# Structural glass - Fracture strength

After summarazing structural analysis and model elaboration in our previous post we now go through the assessment of glass fracture processes and stregth. Fracture mechanics on glass elements are a very complex subject with a vast bibliography and would require too much time to achieve a deep understanding of the whole phenomena. We will try here to summarize main topics to try to explain the whole process in a simple and afordable way.

Introduction

The random glass molecular structure formed by an irregular network of silicon and oxygen atoms with alkaline parts in between (unlike most other construction materials with geometrically regular networks of crystals) has no slip planes or dislocations that allow macroscopic plastic flow before fracture. Therefore, glass will behave as a perfectly elastic material at normal temperature exhibiting brittle fracture. The fact that glass is not capable to yield plastically before fracture means that the fracture strength of glass is very sensitive to stress concentrations. If glass reaches its fracture strength at any point there will be no plastic yielding and therefore no stress redistribution, causing the brittle glass breakage. Therefore, since surface flaws cause high stress concentrations, the assessment of fracture strength of glass must take into account nature and behaviour of this flaws.

Physical principles of fracture strength

We will here first discuss the stress corrosion that causes existing surface flaws to grow slowly in size prior to failure, a phenomenon that is often referred to as "sub-critical crack growth". After that, we will characterize the quasi-static linear elastic fracture mechanics (LEFM) that explain the theoretical basis for mathematical models that determine fracture strength of glass and crack growth used for predictive modelling and structural design (the so called "lifetime prediction models").

Although this quasi-static approach will be suitable for short loading times lifetime prediction models will not be valid to describe dynamic phenomena as glass fracture or response of glass elements to impact loads. Then, to understand the mechanisms and principles of glass behaviour on this situations dynamic fracture mechanics will be introduced. We will cover main considerations that have to be taken into account when assessing this behaviour.

Stress corrosion and sub-critical crack growth

In vacuum, the strength of glass is time-independent. In the presence of humidity, stress corrosion causes flaws to grow slowly when they are exposed to a positive crack opening stress. This means that a glass element which is stressed below its momentary strength will still fail after the time necessary for the most critical flaw to grow to its critical size at that particular stress level. The momentary strength of a loaded glass element therefore decreases with time, even if it is exposed to static loads only.

This relationship between crack velocity propagation and stress intensity will be influenced by a number of aspects. We will summarize here the principal ones.

- As mentioned before, the water content of the surrounding medium strongly influences sub-critical crack growth.

- An increasing temperature causes mainly higher crack velocities.

- The crack velocity generally increases as the pH value of the surrounding medium increases. Furthermore, the pH value has a particularly strong influence on the crack growth threshold K_{th}.

- All parameters of sub-critical crack growth are influenced by the chemical composition of the glass.

- v-K_{I} relationship does not only depend on environmental conditions, but is also strongly loading rate dependent. Stress corrosion requires humidity. If an element is loaded rapidly, the diffusion process is not fast enough, so that a shortage in the supply of water to the crack tip slows down stress corrosion and therefore the sub-critical growth of flaws. Consequently, the v-K_{I} relationship of an element is shifted towards lower crack velocities when loaded rapidly.

- Crack velocity parameters vary widely and depend on several influences, including environmental conditions and the loading rate. Fracture strength predictions for service lives of many years are, therefore, of limited accuracy.

- Structural design will vary on the considerations for the parameters that adjust the mathematical models that simulate the behaviour of glass cracking.

- Strength data from tests at ambient conditions are inevitably dependent on the surface condition and on crack growth behaviour. The large variability of the crack velocity parameters makes it very difficult to obtain accurate surface condition information from tests at ambient conditions. Inaccurate estimation of the crack velocity during testing can yield unsafe design parameters. Testing at inert conditions is, therefore, preferable.

The growth of a surface flaw depends on the properties of the flaw and the glass, the stress history that the flaw is exposed to and the relationship between crack velocity (v) and stress intensity (K_{I}). In relation to that, the ‘classical stress corrosion theory’, involves the chemical reaction of a water molecule with silica at the crack tip: Si-O-Si + H20 -> Si-OH + OH-Si.

According to this theory, the crack velocity scales with the kinetics of this chemical reaction. Its activation energy depends on the local stress and on the radius of curvature at the crack tip. Based on that theory (first order chemical reaction) and in accordance to the observation of the phenomenon, logarithms of crack velocity (v) and humidity ratio (H) are linearly correlated.

