abraded float glass

Journal of the European Ceramic Society 37 (2017) 4197–4206

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Journal of the European Ceramic Society journal homepage: www.elsevier.com/locate/jeurceramsoc

A micromechanical derivation of the macroscopic strength statistics for pristine or corroded/abraded float glass Gabriele Pisano a , Gianni Royer Carfagni a,b,∗ a b

Department of Engineering and Architecture, University of Parma, Parco Area delle Scienze 181/A, I 43100 Parma, Italy Construction Technologies Institute – National Research Council of Italy (ITC-CNR), Viale Lombardia 49, I 20098 San Giuliano Milanese, Italy

a r t i c l e

i n f o

Article history: Received 16 February 2017 Received in revised form 11 April 2017 Accepted 18 April 2017 Available online 11 May 2017 Keywords: Float glass Micromechanics Weibull statistics Abrasion Corrosion

a b s t r a c t The macroscopic strength of float glass is governed by the presence of micro-cracks, whose size, orientation and distribution affects the corresponding statistics. A micro-mechanically motivated model is here proposed, which spells out the connection between crack population and strength statistics, leading to generalized distributions of the Weibull type. Aging in the form of corrosion or abrasion can produce a variation of the defectiveness scenario originally present on the pristine glass surface, and we discuss how such a modification can statistically affect the macroscopic strength. A practical application is made to justify the change in strength experimentally observed passing from the “air” to the “tin” side of float glass. Assuming that the contact with the tin bath and the rollers produce a damage equivalent to the abrasion of the glass surface, we theoretically derive a bimodal Weibull statistics that agrees with the experimental evidence. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction The role of glass in architecture is constantly changing and its application have expanded from that of simple window panes to large load-bearing structures. The mechanical response of glass, linear elastic up to brittle failure, is governed by the presence of micro-defects on its surface, which can be modelled as thumbnail cracks opening in mode I. The macroscopic glass fracture is thus a consequence of the flaw microstructure, since it is associated with the unstable propagation of a crack and the statistical expectation of reaching critical conditions. Due to this inhomogeneity at the microscopic level, glass response is characterized by a considerable spread in the experimentally-measured fracture stress. Micro-cracks arise on the surfaces of soda-lime glass plates during the float production process. In this industrial method, patented by Alastair Pilkington, glass paste is poured on a bed of molten tin forming a floating panel. Very smooth surfaces are obtained while temperature is gradually reduced from 1100 ◦ C down to 600 ◦ C; then, the glass sheet is pulled off by rollers and passes through a lehr where it is gradually cooled. Consequently, the defectiveness scenarios on the side exposed to air (air-side) and on the face in

∗ Corresponding author at: Department of Engineering and Architecture, University of Parma, Parco Area delle Scienze 181/A, I 43100 Parma, Italy. E-mail addresses: [email protected] (G. Pisano), [email protected] (G. Royer Carfagni). http://dx.doi.org/10.1016/j.jeurceramsoc.2017.04.046 0955-2219/© 2017 Elsevier Ltd. All rights reserved.

contact with the tin bath (tin-side) are different. In general, the tinside is more damaged than the air side, especially because of the abrasion produced by the rollers [1]. Glass cutting represents another aspect of primary importance because, even though the technique has been improving during the years, non-negligible additional defects are usually introduced at the borders, which can be schematized as semi elliptical surface flaws and/or quarter elliptical corner cracks. The edge finishing (clean cut, seamed or polished) is certainly a discriminant issue for the evaluation of macroscopic strength of structural glass elements when maximum tensile stresses act at the borders. It should also be mentioned that cracks can grow over time even at stress levels much lower than the critical limit [2], due to a phenomenon usually referred to as static fatigue or subcritical crack growth, which makes the gross material strength dependent upon time and thermohygrometric conditions. A strict factory production control is provided for marketed glass plates, which rejects those panes with defects in transparency. Since the optical and aesthetic properties of glass are determined by the amount of existing flaws, another consequence of the factory production control is that it eliminates those elements that presents cracks whose size is above a certain limit. From a statistical point of view, this is equivalent to a lower truncation in the population of glass strength, i.e., there is a lower bound of glass strength that has been confirmed by experiments [1]. Anyway, the defectiveness scenario that is present on the glass surfaces after the acceptance phase (pristine glass) could

