Finite-element analysis of the residual stresses in tempered glass plates with holes or cut-outs

Research Paper
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Abstract

Due to the increased mechanical strength and with respect to safety, tempered and strengthened glass plates are increasingly employed in modern buildings as architectural and structural components. However, regarding the complete fragmentation by disturbing the equilibrated residual stress state in thermally toughened glass, drillings or cut-outs must be done before quenching the glass. The present paper demonstrates 3D results of the thermal tempering simulation by the Finite Element Method in order to calculate the residual stresses in the area of the holes or cut-outs of a tempered glass plate. A viscoelastic material behavior of the glass is considered for the simulation of the tempering process. The structural relaxation is taken into account using Narayanaswamy’s model. Due to different cooling rates of the convection areas such as edge, chamfer, hole’s inner surface and far-field area, heat transfer coefficients are estimated using experimental data from the literature. It is the objective of the paper to demonstrate the simulation of the residual stresses in tempered glasses with holes or cut-outs and to quantify the amount of temper stresses based on a variation of different geometrical parameters and the local heat transfer coefficient. The residual stresses are thus calculated varying the following parameters: the hole diameter, the plate thickness, different geometries of the cut-outs and heat transfer coefficient.

Keywords

Residual stresses Tempered glass Holes and cut-outs Heat transfer coefficient Finite element simulation 

1 Introduction

The increasing application of tempered glass as a construction element of the structural glazing has substantially increased the demand for structural and architectural flexibility and versatility of glass components. Hence, glass components are applied as architectural and structural elements in buildings but also in furniture production, Fig. 1. In order to ensure an overall transparency so-called point-fixings are used to connect the structural glass and the support structure. Due to the high sensitivity of float glass to concentrated loads and its time dependent strength (Beason and Morgan 1984) float glass is not suitable for bolted connections (Nielsen 2009). Also used as glass furniture components e.g. doors, work surfaces or just from the architectural and artistic point of view glass plates can be provided with various holes and cut-outs which are the weak points of the components. Glass is a brittle material and lacks the capability of yielding at room temperature. The tensile strength of glass is governed by micro cracks in the surface which cannot be avoided during the glass production. This indicates a reduction of the actual engineering strength of ordinary cooled float glass to approximately 50–100 MPa with a large variation in the strength value (Schneider et al. 2016a).

Fig. 1

Examples of glass plates a with holes for point-fixings and b with cut-outs (shown through linearly polarized light)

Float glass is set in to a permanent inner stress state by thermal tempering. Due to a compressive residual stress at the surface balanced with an internal tensile stress the surface flaws will be in a permanent state of compression which has to be exceeded by external loading before failure will occur. Hence, thermal tempering makes the glass considerably more resistant to impact, knocks and bending stresses. However, the thermal tempering requires that subsequent treatments to the glass such as cut-out, grinding, drilling etc. must be carried out before quenching the glass.

The amount of the surface compressive stress strongly depends on the cooling rate and therefore on the heat transfer coefficient between glass and the cooling medium. After the pretreatments (drilling holes, cut-outs, chamfer, edges etc.) have been finished the glass plates are cleaned and moved on rollers in to an oven where they are heated to temperatures above the glass transition temperature \(T_{g}\). Then the glass plates are cooled down rapidly to the ambient temperature and tempered glass is manufactured.
Fig. 2

Sketch of the process line for tempering float glass

Fig. 3

Residual stress distribution in tempered glass. The dotted line represents the qualitatively location where the stress equals to zero near edge and hole

The material behavior of solid glass is assumed to be linear elastic at room temperature. By temperatures above the glass transition temperature, \(T_{g}\), the material behavior changes due to the temperature and time dependency of the glass structure and the viscoelastic material behavior of the glass melt (Narayanaswamy 1971). The combination of these two material behaviors, linear elastic in solid state and viscoelastic in consideration of the structural relaxation in liquid state, is used for the thermal strengthening of the float glass. Considering an “undisturbed” area in an infinite plate with no drillings and cut-outs, the maximum stresses in a pure elastic material are achieved when the maximum temperature difference between the surface and the interior is present (Nielsen 2009).

Glass at temperatures above \(T_{g}\) loses the characteristics of a solid body and becomes softer. Thus, glass melt is unable to withstand stresses and deforms for the relaxation of the stresses in loaded state. By cooling the glass melt, due to the temperature distribution along the thickness where the surfaces are cooler than the interior, the surfaces will solidify first. As the outmost layers, the surfaces, will solidify without generating notable stresses due to the lack of stiffness of the interior. When the interior cools down, solidifies and cools further down it will contract and apply compressive stresses to the surfaces. The temperature of the surfaces is now lower than in the interior and the interior will therefore, in total, contract more during cooling than the surfaces. The surfaces will resist this contraction and thereby end in a permanent state of compression.

Tensile stresses occur in the center of the glass plate due to the equilibrium of the surface compressive stresses. The process line for tempering float glass is shown in Fig. 2. If the residual stress state is disturbed sufficiently, the tempered glass will fragmentize completely. Therefore pretreatments like cut-outs and drilling holes must be done before quenching the glass. The residual stress distribution over the thickness far from edges and any type of openings is showed in Fig. 3. The tensile zone and the compressive zone are separated by the dotted line. The parabolic stress distribution, \(\sigma \left( z \right) \), can be written concerning the mid-plane stress, \(\sigma _m \), with respect to the glass thickness t:
$$\begin{aligned} \sigma \left( z \right) =\sigma _m \left( {1-3\zeta ^{2}} \right) , \quad \zeta =\frac{2z}{t} \end{aligned}$$
(1)

2 FE simulation of the thermal tempering

Previous analyses of the thermal tempering of glass have been concerned with the calculation of residual stresses in glass plates considering the viscous and structural relaxation of glass (Gardon and Narayanaswamy 1970; Narayanaswamy 1978). Numerical simulations of the glass tempering process were carried out by Carre and Daudeville (1996) and Laufs (2000). The computation of residual stresses in a 1D model was carried out by Aronen (2012). Analyses of glass tempering and the computation of the residual stresses in glass plates with holes have been carried out by Schneider (2001) and Nielsen et al. (2010b). The influence of the hole, edge and corner distances in drilled glass plates on the residual stresses at holes after the tempering process has been numerically simulated on a 3D FE-model concerning the different heat transfer coefficients on different convection areas by Pourmoghaddam et al. (2016).