There is another crucial parameter on the crack growth analysis, and this is the stress intensity factor (K_{I}), that will affect the crack propagation velocity enormously. The comparison between K_{I} value and glass stress intensity parameters K_{th}, K_{Ib} and K_{Ic} (ein order of increasing value) will indicate the different phases on crack growth.

Initially, for K_{I} values below threshold stress intensity (For typical soda lime silica glass at a moderate pH value, K_{th} is about 0.2 to 0.3 MPa m^{0.5}) no crack growth occurs.

When K_{I} goes above K_{th} values crack growth starts. Its propagation velocity will be very dependent from the environment and this behaviour, dominated by the activation of the referred chemical reaction, will be adjusted through two empirically obtained parameters S and n, according to next formulation:

$$v=S\ K_I^n$$

This behaviour will go on until K_{I} reaches K_{Ib} parameter, this range (K_{th}-K_{Ib} is known as region I). From this point crack behaviour starts to behave less dependently on environment conditions.

In a narrow region below K_{Ic} value(known as region III) crack velocity will drop in a very steep curve, with values between 0.001 and 1 m/s. For K_{I} values close to glass fracture toughness (K_{Ic}, an intrinsic property of each glass) or even above, crack propagation velocity will not depend on the environment and it will rapidly approach to characteristic crack propagation speed (about 1500 m/s on soda lime silica glasses).

Region I and region III are connected through region II. In this range of stress intensity factor values above K_{Ib} the kinetics of the chemical reaction on glass crack propagation are no longer controlled by the chemical reaction activation, but by the supply rate of water to the crack tip.

In view of the order of magnitude of glass elements in buildings (mm to m), the typical depth of surface flaws (m to mm) and the service life generally required, only the range of extremely slow sub-critical crack growth, region I, is relevant for determining the design life of a glass element. The contribution of regions II and III to an element’s lifetime is negligible.

Crack healing, crack growth threshold and hysteresis effect

The so called crack healing effect, that consists on the increase of the strength on flawed specimens during stress-free phases, is a consequence of two phenomena, the crack growth threshold and the hysteresis effect.

As explained, for stress intensities below K_{th} no significant crack growth takes place. Latest investigations strongly support the hypothesis that this phenomenon may be caused by the expulsion of the alkali compounds out of the glass and that this change on the chemical composition at the tip of the crack is the responsible for the crack growth threshold.

The hysteresis effect, present on alkali containing glasses makes an aged crack not to propagate immediately after reloading. This effect is convincingly explained by the renucleation of the aged crack in a different plane than the original one, as if the path of the crack has to turn around the area just in front of the former crack tip.

As a result of its strong dependence on environmental conditions, crack healing is very difficult to quantify. Therefore it should not be taken into account by design purposes although its favourable influence can be considerable.

Quasi-static fracture mechanics

Linear elastic fracture mechanics (LEFM) provides a good model for describing the brittle fracture of glass. In LEFM, mechanical material behaviour is modelled by looking at cracks. A crack is an idealized model of a flaw having a defined geometry and lying in a plane. It may either be located on the surface (surface crack) or embedded within the material (volume crack). For structural glass elements, only surface cracks need to be considered.

The following sections summarize the main equations used to characterize the mathematical models that simulate LEFM behaviour. The process of obtaining them is obviated given their complexity and length.

Stress intensity and fracture toughness

The theoretical strength of a material is determined by the forces of the interatomic bonds. For a typical silica glass, theoretical strength will be around 32GPa, but in practise the tensile strength of annealed soda lime silica glass is much lower. The explanation of that is the fact that fracture does not start from a pristine surface, but from pre-existing flaws on that surface. Such flaws are not necessarily visible to the naked eye, but they severely weaken brittle solids because they produce very high stress concentrations. As explained before surface flaws in glass grow with time when loaded, the crack growth rate being a function of several parameters.

Based on those assumptions, a static crack can be understood as a reversible thermodynamic system. In the configuration that minimizes the total free energy of the system, the crack is in a state of equilibrium and thus on the verge of extension. The total energy U in the system is:

$$U=UM+US$$

Where UM is the mechanical energy (the sum of the strain potential energy stored in the elastic medium and the potential energy of the outer applied loading system) and US is the free energy expended in creating new crack surfaces.