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be modified by corrosion and/or abrasion phenomena. The attack of alkali solutions leads to the dissolution of external surfaces, producing a leveling of the preexisting defects, whereas an increasing in the porosity could be due to acid corrosion or corrosion by water. On the other hand, abrasion consists in adding new cracks to the pre-existing ones. The typical scenario is that of sand blasting, a common situation for buildings in desert areas. Various experimental campaigns on abraded glass specimens are recorded in the technical literature [3–5], which provide the conclusion that the effect of ageing is that of producing an additional flaws distribution, whose maximum depth can be considered much less than that due to production process. Obviously the sharp grooves produced by a diamond bit, which are much deeper than the largest preexisting cracks so to drive a precise cut, should not be associated with a natural degradation of the material. Structural strength assumes strongly non-deterministic values because of the random nature of shape, size and distribution of surface flaws. Also the state of stress can influence the macroscopic glass strength. To this respect, the uniform equibiaxial state of stress represents the most severe condition for a glass specimen, since in such case there is the 100% probability that the crack axis is perpendicular to the maximum principal direction of stress. The opposite is true for what concerns uniaxial stress state, because the existing crack may not be oriented at right angle to the maximum tensile direction. The macroscopic strength is also size-dependent, because the larger the loaded area is, the higher is the probability of finding a defect of critical size [6]. Furthermore, the annealing treatment at the end of the float production process leads to an initial thermal healing of the cracks previously generated. As shown in [7], the effects of thermal healing are more pronounced on the edges of the glass panes than in the central parts of their surface. The Weibull statistics, based upon the weakest-link-in-thechain concept [8], is generally chosen for interpreting the variability of failure stress values with reference to structural glass specimens. In fact, it is sufficient to reach the critical condition at one crack to produce the catastrophic failure of the specimen. The 2-parameter Weibull distribution is certainly the statistical model most widely used for brittle materials. However, its inability to interpret the experimental data associated with small failure probabilities has been substantiated by many researchers [9–11]. In a previous work [1], the authors have justified the experimental finding by postulating the existence of a lower bound for glass strength due to the aforementioned production control phase, and have consequently proposed to use a left-truncated Weibull distribution to interpret the experimental results. Arguments were presented which supported this hypothesis and, moreover, it was shown that such minimal strength can be reduced, but not annihilated, by natural degradation due to ageing and in-service-related damage. Many works, in which the variability of material strength due to aging is discussed, are available in technical literature, but just a very few of these correlate the gross material response with the underlying micro-crack scenario. Indeed, a limited knowledge exists about crack density, size, shape and orientation, which instead represent relevant parameters to understand the macroscopic mechanical response of glass. Assuming that exhisting cracks do not interact one another, the connection between flaw and macroscopic-strength distributions was inferred by Freudenthal [12] and, more recently, by Batdorf and Crose [13] and Le et al. for silicon mems [14]. The work by Freudenthal [12], in particular, could be considered a milestone for what concerns the micro-mechanical motivation of the 2-parameter Weibull statistics. Here, we present a modified microstructurally-motivated model, able to describe the strong connection between the distribution of crack length, assumed to be power-law shaped, and generalized Weibull statistics. In particular, an upper-truncation of the crack lengths distribution gives rise to a left-truncated Weibull

distribution for macroscopic glass strength. Moreover, we discuss in detail how variations of the defectiveness scenario, due to corrosion and abrasion phenomena, can affect upon the macroscopic strength. Our approach necessarily does not cover all complex phenomena that accompany the change in the crack distribution, such as the crack healing consequent to thermal treatments, or the appearance of residual stresses due to the generation of new micro-cracks, or the crack blunting associated with corrosion. Although the proposed approach could be directly extended to cover such effects, the lack of appropriate experimental data suggests us to postpone further discussion on future works. As a practical application, manipulating the results of a large experimental campaign recorded in the technical literature [15], we use the proposed rationale to interpret the observed difference between the experimentally-measured population of strengths for the air- and tin-side of float glass. 2. Cumulative probability distribution for the size of micro-defects in float glass The macroscopic strength of glass is governed by the distribution of micro-defects on its surface. Such defects can be modelled as thumbnail micro-cracks, whose plane is orthogonal to the glass surface. Failure occurs when the stress intensity factor associated with the dominant micro-crack reaches the critical value. The first assumption in the present theory is that there is a Representative Area Element (RAE) on the glass surface, say A, whose diameter is comparable with the average size of the micro-cracks. The main property of the RAE is that it can host one crack, so that the number of cracks that are contained in a specimen of area A is A/A. Inspecting the glass surface with a microscope, it is possible, at least in principle, to divide the area in sub-areas A, and measure in each of them the size ı of the micro-crack there located. One can thus calculate the corresponding statistics, i.e., the probability of finding, in a specific area A, a microcrack of size ı. We expect that as the number of elements in the population tends to become very large, the statistics tends to a definite probability function. It is reasonable to expect a highly right-skewed distribution of the crack size, meaning that while the bulk of the distribution occurs for fairly small size, there is a small number of critical cracks, of size much higher than the average value, which leads to a very long right-hand-side tail. One of the functions able to interpret this kind of variability is certainly the power law distribution, whose most relevant analytical attribute is that of scale invariance. We assume that, right after the production process, in a material that could be considered “pristine”, the probability density function for the crack size ı in an area A can be written in the form pA,ı (ı) = C ı−˛ ,