This work presents a finite-element analysis of the residual stresses and their development during the tempering process in the area of holes and cut-outs based on 3D FE-models concerning the different heat transfer coefficients of the chamfered holes, cut-outs, edges and the undisturbed far-field area.

2.1 Thermo-mechanical behavior of glass

Thermal tempering of glass consists of cooling very quickly, by air jets, a glass plate that has been heated to approximately \(650\,{^{\circ }}\hbox {C}\). During tempering, the behavior of glass varies quickly around the transition temperature, \(T_{g}\), between the glassy and liquid state. The thermo-mechanical behavior of glass was widely studied in the literature. Kurkjian (1963) demonstrated that glass at high temperatures behaves almost thermorheologically simple (TS). Lee et al. (1965) introduced a viscoelastic model including the TS behavior of glass. The equations which are necessary for the description of the structural relaxation of glass have been discussed by Tool (1946) and by Schwarzl and Staverman (1952). A model for the structural relaxation was proposed by Narayanaswamy (1971). This model is used to calculate the residual stresses in the commercial FE software Ansys 18.1 (ANSYS 2017). The constitutive equations of the thermo-mechanical behavior of glass and a finite element analysis of the inner residual stresses of thin plates have been discussed by Carre and Daudeville (1999). The implementation of a glass tempering 3D-model has been carried out by Nielsen et al. (2010a, b).

The stress relaxation as well as the structural relaxation can be modeled by means of a generalized Maxwell model which can be presented with Prony’s series (Carre and Daudeville 1996; Daudeville et al. 2002; Schneider et al. 2016a, b). The material properties and the parameters of the Prony’s series, at the reference temperature \(T_{ref} =~864\) K, used for the simulations are taken from Carre and Daudeville (1999), see Table 9 in the appendix.
Fig. 4

Heat transfer coefficients; factor of the different convection areas from the experimental results.

(Reproduced with permission from Bernard and Daudeville 2009)

Fig. 5

a Sketch of the “infinite” round plate model with a hole in the center and boundary conditions, b \(5{^{\circ }}\) piece FE-model presenting the residual stress in radial direction \(\sigma _r \) (Pa), \(t = 10\) mm and \(h_{Surface} = 150.7\, \hbox {W/m}^{2}\hbox {K}\), \(T_{0} = 650\,{^{\circ }}\hbox {C}\) (923.15 K) and \(T_{\infty } = 20\,{^{\circ }}\hbox {C}\) (293.15 K)

2.2 Identification of heat transfer coefficients

To calculate the residual stresses at holes, the heat transfers in the cooling process has to be simulated. Hereby the essential influence parameter is the heat transfer coefficient h between glass and the cooling medium. The heat transfer due to radiation is neglected to simplify the calculations. The experimental determination of the heat transfer coefficient is difficult. The convection coefficients in the different areas of perforated plates (far away from edges (far-field), hole, edge and chamfer) have been identified experimentally by Bernard and Daudeville (2009) using a hollow aluminum model. Those experimentally identified heat transfer coefficients are linearly factorized in this work, as shown in Fig. 4, to determine the heat transfer coefficients of glass plates with a hole.

In this work the heat transfer coefficients of glass plates with a hole in the center were determined iteratively by numerical determination of the variable \(\beta \) (initial value \(\beta ~=~1.0\)).

The calculations were carried out by means of an infinite round plate with a hole positioned in the center of the plate. Due to symmetry, only a piece of the plate under an angle of \(5{^{\circ }}\) and half the thickness was modelled, Fig. 5a. Different heat transfer coefficients result in different residual stresses. To have comparable residual stresses for each of the analyzed models the heat transfer coefficients were brought to one stress level. The scaling factor \(\beta \) was varied until a surface compression of approximately 100 MPa was achieved. The heat transfer coefficients were calculated for the different glass thicknesses \(t~=~6\), 10 and 15 mm, the initial temperature of \(T_{0}~=~650\,{^{\circ }}\hbox {C}\) and the ambient temperature of \(T_{\infty }~=~20\,{^{\circ }}\hbox {C}\). The residual stresses after the cooling process are shown in Fig. 5b.
Table 1

Heat transfer coefficient \(h~(\hbox {W/m}^{2}\hbox {K})\), far-field residual surface compression of approx. 100 MPa

Thickness (mm)

6

10

15

\(\beta \)

2.84

1.57

1.11

Heat transfer coefficient h (W/m\(^{2}{K}\))

Edge

204.5

113.0

79.9

Surface

272.6

150.7

106.6

Chamfer

355.0

196.3

138.8

Hole

196.0

108.3

76.6

Fig. 6

a Temperature [\({^{\circ }}\hbox {C}\)] versus time [s], b Stress [MPa] versus time [s] calculated at the surface and at the center of an infinite glass plate with the thickness \(t = 10\) mm; heat transfer coefficient \(h~=~150.7~\hbox {W/m}^{2}\hbox {K}\), \(T_{0} = 650\,{^{\circ }}\hbox {C}\) (923.15 K) and \(T_{\infty } = 20\,{^{\circ }}\hbox {C}\) (293.15 K). Time increment \(\Delta t = 0.005\) s; The dotted curve represents the temperature difference \(\Delta T [{^{\circ }}\hbox {C}]\) between center and surface

The value of the variable \(\beta \), which led to the surface residual stress of \(\sigma _{r,s}^{far} =\sigma _{\theta ,s}^{far} \approx -\,100\) MPa in the far-field area of the plate model, was identified as the factor of the heat transfer coefficients of the different convection areas. However, with the variable \(\beta \) we have assumed a simple relation of the heat transfer coefficient between surface and the hole area, Fig. 4. The results of the calculations are presented in Table 1.

