Effect of molding machine’s stiffness on the thickness of molded glass rings

Ensuring the final center thickness of molded optics within specified tolerance is one standard requirement for precision glass molding. In this study, we mainly discuss why and how the machine's stiffness affects the final thickness of molded glass rings. To this end, molding experiments using displacement control mode are performed at different molding temperatures. Additionally, tailored tests are conducted to determine the mechanical stiffness and thermal deformation of the molding machine. Thermo‐displacement coupled finite element simulations are employed to reveal how the molding behaviors and the thickness evolution depend on the machine's stiffness. Results show that the final thickness deviation is mainly related to the maximum generated machine deformation during pressing, and thus decreases with the machine's stiffness. The influence extent of the machine's stiffness remarkably decreases with the molding temperature, which explains why the experimental final thickness results decrease as well. Besides, compared to the rigid machine assumption, the simulated thickness results under the real machine's stiffness, are well coherent with the experiments, which proves the importance of considering the machine's stiffness. Finally, it is demonstrated that applying a period of position holding after pressing at molding temperature, can effectively reduce the final thickness deviation.

Keywords: machine's stiffness; precision glass molding; thickness; viscoelasticity

INTRODUCTION

Precision glass molding (PGM) has been demonstrated as an efficient and high‐volume fabrication method for types of advanced optics, such as aspherical lens,[1] microlens array,[2] microstructures,[3] and freeform optics.[4] Although glass molding has numerous advantages, several shortcomings are pending to be overcome. For example, besides the limitation of the time and cost consuming mold fabrication,[5] the molded optics could have some minor quality defects, such as dimension or shape deviation,[[6]] birefringence phenomenon,[[8]] refractive index drop.[[10]] Numerous experimental and numerical research studies have been performed to thoroughly investigate these quality defects, which can be mainly attributed to the mismatch of thermal expansion coefficients,[12] residual stress induced by temperature gradient,[8] and volume change due to the structural relaxation of glass.[13]

When considering the quality concern in terms of dimension or shape deviation, most attention is focused on the surface profile or form deviation.[[14]] As for precision optics, the center thickness is as well one important parameter in quality evaluation. Typical manufacturing tolerances for center thickness are: ±0.20 mm for typical quality, ±0.050 mm for precision quality, and ±0.010 mm for high quality.[16]

Research groups have noticed the challenges in controlling the accuracy of the thickness during PGM process and attempted to propose efficient strategies. For example, Ananthasayanam et al[12] pointed out that process parameters, such as molding force and pressing time, must be carefully selected to achieve the same thickness for different molding temperatures. Peter Wachtel et al[17] reported that for a bench‐top glass molding machine, GP‐5000 HT (Dyna Technologies Inc.), pressing 2 mm glass piece to 1 mm target thickness yielded an average thickness of 1.001 mm and a standard deviation of thickness of 0.0055 mm. What's more, the authors instructively proposed that the final thickness was strictly dependent on the press position, which was found to be a function of the molding temperature, soaking time, system structure stiffness, and applied force. In the study of thermoforming mechanism of glass molding, Liu et al[18] further summarized that the final thickness of a lens was determined by the combination of the processing parameters, viscoelastic material parameters of glass, and the dimensional parameters of the lens.

Based on these literatures, the control of the final center thickness for glass molding is a complex issue as reflected in the following aspects. First, the final thickness could be affected by a number of process parameters and material properties parameters. Second, the softened glass during molding is viscoelastic material whose mechanical behaviors are time and temperature dependent, thus further complicates the control and prediction of the final thickness. What's more, because of the thermal deformation of both the molded optics and the molding machine, it is even more difficult to achieve the same thickness for different molding temperatures.

Among the numerous parameters affecting the final thickness, the effect of the molding machine's stiffness has not been given enough attention. Typically, its effect can be neglected when the molding force is relatively low. However, the required molding force can be very high caused by some factors, such as low molding temperature, fast molding rate, and large size of the molded optics. Under these situations, the effect of the molding machine's stiffness on the center thickness has to be well taken care of. Although the effect of the machine's stiffness on the final press position was considered,[17] the mechanism and governing principles have not been fully addressed in the previous studies.

Therefore, this paper aims to investigate the effect of the machine's stiffness on the final thickness of molded optics at different molding temperatures. To achieve this goal, both molding experiments and thermo‐displacement coupled finite element simulations are performed. In order to make sure that equal displacement is applied within the same period during pressing, the displacement control mode is adopted for the molding experiments. Glass rings are used to conduct the thickness study, because the friction coefficient can be obtained simultaneously using numerical friction calibration curve (FCC) method.[19] Additionally, self‐designed tests are conducted to obtain the thermal expansion and mechanical stiffness of the molding machine. The mechanism of the machine's stiffness affecting the final thickness is explored, and the effects on the molding mechanical behaviors and the final thickness at different molding temperatures are addressed in detail. The simulated final thickness results are compared with the experiments. Finally, possible solutions are proposed to reduce the thickness deviation.