(2.1)

where ˛ is the scaling parameter and C is a normalization constant. Observe that the function (2.1) diverges as ı → 0, so that a lower bound ımin shall be imposed. The parameter ımin might be physically interpreted as the size of the physiological defects in glass, i.e., the size of cracks that are naturally present in any glass produced with an industrial process. This definition, somehow artificial at this point since it is introduced to make consistent the power-law assumption (2.1), will be discussed in detail later on. One should observe, however, that strength of glass is governed by large cracks, i.e., ı  ımin , so that what is really important is the shape of the probability function on the right-hand-side tail of the distribution. Consequently, ımin should rather be considered as a material parameter, not necessarily associated directly with the minimum flaw size, whose importance consists in the fact that, in the expression (2.1), it analytically characterizes the statistics of large cracks in the asymptotic limit ı → ∞.

G. Pisano, G. Royer Carfagni / Journal of the European Ceramic Society 37 (2017) 4197–4206

In any case, the physical interpretation of ımin as the size of the physiological cracks, allows to consistently derive a micromecanically-motivated theory. By considering that the probability of finding a crack of size comprised between ı and ı + dı reads pA,ı (ı) dı = C ı−˛ dı, the constant C is given by the normalization requirement that





∞

∞

pA,ı (ı) dı = ımin

C ı−˛ dı = 1 ⇒ C =

ımin

˛−1 ı1−˛ min

.

(2.2)

The integral in (2.2) converges for values of the scaling parameter ˛ strictly higher than unity, a condition that can be naturally assumed for the case of glass. Thus, the probability density function becomes



˛−1 pA,ı (ı) = ımin

−˛

ı

.

ımin

(2.3)

The probability of finding a crack of size equal or higher than ı in the representative area A reads



∞

PA,ı (ı) = ı



˛−1 ımin

−˛

ı



dı =

ımin

1−˛

ı ımin

.

(2.4)

Obviously, Eq. (2.4) is considered to be valid for ı ≥ ımin , because when ı < ımin one has 100 % probability of finding a crack of size higher than ı. Marketed glass plates must guarantee certain optical and aesthetic performance, so that strict factory production controls usually assure that glass with large defects are discarded and not placed on the market. Therefore, as discussed at length in [1], the acceptance phase leads to the refusal of glass with flaws above a certain limit ımax . From a statistical point of view, this leads to an upper-truncation of the population of crack size [1]. Consider than that in a very large sample composed of N glass plates of unitary reference area, the number P0 represents the probability that ı ≥ ımax . After discarding the defective pieces, the number of elements after truncation will be N(1 − P0 ). The relation between the truncated disT (ı) for these remaining elements and PA,ı (ı) tribution curve PA,ı will be [8]





PA,ı (ı) − P0 N

T PA,ı (ı) =



(1 − P0 ) N

where P0 = ımax /ımin obtains T (ı) = PA,ı

1−˛ ı1−˛ max − ı 1−˛ ı1−˛ max − ımin

1−˛

(2.5)

,

and PA,ı (ımin ) = 1. Consequently, one

.

(2.6)

Observe, in passing that the same conclusion could be reached through a new normalization requirement, i.e., by imposing



ımin



 

ımax

pTA,ı ı dı =



⇒C=

ımax

.

˛−1

˛ ı−˛ ımin − ı1−˛ max ımin min

(2.7)





˛−1 ımin − ı1−˛ max

ı˛ min

,

ımin

(2.8)

and the probability of finding a crack of size equal or higher than ı in A takes the form



T PA,ı (ı) =

ı

ımax



˛−1 ımin − ı1−˛ max

ı˛ min

which clearly coincides with (2.6).

We now pass to consider the consequence of the aforementioned assumptions in terms of critical stress, by assuming that the reference unitary surface area, say A0 , consists of N0 elements of area A. Let now PA ( cr ) denote the probability of finding, in A, a crack of a size such that its critical stress is equal or less than  cr . The probability of finding a crack having critical stress between  cr and  cr + d cr reads dPA = dd PA (cr ) dcr . Moreover, for any generic cr state of stress , supposed to be homogeneous in A0 , let (,  cr ) represent the angle containing the normals to all the possible crack planes for which the applied stress component, normal to the crack plane, is higher than the critical value  cr , with 0 ≤ (,  cr ) ≤ . Hence, the probability of failure for a tensile surface area A reads (, cr ) dPA (cr ) dcr .  dcr

dPf (, A) =

ı ımin

−˛

dı =

1−˛ ı1−˛ max − ı

ı1−˛ max

(3.1)

Consequently, the failure probability is given by



∞

Pf (, A) = 0

(, cr ) dPA (cr ) dcr .  dcr

(3.2)

Recalling that the corresponding probability of survivals Ps (, A) = 1 − Pf (, A), for the unitary area A0 , composed by N0 = A0 /A elements of area A, the survival probability is equal to the product of the survival probabilities of each element, that is





∞

Ps (, A0 ) = 1 − 0

(, cr ) dPA (cr ) dcr  dcr

A0 /A

.

