08 5.95E+07 6.00E-10 7.86E+08

Relaxation Modulus

4.34E+08 1.58E+08 2.45E+07 9.00E-10 7.50E+08

3.29E+08 1.55E+08 8.00E+06 2.00E-09 6.72E+08 2.77E+08 1.44E+08 5.28E+06 3.00E-09 6.35E+08 2.51E+08 1.40E+08 3.71E+06 4.00E-09 6.11E+08 2.07E+08 1.27E+08 2.45E+06 5.00E-09 5.92E+08

1.E+05

1.E+06

1.E+07

1.E+08

1.E+09

1.E+10

1.53E+08 1.07E+08 1.14E+06 7.00E-09 5.61E+08 1.48E+08 1.05E+08 9.23E+05 8.00E-09 5.49E+08 1.27E+08 9.41E+07 8.00E+05 9.00E-09 5.37E+08 1.01E+08 7.85E+07 4.28E+05 1.00E-08 5.26E+08

Fig.2 Master Curve (Time)

1E-11 1E-08 1E-05 1E-02 1E+01 Time

12 G' G'' w Gr t Gi τi t Gr(t)

Relative Error 8.85E+08 1.15E+08 5.42E+10 8.50E+08 1.00E-10 1.24E+08 1.00E-10 2.00E-11 1.02E+09 **1.940E+00** 9.47E+08 9.79E+07 2.65E+10 7.35E+08 1.00E-09 2.12E+08 1.00E-09 3.00E-11 9.93E+08 Variance 8.78E+08 1.22E+08 2.52E+10 5.28E+08 1.00E-08 2.17E+08 1.00E-08 4.00E-11 9.76E+08 **9.846E-03** 9.00E+08 1.05E+08 1.23E+10 3.29E+08 1.00E-07 2.25E+08 1.00E-07 5.00E-11 9.63E+08

Frequency Range 8.56E+08 1.20E+08 2.76E+09 5.05E+07 1.00E-05 4.30E+07 1.00E-05 7.00E-11 9.43E+08

Minimum Frequency ωmin 7.35E+08 1.30E+08 5.95E+08 9.62E+06 1.00E-01 3.02E+06 1.00E-01 2.00E-10 8.68E+08 1.00E-01 6.67E+08 1.40E+08 2.76E+08 7.62E+06 1.00E+00 8.86E+05 1.00E+00 3.00E-10 8.38E+08 Maximum Frequency ωmax 5.97E+08 1.50E+08 1.28E+08 7.33E+06 1.00E+01 7.33E+06 Ge 4.00E-10 8.17E+08 1.00E+11 5.88E+08 1.54E+08 1.14E+08 5.00E-10 8.00E+08

Poisson's Ratio 5.15E+08 1.56E+08 5.28E+07 7.00E-10 7.73E+08 4.00000E-01 4.55E+08 1.61E+08 2.76E+07 8.00E-10 7.61E+08

Modulus 3.53E+08 1.55E+08 1.14E+07 1.00E-09 7.40E+08

Relaxation Modulus Approximation <sup>E</sup>

Initial Value 1.81E+08 1.18E+08 1.72E+06 6.00E-09 5.76E+08

Fig.1 Master Curve (Frequency)

1E-01 1E+02 1E+05 1E+08 1E+11 Angular Frequency

Experimental Data Proney Series Approximation

**Figure 19.** Viscoelastic curve fit program using Excel

1.E+05

1.E+06

1.E+07

Storage and Loss Modulus

1.E+08

1.E+09

1.E+10

Input Tensile Test (E) or Shear Test (G)

Takaya Kobayashi, Masami Sato and Yasuko Mihara *Mechanical Design & Analysis Corporation, Japan* 

*Proceedings,* Dassault Systems Simulia Corp., pp. 360-373.

*Mechanical Science and Technology*, Vol. 22, pp. 1483-1489.

*Composites Science and Technology*, Vol.65, pp.621-634.

Emri, I. & Tschoegl, N. W. (1993), Generating line spectra from experimental responses. Part I: Relaxation modulus and creep compliance, *Rheologica Acta*, Vol. 32, pp. 311-322. Kobayashi, T.; Mikami, T. & Fujikawa, M. (2008), Application of Abaqus for Advanced Inelastic Analysis (I: Linear Viscoelastic Materials), *2008 Abaqus Users' Conference* 

Lim, J. H.; Han, M.; Lee, J.; Earmme, Y. Y.; Lee, S. & Im, S. (2008), A study on the thermomechanical behavior of semiconductor chips on thin silicon substrate, *Journal of* 

Shrotriya, P. & Sottos, N. R. (2005), Viscoelastic response of woven composite substrates,

Valdevit, L.; Khanna, V.; Sharma, A.; Sri-Jayantha, S.; Questad, D. & and Sikka, K. (2008), Organic substrates for flip-chip design: A thermo-mechanical model that accounts for

heterogeneity and anisotropy, *Microelectronics Reliability*, Vol.48, pp.245-260.

**Author details** 

G

Auto Manual

Auto Manual

Number of Terms( Prony)

Optimization

**7. References** 

The master curve for epoxy resin material shown in Fig. 8 covers a range of about 12 digits in terms of angular frequency, so the number of terms in the Maxwell model is set to 12. As previously mentioned, positive values for all of Ee, En, and τn are maintained during the calculation. The coefficients of the identified Maxwell model are automatically written into the respective format of the input file for Abaqus, Marc, or LS-DYNA.

**Figure 17.** Smoothing manipulation for relaxation spectra

**Figure 18.** Examples of measured master curve

Application of Thermo-Viscoelastic Laminated Plate Theory to Predict Warpage of Printed Circuit Boards 319

**Figure 19.** Viscoelastic curve fit program using Excel