EXPERIMENTS

Glass ring molding experiments under displacement control mode

Molding experiments are performed from 540°C to 590°C with 10°C as space, by using the Toshiba GMP‐311V machine. Two planar molds made of tungsten carbide (WC) are polished but not coated, with surface roughness of 2 nm in Ra. In this study, the glass samples are prepared in ring shape, with outer diameter (OD) of 24 mm, inner diameter (ID) of 12 mm, and height (H) of 8 mm. The ratio of OD:ID:H = 6:3:2 is coherent with most experiments in which the FCC method is adopted to estimate the forming‐interface friction behavior. The material of glass ring is P‐SK57 whose glass transition temperature Tg is 493°C.

As illustrated in Figure , the whole molding process includes heating, soaking, pressing, cooling, and releasing stages. The glass ring and molds are both heated up to the specified molding temperature by infrared radiation. After that, the molding chamber is soaked for 300 seconds, in order to achieve uniform temperature distribution within the glass specimen. During pressing, a linear ramp displacement control mode is employed. The top mold is fixed, meanwhile the bottom mold is driven to compress the glass ring at a rate of 2 mm/min approximately. To be noted, in the early pressing stage, the position at which the molding force increases abruptly, indicates the beginning of molding, and is defined as the touching position. Herein, the reduction in thickness during pressing is calculated by subtracting the ending position from the touching position and expected as 4.0 mm. However, the touching position is affected by the thermal expansion of both the molding machine and the glass ring, and thus dependent on the molding temperature. Therefore, the ending position setup is altered accordingly to make sure that the reduction in thickness is equal for different molding temperatures.

During cooling, the position of the bottom mold is set the same with the end of pressing stage. Nitrogen is purged to cool the chamber to 400°C with a slow cooling rate of 0.5°C/s, and then the bottom mold is released. A fast cooling rate of 1°C/s is followed to cool the chamber to 200°C. Finally, the chamber is opened, and the glass ring is taken out to be cooled to room temperature. The thickness of the compressed glass ring is measured with caliper at six equal‐spaced locations along the radial direction, and then an average value is calculated.

Determination of molding machine's thermal deformation

Although both molds are set fixed during cooling, the thermal shrinkage could change their expected positions and thus affects the final thickness. To account for this effect in the following simulation, a specially designed thermal cycle test (Figure A) is conducted to determine the thermal deformation of the molding machine. In the test, the highest temperature is set as 800°C and reached within 250 seconds, then kept constant for 240 seconds, and finally slowly cooled to 220°C. The molds are touching each other at the start of the measurement with an initial force of 1 KN. In order to keep this force constant throughout the experiment, the molds have to move apart or closer to equally eliminate the influence of thermal expansion or shrinkage. In other words, the recorded displacement of the bottom screw jack is exactly the thermal deformation of the molding machine, and plotted against the temperature of the molds as shown in Figure B. The thermal expansion and shrinkage rate of the molding machine are both determined as about 1.0 µm/°C by calculating the slopes of the segmental curves.

Determination of molding machine's mechanical stiffness

The measurement of machine's stiffness is a must for this study and realized through the tailored compression tests. Herein, the machine's mechanical stiffness refers to the total equivalent stiffness of the composing parts, which include the molds, the structures to support the molds, the machine's framework, and others. As shown in Figure A, the two molds are directly compressing each other throughout a loading‐unloading cycle at specified temperature. The test is performed under the force control mode, where the molding force linearly increases to 30 KN, holds for seconds, and then linearly decreases to zero. The recorded machine deformation against the applied force (Figure B) are adopted to calculate the stiffness. Tests are conducted at five different temperatures, but no obvious difference in the force‐deformation curves is observed. This indicates that the molding machine's stiffness is very thermally stable, which is probably attributed to the usage of these excellent thermal‐enduring materials, such as silicon carbide (SiC) and tungsten carbide (WC). Finally, the real mechanical stiffness of the molding machine is calculated as 35.5 KN/mm during loading based on the slope of the deformation‐force curves.

FINITE ELEMENT SIMULATION

Setup of finite element analysis model

Thermo‐displacement coupled finite element simulations are performed in Abaqus, in order to predict the final thickness of the molded glass ring. To reduce the computation costs, two‐dimensional axisymmetric model is adopted. Herein, only the core parts, namely, the glass ring and WC molds are modelled (Figure A). These parts are deformable and meshed with four‐node quadrilateral thermal‐displacement coupled elements (CAX4T). About 4000 elements are generated for the glass ring with average edge size of 0.1 mm. A careful edge seed control strategy is applied for the glass ring to handle the large deformation and the high distortion of mesh during molding.

Most importantly, the mechanical stiffness of the molding machine is modelled using the special interaction type—the springs connected to the ground.[20] Here, the bottom mold is connected to the springs. If the stiffness of each spring is ks and the number of nodes on the bottom mold connected to the springs is N, the equivalent total stiffness of theses springs is calculated as $K_{\text{M}}=N·k_{\text{s}}$ .