2.3 Methodology

The goal of FE simulations of quenching is to obtain residual stresses of soda–lime–silica glass plates. The radiation effect is neglected. The transient finite element simulation of the residual stresses is carried out in two steps. First, the temperature history during the tempering process is determined in a transient temperature calculation using the 3D-20-Node thermal solid element SOLID90 (available in FE-Program Ansys 18.1), which is a thermal element with temperature as the only degree of freedom. In the second step the temperature change over time is put in terms of load steps on a structural mechanical model with the material properties listed in Table 8 in the appendix as well as the parameters of the Prony series, Table 9 in the appendix, to simulate the viscous and structural relaxation of glass during the tempering process. Thereby the thermal element SOLID90 is changed in to the 3D-20-Node viscoelastic solid element VISCO89.

In Fig. 6, a temperature time diagram (Fig. 6a) and the resulting stress time diagram (Fig. 6b) of a 10 mm thick glass plate during the cooling process is shown. The initial temperature \(T_{0}\) is \(650\,{^{\circ }}\hbox {C}\) (equates to 923.15 K) and the ambient temperature \(T_{\infty }\) is \(20\,{^{\circ }}\hbox {C}\) (equates to 293.15 K).
Fig. 7

Temperature difference between center and surface for different plate thicknesses t; constant heat transfer coefficient of \(h~=~144\hbox { W/m}^{2}\hbox {K},\, T_{0} = 650\,{^{\circ }}\hbox {C}\) (923.15 K) and \(T_{\infty } = 20\,{^{\circ }}\hbox {C}\) (293.15 K). Time increment \(\Delta t\) = 0.2 s

The temperature difference between center and surface varies dependent on the plate thickness, Fig. 7. As can be seen in Fig. 7, the temperature difference raises similarly for all thicknesses during the first second. After that the mid-plane temperature starts to decrease for thin glass and later for thicker glass. Hence, the thicker the glass plate the later the maximum of the temperature difference max \(\Delta T\) occurs.
Fig. 8

Time increment convergency for the first cooling step \(t_{1}\) using an infinite plate, thickness: \(t~= 6\) mm, \(t~= 10\) mm and \(t~= 15\) mm. Heat transfer coefficients according to Table 1. Deviation from a model using the time increment \(\Delta t~=~0.005\) s. \(T_{0} = 650\,{^{\circ }}\hbox {C}\) (923.15 K) and \(T_{\infty } = 20\,{^{\circ }}\hbox {C}\) (293.15 K)

2.3.1 Convergence study: time step

Due to a higher heat transfer coefficient (cooling rate), which is needed for the thermal tempering of thinner glasses, different time stepping is required for different thicknesses. The cooling time is divided into two cooling steps in terms of time steps. In the first cooling step \(t_{1}\) the glass is cooled down in small time increments \(\Delta t_{1}\) to a temperature below an assumed glass transition temperature \(T_{g} = 550\,{^{\circ }}\hbox {C}\). In the second cooling step \(t_{2}\) the solified glass is cooled down to the ambient temperature \(T_{\infty }\) in larger time increments \(\Delta t_{2}\) in order to save computing time. The time stepping significantly influences the accuracy of the residual stresses. Time stepping with small increments is necessary to simulate the rapid cooling of the glass with the initial temperature \(T_{0}\). As it is shown in Fig. 6a, the temperature time curve steeply declines at the beginning of the cooling process and afterwards levels off at the ambient temperature \(T_{\infty }\). Hence, the time stepping with small increments is needed at the beginning of the cooling process until the model, which was heated by an initial temperature \(T_{0}\) at approximately \(100\,{^{\circ }}\hbox {C}\) above \(T_{g}\), achieves a temperature below the glass transition temperature at the surface as well as at the mid-plane. In Fig. 8 the convergency of the time increments for the first cooling step \(t_{1}\) is presented. The mid-plane and surface residual stresses are calculated with different time increments \(\Delta t\) and variable thicknesses using a 2D-axisymmetric model with 2D-8-Node Thermal Solid elements (PLANE77) for the temperature calculations, which were changed in to 2D 8-Node Viscoelastic Solid elements (VISCO88) for the stress calculations.

The solutions are compared to a calculation with the time stepping of \(\Delta t = 0.005\) s. For the first cooling step \(t_{1}\) the residual stresses at the surface and in the mid-plane converge at a time increment of \(\Delta t = 0.1\) s. After the whole glass plate is cooled down below the transition temperature and solifies, in the second cooling step \(t_{2}\) larger time increments can be set for cooling the glass plate down to the ambient temperature. In Table 2, the time increments of the two cooling steps for three different thicknesses are listed.
Table 2

Cooling steps and time stepping dependent on the plate thickness

Thickness (mm)

Cooling step

Cooling time (s)

Time increment \(\Delta t_{i}\) (s)

6

\(t_{1}\)

15

0.1

\(t_{2}\)

380

5.0

10

\(t_{1}\)

35

0.2

\(t_{2}\)

800

10.0

15

\(t_{1}\)

80

0.5

\(t_{2}\)

2000

50.0

Cooling time values are valid for heat transfer coefficients according to Table 1

The time increments of the second cooling step are freely selected. However, experience has shown that, in order to simulate the parabolic stress distribution along the plate thickness, the second cooling step \(t_{2}\) below the glass transition temperature cannot be carried out in one step. The difference between the numerical simulation and the parabolic function according to the Eq. (1) of the residual stresses in the far-field area for the thickness of 10 mm is shown in Fig. 9.