(3.3)

This formula can be specialized and simplified for paradigmatic states of stress. 3.1. Equibiaxial stress state The equibiaxial state of stress generally represents the reference static state for characterizing the bending strength of glass plates. This is due to the fact that this ideal condition makes the material strength independent upon the orientation of micro-cracks, since the maximum tensile stress is always at right angle with crack plane. In this situation, when  eqb denotes the applied equibiaxial stress, one has ( eqb ,  cr ) =  when  eqb =  cr and ( eqb ,  cr ) = 0 when  eqb <  cr . Hence, Eq. (3.3) becomes



Ps (eqb , A0 ) = 1 −

eqb

dPA (cr ) dcr dcr

A0 /A

.

(3.4)

Observe that the upper limit of integration interval is  eqb , since the probability of failure is null when  cr >  eqb .

−˛

ı

3. From the statistics of micro-defects to macroscopic strength of glass

0

In this way the probability density function becomes pTA,ı (ı) =

In the following we will show that the distribution (2.4) will give rise to a 2-parameter Weibull distribution for the macroscopic glass strength, whereas (2.6) will provide a truncated Weibull distribution. The importance of both statistics in the interpretation of the macroscopic properties of glass has been discussed in [1,11] and they will both considered in the sequel.



Cı−˛ dı = 1

ımin

4199

− ı1−˛ min

,

(2.9)

3.1.1. 2-Parameter Weibull distribution Assume that the probability of finding a micro-crack equal or higher than ı is given by (2.4). This expression, emphasizing the importance of A, can be re-written in the more convenient form



 

PA,ı ı = A 0 =

ımin (A)

1/˛∗

=



ı 0

−˛∗

,

ımin A0 /N0

1/˛∗ ,

(3.5)

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where ˛* = ˛ − 1. From the expression of the Stress Intensity Factor (SIF) governing the crack growth in mode I, one has KI = ⊥ Y

K 2 1 I



ı ⇒ ı =



⊥ Y



 1 KIc 2 

cr Y

,

Ps (eqb , A0 ) =

 1 − A

eqb

0



= A cr

⇒ Ps (eqb , A0 ) =



d dcr



√

∗ Y 0 2˛ , KIc

(3.7)

√

∗ A0 /A Y 0 2˛ cr KIc



eqb

1 − A

2˛∗ A0 /A

lim (1 + h a)

h→0



= exp(a),

one obtains



lim Ps (eqb , A0 ) = lim 1 − hA0



= exp −A0

eqb

2˛∗ 1/h

.



Ps (eqb , A0 ) =

∗

1 − A

∗

2˛ eqb − 02˛

A0 /A

.

∗

2˛ 0,lt

(3.15)

By posing again A0 /A = 1/h, for large values of A0 one has h → 0 and



∗

∗

2˛ eqb − 02˛



.

∗

2˛ 0,lt

h→0

(3.16)

Finally, reasoning as in Section 3.1.2, the survival probability for a generic surface area A takes the form



∗

∗

2˛ eqb − 02˛ ∗

2˛ 0,lt



.

(3.17)

This represents a left-truncated Weibull distribution, whose importance in the statistical evaluation of the macroscopic glass strength has been evidenced in [1]. 3.2. Uniaxial stress state

eqb

.

0,2p

(3.10)

A/A0

⇒ Pf (eqb , A) = 1 − exp −A

eqb

cr /unx ,

where is the angle between the maximum principal stress axis and the normal to the crack plane. It is clear from Fig. 1 that the angle ( unx ,  cr ) containing the normals to all the orientations for which  ⊥ ≥  cr is two times the angle cr , so that, from (3.18), one has



(unx , cr ) = 2 cr = 2 arccos





(3.18)

2˛∗ 

Ps (eqb , A) = Ps (eqb , A0 )



dcr

⊥ = unx cos2 ⇒ cr = unx cos2 cr ⇒ cr = arccos

Equivalently, for a generic surface area A, the survival probability reads



A0 /A

The state of stress due to three- and four-point bending tests is generally schematized as uniaxial ( =  unx ). In this case, the component of the stress orthogonal to the crack plane reads

0,2P

h→0



∗

2˛ 0,lt



(3.9)



h→0

∗

∗

2˛ −  2˛ cr 0

Observe that the critical stress  0 , correlated with the maximum value of the crack size ımax , represents the lower limit of integration interval. Eq. (3.14) can be rearranged as

Pf (eqb , A) = 1 − exp −A

,

0,2p

(3.8)

1/h

 d dcr

(3.14)

lim Ps (eqb , A0 ) = exp −A0

√ having defined 0,2p = KIc /(Y 0 ). Indicating with 1/h = A0 /A, for very small values of A one has h → 0. Since



1 − A

eqb

(3.6)

and the survival probability (3.4) becomes



Ps (eqb , A0 ) =



0

where Y is the geometric factor depending upon crack shape (Y = 2.24/ for half-penny-shaped thumbnail cracks) and  ⊥ is the component of stress perpendicular to the crack plane. When the critical conditions are reached  ⊥ =  cr and KI = KIc , so that the probability of finding a crack having critical stress equal or lower than  cr reads PA (cr ) = PA,ı



2˛∗ 

(3.11) arccos

,

0,2p

=

which clearly represents the well-known 2-parameter Weibull distribution.





cr unx



⇒

(unx , cr ) 

cr /unx

/2

.