This concept is extended to provide a means of characterizing a material in terms of its brittleness or fracture toughness. It was here introduced the concept of the stress intensity factor K, which represents the elastic stress intensity near the crack tip. The stress intensity factor for mode I loading, K_{I}, is given by:

$$K_I=Y\ \sigma_n\ \sqrt{\pi\ a}$$

Where n is the nominal tensile stress normal to the crack’s plane, Y is a correction factor, and a represents the size of the crack (i. e. the crack depth or half of the crack length).

The correction factor Y depends on the crack’s depth and geometry, the specimen geometry, the stress field and the proximity of the crack to the specimen boundaries. While the dependence on the specimen geometry, the stress field and the crack depth is small for shallow surface cracks and can generally be ignored, the dependence on the crack geometry and the proximity to boundaries is more significant. Y is therefore often called the geometry factor . A long, straight-fronted plane edge crack in a semi-infinite specimen has a geometry factor of Y = 1.12. For half-penny shaped cracks in a semi-infinite specimen, the geometry factor is in the range of 0.637 to 0.713, depending on the approach used.

The fracture toughness K_{Ic} can be considered to be a material constant. It does not depend significantly on influences other than the material itself. A value of K_{Ic} = 0,75 MPa m^{0,5} can be used for all practical purposes.

Heat-treated glass

The in-plane surface stress normal to a crack’s plane, also known as the crack opening stress, is the sum of the surface stress due to actions (with an usual positive value when assessing glass breakage), the residual surface stress due to heat treatment (with an usual negative value) and the surface stress due to external constraints or prestressing (with an usual negative value). A crack can only grow if exposed to tensile stress (positive value of the sum).

From this assumption follows that the fracture strength of heat-treated glass is the sum of the absolute value of the residual (compressive) surface stress and of the strength of the glass itself, called inherent strength henceforth. Only the latter is influenced by sub-critical crack growth and depends, therefore, on time and environmental conditions as seen before. The residual stress is constant.

Inert strength

The stress causing failure of a crack of depth a, the critical stress 𝜎c can be obtained as:

$$\sigma_c\left(t\right)=\frac{K_{Ic}}{Y\ \sqrt{\pi\ a(t)}}$$

The critical stress represents the resistance of a crack to instantaneous failure (i.e. failure that is not triggered by sub-critical crack growth) and is therefore called inert strength henceforth.

The depth of a crack failing at the stress 𝜎_{n}, the critical crack depth a_{c}, is:

$$a_c\left(t\right)=\left(\frac{K_{Ic}}{\sigma_n\left(t\right)Y\ \sqrt\pi}\right)^2$$

Lifetime of a single flaw

Starting from the ordinary differential equation of crack growth:

$$v=da/dt=v_0\left(K_I/K_{Ic}\right)^n$$

We will, through some basic assumptions and mathematical calculation obtain the next formulation of general validity:

$${\widetilde{a}}_c\left(\tau\right)=\left[\left(\frac{\sigma_n\left(\tau\right)\ Y\ \sqrt\pi}{K_{Ic}}\right)^{n-2}+\\\frac{n-2}{2}\ v_0K_{Ic}^{-n}\ \left(Y\ \sqrt\pi\right)^n\ \int_{0}^{\tau}{\sigma_n^n(\widetilde{\tau})d\widetilde{\tau}}\ \right]^\frac{2}{2-n}$$

$${\widetilde{a}}_c\left(\tau\right)=\left[\left(\frac{\sigma_n\left(\tau\right)\ Y\ \sqrt\pi}{K_{Ic}}\right)^{n-2}+\frac{n-2}{2}\ v_0K_{Ic}^{-n}\ \left(Y\ \sqrt\pi\right)^n\ \int_{0}^{\tau}{\sigma_n^n(\widetilde{\tau})d\widetilde{\tau}}\ \right]^\frac{2}{2-n}$$

Where ã_{c}(τ) is the initial depth of a crack that fails at the point in time 𝜏 when exposed to the crack-opening stress history 𝜎_{n}(𝜏). As the momentary stress 𝜎_{n}(𝜏) is not monotonously increasing (could be some moments through lifetime of the flaw where stresses drop), the minimum initial crack depth which is relevant for design, does not necessarily occur at the end of the stress history but may occur at any time during its lifetime. Nevertheless, we are able to assure that a crack does not fail if its initial depth is below this minimum. In addition, through the study of this formulation It can be seen that the strength of cracks is strongly time-dependent. Furthermore, the long-term strength of cracks with an initial depth in the order of 100m or more is low.