The mechanical and thermal properties of WC and P‐SK57 glass are listed in Table. Viscoelastic stress relaxation behavior of P‐SK57 glass is modeled with the generalized Maxwell model in Prony series,[21] and the temperature dependence is characterized with the thermal rheological simple model using William‐Landel‐Ferry Equation.[22] In addition, structural relaxation using Tool‐Narayanaswamy‐Moynihan[23] model is included, in order to precisely predict the thermal deformation of glass ring. These relaxation parameters of P‐SK57 have been measured and reported in Ref [9] and listed in Table.

Mechanical and thermal material properties of P‐SK57 glass and tungsten carbide (WC) molds

Thermo‐mechanical properties	P‐SK57	WC
Modulus of elasticity (GPa)	93	690
Poisson's Ratio	0.249	0.22
Density (Kg/m3)	3010	15 800
Thermal conductivity (W/m·K)	Temperature dependent30	84.02
Coefficient of thermal expansion	9.2 × 10−6	5.5 × 10−6
Specific heat (J/Kg·K)	Temperature dependent30	314

Viscoelastic properties of P‐SK57 glass

Stress relaxation properties	Structural relaxation properties
WLF shift constants	C1	C2	Structural relaxation terms31	wi	$τvis$
	45.46	847.482		0.006369370	0.000002012
Prony series	gi	τi		0.078466102	0.000239586
0.008	0.00028		0.212515931	0.001197567
0.035	0.014		0.025472290	0.000032643
0.11	0.235		0.251525329	0.012365678
0.255	2.34		0.425328864	0.004468109
0.354	17.5	H/R	84396.5
0.237999	83.4	Fraction parameter x	0.798
Reference temp (°C)	491	Reference temp (°C)	572
		CTE of liquid glass $αl$	62.3 × 10−6

In the simulations, all the stages same with the real molding process are included except the soaking stage, since the temperature distribution in this stage is assumed uniform. A time‐dependent displacement boundary condition is applied to the top surface of the top mold (Figure B). In detail, during pressing stage the top mold moves down 4.0 mm within 140 seconds; during cooling stage it moves upward at rate of 1.0 µm/°C which is the thermal shrinkage rate of the molding machine; once released, it moves upward to the original position. As the mold material WC is excellent thermal conductor, the temperature non‐uniformity within molds is neglected. Besides, the temperature difference between the top and bottom mold is small, so the same prescribed time‐dependent temperature boundary condition is applied for both molds. As for the glass ring, the time‐dependent temperature boundary condition, is applied only for the heating and pressing stage. During the cooling and releasing stage, the temperature distribution within glass ring is automatically predicted by the software.

Hard contact interactions between the molds and glass ring are established to characterize the normal contact behavior, where the contact edges of molds are defined as master surfaces, and all the outer edges of glass ring are defined as slave surfaces. The heat transfer between the molds and glass ring is modeled using gap‐dependent thermal contact model. When the molds and glass make contact, the contact conductance hc is set as 2500 W/m2·K.[24] When there is a physical gap between them, the gap conductance hg is calculated using[25] $h_{\text{g}}=k_{{\text{N}}_2}/d$ , where $k_{{\text{N}}_2}$ is the thermal conductivity of nitrogen gas (0.024 W/m2·K), and d is the physical gap value.

Coulomb friction model with penalty friction is adopted to characterize the tangential friction behavior of the mold/glass interface. The friction coefficients are deduced using the numerical FCC method. The experimental data pairs (Figure A), reduction in inner diameter vs reduction in height, are scattered to evaluate the friction coefficients. When the friction coefficient is high, the calibration curves are too close to be distinguished (Figure B), therefore, a sub‐region plot is added in Figure C. It is observed that the deduced friction coefficients fluctuate around 0.60. As the difference is slight for different molding temperatures, the friction coefficient of 0.6 is adopted to characterize the tangential friction behavior between the P‐SK57 glass and the uncoated WC mold. Although the friction coefficient's dependence on temperature was reported for glass molding,[26] some references also showed that the dependence effect was not obvious. For example, these friction coefficients, such as 0.77 at 560°C, 0.73 at 580°C[27] and 0.70 at 572°C,[9] were concluded for the same molding interface.

Verification of the simulated molding force

The reliability of the finite element model is testified by comparing the simulated reaction force behavior with the experimental molding force. Herein, the real machine's stiffness is used, which means that the total equivalent stiffness of the connected springs is 35.5 KN/mm. As shown in Figure , the simulation results generally agree well with the experiments at both 540°C and 550°C. Therefore, it is verified that the developed finite element model is reliable to predict the real molding process. The difference between the simulated reaction force and the experimental molding force could be attributed to the integrated effects of several aspects, such as the mesh distortion, the nonlinear viscoelasticity behaviors, and the accuracy of both the viscoelastic parameters and the friction coefficient. Moreover, the relaxation oscillation phenomenon of borosilicate glass reported in Ref [28] could also be responsible for the difference.

