Tensile behaviour of soda-lime-silica glass and the significance of load duration

1. Introduction

In civilian infrastructure, soda-lime-silica glass is used in a wide range of applications, such as windshields, load-bearing glass beams, residential windows, large glass plates covering whole building facades, and many more. To ensure improved performance, these applications are typically based on post-processed flat glass (also denoted ‘float’ or ‘annealed’ glass) either in the form of laminated glass, tempered glass, or a combination of both. Depending on the application, the glass is exposed to different load history during its lifetime, which can be a period of stress as long as several years to as low as a few microseconds. Therefore, it is important that a detailed level of knowledge about the material dependent constitutive relation and the failure criteria has been developed over a wide range of strain rates (see Fig. 1). This is essential for the design of glass structures that will have a well-defined service lifetime.

Examples of the typical long/short term loads found to act on glass structures are, snow loads [[1], [2], [3], [4]], wind loads [[5], [6], [7], [8], [9], [10], [11]], seismic loads [[12], [13], [14], [15], [16], [17], [18]], wind-born debris impact [[19], [20], [21], [22], [23], [24]], ballistic (hail and bullets) impact [[25], [26], [27], [28], [29], [30], [31]], and blast loads (accidental and man-made) [[32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58]]. In all cases, the load duration as well as the rate of loading affects the static and dynamic glass strength significantly. The rate-of-loading effect also impacts other structural materials used by civil engineers, see e.g. Refs. [[59], [60], [61]]. Nevertheless, the affect on strength is only significant when the duration or rate-of-loading changes by more than one order of magnitude.

In the past, research on glass strength focused on long-term behaviour (static to quasi-static loading) and the effects of sub-critical crack growth (described in Sec. 2), showing that the strength of glass is being sensitive to the applied load duration. Even though glass is generally acknowledged to exhibit a higher dynamic strength, the available data, however, is quite limited for higher loading rates relevant to impact and blast loading events [49,53,55,62,63]. In general, this is attributed to the fact that dynamic load experiments are much more complicated to perform and interpret, because several parameters such as the measurement equipment, specimen geometry, and stress-wave propagation effects, must be given careful consideration.

To provide a broad overview of the time-dependent tensile behaviour of soda-lime-silica glass, it is the aim of this paper to review all available literature (to the best of the authors' knowledge) concerning experimental investigations on the mechanical material properties, such as strength and stiffness, over a wide range of load duration and rate of loading. No limits are set on the magnitude of the load duration and rate of loading to be considered. Furthermore a brief overview of applied experimental techniques is presented. In closing, a direct comparison is made between the collected strength data and various available Standards defining the load duration dependence on glass strength.

2. Fundamental aspects of time- and rate-dependent failure

Typically, glass is a linear elastic, isotropic material that exhibits brittle failure. Before post-processing, such as chemical treatments or thermal/chemical tempering, glass strength is essentially governed by the surface flaws (or surface cracks) located on the tensile loaded surface. The compressive strength of glass is much higher, and usually not important in structural applications; therefore, not considered in this paper. As with many other brittle materials, glass will fail instantaneously after reaching a critical value for the stress intensity at the tip of one surface crack. However, because of the characteristic differences in surface flaws, glass strength is not considered to be a material constant, and the size of the glass element, the load history (intensity and duration), the residual stress, and the environmental conditions, also affect significantly the ultimate strength exhibited. Moreover, glass has a unique characteristic where its atomic structure reacts with moisture from the environment [[64], [65], [66]]. As a result, sub-critical crack growth1 effects are observed under normal environmental conditions where a level of humidity is present and surface flaws grow under a constant tensile load (see e.g. Refs. [[67], [68], [69]]). This is directly related to the phenomenon of long-term loads leading to a distinct reduction in strength. The first observation of static fatigue was published in 1899 by the French scientist Grenet [70], who also explained the unexpected fracture of filled champagne bottles by the delayed failure of glass2.

Taking into account the effect of load duration on glass strength requires a known corrosive crack behaviour, characterised through the crack velocity, v, and the stress intensity factor for mode I loading, KI. Extensive studies have looked at the effect of different environmental conditions [[72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84]], and within the construction industry water appears to be decisive. A schematic representation of the relationship between v and KI is given in Fig. 2 with four essential regions highlighted:

0
In region 0, no measurable sub-critical crack growth effects occur below Kth. For soda-lime-silica glass this value can range from 0.14 to 0.28 MPa m1/2 in water [79,83,[85], [86], [87], [88], [89], [90]] and from 0.37 to 0.39 MPa m1/2 in air (50% RH) [86,88], at crack velocities as low as 10−14 m/s.
I
In region I, the crack velocity is essentially governed by the molecular mechanisms of stress corrosion at the crack tip. The crack growth rate thus depends on the applied stress intensity and relative humidity. A measure of how reactive water molecules are within the glass lattice, is characterised by the slope of the curve. For region I, the fracture behaviour in water at 25 °C can be characterised with a minimum crack velocity of around 10−10 m/s [74]. While in very dry air (0.017% RH) a maximum crack velocity of 10−7 m/s is observed, the maximum crack velocity in saturated air (100% RH) increases to 10−4 m/s [73].
II
In region II, sub-critical crack growth is still influenced by the chemical reactivity of the surrounding environment, but is independent of the stress intensity. As the crack velocity is directly proportional to relative humidity, a plateau is formed in the v(KI)-curve. The constant crack velocity range narrows as relative humidity increases, which emerges from the findings of Wiederhorn [73].
III
In region III, the v(KI)-curve increases rapidly approaching the limit for an inert environment. At this point, it is no longer possible for the surrounding reactive environment to follow the crack tip, thus resulting in an environment independent crack velocity between 10−3 m/s and 1 m/s. After reaching the fracture toughness of the glass, KIc, crack growth becomes unstable leading to failure of the glass element. For soda-lime-silica glass KIc is estimated to be between 0.72 and 0.82 MPa m1/2 at room temperature [91].