2.3.2 Convergence study: mesh density

The accuracy and the computation time of the residual stress calculation is dependent on the time stepping of the transient calculation as well as on the element size of the FE mesh respectively the mesh density in the observation area. In order to save computation time the FE mesh of the model is divided into a fine meshed observation area with elements of equal edge length and an area with a coarse mesh, see Fig. 10.
Fig. 9

The parabolic function (Eq. 1) and the FE simulation of the residual stresses along half the thickness of the glass plate in far-field area, \(t = 10\) mm, \(h = 150.7\hbox { W/m}^{2}\)K, \(T_{0} = 650\,{^{\circ }}\hbox {C}\) (923.15 K) and \(T_{\infty } = 20\,{^{\circ }}\hbox {C}\) (293.15 K), the FE-Simulation is led on symmetrically by the dashed line

Fig. 10

Infinite plate (\(t~=~10\) mm) with a fine meshed observation area with n elements of equal size

Fig. 11

Convergency for the mesh density around the observation point at the surface of the plate (\(t~=~10\) mm). Deviation from a model using 200 20-node elements in the observation area

A convergence study for the number of elements along the thickness is carried out in Nielsen (2009). The convergence analyses showed that, if the elements are biased towards the surface, 10 elements through half the thickness provide an accuracy of 30 unbiased elements. Thus, the elements become thicker from the surface to the center. For a bias factor of 4 the element at the center is 4 times thicker than the element at the surface. This saves computation time and ensures accuracy for the simulation of the parabolic stress distribution along the thickness. The convergence study of the mesh density around an observation point is carried out on a 3D infinite plate (\(t~=~10\) mm) with a fine meshed observation area. The edge length of the elements at the surface in the observation area is chosen equal to the element thickness at the surface.

The bias factor along the thickness is set to 4. The symmetry of the model is used for the calculations, see Fig. 10.

In Fig. 11, the convergency of the mesh density around the observation point at the surface of the plate is presented. The mesh consists of 3D-20-node hexa hedron elements (see chapter 2.3). The solutions are compared to 200 elements of equal size in the observation area.
Fig. 12

Comparison of the FE model (lines) with experimental data [points (Gardon 1965)] for mid-plane residual stresses with different initial temperatures and heat transfer coefficients, \(t = 6.1\) mm; dotted lines show the FE-results for heat transfer coefficients \(h~=~13\hbox { W/m}^{2}\)K and \(h~=~109~\hbox {W/m}^{2}\)K according to Gardon (1965)

2.3.3 Comparison of stress calculations

The comparison of the numerically calculated residual stresses in the far-field area of an infinite model for different heat transfer coefficients h with experimentally measured values by Gardon (1965) is shown in Fig. 12. In this case the glass thickness is 6.1 mm.

2.4 Variation of geometrical parameters for typical geometries of holes and cut-outs

The described FE model is used to calculate residual stresses for glass plates with typical geometries of holes and cut-outs. The variation of geometrical parameters for openings and cut-outs, which are considered in the calculations, is shown in Fig. 13. The plates have the same dimension of 800 mm \(\times \) 800 mm with a thickness of 10 mm.

The cut-outs (a)–(c) in Fig. 13 are in the corner area. The 20 mm hole with the 2 mm slot (Fig. 13d) and the 10 mm hole (Fig.  13e) are located at one edge and far away from the other edges. The edges are chamfered at an angle of \(45{^{\circ }}\).

The influence of the hole, edge and corner distances on the minimum residual compressive stresses at holes after the tempering process has been demonstrated in Pourmoghaddam et al. (2016). In this work the calculation of the residual stresses for an infinite round plate with a hole in its center (Fig. 5a) is carried out to analyze the influence of holes with typical diameters on the residual stresses in an infinite plate.
Fig. 13

Variation of geometrical parameters for the calculations, glass plates (800 mm \(\times \) 800 mm \(\times \) 10 mm) with openings and cut-outs, all distances in (mm), cut-outs are shown from an upper view

The variation of the hole geometries based on the thickness, t, and the hole diameter, Ø, are stated in Table 3.

3 Results of numerical simulations

The results of the numerically simulated residual stresses for glass plates with the variation of geometrical parameters for typical geometries of holes and cut-outs according to chapter 2.4 are presented. The calculations were carried out in the finite element program Ansys 18.1.

3.1 Residual stresses at holes

The numerical calculations are carried out on the “infinite” round plate shown in Fig. 5a. The heat transfer coefficients in Table 1 are set on the convection areas. Thus, the surface compression far away from the hole and the edge (far-field) in radial and in tangential direction is approx. 100 MPa. The initial temperature is set to \(T_{0} = 650\,{^{\circ }}\hbox {C}\) (923.15 K) and the ambient temperature to \(T_{\infty } = 20\,{^{\circ }}\hbox {C}\) (293.15 K). In Fig. 14, the tangential stress in hole area is presented for the plate thickness of t = 6 mm and the hole diameter of Ø = 15 mm.

