4. Finite element analysis of ultrasonic vibration-assisted hot glass embossing process

Glass molding, also hot embossing, is a replicative process that allows the production of high-precision optical components from glass without grinding and polishing. Many researchers have been studying the glass molding process using finite element analysis. However, very few studies have focused on the hot glass embossing process assisted by ultrasonic vibration. Since the only difference between the conventional process and the ultrasonic vibration-assisted process is in the embossing stage, the glass model for the embossing stage should be created. This model could not only describe the glass behavior under embossing force but also express the effect of ultrasonic vibration. Standard linear solid (SLS) model, one kind of viscoelastic models, which combines a Maxwell model and a spring in series, was proposed for the glass deformation behavior during the embossing stage [8] (as shown in Figure 17). Substituting complex strain and complex stress from Eq. (5) into constitutive equation [8]:

$$\sigma = \left[ \frac{E\_0 \eta\_1}{E\_1} - \frac{\nu}{L} \frac{(E\_0 + E\_1)}{\left(\frac{E\_1}{\eta\_1} - \mathbf{1}\right)} \right] \exp\left(-\frac{E\_1}{\eta\_1} \varepsilon\right) + \frac{\nu}{L} \frac{(E\_0 + E\_1)}{\left(\frac{E\_1}{\eta\_1} - \mathbf{1}\right)} \exp\left(-\varepsilon\right) + E\_0 \varepsilon - \frac{E\_0 \eta\_1}{E\_1} \tag{10}$$

where E0, E1, and η<sup>1</sup> are constants, determined by fitting experimental data; v is the embossing speed; and L is the initial height of glass sample. Storage and loss moduli can be calculated as

Noise and Vibration Control - From Theory to Practice

$$\mathbf{G}' = \mathbf{E}\_1 \left[ \frac{\left(o\eta\eta\_1^2\right)}{E\_1^2 + \left(o\eta\eta\_1\right)^2} \right]; \mathbf{G}' = \mathbf{E}\_1^2 \left[ \frac{o\eta\_1}{E\_1^2 + \left(o\eta\_1\right)^2} \right] \tag{11}$$

As an alternating stress is applied to the material specimen, it is similar to an elastic solid which its elastic properties are exhibited, and the dynamic viscosity decreases significantly. The above reason shows that ultrasonic vibration could be

After inputting the above models to simulation, simulated results could be used to verify the value of proposed models. As shown in Figures 19 and 20, the agreement between simulated results and experimental data proved that finite element analysis would have an important role in analyzing and predicting the effect of

assumed to improve the forming process [15].

Ultrasonic Vibration-Assisted Hot Glass Embossing Process

DOI: http://dx.doi.org/10.5772/intechopen.86546

3D profile of pyramid microstructures after hot embossing process [10].

Plot of glass maximum filling depth against amplitude [14].

Figure 19.

Figure 20.

29

where ω is the angular frequency of ultrasonic vibration. Substituting G<sup>00</sup> in Eq. (11) into Eq. (9), the amount of heat created by ultrasonic vibration could be determined.

Another viscoelastic model, which has been proposed for the glass deformation behavior during the hot embossing stage, is the Generalized Maxwell model, as shown in Figure 18 [14]. Basically, Generalized Maxwell model is similar to SLS model, except having more the elements of Maxwell model.

The time-dependent response is characterized by the deviatoric terms as

$$
\sigma = \int\_0^t \mathbf{2G}(t-\tau) \frac{de}{d\tau} d\tau \tag{12}
$$

The above integral is evaluated for current time t on the basis of past time τ. G(t � τ) is not a constant value, but it is represented by a Prony series, as described by

$$\mathbf{G}(\mathbf{t} - \mathbf{r}) = \mathbf{G}\_0 \left[ \sigma\_{\infty} + \sum\_{i=1}^{n\_G} a\_i \exp\left(-\frac{t}{\sigma\_i}\right) \right] \tag{13}$$

where τ<sup>i</sup> is the relaxation time, α<sup>i</sup> the weight factor, nG the number of Generalized Maxwell model units, and G<sup>0</sup> the initial modulus. The dynamic viscosity η <sup>∗</sup> can be calculated correspondingly, as demonstrated by

$$\boldsymbol{\eta}^{\*} = \sum\_{i=1}^{n\_G} \mathbf{G}\_i \frac{\eta\_i}{\mathbf{1} + \alpha\_i^2 \tau\_i^2} \tag{14}$$

Figure 17. Standard linear solid model.

Figure 18. Generalized Maxwell model.

Ultrasonic Vibration-Assisted Hot Glass Embossing Process DOI: http://dx.doi.org/10.5772/intechopen.86546

As an alternating stress is applied to the material specimen, it is similar to an elastic solid which its elastic properties are exhibited, and the dynamic viscosity decreases significantly. The above reason shows that ultrasonic vibration could be assumed to improve the forming process [15].

After inputting the above models to simulation, simulated results could be used to verify the value of proposed models. As shown in Figures 19 and 20, the agreement between simulated results and experimental data proved that finite element analysis would have an important role in analyzing and predicting the effect of

G<sup>0</sup> ¼ E<sup>1</sup>

Noise and Vibration Control - From Theory to Practice

determined.

as described by

Figure 17.

Figure 18.

28

Generalized Maxwell model.

Standard linear solid model.