RESULTS AND DISCUSSIONS

Mechanism of molding machine's stiffness affecting final thickness

In order to clarify why the finite machine's stiffness affects the actual deformation of glass ring, efforts are taken to study the deformation evolution behaviors of the glass ring and molding machine during pressing and cooling stages. The effect of machine's stiffness on thickness should be the most obvious at 540°C as the applied molding force is the maximum. Therefore, the simulation results of 540°C with the real machine's stiffness are first presented in this section.

Figure A schematically illustrates the deformation evolution behaviors of both the glass ring and the molding machine during pressing and cooling stage. Figure B presents the time history plots of the applied displacement on the top mold $s_{\text{total}}$ , the axial deformation of the glass ring $s_{\text{glass}}$ , and the axial deformation of the molding machine $s_{\text{machine}}$ . Herein, $s_{\text{machine}}$ is presented by the nodal displacement of the springs, and calculated as $s_{\text{machine}}=F/K_{\text{M}}$ , where F is the molding force, and KM is the stiffness of the molding machine.

The evolution principles at different moments are summarized as follows: (a) during pressing stage, the machine's deformation increases all the time, because the molding force keeps growing under ramp displacement. Since the total displacement is partially assigned to the machine's deformation, the actual axial deformation of glass ring is calculated as $s_{\text{machine}}=F/K_{\text{M}}$ , therefore less than the expected value. The maximum deformation of molding machine can be as much as 0.76 mm at 540°C, as the maximum molding force is about 27 KN at the end of pressing stage; (b) During the early cooling stage, the reaction force between glass ring and molds significantly drops, because of the stress relaxation behavior of viscoelastic glass. As a result, the molding machine recovers from the compressive state and thus continues compressing the glass ring. Therefore, the machine's deformation begins to decrease, while the compressive deformation of glass ring continues growing; (c) When the temperature within glass is below the transition temperature and glass enters the elastic solid state, the glass ring's compressive deformation stops growing though the reaction force is not zero. The machine's deformation still decreases slowly as the top mold retracts back due to thermal shrinkage; and (d) Once the top mold is released, the reaction force drops to zero so does the machine's deformation.

Based on the above analysis, the reason why the thickness reduction of glass ring is less than the expected value, can be attributed into two aspects: first, significant machine deformation is generated during pressing stage, which is affected by both the maximum molding force and the machine's stiffness. During cooling stage, the generated machine deformation cannot be fully recovered before the glass ring enters the elastic solid state, and thus permanent thickness deviation occurs.

Effect of molding machine's stiffness on final thickness

To quantitatively investigate the effects of the machine's stiffness on the molding behaviors and the final thickness of glass ring, simulations are performed under different machine's stiffness values from 10 to 500 KN/mm. The results for molding temperature of 540°C are presented first for purpose clarification, followed by results and discussion on different molding temperatures.

Figure A,B present the time history plots of the reaction molding force between molds and glass ring, and the generated machine deformation under different machine stiffness values. Interestingly, as shown in Figure A, the reaction force increases with the machine's stiffness, what's more, it decreases more rapidly under larger machine's stiffness. For brevity, the reasons are explained in Appendix A using the standard linear solid viscoelastic model.[29] It is easily understood that the generated machine deformation decreases with the machine's stiffness as shown in Figure B. To present the results clearer, the quantitative relations for the maximum reaction force and the maximum machine deformation vs the machine's stiffness are presented in Figure C.

The thickness evolution curves are plotted under different machine's stiffness values for pressing and cooling stage, respectively (Figure A,B). The thickness reduction of glass ring remarkably increases with the machine's stiffness during pressing stage, but slightly decreases with the machine's stiffness during cooling stage (Figure C). The reason can be explained by that under smaller machine's stiffness, not only is more machine deformation generated during pressing, but also more is recovered during cooling stage as shown in Figure B. Eventually, the relation for the final thickness's dependence on the machine's stiffness is quantitatively presented in Figure D. It is found that the final thickness decreases in a power function manner with the machine's stiffness.

Furthermore, the influence extent of the machine's stiffness on both the mechanical behaviors and the glass ring's thickness, is compared for different molding temperatures, as presented in Figure A‐E. Overall, the conclusions for the other molding temperatures are similar with 540°C. However, the influence extent of the machine's stiffness is observably weakened by increasing the molding temperature, because the molding forces are strongly dependent on temperature (Figure A). Particularly, as shown in Figure E, at lower molding temperatures (540‐560°C), the influence of the machine's stiffness on the final thickness is very evident, but turns out not distinguishable at higher molding temperatures (570‐590°C). What's more, based on the trend similarity of these curves in Figure B,E, it can be concluded that the final thickness of glass ring is mainly related to the maximum machine deformation.