The residual stresses in the hole area and close to the hole at the in-plane surface are evaluated along the paths P1 (hole), P2 (chamfer) and P3 (surface), shown and described in Fig. 15. The results are compared to the tangential residual stress at the surface of the plate in far-field area \(\sigma _{\theta ,s}^{far} \):
$$\begin{aligned} \psi _\theta =\frac{\sigma _\theta }{\sigma _{\theta ,s}^{far} }\approx \frac{\sigma _\theta }{-\,100\,\mathrm{MPa}} \end{aligned}$$
(2)
\(\sigma _\theta \) is the tangential residual stress, which is calculated numerically. \(\psi \) in Eq. (2) is the factor for the comparison of the residual stress results to the surface compressive stress of approx. 100 MPa far away from the hole and the edge. This is the target residual stress in the far-field area for the identification of the heat transfer coefficients (see chapter 2.3).
Due to the dividing of the residual stresses evaluated along the paths P1, P2 and P3 by the far-field residual surface compressive stress (negative value) the factor \(\psi \) is positive in the case of compressive stresses and negative in the case of tensile stresses. In Fig. 16, the maximum and the minimum values of the tangential residual stresses at the inner surface of the hole are shown. The tangential residual stress results along the path P1 (Fig. 15) at the inner surface of the hole compared to the far-field surface compressive stress \(\psi _\theta ^{hole} \) are shown in Fig. 17 for three different thicknesses and four different hole diameters as described in Table 3. It was observed that the tangential residual compressive stress at the hole’s inner surface increases from the mid-plane of the hole’s inner surface to the bottom of the chamfer. For the 6 mm thick plate and the hole diameter of 8 mm, for example, the surface compressive stress increases from \(\psi = 0.79\) at the beginning of the path P1 at the mid-plane of the hole to \(\psi = 1.30\) at the bottom side of the chamfer.
Table 3

Variation of the hole geometries, thickness t (mm) and diameter Ø (mm)

Thickness t (mm)

Diameter Ø (mm)

6

8

15

50

100

10

8

15

50

100

15

8

15

50

100

Fig. 14

tangential residual stress \(\sigma _\theta \) (Pa) in hole area, \(t = 6\) mm and \(\beta ~=~2.84\)

Fig. 15

Path directions, P1 (hole): from mid-plane to the bottom side of the chamfer, P2 (chamfer): from the bottom side of the chamfer to the upper side of the chamfer and P3 (surface): from the upper side of the chamfer away into the field (30 mm)

Fig. 16

Min./Max. value of the tangential residual stress at the inner surface of the hole compared to the far-field surface compression of 100 MPa (Eq. 8). Maximum value (mid-plane), minimum value (bottom side of the chamfer)

Fig. 17

Tangential residual stress at the inner surface of the hole along the defined path P1 (Fig. 15) compared to the far-field surface compression of 100 MPa (Eq. 8). Variation of the hole diameter Ø. a \(t = 6\) mm, b \(t = 10\) mm and c \(t= 15\) mm

Table 4

FE-Results of the tangential residual stress factor \(\psi _\theta ^{hole} \) [−] in the hole area, mid-plane (maximum)/bottom side of the chamfer (minimum)

t (mm)

\(\phi 8\)

\(\phi 15\)

\(\phi 50\)

\(\phi 100\)

6

0.79/1.30

0.89/1.27

0.96/1.21

0.97/1.19

10

0.55/1.48

0.73/1.44

0.90/1.38

0.94/1.35

15

0.38/1.57

0.62/1.52

0.87/1.47

0.92/1.44

Fig. 18

Tangential residual stress at the chamfer along the defined path P2 (Fig. 15) compared to the far-field surface compression of 100 MPa (Eq. 8). Variation of the hole diameter Ø. a t = 6 mm, b \(t = 10\) mm and c \(t = 15\) mm

Table 5

FE-Results of the tangential residual stress factor \(\psi _\theta ^{Chamfer} \) [−] at the chamfer, minimum value along the chamfer/top side of the chamfer

t (mm)

\(\phi 8\)

\(\phi 15\)

\(\phi 50\)

\(\phi 100\)

6

1.36/1.22

1.35/1.24

1.32/1.25

1.31/1.24

10

1.49/1.32

1.47/1.37

1.44/1.40

1.42/1.39

15

1.57/1.33

1.54/1.40

1.53/1.48

1.51/1.49

At the inner surface of the hole the maximum tangential residual stress occurs at the mid-plane. The minimum tangential residual stress in the hole area occurs at the chamfer. The minimum value of the tangential residual stress at the inner surface of the hole is evaluated at the bottom of the chamfer.

Figure 16 shows, that the variation of the diameter of the hole has a significant influence on the maximum value of the residual stress resp. the minimum value of the residual compressive stress at the mid-plane of the inner surface of the hole. For the 15 mm thick plate, the residual compressive stress at the mid-plane of the hole decreases approx. 41% from a large diameter of 100 mm to the small diameter of 8 mm, where the compression at the bottom side of the chamfer increases approx. 8%.