ωη<sup>1</sup> <sup>2</sup> � �

" #

<sup>1</sup> <sup>þ</sup> ωη<sup>1</sup> ð Þ<sup>2</sup>

; G<sup>00</sup>

Another viscoelastic model, which has been proposed for the glass deformation behavior during the hot embossing stage, is the Generalized Maxwell model, as shown in Figure 18 [14]. Basically, Generalized Maxwell model is similar to SLS

where ω is the angular frequency of ultrasonic vibration. Substituting G<sup>00</sup>

The time-dependent response is characterized by the deviatoric terms as

2G tð Þ � τ

nG i¼1

The above integral is evaluated for current time t on the basis of past time

τ. G(t � τ) is not a constant value, but it is represented by a Prony series,

where τ<sup>i</sup> is the relaxation time, α<sup>i</sup> the weight factor, nG the number of Generalized Maxwell model units, and G<sup>0</sup> the initial modulus. The dynamic viscos-

> <sup>η</sup> <sup>∗</sup> <sup>¼</sup> <sup>∑</sup> nG i¼1 Gi

G tð Þ¼ � τ G<sup>0</sup> σ<sup>∞</sup> þ ∑

ity η <sup>∗</sup> can be calculated correspondingly, as demonstrated by

de dτ

<sup>α</sup><sup>i</sup> exp � <sup>t</sup>

� � � �

ηi <sup>1</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup> i τ2 i

τi

Eq. (11) into Eq. (9), the amount of heat created by ultrasonic vibration could be

<sup>¼</sup> <sup>E</sup><sup>2</sup> 1 ωη<sup>1</sup>

" #

<sup>1</sup> <sup>þ</sup> ωη<sup>1</sup> ð Þ<sup>2</sup>

dτ (12)

(11)

in

(13)

(14)

E2

E2

model, except having more the elements of Maxwell model.

σ ¼ ðt 0

Figure 19. 3D profile of pyramid microstructures after hot embossing process [10].

Figure 20. Plot of glass maximum filling depth against amplitude [14].

process factors, such as temperature, speed, amplitude, frequency, etc. in improving the quality of the final glass products.

References

[1] Worgull M. Hot Embossing: Theory and Technology of Microreplicaiton. Oxford: William Andrew; 2009

DOI: http://dx.doi.org/10.5772/intechopen.86546

Ultrasonic Vibration-Assisted Hot Glass Embossing Process

[10] Nguyen L. Finite Element Analysis of the Ultrasonic Vibration-Assisted Hot Glass Embossing Process. [dissertation]

[11] Nguyen L, Wu M-H, Hung C. Effect

[12] Nguyen L, Wu M-H, Hung C. Effect of Ultrasonic Vibration on Increasing Embossing Speed during Hot Glass Embossing Process. In: Proceedings of the First International Conference on Materials, Machines and Methods for

microstructure hot glass embossing process. Journal of Science and Technology. 2017;122:17-21

Hsinchu: National Chiao Tung

of ultrasonic vibration on

Sustainable Development

and Materials. 2019;889:71-79

Wang X. Improvement of glass formability in ultrasonic vibration assisted molding process. International Journal of Precision and Manufacturing.

Wang X. Mechanism study on microgroove forming by ultrasonic vibration assisted hot pressing. Precision

Engineering. 2016;46:270-277

[16] JinyaoChen Y, Chen H, Lia S-Y, Jowc J. Physical and chemical effects of ultrasound vibration on polymer melt in extrusion. Ultrasonics Sonochemistry.

2018;1:98-105

2017;18(1):57-62

2010;17(1):66-71

(MMMS2018). Da Nang; 18-19 May

[13] Nguyen L, Wu M-H, Hung C. Effect of ultrasonic vibration on increasing embossing speed during hot glass embossing process. Applied Mechanics

[14] Zhou T, Xie J, Yan J, Tsunemoto K,

[15] Xie J, Zhou T, Liua Y, Kuriyagawa T,

University; 2015

[2] Tsai Y-P. Ultrasonic Vibration-Assisted Optical Glass Hot Embossing Process. [dissertation] Hsinchu: National Chiao Tung University; 2013

[3] Hung J-C, Tsai Y-P, Hung C. Development of a new apparatus for ultrasonic vibration-assisted glass hot

embossing process. Precision Engineering. 2013;37(1):222-227

[4] Gardner L, Insausti A, Ng KT, Ashraf M. Elevated temperature material properties of stainless steel alloys. Journal of Constructional Steel Research. 2010;66(5):634-647

[5] Nguyen LP, Tsai Y-P, Hung J-C, Hsieh Y-C, Hung C. Finite element analysis of an ultrasonic vibration device at high temperatures. Journal of the Chinese Society of Mechanical Engineers. 2016;37(3):193-200

[6] Zarzycki J. Glasses and the Vitreous State. New York: Cambridge; 1991

[7] Tsai Y-P, Hung J-C, Yin L-C, Hung C. Ultrasonic vibration-assisted optical glass hot embossing process. The International Journal of Advanced Manufacturing Technology. 2012;

[8] Nguyen LP, Hao K-C, Su Y-H, Hung C. Modeling the embossing stage of the ultrasonic-vibration-assisted hot glass embossing process. International Journal of Applied Glass Science. 2015;

[9] Tsai Y-C. Finite Element Analysis on the Optical Glass Molding Process. [dissertation] Hsinchu: National Chiao

60(9-12):1207-1213

6(2):172-181

31

Tung University; 2010