Temperature dependence of the molded glass ring's final thickness

As concluded above, the influence of the machine's stiffness on the final thickness, is strongly dependent on the molding temperature. Based on this, it is ready to explain the experimental thickness results. Figure presents the images of the molded glass rings at different molding temperatures. Although the thickness differences among these rings are not easily captured by eyes, they can be reflected somehow from the top‐view appearances. As outlined with a red circle, the inner diameters of the molded rings are remarkably different between 540 and 590°C. The specific experimental final thickness results are listed in Table.

Summary of the experimental and simulated final thickness results of the molded glass rings

Molding temperature (°C)	Final thickness (mm)	Error (mm)
Experiment x0	x1: simulation 35.5 KN/mm	x2: simulation rigid	x1 − x0	x2 − x0
540	4.56	4.67	4.04	0.11	−0.52
550	4.23	4.22	4.04	−0.01	−0.19
560	4.08	4.07	4.05	−0.01	−0.03
570	4.03	4.06	4.06	0.02	0.02
580	4.05	4.08	4.08	0.03	0.03
590	4.07	4.09	4.09	0.02	0.02

Simulations with the real machine's stiffness are performed to predict the final thickness values at these tested molding temperatures. The thickness evolution behaviors are different as shown in Figure A,B. The achieved thickness reduction during pressing increases with the molding temperature. As compared in Table and Figure D, the simulated results of the final thickness with the real machine's stiffness, are in good agreement with the experiments. The error at 540°C could be attributed to the extremely large molding force. Overall, the final thickness results decrease with the molding temperature, and significantly deviate from the target at low molding temperatures. The surprising deviation exactly triggered the research motivation of this paper. As shown in Figure C, the maximum experimental molding forces decrease exponentially from more than 20 KN to about 200 N with the molding temperature, so the maximum machine deformation significantly decreases as well. By comparing Figure C,D, it is confirmed that the obvious decreasing trend of the experimental thickness results is determined by the maximum generated machine deformation.

Moreover, the simulated results under the ideal rigid machine assumption are compared to those of the real machine stiffness. First, most of the differences are observed at low molding temperatures, while negligible at high molding temperatures. To be mentioned, under the rigid machine assumption, the final thickness results slightly increase with the molding temperature, mainly due to that more springback of the glass ring is achieved during cooling at higher molding temperature. The reason is discussed in Appendix B. The springback phenomenon can also be found in the results of high molding temperatures as shown in the subplot of Figure B.

To sum, under the ramp displacement control mode, the obvious thickness deviation at low molding temperature is caused by the large molding force and the corresponding remarkable machine deformation. Therefore, consideration of the machine's stiffness is of great importance in accurately determining the final thickness of molded optics when the molding force is large.

Approaches to reduce the thickness deviation

Since the thickness deviation is remarkable at low molding temperature, possible approaches to reduce the deviation are discussed in the last section. The thickness deviation is mainly determined by the maximum machine deformation during pressing. Therefore, in order to reduce the thickness deviation, the core idea is to reduce the maximum machine deformation, which is affected by both the machine's stiffness and the maximum molding force. When the machine's stiffness is fixed, the practical approach is to reduce the maximum molding force. Herein, two approaches are proposed, that is, (a) reducing the molding rate and (b) applying a period of holding action after pressing at molding temperature. The process diagrams are shown in Figure , where both process durations are extended by 180 seconds.

The final thickness results in Figure A indicate that both approaches can reduce the thickness deviation. Reducing the molding speed can decrease the maximum molding force, because the mechanical behavior of viscoelastic glass is strain‐rate dependent. But the molding force is still considerably large (Figure B). By contrast, applying 180 seconds after pressing at molding temperature, the reaction force drops to a very small value (Figure C), because of the stress relaxation behavior of viscoelastic glass. Meanwhile, the deformation of glass gradually creeps to the target. As a result, the effect of machine deformation will be negligible after holding stage. Therefore, as clearly indicated in Figure A, the approach of applying a holding period after pressing, is more effective in reducing the thickness deviation than using a slower molding rate.

Furthermore, the thickness deviation's dependence on the duration of holding action is investigated for different molding temperatures. The experimental results are presented in Figure B. Overall, extending the length of holding time produces less thickness deviation, because the deformation of glass would creep closer to the target with more holding time. What's more, since the stiffness‐induced deviation is much greater at low molding temperature, the effect of extending holding duration to reduce the thickness deviation is naturally more obvious. To be noticed, although the position holding approach can remarkably reduce the thickness deviation, it could bring out serious drawbacks on the quality of molded optics if the holding period is too long. More practically, the final thickness of molded optics can be adapted by changing the stop position through the experimental trial and error method.

CONCLUSIONS

The paper mainly discusses the effect of the molding machine's stiffness on the final thickness of molded glass rings, and provide possible solutions to eliminate the adverse effect. First, significant machine deformation is generated during pressing because of the large molding force, and cannot be fully recovered before glass enters the elastic solid state, which explains the mechanism of machine's stiffness affecting the final thickness. The final thickness deviation decreases with the machine's stiffness, because the greater the machine's stiffness is, the less machine deformation is generated.