The difference between the tangential residual stress at the mid-plane of the hole and the tangential residual stress at the bottom side of the chamfer increases with the thickness of the plate and decreases with the diameter of the hole, see Fig. 17. In Table  4, the minimum and the maximum values of the tangential residual stress factor \(\psi \) are listed dependent on the diameter of the hole and the thickness of the plate.
Fig. 19

Tangential residual stress close to hole at in-plane surface along the defined path P3 (Fig. 15) compared to the far-field surface compression of 100 MPa (Eq. 8). Variation of the hole diameter Ø. a \(t = 6\) mm, b \(t = 10\) mm and c \(t = 15\) mm

Table 6

FE-Results of the maximum value of the tangential residual stress factor \(\psi _\theta ^{surface} \) [−] at in-plane surface close to hole, the maximum residual stress value along the path P3 was evaluated

t (mm)

\(\phi 8\)

\(\phi 15\)

\(\phi 50\)

\(\varphi 100\)

6

0.96

0.97

0.98

0.99

10

0.94

0.96

0.97

0.98

15

0.92

0.95

0.97

0.98

In Fig. 18, the tangential residual stress results along the chamfer (Path P2) compared to the far-field surface compressive stress \(\psi _\theta ^{chamfer} \) are shown. The tangential residual stress in the hole area has its minimum in the chamfer area. The minimum value of the residual tangential stress occurs close to the bottom side of the chamfer for smaller hole diameters and thicker plates. The location with the minimum residual stress moves away from the bottom side of the chamfer for bigger diameters and thinner plates. In Table 5, the minimum values of the tangential residual stress factor \(\psi \) along the chamfer and the values at the top of the chamfer are listed.
Fig. 20

Tangential stress development during the tempering process at the inner surface of the hole (Ø = 8 mm) evaluated at the mid-plane of the hole and the bottom side of the chamfer. \(\psi \) is negative for tensile stresses and positive for compressive stresses (Eq. 8). a \(t~=~6\) mm, b t = 10 mm, c t = 15 mm

In Fig. 19, the tangential residual stress results close to the hole at in-plane surface along the defined path P3 (Fig. 15) compared to the far-field surface compressive stress \(\psi _\theta ^{surface} \) are shown. Path P3 begins at the upper side of the chamfer and ends 30 mm away from it. The influence of the hole on the residual stresses decreases with greater distance to the hole. Thus, the residual surface compressive stress for each of the four diameters decreases from the value at the upper side of the chamfer to the value of the far-field compression of 100 MPa for greater distances from the hole. As it can be observed in Fig. 19a–c, the distance of influence is greater for thicker plates. One can say that the influence of holes in tempered glasses on the residual stresses increases with the thickness of the plate. There is also an influence of the diameter of the hole on the residual stress distribution close to the hole. For the bigger diameters the tangential residual stress converges to the far-field compression of 100 MPa without any extreme values. However, the residual stress for a glass plate with the small diameter of Ø = 8 mm, for example, has a maximum value in an area less than 5 mm close to the hole before the tangential residual stress converges to the far-field compressive stress. The maximum residual stress at the in-plane surface close to hole is equal to the minimum residual compressive stress. The minimum values of the residual compressive stress are more pronounced in the case of small diameters in thicker glass plates, see Fig. 19.

In Table 6, the maximum values of the tangential residual stress factor \(\psi \) close to the hole at in-plane surface along the path P3 are listed.
Fig. 21

Time increment convergency for the maximum surface tensile stress in the beginning of the cooling (Time \(\le \) 2 s), thickness: \(t~=~6\) mm, \(\beta ~= 2.84\). Deviation from a model using the time increment \(\Delta t~=~0.001\) s. \(T_{0}~=~650\,{^{\circ }}\hbox {C}\) (923.15 K)

Fig. 22

Tangential Stress development during the first 10 s of the tempering process in the hole area. Plate thickness \(t~= 6\) mm. \(\beta ~=~2.84\) for yielding a far-field residual surface compressive stress of 100 MPa. \(\psi \) values below 0 are tensile stresses. a Bottom side of the chamfer. Maximum tensile stress of 19.1 MPa \((\psi =-\,0.19)\) for the hole diameter Ø = 100 mm. b Mid-plane of the hole. Maximum tensile stress of 9.7 MPa \((\psi =-\,0.10)\) for the hole diameter Ø = 100 mm

Due to the fracture of the glass plates in case of the propagation of micro cracks under tensile stress the compressive stress at the surface after the tempering process is crucial with regard to the strength of the tempered glass plates. However, due to the tensile stress development at the beginning of the tempering process glass plates with holes or cut-outs can fail during the tempering process. The weak points of the glass plates during the process are not necessarily the same weak points of the glass plates after the tempering process. Figure 20 shows the development of the tangential residual stresses during the tempering process at the bottom of the chamfer and at the mid-plane of the hole. At the beginning of the tempering process tensile stresses occur at the inner surface of the hole. After a few seconds the stress status switches to compression where the compressive stresses at the bottom of the chamfer increase steeper than the compressive stresses at the mid-plane of the hole, Fig. 20. Thus, the mid-plane of the hole’s inner surface is the weak point in the hole area. In contrast it was observed that in the first seconds of the tempering process the chamfer is the weak point in the hole area. For the simulation of the stress development at the beginning of the process (Time \(\le \) 2 s.) a time increment convergence study was carried out for the plates with the thickness of 6, 10 and 15 mm, see Fig. 21. The calculations were then carried out for a plate with the thickness of 6 mm in three steps with a time increment of \(\Delta t~=~0.01\) s for the first two seconds to get the maximum tensile stress on surface. Thereby, the maximum surface tensile stress deviation from a model using the time increment \(\Delta t~=~0.001\) s is about 1%. As it is shown in Fig. 21, it should be noted that the mentioned stress deviation increases steeply with greater time increments than 0.01 s. Concerning a thickness of 6 mm, the maximum surface tensile stress deviation is about 8% for a time increment of \(\Delta t~=~0.1\) s and about 30% for \(\Delta t~=~1\) s.

In the second step the time increment was changed to \(\Delta t~=~0.05\) s (2 s < Time \(\le \) 5 s) and carried on with \(\Delta t~=~0.1\) s to the end of the calculation (Time = 10 s).