(3.19)

Note that when  →  cr , one has that cr → 0 and ( unx ,  cr )/ → 0, whereas when  unx   cr , then cr → /2 and ( unx ,  cr )/ → 1. Moreover, when  unx <  cr the probability of failure is null.

3.1.2. Left-truncated Weibull distribution Assuming the crack size distribution of Eq. (2.6), from the expression (3.6) of the stress intensity factor one has ı=

1 

K 2 Ic Ycr

,

ımax =

1 

K 2 Ic

Y0

,

ımin =

1 

K 2 Ic

Yk

,

(3.12)

where  0 and  k are the critical stresses correlated to ımax and ımin , respectively. Thus, the probability of finding a crack having critical stress equal or lower than  cr reads PA (cr ) =

∗

∗

2˛∗

2˛∗

2˛ −  2˛ cr 0

k



− 0



∗

∗

= A ∗

2˛ −  2˛ cr 0

(0,lt ) ∗

2˛∗

,

(3.13)

1/2˛∗

. Hence, the survival probawhere 0,lt = A k2˛ − 02˛ bility for an equibiaxially-stressed reference area A0 takes the form

Fig. 1. Angle within which the crack must lie for fracture occurrence.

G. Pisano, G. Royer Carfagni / Journal of the European Ceramic Society 37 (2017) 4197–4206

By statistically interpreting the population of micro-cracks through a power-law function, PA ( cr ) is given by Eq. (3.7) and the survival probability (3.3) becomes

⎡ Ps (unx , A0 ) =

⎣1 −



arccos

unx

A

cr

 cr /unx

/2

0

d dcr



A0 /A √

∗ Y 0 2˛ . dcr KIc

(3.20)

After some analytical manipulation, Eq. (3.20) takes the form

 Ps (unx , A0 ) =

 1 − A

unx 0,2p

2˛∗



 A0 /A

12 + 2˛∗ 1   ∗ 2 ˛ 1 [2˛∗ ] 2

K √ Ic Y 0

is the scale

factor of the distribution. Setting again A0 /Aı = 1/h, for A0  A so that h → 0, one obtains from (3.9)





Ps (unx , A0 ) = exp −A0

unx 0,2p

2˛∗





1 ∗ 1 2 + 2˛   2 ˛∗ 1 [2˛∗ ] 2

.

(3.22)

In conclusion, the probability of failure for a float glass plate subjected to an uniaxial stress state on a generic area A reads





Pf (unx , A) = 1 − exp −Kunx A

unx 0,2p

2˛∗  .

(3.23)

where



Kunx



1 ∗ 1 2 + 2˛   = , 2 ˛∗ 1 [2˛∗ ] 2

4. Possible modifications of micro-defects during service life and consequent variation of mechanical properties The defectiveness scenario on the surface of glass continuously changes during the service life due to various phenomena, such as ageing, abrasion or subcritical crack growth. There are a few contributions in the technical literature that record experimental campaigns where specimens have been tested after aging, etching or conditioning, in order to artificially reproduce the natural degradation. To our knowledge, these data have never been analyzed with a micro-mechanically motivated approach. Our aim, here, is to correlate the microscopic change in the crack population with macroscopic variations of mechanical properties. 4.1. Glass corrosion/etching

, (3.21)

where is the Euler’ Gamma Function and 0,2p =

4201

(3.24)

is the coefficient that defines the effective area Aeff,unx = Kunx A, i.e, the area that is statistically equivalent, in term of strength, to an uniform equibiaxial state of stress. The values of the coefficient Kunx , which is much lower than the unity, is shown in Fig. 2 as a fucntion of ˛* in the representative range for float glass. The upper-truncation of the power law distribution can be handled in the same manner as described in Section 3.1.2. However, the resulting expression is quite complicated and it is not recorded here for the sake of briefness.