Lifetime of a glass element with a random surface flaw population

As stated in previous posts, glass elements surface usually contains large number of flaws not necessarily visible to naked eye which distribution can be represented by statistical approach, namely, a random surface flaw population (RSFP). The mathematical relations stated on previous section need then to be extended to describe glass elements in which resistance is governed by RSFP.

Through some additional hypothesis and mathematical operations, starting from the application of the last sections formulation to a constant, uniform and uniaxial stress scenario and extending the formulation to non-uniform, biaxial stress fields we are able then to introduce time dependent loading and sub-critical crack growth on formulation. This way we achieve an expression for the time-dependent failure probability of a general glass element that takes sub-critical crack growth, non-homogeneous time-variant biaxial stress fields, arbitrary geometry and arbitrary stress histories into account:

$$P_f\left(t\right)=1-exp\left\{-\frac{1}{A_0}\int_{A}^{\ }{\frac{2}{\pi}\int_{\varphi=0}^{\pi/2}{\left[\underset{\tau\in\left[0,t\right]}{max}{\left\{\left(\left(\frac{\sigma_n\left(\tau,\vec{r},\varphi\right)}{\theta_0}\right)^{n-2}+\\\frac{1}{U\ \theta_0^{n-2}}\int_{0}^{\tau}{\sigma_n^n\left(\widetilde{\tau},\vec{r},\varphi\right)d\widetilde{\tau}}\right)^\frac{1}{n-2}\ \right\}}\right]^{m_0}dAd\varphi}}\right\}$$

Where:

A_{0}: Unit surface area (A_{0} = 1m^{2})

A: Surface area of the glass element (both faces)

t: Point in time

𝜎_{n}(𝜏,r,𝜑): In-plane surface stress component normal to a crack of orientation 𝜑 at the point r(x, y) on the surface and at time 𝜏

𝜃_{0}, m_{0}: Surface condition parameters to be determined from experiments

U: Combined coefficient containing parameters related to fracture mechanics and sub-critical crack growth

K_{Ic}: Fracture toughness

v_{0}, n: Crack velocity parameters

Y: Geometry factor

Simplification for structural design

The following simplifications are appropriate for the vast majority of common structural glass design tasks:

- Calculating the failure probability on the basis of the risk integral is an approximation of sufficient accuracy.

- The crack growth threshold can be neglected.

- An equibiaxial stress field may be assumed.

These assumptions enable the model to be simplified substantially.

$$P_f\left(t\right)=1-exp\left\{-\frac{1}{A_0}\left(\frac{t_0}{U\ \theta_0^{n-2}}\right)^\frac{m_0}{n-2}\int_{A}^{\ }{\left[\sigma_{1,t_0}\left(t,\vec{r}\right)\right]^\frac{n\ m_0}{n-2}dA}\right\}$$

Adding to that a series of transformations and additional assumptions allow to transform the obtained equations to a manageable set of equations from the design and engineering point of view.

$$\bar{\sigma}=A_0^{-1/\bar{m}}\ {\breve{\sigma}}_{t_0}\ {\bar{A}}^{1/\bar{m}}$$

$$\bar{A}=\int_{A}^{\ }{c\left(\vec{r}\right)^{\bar{m}}dA}$$

$${\breve{\sigma}}_{t_0}$$ is the t_{0} second equivalent representative stress. The equivalent area Ā (also known as the effective area) is the surface area of a glass element that fails with the same probability, when exposed to the uniform representative stress $${\breve{\sigma}}$$, as an element with surface area A when exposed to the non-uniform stress field σ(r). Ā can be defined for ambient and inert conditions alike, with $$\bar{m}$$ being nm_{0}/(n-2) and m_{0} respectively.

It should, however, be noted that this last equation is valid only if Ā and therefore the stress distribution function c(𝜏,r) are constant during lifetime. If this conditions are violated, more complex and costly formulations should be solved.