In the hole area the maximum tensile stress at the beginning of the tempering process occurs at the bottom side of the chamfer. This is because the chamfer cools faster during the tempering process than the mid-plane of the hole. At the bottom side of the chamfer the switch of the stress status into compressive stresses also depends on the diameter of the hole, see Fig. 22a. The smaller the hole diameter, the earlier the stress status switches from tensile stress into compressive stress. For the hole diameter of 8 mm (t = 6 mm) the stress state changes after approx. 5 s and for the hole diameter of 100 mm the stress state changes after approx. 10 s.

3.2 Residual stresses at cut-outs

In addition to glass plates with holes also glass plates with cut geometries can be produced as “safety glasses” by means of the thermal tempering. The cut-outs must be carried out before quenching the glass. Thus the stress development along the thickness of the glass plate is disturbed in the area of the cut-outs. This is due to the fact that the cooling process resp. the heat transfer coefficient of the convection surfaces differ from each other by the type of geometry (Daudeville et al. 2002; Pourmoghaddam et al. 2016). Some of the usual cut-out geometries have been shown in Fig. 13. Finite element simulations of the tempering process of glass plates with such cut-out geometries have been carried out using the same method. The heat transfer coefficients were set on the convection surfaces using the factors of the different convection areas for a glass plate with a hole as it was sketched in Fig. 4 with the following assumptions: The cut-out geometries (a)–(c) (Fig.  13) were treated as edges with chamfer and the cut-out geometries (d) and (e) were treated as holes with chamfer. Thus, for the previous calculation of the heat transfer coefficients the factor \(\beta \) was determined. For a glass plate with the thickness of 10 mm and a far-field residual surface compression of approx. 100  MPa the factor \(\beta \) was determined to 1.57 (see Table 1). Here, the far-field residual surface compression in two perpendicular directions at nine different locations far away from the edge and the cut-outs on each specimen (a)–(e) were determined using a scattered light polariscope (SCALP) developed by GlasStress Ltd. in Tallin (Estonia), based on the photoelastic response of glass (Aben et al. 2008). The far-field residual surface compression in the specimens (a), (c) and (e) is approx. 90 MPa and in specimen (b) and (d) approx. 82 MPa, see Table 7. The convection factor \(\beta \) was varied until the measured far-field residual surface stress was achieved. Thus \(\beta \) was determined to 1.37 for a far-field residual surface compression of approx. 90  MPa and 1.23 for specimen (b) and (d) with the measured far-field compression of approx. 82 MPa.
Table 7

Measured far-field residual surface stress, average of nine different locations in two perpendicular directions, variable \(\beta \)

Cut-out geometry

a)

b)

c)

d)

e)

\(\sigma _s^{far} \)

\(-\) 89.5

\(-\) 82.3

\(-\) 89.1

\(-\)82.6

\(-\) 89.1

\(\beta \)

1.37

1.23

1.37

1.23

1.37

Fig. 23

Corner of a tempered glass plate shown through circular polarized light. The isochromats (difference between the two principal stresses) are presented here

Fig. 24

Cut-out areas of the tempered glass plates shown through linearly polarized light. The isochromats \(+\) isoclines are presented here

Fig. 25

FE-simulation of a the principal tensile stresses (\(\sigma _1 )\) (Pa) at the beginning of the tempering process (in this case after 4 s) and b the principal compressive stresses (\(\sigma _3 )\) (Pa) after the tempering process (after 1000 s) in the hole area near the edge with a slot of 2 mm (Model d)

In Fig. 23, the corner of a tempered glass plate is shown through circular polarized light. For a better illustration, the plates with the different cut-out geometries were observed under linearly polarized light (Fig. 24).

In Fig. 27 (see appendix) the principal stresses in the cut-out areas at mid-plane and at the surface are shown. Regarding the evaluation of the 3D principal stresses it was distinguished between the two relevant principal stresses \(\sigma _1 \) (principal tensile stress) and \(\sigma _3 \) (principal compressive stress). According to amount the second principal stress \(\sigma _2 \) is between the other two principal stresses and can be neglected. Negative values represent compressive stresses and positive values represent tensile stresses. Hence, the weak points are those with the maximum stress. According to our FE calculations we could observe that the weak points during the beginning of the tempering process are exactly the areas where minimum stress respectively maximum compressive stress in amount occurs after the tempering process is terminated. Figure 25 shows the principal tensile stresses at the beginning of the tempering process after 4 s versus the principal compressive stresses after the tempering process in the hole area of model d. In model d the minimum residual stress resp. the maximum principal compressive stress of approx.

156 MPa after the tempering process occurs at the hole edge (Fig. 25b). However, the maximum principal tensile stress of approx. 14 MPa occurs at the same point in the beginning of the tempering process (Fig. 25a). In Fig. 28 (see appendix), the FE-simulation of the principal tensile stress \(\sigma _1 \) in the first 0.4 s of the tempering process is shown compared to the principal compressive stress \(\sigma _3 \) at the end of the tempering process in the area of the cut-outs of all models.

3.2.1 Comparison with photoelastic measurements

For the validation of the used finite element method to calculate the residual stresses developing in the glass plates with cut-outs during the tempering process the FE-results were evaluated along the paths shown in Fig. 26 (dotted lines) and were compared to photoelastic measurements of the specimens a) to e). The residual surface stresses of the specimens in the area of the cut-outs were measured at the points (plus signs) shown in Fig. 26 using the SCALP (scattered light polariscope).
Fig. 26

FE evaluation paths (dotted lines) and photoelastic measurement points (horizontal and vertical)

The stresses in X- (horizontal) and in Y- (vertical) direction were measured and compared to the calculated residual stresses of the same directions, see Fig. 29 in the appendix. The measured results of the residual stresses at cut-outs were compared to the measured residual stress at the surface of the plate in the far-field area \(\sigma _s^{far} \), see Table 7. And the FE-results of the residual stresses at cut-outs were compared to the numerically determined far-field residual surface compression, which was calculated for each specimen. We note parenthetically that the calculated far-field residual surface stresses are approximately comparable to the measured values in Table 7. Thus \(\uppsi \) was evaluated for the calculated as well as for the measured residual stresses according to the Eq. (2).