In general, the determination of the design lifetime of a glass element is based on region I, while the contribution from region II and III can be neglected for quasi-static loads. A good approximation of sub-critical crack growth in region I is described by the following empirical power law3 originally proposed by Evans and Wiederhorn [92]: $v = \frac{d a}{d t} = v_{0} {(\frac{K_{I}}{K_{Ic}})}^{n}$ where v0, on a logarithmic scale, represents the ordinate of KIc and the exponent n defines the slope of the curve. When the value of the crack growth parameter n is high, it indicates that the chemical reactivity at the crack tip is reduced due to a decrease in humidity, resulting in slower crack growth. The opposite is true when n is low. The well-established theory of linear elastic fracture mechanics (LEFM) defines the stress intensity factor, KI, with respect to a geometry (correction) factor Y (0.637 for half penny shaped cracks and 1.12 for edge cracks in semi-infinite plates) and a crack (or flaw) depth, a (see e.g. Ref. [93]): $K_{I} = σ \cdot Y \sqrt{π a}$ where σ is the stress acting normal to the crack plane. Inserting Eq. (1), (2) into Eq. (1) and assuming the ordinary differential equation to be valid over the full range of KI with a constant n, the method of variable separation yields: $\int_{0}^{t_{f}} σ^{n} (t) d t = \frac{2 K_{Ic}^{n}}{(n - 2) a_{i}^{(n - 2) / 2} \cdot v_{0} {(Y \sqrt{π})}^{n}}$

With a given stress history, σ(t), and neglecting the crack growth threshold, Kth, this equation can be used to estimate the time-to-failure of a crack (or flaw) with an initial depth, ai.

2.1. Crack resistance at constant stress

For a constant applied stress, i.e. $σ (t) = σ_{f, s}$ , as shown in Fig. 3a, a static crack resistance for any given load duration is found by inserting the stress history into Eq. (3): $σ_{f, s} = α \cdot t_{f, s}^{- 1 / n}$ where $α = {[\frac{(n - 2) a_{i}^{(n - 2) / 2} \cdot v_{0} {(Y \sqrt{π})}^{n}}{2 K_{Ic}^{n}}]}^{- 1 / n}$ with tf,s being the time to failure or lifetime of a given initial crack exposed to σf,s.

It follows from Eq. (4) that for two identical cracks (ai, Y) found on two identically sized glass elements, index 1 and 2, for identical conditions (v0, n, KIc), the interrelationship between the constant applied stresses and lifetimes can be expressed as: $\frac{σ_{f, s, 2}}{σ_{f, s, 1}} = {(\frac{t_{f, s, 1}}{t_{f, s, 2}})}^{1 / n}$

This result can be used to determine n, since it is independent of v0.

2.2. Crack resistance at constant stress rate

In the case of a constant stress rate, $\dot{σ}$ , commonly used for strength testing glass, the applied stress increases linearly with time as illustrated in Fig. 3b: $\dot{σ} = \frac{d σ}{d t} = c o n s t . \Rightarrow σ (t) = \dot{σ} \cdot t$

Similar to σf,s, a dynamic crack resistance, σf,d, can then be found by inserting Eq. (6) into Eq. (3): $σ_{f, d} = β \cdot {\dot{σ}}^{1 / (n + 1)}$ where $β = {[α^{n} \cdot (n + 1)]}^{1 / (n + 1)}$ Here, the constant β is defined such that it depends on α from tests with constant applied stress (see Eq. (4)). By doing so, one could use data from constant stress rate tests to describe crack growth in cases with constant applied stresses. The reverse approach is also possible when data from tests with constant applied stresses is available and an estimate of crack growth at constant stress rate is needed. However, both approaches implies that cracks and conditions must be identical to be able to perform the conversion.

The use of Eq. (7) was first suggested by Charles in 1958 [94] to describe rate dependent crack growth at constant temperature in his dynamic load experiments on glass. For low stress rates, it is sufficient to only consider the contribution from region I. However, for dynamic events over short times, such as impact loads and blast waves, region II and III seem to affect the design lifetime significantly according to Kuntsche [95].