Discussion

On the last sections we describe a lifetime prediction model with a greater complexity than semi-empirical models but with crucial advantages over these as it does not contain simplifying hypotheses which would restrict its applicability to special cases. Furthermore, all formulation is stated through parameters that represent physical properties, having a clear signification and values that do not depend on the experiments used for their determination. The fact that both single surface flaw or random surface flaw population models can be used allows the assessment of hazard scenarios that involve surface damage or the adjustment of those model parameters with data from quality control measures or research. Finally, the material strength rightly converges on the inert strength for very short loading times or slow crack velocity.

Dynamic fracture mechanics

The glass lifetime prediction model previously explained gives us information regarding fracture strength during its lifetime, but it does not model glass behaviour after this fracture strength is reached. It is very useful to properly characterize post-breakage glass behaviour when we try to understand failure on glass elements.

When loading condition of glass reaches a state where K_{I} > K_{Ic}, an excess of energy exists on glass element that turns the fracture unstable, generating what is known as a dynamic fracture. Under this circumstances crack propagation velocity dramatically increases reaching values between 1500-2500 m/s for common glass. There are two ways of achieving a dynamic fracture:

1. The flaw reaches an instability point as a result of whether flaw depth or applied load increases K_{I} value until it reaches critical value K_{Ic}. In this situation, although static loads are acting, a dynamic fracture behaviour will be achieved.

2. Applied load has a sudden time increase (such as impact loads)

From this point, crack will grow, generating ramifications and spreading through the glass. The ramification process of cracks is a complex process for which there is no a clearly supported theory and it is far beyond this post scope.

Laboratory testing procedures

On the following sections basic information from main testing procedures regarding crack propagation velocity and glass strength assessment will be summarized.

Testing procedures for crack velocity parameters

Following methods are vastly used for crack propagation velocity assessment on glass elements.

- Direct measurement of the growth of large through-thickness cracks

Before measurement of indentation flaws became generalized (see below) this was the most usual way to assess the parameter defining crack propagation velocity. In this method the growth of a large through-thickness crack is directly measured as a function of the stress intensity factor. This can be done both visually or using sound waves. Main problem with this method is that study if this kind of cracks does not necessarily represent real behaviour of glass elements with surface flaws, that are the most relevant from structural design point of view.

- Direct or indirect measurement of the growth of indentation flaws

Due to the fact that indentation flaws are much similar to surface flaws present on glass elements, those are much more representative if we compare them with the ones generating real element failure. Advantage of using indentation flaws is that, comparing with natural flaws, its behaviour at breakage is well known and therefore obtaining reliable parameters from its characterization is much easier.

Testing procedures for strength data

- Static long-term tests

Also known as static fatigue tests, they are usually reproduced enforcing a bending state through 4 points (4PB), 2 supports and 2 load points. Main advantage is that enforced bending state is very similar to the one that glass element will be submitted to when permanent loads act during its lifetime. Main counterpart is that, if not properly designed, tested element could last years to fail or even not to fail.

- Dynamic fatigue tests

The dynamic fatigue term refers to tests with constant load rate, constant stress rate or cyclical load testing. 4 point tests (4PB), as the one stated on static tests, or coaxial double ring (CDR) are the most widely used.

On 4PB test the sample is subjected to a uniaxial stress field (σ_{1}≠0, σ_{2}=0). On the contrary, for CDR test biaxial stress field with equal intensities is used (σ_{1}=σ_{2}). Use of those methods is based on their simplicity and the fact that both are able to produce sample failiure even for element with few surface flaws.

Final conclusion

The assessment of glass strength, as previously explained, is not an easy task due to all the parameters which it depends from. Through this posts we provided information regarding fracture mechanism on glass elements and the parameters of lifetime prediction models. More simplified approaches are held on main standards and recommendations which are very useful for glass elements design.

Regarding parameters gathered on the summarized models, we have previously stated typical values for many of them, such as Y, n, ν_{0}, K_{th} y K_{Ic}. To pick up different or more adjusted values for these parameters, as for the ones characterizing surface flaws on glass (θ_{0} y m_{0}) it is available many documentation both theoretical and empirical. In case of pretending to adjust a lifetime prediction model it would be convenient to search bibliography that allows to adjust those parameters to the case studied on the most accurate way.

Next Friday we will release our last post on our structural glass series, regarding conections in glass elements.