Due to the low accuracy of the SCALP measurements closer to the edge, there are differences between simulated and measured results with regard to the stress peaks next to the edge, see Fig. 29 (model d) Y-direction and (model e), X- and Y-direction. However, qualitative results of the simulated residual stresses in plates with cut-outs could be compared very well to photoelastic measurements.

4 Conclusion and future research

4.1 Summary and conclusion

For the simulation of the tempering process the FE-model of an infinite glass plate with a cylindrical hole with four different diameters or the FE-model of a glass plate with cut-outs with five different cut-out geometries was given an initial temperature of \(T_0 ~=~650\,{^{\circ }}\hbox {C}\) and cooled down to the ambient temperature of \(T_\infty ~=~20\,{^{\circ }}\hbox {C}\).

A convergence study was carried out for defining the time increments \(\Delta t\) which numerically has a significant influence on the development of the residual stresses. The resulting time-temperature curve was put in terms of load steps on a structural mechanical model to calculate the stress response due to the tempering process. The cooling process was simulated using different heat transfer coefficients for different convection areas of edge, chamfer, hole and undisturbed area.
Fig. 27

FE-simulation of a the principal tensile stress (\(\sigma _1 )\) (Pa) at the mid-plane and b the principal compressive stress (\(\sigma _3 )\) (Pa) at the surface after the tempering process

Fig. 28

FE-simulation of (left) the principal tensile stress \((\sigma _1 )\) in (Pa) after 0.4 s and (right) the principal compressive stress \((\sigma _3 )\) in (Pa) at the end of the tempering process in the area of the cut-outs, plate thickness \(t~=~10\) mm

Fig. 29

Comparison of the FE calculated residual stresses at defined paths in cut-out areas to the photoelastic measurements (see Fig. 26). Left: Stresses in X-direction (horizontal), right: Stresses in Y-direction (vertical). The residual stress factors \(\psi _x \) and \(\psi _y \) are compared to the measured far-field surface compression (see Table 7)

The different cooling behavior of the convection areas was considered based on the experimental data in Bernard and Daudeville (2009). The different heat transfer coefficients were determined, accordingly, for yielding a surface compression of 100 MPa using a scaling factor \(\beta \), which was calculated iteratively for different thicknesses. Hence, the scaling factor \(\beta \) takes the target surface compression (here 100 MPa) after the tempering process as well as the thickness of the plate into account. The influence of the plate thickness and the diameter of a hole in an infinite glass plate were shown. The mid-plane residual stress at the inner surface of a hole is critical for small hole diameters. The mid-plane residual compression at the inner surface of a hole decreases with smaller hole diameters and higher thicknesses. The value decreases to approx. 38% of the far-field residual stress for a 15 mm thick glass plate and a small hole diameter of 8 mm. In comparison to that the residual compression at the chamfer level is 157% of the far-field residual stress. Furthermore the tensile stress development at the beginning of the tempering process in the hole area was shown for a plate thickness of 6 mm and different hole diameters of 8, 15, 50 and 100 mm. The FE simulation of the first 10 s of the tempering process showed that the maximum tensile stresses in the beginning of the process occur at the chamfer level. The same effect was observed for glass plates with cut-outs. Maximum tensile stresses in the beginning of the process occur at the same positions with the maximum compression after the tempering process is terminated. For glass plates with cut-outs the geometry of the cut-outs significantly influences the residual stress development. The residual stresses of five typical cut-out geometries in glass plates were numerically calculated. The finite element results of the residual stresses in the area of the cut-outs were compared to photoelastic measurements using a scattered light polariscope (SCALP).

Nevertheless, based on the results in this work further investigations and model optimizations are required for an accurate simulation of the beginning of the tempering process.

4.2 Future research

The occurrence of the short-term tensile stresses at the surface of the glass plates with holes and unfavorable cut-out geometries in the first seconds of the cooling process, which was shown in this work, can lead to failure at edges and in case of surface damages. However, for a quantitative analysis of the glass failure during the tempering process there is a lack of deeper knowledge of the glass strength in the appropriate temperature range up to the glass transition temperature. Therefore an optimization and an improvement of the process and the system technology are only possible to a limited extent. This is needed for the right choice of the system configuration parameters such as the required initial temperature above the glass transition temperature, the cooling rate, nozzle configuration, nozzle diameter, the distance between the nozzles and the glass surface as well as the roller distances. A closer investigation of the FE-simulation of the tempering process and its accuracy in the first seconds of the process will be an objective of our future research. Due to the significant influence of the time increments on the stability and the accuracy of the FE-model and the high instability of the glass structure at temperatures near the glass transition temperature a convergence study for the determination of the time increment \(\Delta t\) has to be carried out for the first seconds of the tempering process. We also intent to experimentally investigate the flexural strength of float glass at high temperatures. Therefore we have planned to carry out double ring bending tests on glass plates in a defined temperature range from room temperature up to the glass transition temperature. With regard to the tempering process, a quantitative investigation of the temperature-dependency of the flexural strength of float glass resp. improvement of our knowledge about the material behavior of float glass at high temperatures up to the glass transition temperature is an objective of our future research.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Technical University of DarmstadtDarmstadtGermany

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