Also, Eq. (7) can be used to describe the interrelationship dependence between two identical cracks (ai, Y) found on two identically sized glass elements, index 1 and 2, in identical conditions (v0, n, KIc) loaded at constant stress rates ${\dot{σ}}_{1}$ and ${\dot{σ}}_{2}$ : $\frac{σ_{f, d, 2}}{σ_{f, d, 1}} = {(\frac{{\dot{σ}}_{2}}{{\dot{σ}}_{1}})}^{1 / (n + 1)}$

Since Eq. (8) also is independent of v0, it is commonly used to determine n from experiments, by plotting the failure stress as a function of the stress rate on a logarithmic scale, resulting in a slope of 1/(n + 1). However, this is only valid in limited cases in which flaws, conditions, and v0 are identical during all tests. According to Haldimann et al. [96], v0 can be strongly stress rate dependent, which is why this method should be used with caution.

It has been shown that the v(KI)-curves, as illustrated in Fig. 2, successfully describe the variations observed in time-to-failure at constant applied tensile stress. Expanding on this, these curves can be used to examine the characteristics of dynamic failure at a constant stress rate; this was first reported by Evans in 1974 [85] and later reexamined by Evans and Johnson [97] and Chandan et al. [98]. These authors suggest that by integrating over the 4-regions of the sub-critical crack growth velocity curve, a new curve is obtained where fracture strength is a function of stress rate (the corresponding four regions are now A, B, B’ and C), as depicted in Fig. 4.

At low stress rates, region A, glass strength begins to increase from a minimum value, σ0. For an increasing stress rate, region B, there exists a simple logarithmic relationship between σf,d and $\dot{σ}$ , which is identical to that derived by Charles [94] (see Eq. (7)). At still higher stress rates, region B’, strength varies in a non-trivial manner with stress rate. Lastly, at the highest stress rate, region C, strength is independent of stress rate due to the absence of sub-critical crack growth effects.

Evans and Johnson [97] described the transition between the different regions analytically and presented strength data at stress rates ranging from 2 ⋅ 10−4 to 3 ⋅ 101 MPa s−1. However, their results did not provide any conclusive evidence of distinctly different regions. A few years later, Chandan et al. expanded on this work to try and characterise the two higher regions, B’ and C, using stress rates between 5 ⋅ 10−1 and 2 ⋅ 106 MPa s−1. However, the work was unsuccessful because the measured strength data would fit region B only and no evidence was found of the two higher regions, confirming the general agreement on the applicability of Eq. (7).

As shown, the fracture resistance of glass, for a surface crack (or flaw) loaded at either constant stress or constant stress rate, can in both cases be described by an empirical power law, in which the stress intensity at the crack tip is given by LEFM. In either load case, the load duration, or time to failure, is a fundamental parameter governing the fracture resistance of a glass element. For constant applied loads, the load duration is a common measure when investigating material characteristics, whereas it is the rate of loading that is of more importance when considering a constant loading rate. However, the constant loading rate is inversely proportional to the load duration, allowing a direct comparison between these two types of load configurations. The following sections present, and discuss, the study undertaken to understand better the behaviour of soda-lime-silica glass to tensile loads, using the two load configurations. First, the focus of this work is on the various applied experimental techniques, followed by a discussion of the material characteristics measured.

3. Experimental techniques

A variety of experimental techniques have been employed by researchers to investigate the mechanical properties of glass for various load conditions, using strain rates typically expected in practice. The typical strain rates found in engineering applications, range from creep loading (<10−6 s−1), through the quasi-static case of around 10−6 s−1-10−5 s−1 to an intermediate range (≈10−4 s−1-1 s−1) covering structural dynamics imposed by wind and seismic loading, to even higher levels (>1 s−1) that include debris impact and blast loads, as shown in Fig. 1. Other relevant load conditions that have been used are constant loads, e.g. similar to snow load or dead-weight, also considered as static loads, where strength is dependent on load duration, rather than the loading rate.

In general, two different test methods can be identified, which are used to study the mechanical properties of glass:

1.
Static fatigue test (σ = const.) – used to investigate failure that would occur over time when a constant stress is applied, see Fig. 3a. This method is used to measure the sub-critical crack growth effects that result in delayed fracture. The advantage of such tests is that the test conditions are representative of typical scenarios of long-term loading. However, a disadvantage is that they can be extremely time-consuming.
2.
Dynamic fatigue test ( $\dot{σ} = const.$ ) – used to investigate fracture that would occur at a constant stress/strain rate, see Fig. 3b. This is a common method used to investigate the rate-dependent tensile strength, and often used to identify the crack growth parameter, n, using Eq. (8).

Tensile strength is a critical parameter when designing load carrying glass structures. To study it by means of the above mentioned methods, several experimental setups have been employed to test materials on an engineering scale; the most common are the three-point and four-point bend test, and the axisymmetric bending (coaxial double ring tests). The load configurations and resulting stress distributions are illustrated in Fig. 5.

However, other, albeit rather rare techniques have also been reported, such as uniaxial tension and diametral compression tests. For the direct investigation of sub-critical crack growth effects in glass, the double cantilever beam test is a widely used technique. A brief introduction to each technique is provided below. Local characterisation of a material can also be carried out by e.g. indentation techniques such as the Vickers indentation; however, these methods are not within the scope of this review paper.