Moreover, the influence extent of the machine's stiffness is observably weakened by increasing the molding temperature. Therefore, the final thickness deviations overall decrease with the molding temperature. Compared to the rigid machine, the simulated thickness results under the real machine's stiffness, are considerably coherent with the experiments, which proves the importance of considering the machine's stiffness. Finally, applying a period of holding action after pressing at molding temperature, can effectively reduce the final thickness deviation. The effect of extending the holding period to reduce the thickness deviation is obvious at low molding temperature.

Work in this paper would be helpful in achieving better control of the final thickness of molded optics with consideration of the molding machine's stiffness. Also, it is instructive in designing the proper stiffness of the molding machine.

ACKNOWLEDGMENT

The authors are grateful for the financial support from the Natural Science Foundation of Anhui Province, China (No. 1808085ME118), and the Innovation and Technology Commission of HKSAR Project (No. GHP/043/14SZ).

APPENDIX

A. INFLUENCE OF THE MACHINE'S STIFFNESS ON THE VISCOELASTIC GLASS'S STRESS RESPONSE

As shown in Figure (A), a constitutive mechanical model is established to explain why the reaction molding force increases during pressing but drops more quickly during cooling with the machine's stiffness. Herein, the molding machine and the glass sample are connected in series. The machine is presented by the spring with stiffness of EM, and glass is presented by the standard linear solid (SLS) viscoelastic model using Maxwell form. The Maxwell arm composed of spring E2 and dashpot η is labeled with sub‐index m.

GRAPH: A1 (A) The constitutive mechanical model composed of the molding machine and the standard linear solid (SLS) viscoelastic glass connected in series, (B) the step function strain input

Based on the constitutive model, equations are listed as:

$$\begin{array}{c}\epsilon ={\epsilon }_{\text{M}}+{\epsilon }_m\end{array}$$

$$\begin{array}{c}\sigma ={\sigma }_m+{\sigma }_1\end{array}$$

$$\begin{array}{c}{\sigma }_1=E_1{\epsilon }_1\end{array}$$

$$\begin{array}{c}{\epsilon }_{\text{M}}=\sigma /E_{\text{M}}\end{array}$$

$$\begin{array}{c}\stackrel{.}{{\epsilon }_m}=\stackrel{.}{{\sigma }_m}{E}_2+{\sigma }_m\eta \end{array}$$

By eliminating the unknown parameters, we can get the Laplace transform of the system response in the following way:

$$\begin{array}{c}\overline{\sigma }=E_{\text{M}}\left(E_1+E_1\tau s+E_2\tau s\right)E_1+E_{\text{M}}+\left(E_1+E_2+E_{\text{M}}\right)\tau s\overline{\epsilon }\end{array}$$

where $\tau =\eta /E_2$ refers to the stress relaxation time of the Maxwell arm.

When a step function strain input (Figure (B)) is applied ( $\overline{\epsilon }={\epsilon }_0/s$ ), the corresponding stress response is obtained by using the inverse Laplace transform and given as:

$$\begin{array}{c}\sigma \left(t\right)={\epsilon }_0\left[E_{\text{M}}{E}_1{E}_{\text{M}}+E_1\right.\\ \left.+E_{\text{M}}^2{E}_2\left(E_{\text{M}}+E_1+E_2\right)\left(E_{\text{M}}+E_1\right)exp\left(-E_{\text{M}}+E_1{E}_{\text{M}}+E_1+E_{2}t\tau \right)\right]\\ ={\epsilon }_0\left[E_{1}1+E_1/E_{\text{M}}\right.\\ \left.+E_2\left(1+E_1/E_{\text{M}}+E_2/E_{\text{M}}\right)\left(1+E_1/E_{\text{M}}\right)exp\left(-t{\tau }^{\prime }\right)\right]\\ \end{array}$$

where ${\tau }^{\prime }=\tau \left[1+E_2/\left(E_{\text{M}}+E_1\right)\right]$ is the new stress relaxation time of the system.

Based on Equation (A.7), the maximum reaction force ${\sigma }_{max}$ at the end of pressing stage is calculated as:

$$\begin{array}{c}{\sigma }_{max}=\sigma \left(t=0\right)={\epsilon }_{0}1+E_1/E_{\text{M}}\left[E_1+E_{2}1+E_1/E_{\text{M}}+E_2/E_{\text{M}}\right]\end{array}$$

Observed from Equation (A.8), ${\sigma }_{max}$ increases with the machine's stiffness EM, which explains why the reaction molding force increases with the machine's stiffness during pressing stage. To be mentioned, if the machine's stiffness is ideally infinite, the result of ${\sigma }_{max}\left(E_{\text{M}}=\infty \right)$ will be ${\epsilon }_0\left(E_1+E_2\right)$ .