During its service life, float glass is subjected to corrosion phenomena caused by reactions between surface and gases in atmosphere. Corrosion is often noticed as visible degradation of the glass surface, accompanied by whitening, iridescence, loss of transparency and weight loss, due to uniform glass dissolution. Indeed, artificial corrosion through etching, by using hydrofluoric acid, is also a well-known efficient way for strengthening glass [16,17]. More precisely, as pointed out by Kolli et al. [18], the strengthening of corroded glass panes is due to the simultaneous decreasing of micro-cracks length and the blunting of the crack tip. The proposed model could directly consider the effect of crack blunting by introducing a proper modification of the stress intensity factor, but the quantification of this effect is difficult to obtain. Therefore, at least at a first order approximation, we shall combine these two effects by assuming that surface dissolution due to glass corrosion leads to an equivalent constant length reduction for all micro-cracks constituting the defectiveness scenario. We will indicate with the suffix “1” and “2” the glass states before and after the etching treatment, respectively. We then consider the effects of etching by assuming a constant reduction of the crack lengths, i.e., a generic crack of size ı1 becomes of size ı2 = ı1 − ε, where ε represents the etch depth. For what concerns the size of the physiological cracks, one has consequently ımin,2 = ımin,1 − ε, supposing ε < ımin,1 . Clearly, under the aforementioned assumptions, the probability of finding a crack of size ı in state “2” is equal to the probability of finding a crack of size ı + ε in state “1”. Consequently, by taking a non-truncated power-law function (2.4) for representing crack size population of the state “1” and setting, as in (3.5), ˛* = ˛ − 1, the new probability function for the state “2” reads

 (2) PA,ı (ı)

=

ı+ε ımin,1

−˛∗

 = A

ı+ε 0

−˛∗ ,

(4.1)

∗

where 0 = ımin A−1/˛ . By recalling the equation for the critical SIF, one can write ı=

1 

K 2 Ic Ycr

,

ε=

K 2

1 

Ic

Yε

,

(4.2)

where  ε is the critical stress correlated to the etch depth ε. After some calculations, function (4.1) becomes

 (2) PA (cr )

Fig. 2. Values of Kunx as a function of ˛* .

= A

2˛∗ 0,2p



cr 2 / 2 1 + cr ε

,

(4.3)

where 0,2p is the same as (3.8). For the sake of simplicity, we here refer to an equibiaxial stress state. From Eq. (3.4), the survival probability for an etched float glass pane subjected to an uniform

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∗

Fig. 3. (a) Effect of 10 ␮m-deep surface dissolution upon the 2-parameter Weibull distribution (ε = 10 ␮m; ˛* =2.5; 0,2p = 1000 MPa mm1/˛ ); (b) effect of 10 ␮m surface ∗ dissolution upon the left-truncated Weibull distribution (ε = 10 ␮m; ˛* = 2.5; 0,2p = 1000 MPa mm1/˛ ;  0 = 40 MPa).

equibiaxial stress state in the reference area A0 takes the form

ımax . With the same arguments as before, one obtains

(2)

Ps (eqb , A0 )

⎡

= ⎣1 − A



eqb

0

 d dcr

⎤A0 /A

2˛∗ 0,2p



cr

dcr ⎦

2 / 2 1 + cr ε

T (2) Pf (eqb , A)=1 − exp

(4.4) .

By taking again A0 /A = 1/h, for small value of A one has h → 0 and, from (3.9),

⎡

⎛

⎢

Ps (eqb , A0 ) = exp ⎣−A0 ⎝ (2)

eqb

 0,2p

2 / 2 1 + eqb ε

⎞2˛∗ ⎤ ⎠ ⎥ ⎦,

(4.5)

so that the probability of failure for a generic uniformly loaded area A reads

⎡

⎛

⎢

Pf (eqb , A) = 1 − exp ⎣−A⎝ (2)

 0,2p

eqb 2 / 2 1 + eqb ε

⎞2˛∗ ⎤ ⎠ ⎥ ⎦.

(4.6)

⎧ ⎡   2˛∗⎤⎫ ∗ 2˛∗ 2 ⎬ ⎨ eqb − 02˛ 1 eqb ⎦ . −A⎣ 1 − ∗ 2 ε2 2˛ ⎭ ⎩ 0,lt (4.9)

Fig. 3 shows how the statistical distribution of strength varies by passing from the state “1” to “2”. Here, the reference values for the power law function parameters are ˛* =2.5 and ∗ 0,2p = 1000 MPa mm1/˛ , whereas for the upper-truncated power ∗ law distribution we have assumed ˛* =2.5, 0,2p = 1000 MPa mm1/˛ and  0 = 40 MPa. The assumed value for the etch depth is ε = 10 ␮m, from which  ε = 187.62 MPa. For the sake of simplicity, we have considered an uniformly equibiaxial state of stress acting on a unitary area A0 = 1 m2 . As it is evident from Fig. 3, the effects of etching upon strength distribution are much more evidents for the highest fractiles, associated with small crack lengths. This is not difficult to explain, since a 10 ␮m cracks leveling strongly affects the smallest cracks, whereas it is almost negligible for largest flaws. 4.2. Abrasion

Note that such an equation is valid for ε ı, hence for  ε   eqb , 2 / 2 1. Consequently, it is possible to expand in Taythat is eqb ε



lor series 1/

2 / 2 1 + eqb ε

= 1 − 1/2



2 / 2 eqb ε

+o



2 / 2 eqb ε

, so

that the probability of failure for the state “2” becomes

(2)