Three-point bending: The glass sample is loaded at three points as shown in Fig. 5a. The load points are typically placed symmetrically over the test sample. In order to minimise membrane stresses, two of the applied forces (typically the reaction points) are roller contacts, allowing for horizontal movement. The setup introduces a linearly varying tensile stress on the bottom surface between the two roller supports, with a maximum value below the central load point, as shown in Fig. 5a. In this configuration, the probability of finding a critical favourably located flaw, in the relatively small area of highest tensile stress, is low. It is, therefore, important to locate the origin of fracture in order to precisely determine the strength of the test sample. Furthermore, the distribution of stress can complicate the statistical analysis of the data.

Four-point bending: This configuration is similar to the previous test with the difference that there are four contact points as shown in Fig. 5b. Again the contact points are typically placed symmetrically and three of the applied forces are rolling contacts (free horizontal movement) in order to minimise membrane stresses. An advantage over the three-point bend test is the relatively larger area of maximum tensile stress as shown in Fig. 5b. However, both the three- and four-point bend tests produce undesirable high tensile stresses at the sample edges. Since there is a higher probability of finding a critical favourable flaw at the edge of the sample as compared to surface areas far from the edge (a consequence of glass cutting and handling), it can be challenging to determine the true surface strength of the glass specimen using these two bend test methods.

Axisymmetric bending: In order to minimise undesirable edge effects, the axisymmetric bend configuration can be employed. Such tests are usually carried out in a ring-on-ring test setup where two concentric rings sandwich a flat sample: a support ring on which a sample of glass is placed and then loaded using a smaller load ring. The circular geometry gives rise to rotationally symmetric stresses, where the maximum surface stress is nearly uniformly distributed within the area of the load ring. This is similar to the four-point bend test, but with the exception that this configuration reduces significantly the stress at the specimen edges, resulting in strength data directly related to the surface, see Fig. 5c. However, due to non-linear effects, the acceptable specimen size in the test is limited; to overcome the limit more complicated test setups have been suggested as in EN 1288-2 [99].

Other loading techniques: Although it is often more convenient to test glass in bending due to the relatively simple support and load configuration, some researchers have used alternative techniques to study the tensile behaviour of glass. One of them is the uniaxial tensile test, which is a direct and fundamental technique used to characterise the pure tensile behaviour of a solid. However, for the test the tensile grips to the sample must be chosen carefully since large stress concentrations at the grip point would be highly undesirable. This may explain the infrequent reporting of direct tensile tests. There is also the diametral compression test. It is used for indirect measurements of the tensile strength of brittle materials, such as rock like materials and concrete. By placing a cylindrical specimen in diametral compression, as shown in Fig. 6, a broad region of tensile stress is produced, with a narrow region of compressive stress produced at the ends of the sample, where the load is applied. The maximum tensile stress is located at the centre of the sample. Unfortunately, this method tends to overestimate tensile strength [100], and together with a load configuration that is far from the typical bending induced tensile stresses, it explains why this method is seldom used. Another, frequently reported test is the double cantilever beam technique (DCB). However, its intention is not to determine the strength of a material, but rather to characterise crack growth in the pure mode I opening configuration. By applying a constant tensile load perpendicular to a crack surface of known dimensions, in a predefined sample geometry (see Fig. 8), the crack velocity is measured where the stress intensity at the crack tip can be calculated for any measurable length of the crack. There are several other methods used to measure stable and unstable crack growth data over quite a wide range of crack velocities. In this paper the reviewed articles have only used the DCB method and as such it is the only method presented.

The discussion above provides a brief outline of the more common load configurations. It is, however, more complete to also consider the implications of how the load is actually applied to a glass specimen; that is, what are the implications of static and dynamic loads.

3.1. Static fatigue tests

In general, static fatigue data is gathered by measuring the time-to-failure of a number of samples at different constant applied stresses, as shown in Fig. 3a. A summary of the failure times achieved in the published works is shown in Fig. 7, while the detail of a review of the tests can be found in Appendix A, Table A.1.

It is relatively straight forward to implement a test on glass using a constant load configuration. Grenet [70] in 1899 was one of the first to report performing such a load test. He used a bucket filled with water, hung from the centre of a glass plate, which was supported along two of its four edges (three-point bending), to study time dependent failure. Since then, the use of ‘dead-weight’ loads has been extensive; in most cases, the weight is applied by either filling containers with e.g. sand [103] or weights, attached to a lever arm that applies the load to a specimen in a three- or four-point bend test configuration [[104], [105], [106], [107], [108], [109]].

This simple static configuration allows users to perform measurements over weeks to months (see e.g. Ref. [110]). However, failure over shorter periods is only possible for much higher loads. Typically, this then produces undesirable inertial effects, which places a lower limit on the static load duration. This explains why the majority of published data summarised in Fig. 7 are at failure times above 1 s. Baker and Preston [103] and, Mould and Southwick [105] have both reported measurements below this limit using an apparatus constructed from a loudspeaker, which is capable of applying loads without inertia effects. Thereby reaching static fatigue failure times as low as approx. 2.5 ms.

Even though simple ‘dead-weight’ load configurations have proven to be highly useful, since the 1980s a new class of ‘universal testing machines’ have been developed. Using integrated systems with efficient mechanical actuators and computer software, these testing machines have greatly improved the study of the static fatigue process [[111], [112], [113], [114]]. This allowed highly controllable load duration in the range of 4 s to almost 1 d.