Correspondingly, the maximum generated machine deformation ${\epsilon }_{\text{M}}^{max}$ is given by:

$$\begin{array}{c}{\epsilon }_{\text{M}}^{max}={\sigma }_{max}{E}_{\text{M}}={\epsilon }_0{E}_{\text{M}}+E_1\left[E_1+E_{2}1+E_1/E_{\text{M}}+E_2/E_{\text{M}}\right]\end{array}$$

As Equation (A.9) is mainly governed by $1/\left(E_{\text{M}}+E_1\right)$ , ${\epsilon }_{\text{M}}^{max}$ decreases with $E_{\text{M}}$ , that is, the maximum generated machine deformation decreases with the machine's stiffness.

Secondly, during the early cooling stage, the stress relaxation rate is indicated by the new relaxation time ${\tau }^{\prime }=\tau \left[1+E_2/\left(E_{\text{M}}+E_1\right)\right]$ . As observed, the new relaxation time ${\tau }^{\prime }$ decreases with $E_{\text{M}}$ , which explains why the reaction molding force drops more quickly with the increasing machine's stiffness during cooling.

B. RECOVERY RESPONSE OF THE STANDARD LINEAR SOLID VISCOELASTIC MODEL

To explain why more springback is generated at higher molding temperature, the recovery response of the standard linear solid (SLS) viscoelastic model using Maxwell form is investigated (Figure (A)). The Laplace transform of the SLS model is expressed as[29]:

$$\begin{array}{c}\overline{\sigma }=\left(E_1+E_{2}ss+1\tau \right)\overline{\epsilon }\end{array}$$

where $\mathrm{\tau }=\eta /E_2$ is the stress relaxation time.

GRAPH: A2 (A) The standard linear solid (SLS) viscoelastic model using Maxwell form, (B) the rectangular pulse stress input

The rectangular pulse stress input (Figure (B)) can be expressed as $\sigma \left(t\right)={\sigma }_0\left[u\left(t\right)-u\left(t-t_0\right)\right]$ , and its Laplace transform is given by:

$$\begin{array}{c}\overline{\sigma }={\sigma }_{0}s\left(1-e^{-t_{0}s}\right)\end{array}$$

Then the Laplace transform of the strain response $\overline{\epsilon }$ is calculated as:

$$\begin{array}{c}\overline{\epsilon }=\frac{\overline{\sigma }}{\left(E_1+E_{2}ss+1\tau \right)}=\frac{{\sigma }_{0}s\left(1-e^{-t_{0}s}\right)}{\left(E_1+E_{2}ss+1\tau \right)}\end{array}$$

By applying the inverse Laplace transform, the time domain strain response can be obtained as the following segments: during the creep stage when $0<t<t_0$ ,

$$\begin{array}{c}\epsilon \left(t\right)={\sigma }_0\left[1E_1-E_2{E}_1\left(E_1+E_2\right)exp\left(-E_1{E}_1+E_{2}t\tau \right)\right]\end{array}$$

During the recovery stage when $t\ge t_0$ ,

$$\begin{array}{c}\epsilon \left(t\right)={\sigma }_0{E}_2{E}_1\left(E_1+E_2\right)\left[exp\left(E_1{E}_1+E_2{t}_0\tau \right)-1\right]exp\left(-E_1{E}_1+E_{2}t\tau \right)\end{array}$$

Based on the recovery Equation (A.14), we can figure out that the recovery rate is determined by the stress relaxation time. The stress relaxation time decreases significantly with the molding temperature. Therefore, more strain will be recovered when the stress relaxation time is decreased, which mainly explains why more springback is generated at higher molding temperature.