Pf (eqb , A) = 1 − exp

⎧ ⎨



−A

⎩

 eqb 0,2p

 1 1− 2

2 eqb

2˛∗ ⎫ ⎬

ε2

⎭

, (4.7)

which represents a new generalized Weibull distribution. On the other hand, if one assumes that the population of crack length in the pristine state “1” is interpreted by an upper-truncated power law function of the type (2.6), in the state “2” the probability that a crack is correlated with a critical stress equal or lower than  cr reads

⎡

T (2)

PA (cr ) =

A ∗

2˛ 0,lt

⎣ 

⎤

2˛∗ cr 2 / 2 1 + cr ε

2˛∗

− 0

⎦,

(4.8)

where 0,lt is the same used in Eq. (3.14) and  0 represents the critical stress associated with the maximum allowable crack length

The defectiveness scenario of the pristine glass can also be modified by abrasion. Differently from corrosion/etching, which consists in a uniform “dissolution” of the glass surface, abrasion consists in adding new cracks to the pre-existing ones. The new flaws are clearly strongly affected by the cause that produce the abrasion itself. The technical literature records experimental campaigns on specimens that have been artificially scratched in order to reproduce the effect of natural abrasion. Durchholz et al. [3] statistically analyzed the results from float glass plates treated by dropping corundum (Al2 O3 ) on them, tested according to the set-up of EN1288-2 [19]. Interpolating the data with a 2-parameter Weibull distribution, they showed that damaging leads to a lower scale parameter and to a much greater scale exponent, i.e., the highest strengths of the population are lowered, but the dispersion is reduced. Moreover, minimum strength values remained of the same order as those obtained by analyzing the pristine float glass. Other scientists [4] analyzed the influence of sandblasting, showing that the strength diminishes in general. More precisely, an increase of the sandblasting time reduces the strength at high probability of failure, but the opposite is true at low probabilities. Such a finding was confirmed by Wang et al. [5]. In both the aforementioned studies, it was observed that the maximum crack length due to

G. Pisano, G. Royer Carfagni / Journal of the European Ceramic Society 37 (2017) 4197–4206

exposition to sandblasting was approximately 35 ␮m. Considering that the length of the largest cracks in the pristine material, just after the industrial production, are of the order of 150–250 ␮m, it can be argued that flaws due to the extreme condition of abrasion are much smaller. It should also be mentioned that when glass and ceramic degrade as consequence of abrasion, a residual stress state may appear [20,21] that may affect the crack propagation. However, Datsiou and Overend [22] have recently experimentally verified that the surface compression for naturally abraded (aged) glass is very close to that measured on the surface of as-received annealed float glass, i.e., of the order 2.31 ± 0.65 MPa. This subtle effect could be directly considered in the proposed approach by introducing a “negative” Stress Intensity Factor (SIF), that plays the role of a threshold for fracture activation, and sums up with the SIF induced by the applied loads. However, since a quantification is difficult, while the experimental measurements indicate that the induced state of residual stress is mild, at least as a first order approximation we will neglect this effect here, referring its consideration in future work. Therefore, we shall assume that two different surface cracks populations, i.e., the pre-existing one and the one induced by abrasion, are super-imposed and that, at least as a first order approximation, they are both interpreted by power-law distributions of the same type of (2.3), that is pre

pA,ı (ı) =

˛1 − 1 ımin,1



−˛1

ı ımin,1

˛2 − 1 ımin,2

pabr (ı) = A,ı

,



−˛2

ı

 =

−˛∗



1

ı

abr PA,ı (ı)

,

ımin,1

=

ımin,2 (4.10)

−˛∗

≤ PA,ı (ı) =

 1−



ı

−˛∗ 1



.

ımin,2

 1−

ımin,1



ı

(4.11)

−˛∗ 2



,

ımin,2

(4.12)

which implies that the probability of finding a crack of size higher than ı is



PA,ı (ı) =

≤ 1 − PA,ı (ı)

ı

−

=

−(˛∗ +˛∗ ) 1 2

−˛∗

−˛∗

−˛∗

1

ı ımin,1

0,1m =

0,3m =

ımin,1 (Am )

 +

ı 0,1m

1

1

2

(ımin,2 )

,

1/˛∗

2

˛∗ /(˛∗ +˛∗ ) 2

1

2

1/(˛∗ +˛∗ ) 1

.