3.1.1. Crack velocity measurements

Since it is common practice to measure crack velocities at constant applied load, the study of crack growth can be considered as a subgroup of static fatigue tests. Therefore it is included in this review. From the literature reviewed here, it is found that the double cantilever beam technique is employed in most work. In this technique a glass specimen with an initial edge crack of known dimensions is prepared and loaded at constant load perpendicular to the crack surfaces, as shown in Fig. 8. In this specimen/load configuration the stress intensity of the crack is well defined, Eq. (2), and therefore, can be calculated at each point the velocity of the crack is measured, resulting in a v(KI)-curve and an estimate of the sub-critical crack growth parameters, KIc, v0, and n, defined in Eq. (1) (also see Fig. 2).

3.1.2. Definition of relative applied stress

The static fatigue data of soda-lime-silica glass reviewed here are reported for failure loads presented as either an applied mass (lb. or kg) or an applied stress (psi or MPa). In the following assessment and discussion, all reviewed data are converted to a relative applied stress by normalising with respect to a ‘60-second’ strength, $σ_{t_{0}} (t_{0} = 60 s)$ , using the relationship defined in Eq. (5). Therefore, the crack growth constant α vanishes, and only the relative glass strength is highlighted, without showing absolute strength values for the various specimen sizes provided in Table A.1.

3.2. Dynamic fatigue tests

All the reported dynamic fatigue data are measurements of the failure load of a number of samples at different constant loading rates. The detail of the experiments in these previous tests are given in Appendix A, Table A.2, while Fig. 9 is a summary of the range of loading rates achieved, which also indicates the strain rates that can be expected for the different experimental techniques. However, it should be noted that the strain rate not only depends on the experimental technique but also on the specimen geometry under investigation.

Typically, universal testing machines are used to strength test glass in the quasi-static load regime at constant stress rates near 2 MPa s−1 (see e.g. Refs. [99,115,116]), corresponding to strain rates of around 10−5 s−1 at a Young's modulus of 70 GPa. The summary of data in Fig. 9 highlights this point since most of the published works before 2010 are for lower strain rates from 10−9 s−1 to about 10−3 s−1. By using fast valves and gas reservoirs, to pressurise an oil reservoir (pneumatic-hydraulic system), higher strain rates as high as 101 s−1 have been reported [[117], [118], [119]]. However, it is difficult to implement a direct feedback control system for constant strain rates over a short duration, which is why the strain rate depends more on the relationship between machine and specimen stiffness. Strain rates at about 101 s−1 were also reported by Chandan et al., in 1978 [98]. The authors used a pendulum impact machine to test Charpy sized glass specimens in three-point bending – this setup is traditionally used to measure the impact resistance of many other materials. Pal and Pennington [120] employed a drop test setup, releasing weights between 44 and 84 kg, from drop heights of 1.5–3.0 m, resulting in impact velocities of about 5–7 m/s. They undertook dynamic impact tests of rectangular glass plates, of dimensions 812.8 × 685.8 × 2.2 mm3, which were simply supported along all four edges. Since the samples were large in area and thin, the measured strain rates (around 10−5 s−1) were quite low.

Much higher impact velocities, and thus, higher strain rates, can be achieved when using a gas gun driven test setup, such as a split Hopkinson pressure bar (see e.g. Ref. [121]), in which more recently strain rates up to 102 s−1-103 s−1 have been reported in a split tensile test configuration [101,102]. To the authors' best knowledge, these are the highest strain rates reported when characterising the tensile behaviour of soda-lime-silica glass.

In addition, other, less conventional dynamic fatigue load tests have been reported in the literature reviewed here. To these tests belong the investigations conducted by Borchard in 1937 [122], who studied the effect of loading rate on the strength of 1/2-L glass bottles. The author performed his measurements by pressurising the internal volume of the bottle at a constant rate until glass failure, where lower strain rates from about 10−9 to 10−6 s−1 were obtained. One order of magnitude higher strain rates were reported by Thompson and Cousins in 1949 [123] using explosive charges (black powder) placed inside a box, to which a glass pane of size 355.6 × 482.6 × 2.3/3.1 mm3 was mounted, where the glass was constrained along its four edges to the box. In the 1980s, similar boundary conditions were reported in the test setups employed by Johar [124,125] and Pal and Pennington [120] to investigate large sized window glass panes at low strain rates, from about 10−11 to 10−7 s−1. An air chamber on one side of the constrained glass pane was pressurised by moving a plunger, until failure of the glass.