1 Yi AY, Jain A. Compression molding of aspherical glass lenses‐a combined experimental and numerical analysis. J Am Ceram Soc. 2005 ; 88 (3): 579 – 86. 2 Huang C‐Y, Hsiao W‐T, Huang K‐C, Chang K‐S, Chou H‐Y, Chou C‐P. Fabrication of a double‐sided micro‐lens array by a glass molding technique. J Micromech Microeng. 2011 ; 21 (8):085020/1–6. 3 Zhou TF, Liu XH, Liang ZQ, Liu Y, Xie JQ, Wang XB. Recent advancements in optical microstructure fabrication through glass molding process. Front Mech Eng. 2017 ; 12 (1): 46 – 65. 4 Krevor DH, Pongs G, Bresseler B, Bergs T, Menke G, Beich WS, et al. Precision molding of advanced glass optics: innovative production technology for lens arrays and free form optics. Polymer optics and molded glass optics: design, fabrication, and materials II. San Diego, California, USA: SPIE; 2012 :p. 84890I‐1/8. 5 Dambon O, Wang F, Klocke F, et al. Efficient mold manufacturing for precision glass molding. J Vac Sci Technol B. 2009 ; 27 (3): 1445 – 9. 6 Wang F, Chen Y, Klocke F, Pongs G, Yi AY. Numerical simulation assisted curve compensation in compression molding of high precision aspherical glass lenses. J Manuf Sci Eng. 2009 ; 131 (1):011014/1–6. 7 Tsai YC, Hung C, Hung JC, Tseng SF. Finite element prediction of the lens shape of moulded optical glass lenses. Proc Inst Mech Eng, Part B. 2011 ; 225 (B2): 224 – 34. 8 Chen Y, Yi AY, Su L, Klocke F, Pongs G. Numerical simulation and experimental study of residual stresses in compression molding of precision glass optical components. J Manuf Sci Eng. 2008 ; 130 (5):051012/1–9. 9 Tarkes DP, Vu AT, Krishnamurthi R, Puneet M, Liu G, Dambon O. Birefringence measurement for validation of simulation of precision glass molding process. J Am Ceram Soc. 2017 ; 100 : 4680 – 98. Su LJ, Yi AY. Finite element calculation of refractive index in optical glass undergoing viscous relaxation and analysis of the effects of cooling rate and material properties. Int J Appl Glass Sci. 2012 ; 3 (3): 263 – 74. Fotheringham U, Baltes A, Fischer P, et al. Refractive index drop observed after precision molding of optical elements: a quantitative understanding based on the Tool‐Narayanaswamy‐Moynihan model. J Am Ceram Soc. 2008 ; 91 (3): 780 – 3. Ananthasayanam B, Joseph PF, Joshi D, et al. Final shape of precision molded optics: part II‐validation and sensitivity to material properties and process parameters. J Therm Stresses. 2012 ; 35 (7): 614 – 36. Jain A, Allen YY. Finite element modeling of structural relaxation during annealing of a precision‐molded glass lens. J Manuf Sci Eng. 2006 ; 128 (3): 683 – 90. Ananthasayanam B, Joseph PF, Joshi D, et al. Final shape of precision molded optics: Part I‐computational approach, material definitions and the effect of lens shape. J Therm Stresses. 2012 ; 35 (6): 550 – 78. Zhou J, Li LH, Gong F, Liu K. Quality dependence study on dimensions for plano‐concave molded glass lenses. Int J Appl Glass Sci. 2017 ; 8 (3): 266 – 75. Fischer RE, Tadic‐Galeb B, Yoder PR, Galeb R. Optical system design. NewYork, NY : McGraw Hill ; 2000. Wachtel P, Mosaddegh P, Gleason B, Musgraves JD, Richardson K. Performance evaluation of a Bench‐Top precision glass molding machine. Adv Mech Eng. 2013 ; 2013 : 1 – 12. Liu WD, Zhang LC. Thermoforming mechanism of precision glass moulding. Appl Opt. 2015 ; 54 (22): 6841 – 9. Ananthasayanam B, Joshi D, Stairiker M, Tardiff M, Richardson KC, Joseph PF. High temperature friction characterization for viscoelastic glass contacting a mold. J Non‐Cryst Solids. 2014 ; 385 : 100 – 10. Dassault Systèmes Abaqus/CAE User's guide. Creating springs and dashpots connecting points to ground. Version 6.14, Vol 37.5. Scherer GW. Relaxation in glass and composites. New York, NY : John Wiley & Sons ; 1986. Williams ML, Landel RF, Ferry JD. The temperature dependence of relaxation mechanisms in amorphous polymers and other glass‐forming liquids. J Am Chem Soc. 1955 ; 77 (14): 3701 – 7. Narayanaswamy OS. A model of structural relaxation in glass. J Am Chem Soc. 1971 ; 54 (9): 491 – 8. Vu AT, Vu AN, Liu G, et al. Experimental investigation of contact heat transfer coefficients in nonisothermal glass molding by infrared thermography. J Am Ceram Soc. 2019 ; 102 (4): 2116 – 34. Madhusudana CV, Ling FF. Thermal contact conductance. New York, NY : Springer ; 1996. Sarhadi A, Hattel JH, Hansen HN. Evaluation of the viscoelastic behaviour and glass/mould interface friction coefficient in the wafer based precision glass moulding. J Mater Process Technol. 2014 ; 214 (7): 1427 – 35. Fei W. Process simulation of precision glass molding. RWTH Aachen. PhD Thesis; 2013. Liu WD, Zhang LC, Mylvaganam K. Relaxation oscillation of borosilicate glasses in supercooled liquid region. Sci Rep. 2017 ; 7 :15872/1–8. Roylance D. Engineering viscoelasticity. Cambridge, MA : Department of Materials Science and Engineering, Massachusetts Institute of Technology. 2001 ; 2139 : 1 – 37. Li L, He P, Wang F, et al. A hybrid polymer‐glass achromatic microlens array fabricated by compression molding. J Opt. 2011 ; 13 (5):055407/1–11. Gaylord S, Ananthasayanam B, Tincher B, et al. Thermal and structural property characterization of commercially moldable glasses. J Am Ceram Soc. 2010 ; 93 (8): 2207 – 14.

By Jian Zhou; Lihua Li; Man Cheung Ng and Wing Bun Lee

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