3.2.1. Definition of loading rate and relative strength

In the literature reviewed here the dynamic fatigue data in each publication are presented differently. That is, the loading rate has been related to the applied mass, the stress rate (slope of the stress-time curve as in Fig. 3b), strain rate, and/or at some cases with respect to the rate of travel of a piston actuator. In the following discussion of the data, no distinction is made between the different methods used to measure the loading rate. Here, the reported data have been converted to strain rate, using a Young's modulus of E = 70 GPa (unless otherwise stated) for comparison purposes: $\dot{ε} = \frac{\dot{σ}}{E}$

In the case of axisymmetric bending, Poisson's effects are present, which are accounted for by the following expression, using a Poisson's ratio of ν = 0.23: $\dot{ε} = \frac{\dot{σ}}{E} (1 - ν)$

Moreover, crack properties (as defined by β in Eq. (7)) and specimen size effects, have been made dimensionless using Eq. (8) by normalising dynamic strength with respect to a static strength, $σ_{{\dot{ε}}_{0}}$ , interpolated or extrapolated at a strain rate of ${\dot{ε}}_{0} =$ 2.86 ⋅ 10−5 s−1 ( $\propto \dot{σ} =$ 2.0 MPa s−1 for E = 70 GPa), which means that only the rate dependence of the glass strength dominates and other effects are removed.

4. Investigated material characteristics

Over a century numerous tests have been carried out to investigate the behaviour of glass under various test conditions. Whether the tests were looking at crack growth or strength, glass specimens were either subjected to a number of constant loads or constant loading rates, using experimental techniques as discussed in the previous sections. Therefore, the published data of these tests provide information about load duration effects and the effect of test environment.

This section reviews the results from past investigations that looked into the tensile behaviour of soda-lime-silica glass. All the data presented satisfy the inclusion (search) criteria defined in Appendix B, and no limits were set with respect to the acceptable loading rate. These data are available in digital format in the online repository DTU Data [126]. Results from static fatigue tests have confirmed, generally, that increasing the applied load decreases the time to failure. Similar behaviour applies to the dynamic fatigue data, where the strength increases when the loading rate is increased (i.e. decreasing load duration). In both cases, the data follow well a linear line of best fit when using a log-log plot.

In the following sections, the test results are subdivided into the two categories of static and dynamic fatigue. Furthermore, all of the compiled strength data (from the literature review) are compared to the existing definition of glass strength as typically accepted in the Standards. A few of the reviewed works are highlighted here since they provide further detail to explain the connection between loading rate and Young's modulus.

4.1. Load duration effects on strength (static fatigue)

4.1.1. Static fatigue data

The static fatigue data (constant stress) is presented in Fig. 10, as a plot of strength (normalised to a ‘60-second’ strength, $σ_{t_{0}}$ , as explained in Sec. 3.1.2) as a function of time to failure. A summary of the key parameters related to test condition, applied minimum and maximum stress, and sub-critical crack growth parameters, are given in Appendix A, Table A.3. A common outcome from all the data reviewed, is that the glass strength decreases when the load duration increases, which is a result that is not dependent on test method and/or specimen geometry. This relationship can be described using a power law as defined by Eq. (4), where the time to failure is related to Eq. (1), which defines crack growth through the crack tip stress intensity factor, KI, Eq. (2).

Failure times span from 2.5 ms (Mould and Southwick [105]) to 270 d (Shand [110]), resulting in a strength increase of about 100% and a reduction of about 50%, respectively, with respect to $σ_{t_{0}}$ . However, a greater number of strength data exists for load duration between 1 s and 1 d, which highlights typical results of experiments; for a very short duration, inertia effects must be considered, and for a very long duration space and patience are needed.

For times lower and higher than t0 = 60 s, it is clear that the data in Fig. 10 diverges from the general trend seen. However, most of divergent data should not be treated as a scatter. In this plot the test results obtained for different test conditions show deviations, because of the sub-critical crack growth effect, which is highly dependent on environmental conditions, such as temperature and humidity (see Sec. 2). This results in a change of the log-log linear slope.

The results which exhibit the most significant divergence from the trend are those obtained by Gurney and Pearson [127], who investigated the static fatigue behaviour of soda-lime-silica glass in air and vacuum. In the case of the latter n-values where higher, between 40 to around 135 (see Table A.3), confirming that the crack growth rate strongly depends on the environment. This dependence has been validated further, by investigations that have varied humidity in the surrounding air, up to liquid water, which showed that n, between 14.1 and 20.3, decreases with increasing water content [106,108,[128], [129], [130], [131], [132]]. In predicting the design lifetime of a glass element, a constant value of n = 16 is a reasonable and conservative choice [96].

Other, more extreme, variations in test conditions have been used by Vonnegut and Glathart [133], who studied the effect of temperature on the strength and fatigue of scratched soda-lime-silica glass rods, between −190 °C and 520 °C. They found a very strong dependence on temperature, and between 100 °C and 200 °C, the strength was at a minimum and the fatigue process dominated. When the temperature was outside this range, lower or higher, the effect of delayed failure was less, which was explained by a low reaction activity at the lower temperatures and evaporation of water at the higher temperatures. Complete control of the surrounding atmosphere is extremely difficult. Therefore, in 1958 Charles [129] performed additional experiments at temperatures between −170 °C and 242 °C, where a constant atmosphere of saturated water vapour was used in the tests above 0 °C, and below 0 °C the atmosphere was adjusted from saturated water vapour to low humidity air. Similar to Vonnegut and Glathart, the strongest delayed failure effect together with lowest strength was found at a temperature around 150 °C, while the effect was weaker for higher temperatures, also indicated by an increased n-value in Table A.3.

In addition to environmental conditions, the strength of glass also strongly depends on the surface condition of the glass. The effect of removing surface scratches on glass rods, using an acid solution, has been studied by Ritter and Vrooman [134] and Ritter and Sherburne [107], both concluding that acid-etched glass is less susceptible to static fatigue, where for n an increased value between 31.0 and 37.6 was found. The opposite effect was investigated extensively by Mould and Southwick [105], who tested microscope slides under controlled ambient conditions with six different surface abrasions produced by grit blasting and emery cloth. These tests resulted in the same general fatigue behaviour, however, with differences in slope. Taking a closer look at the data, it is clear that the abrasions produced from the emery cloth are more susceptible to static fatigue than those produced from grit blasting.

Other studies looking into glass strength and surface condition effects have been conducted by Shand [135], Chantikul et al. [111], and Sglavo and Green [113,114], all of these works introduced local surface cracks either in the form of cleaved cracks or through Vickers indentation. The procedure of crack formation can introduce a residual stress field around the crack, which in all studies has been removed by annealing the glass at 520 to 540 °C. Both as-received and annealed conditions have been tested and compared. While Shand showed that the annealed specimens with cleaved cracks exhibit shorter lifetimes compared to the as-received specimens, the other authors have reported an opposite behaviour for specimens with Vickers indentation cracks. However, no explanation for the effect of annealing was given for the data from the investigation with cleaved cracks. For the Vickers indented specimens, the increase in lifetime after the annealing process is assumed to be caused by the removal of the indentation residual stress field.

The works summarised in Fig. 10 are in agreement about the general static fatigue behaviour of soda-lime-silica glass. Nevertheless, from the summary the influence of environmental conditions is not apparent. To highlight these environmental effects, Fig. 11 are two plots which divide the relevant data from Fig. 10 into test environments (around room temperature) important for the construction industry: (a) air (40–80% RH) and (b) water (liquid or 100% RH). To estimate the sub-critical crack growth parameter n, the extreme data points have been excluded by only considering values within the interval from the 2.5th to the 97.5th percentile, i.e. the interval of values containing the central 95% of the data. Since these data points are based on different sample sizes, the method of weighted least squares (WLS) is used for the linear regression with $\sqrt{N}$ as the weight factor, where N is the sample size provided in the digital datasets [126] (e.g. N = 1 for data points representing single measurements). Thus, for both test environments (water content) the value for n listed in Table 1 is determined from the resulting slope 1/n according to Eq. (5). As expected, soda-lime-silica glass is less susceptible to static fatigue in humid air than in water, which is also indicated by a lower gradient of the curve in Fig. 11a. This can be explained by the fact that in humid air, less water molecules react with the atomic structure of glass, as compared to a water environment (liquid or 100% RH), resulting in a slower crack growth and thus a longer life time. The values of n further confirm that the most accepted value of n = 16 is a reasonable and conservative estimate for the design of float glass used in buildings.

Estimation of initial crack length range from static fatigue tests using the strength model proposed by Overend and Zammit [

Test condition	n	$a_{i}^{\min}$	$a_{i}^{\max}$
Air (40–80% RH)	21.2	38.1 μm	0.45 mm
Water (liquid or 100% RH)	16.7	173.7 μm	1.41 mm

According to the model proposed by Overend and Zammit [136] to determine the tensile strength of float glass, it is unlikely that the strength will continue to increase or decrease constantly for a load duration approaching zero or infinity, respectively. For very short duration tests the strength will approach the inert strength of glass and for a very long duration the threshold strength becomes the limiting factor, as also seen from the v(KI)-curve in Fig. 2. This leads to a strength interval, which can be expressed as: $\frac{K_{th}}{Y \sqrt{π a_{th}}} ⩽ σ_{f, s} ⩽ \frac{K_{Ic}}{Y \sqrt{π a_{f}}}$ where ath and af are the threshold crack size and the critical crack size, respectively. The 2.5th and 97.5th percentile values are used to establish the two asymptotes shown in Fig. 11. In the literature review it was found that in most studies the inert strength of glass is unknown. Therefore, the data here are normalised with respect to a ‘60-second’ strength, $σ_{t_{0}}$ , see Sec. 3.1.2. For glass tested in air, 40–80% RH, the relative strength is estimated to approach the asymptotes at 0.56 and 1.51. These values are in line with the asymptotes determined by Overend and Zammit [136] for several initial crack sizes. Similar good agreement to the results from Overend and Zammit is found for the glass tested in water where the estimated asymptotes are located at 0.66 and 1.74.

Additional information can be drawn from Fig. 11 by applying the strength model proposed by Overend and Zammit [136]. For example, on the glass there is a distribution of surface flaws, which vary in length. An estimate of the minimum and maximum initial length of the flaws, $a_{i}^{\min}$ and $a_{i}^{\max}$ , respectively, can be calculated using Eqs. (12), (13) and (13). The times ta and tb are the intersection points between the asymptotes and the linear regression curve, as labelled in Fig. 11. $a_{i}^{\min} = \frac{t_{a}}{2} (n - 2) v_{0}$ $a_{i}^{\max} = \frac{t_{b}}{2} (n - 2) v_{0} {(\frac{K_{th}}{K_{Ic}})}^{n}$

The calculated minimum and maximum initial surface crack length are given in Table 1, and for both test environments they are determined by assuming a sub-critical crack growth limit Kth = 0.25 MPa m1/2 [137] and a fracture toughness KIc = 0.75 MPa m1/2 [96]. However, a distinction is made between the crack velocities applied. For the glass tested in air v0 = 6 mm/s, which can be considered a conservative estimate for in-service conditions of float glass in buildings, and for the glass tested in water v0 = 30 mm/s is representative [96].

The minimum to maximum length ranges from micrometres to millimetres. Furthermore, it appears from the values given in Table 1 that for some of the glass specimens tested in water the initial crack length might have been larger compared to those tested in moist air. Nevertheless, the estimated crack length range, for both test environments, is comparable in magnitude and an overlap is present in the millimetre range. According to Petzold et al. [138] these dimensions correspond to micro-cracks from processing, up to visible flaws typically arising from handling and ageing. Relating these crack origins to the glass samples from the studies, i.e. with respect to handling and surface treatment, the magnitude of the estimated initial crack lengths appear reasonable. Hence, the values given in Table 1 provide a suitable basis for the determination of the design lifetime of a glass element using Eq. (4).

4.1.2. Crack velocity data

Crack velocity data can also be used to describe the static fatigue behaviour of soda-lime-silica glass (see e.g. Wiederhorn and Bolz [74]). Although, published crack velocity experiments do not provide direct strength measurements, the data from region I in v(KI)-curves can be used to estimate the sub-critical crack growth parameters n and v0 (see Sec. 2). A summary of the parameters determined from the literature is provided in Table 2. It is clear that the average value of n is slightly larger for glass tested in moist air as compared to water immersion, which agrees well with the static fatigue data. Furthermore, the listed values for v0 confirm that this parameter is also affected by water content in the environment, as an increase is seen for most of the investigations conducted in water. This behaviour has already been accounted for in the estimate of the initial crack lengths conducted in the previous subsection, using conservative values proposed in the literature, see Sec. 4.1.1.

Table 2. Summary of sub-critical crack growth parametersa, n and v0, from crack velocity tests on soda-lime-silica glass divided into the test environments (a) air and (b) water. An averaged value of each parameter is provided at the bottom of the table.

(a) Test environment: air
Reference	Test cond.	n	v0
Reference	Test cond.	[−]	[mm/s]
Wiederhorn [72]	Moist air	22.3	–
Wiederhorn [73]	30% RH	22.7	1.8
Kerkhof et al. [78]	50% RH	18.1	2.4
Gehrke et al. [82]	50% RH	16.7	0.9
Ullner [84]	Air	19.6	2.5
Dwivedi and Green [139]	64–78% RH	20.4	0.3
Average:		20.0	1.6

(b) Test environment: water
Reference	Test cond.	n	v0
Reference	Test cond.	[−]	[mm/s]
Wiederhorn [72]	Water	17.1	–
Wiederhorn [73]	Water	17.9	4.3
Wiederhorn [73]	100% RH	21.0	3.8
Wiederhorn and Bolz [74]	Water	17.2	35.3
Wiederhorn and Johnson [75]	Water	16.9	6.8
Freiman [76]	Water	15.6	6.6
Kerkhof et al. [78]	Water	16.0	50.2
Simmons and Freiman [79]	Water	17.9	12.7
Wiederhorn et al. [80]	Water	17.4	9.7
Gehrke et al. [82]	Water	15.5	3.1
Gehrke et al. [83]	Water	19.0	1.8
Singh and Shelty [140]	Water	11.8	2.5
Ullner [84]	Water	18.4	14.9
Average:		17.0	12.6

a: Determined from v(KI)-curves using the relationship given by Eq. (1) and assuming a fracture toughness of KIc = 0.75 MPa m1/2.

Based on the relationship given in Eq. (5), which assumes identical cracks for identical conditions, a static fatigue curve from crack velocity experiments can be estimated using n only and neglecting the asymptotes defined by the inert and threshold strengths: $\frac{σ_{f, s}}{σ_{t_{0}}} = {(\frac{t_{f, s}}{t_{0}})}^{- 1 / n}$ with $σ_{t_{0}}$ being the strength at a load duration of t0 = 60 s. For each of the studies listed in Table 2 a static fatigue curve was estimated and compared to the static fatigue data from Fig. 11. The comparison is shown in Fig. 12 divided into the test environments (a) air and (b) water.

The estimated static fatigue curves in air are in good agreement with static fatigue data obtained in the same test environment. Variations in slopes are seen, however, they are within the scatter of static fatigue data. Thus, the n-values from crack velocity experiments in air provides a good estimate for the static fatigue behaviour of soda-lime-silica for a load duration where a constant decrease in strength can be assumed.

A good agreement between estimated curves and static fatigue data is seen for the experiments conducted in water as well. Although, it seems like the estimates based on data from Wiederhorn [73] (for 100% RH) and Gehrke et al. [83] underestimate the strength for a shorter load duration than 1 s, the variation in slope is still within scatter seen in the static fatigue data. The largest deviation seen, however, is the estimate determined from data published by Singh and Shelty [140], which results in n = 11.8. The increase in slope is most likely related to the experimental setup where crack velocities were obtained from cylindrical specimens in a diametral compression test.