(4.15)

2

For a homogeneous equibiaxial state of stress  eqb , the same argument of Section 3.1.1 can be repeated. Substituting (4.14) into Eq. (3.4) and performing the limit as per (3.9), the counterpart of the cumulative probability of failure (3.11) becomes

Pf (eqb , A)



= 1 − exp



2˛∗1

eqb

−A

+

0,2p,1m

2˛∗2

eqb

−

0,2p,2m

eqb

2(˛∗1 +˛∗2 )  ,

0,2p,3m

(4.16)

where 0,2p,1m =

KIc , √ Y 0,1m

0,2p,3m =

KIc . √ Y 0,3m

0,2p,2m =

KIc , √ Y 0,2m (4.17)

It is evident that the resulting statistical function is much more complex than the classical 2-parameter Weibull distribution. A more refined analysis can be obtained as in Section 3.1.2, by interpreting the variability of the crack length population in the pristine glass with an upper-truncated power law distribution. In this conditions, the two probability functions read T,pre pA,ı (ı)

=

(ı) = pabr A,ı

1−˛

˛2 − 1 ımin,2



−˛1

ı

,

ımin,1

−˛2

ı

.

ımin,2

(4.18)

For the case at hand, recalling the expression (3.12) for the SIF and the definition of  0 and  k of (3.12), the cumulative function describing the probability of finding a crack correlated to critical stress equal or lower than  cr is given by

ımin,2

 (4.13)

˛

1 ımin,1 − ımax1 ımin,1

PA (cr ) = Am

.



˛1 − 1



2

PA (cr ) = Am

2˛∗2

cr

when cr < 0 ,

0,2p,2m 2˛∗

cr

1



2˛∗

− 0

2˛∗

1

+

2˛∗2

cr 0,2p,2m

1 0,lt,1m

2˛∗

−

0

1

2˛∗

cr

2

2(˛∗ +˛∗ )

− cr

1



(4.19)

2

2(˛∗ +˛∗ )

1 2 0,lt,3m

when cr ≥ 0 ,

In general, the number of pre-existing cracks N0,1 is different from the number of cracks N0,2 induced through abrasion or, equivalently, the representative surface elements A1 and A2 for the pre-existing and abrasion-induced cracks, respectively, are different. This renders a derivation similar to that obtained through Eq. (3.4) not straightforward. As an approximation, we may consider an average value of the representative surface element Am so that, similarly to (3.5), one can write

PA,ı (ı)=Am

˛∗ /(˛∗ +˛∗ )

(Am )

1 2 ımin,1 ımin,2



(Am )

1

(ımin,1 )

−˛∗ 

ı

ımin,2

0,2m =

,

1/˛∗

2

ı

Superimposing the two effects, the cumulative distribution for the probability of finding a crack lower than ı is given by



where

.

Similarly to (2.4), setting ˛∗1 = ˛1 − 1 and ˛∗2 = ˛2 − 1, the probabilities of finding a crack of size equal or higher than ı in the representative area element A reads pre PA,ı (ı)

4203

−˛∗  1

+

ı 0,2m

−˛∗  2

−

ı 0,3m

−(˛∗ +˛∗ )  1

where





2˛∗

2˛∗

1/2˛∗

2˛∗

2˛∗

0,lt,1m = Am k,1 − 0 1





0,lt,3m = Am 0

1

1

1

− k,1

1

2˛∗

2

k,2



,

1 2˛∗1

0,2p,2m = k,2 Am

1 2(˛∗1 + ˛∗2 )

,

. (4.20)

2

, (4.14)

Using the same arguments as above, one obtains the cumulative probability of failure in the form

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G. Pisano, G. Royer Carfagni / Journal of the European Ceramic Society 37 (2017) 4197–4206



−A



2˛∗

eqb1 − 0

1

2˛∗

2˛∗

2 0,lt,2m

 

2˛∗

eqb2 − 0

1

−

2(˛∗ +˛∗ )

+ eqb 1

2

(0,lt,3m )

2

 ⎤⎫ 2 ⎬ ⎦ ⎭

2(˛∗ +˛∗ )

− 0

1

2(˛∗ +˛∗ ) 1

−

2

=

1−˛

˛

1 ımin,1 − ımax1 ımin,1

˛2 − 1

(ı) = pT,abr A,ı

1−˛

,

ımin,1



˛

−˛2

ı

.

ımin,2

2 ımin,2 − ımax2 ımin,2

(4.22)

Observe that taking the same upper bound ımax of crack length for both the distributions is equivalent to assume that the truncation is due to the same production control. Similar calculations to those developed above lead to the probability of finding a crack with length equal or larger than ı in the form



PA,ı (ı) =

−˛∗

−˛∗

ımax1 − ı −˛∗

1

−˛∗

1 ımax1 − ımin,1

−˛∗

−

−˛∗

−˛∗ 1

−˛∗

−˛∗

ımax2 − ı

1

−˛∗ 1

2

2 ımax2 − ımin,2

−˛∗

−˛∗

ımax1 − ı

−˛∗

−˛∗ 2

2

−˛∗

2 ımax − ımin,1 ımax − ımin,2

.

(4.23)

If the shape of the cracks in the two populations is the same, so that Y1 = Y2 = Y, one obtains

(

Pf (eqb , A) =

1 − exp

−







−A

2(˛∗ +˛∗ ) 1 2

2˛∗

1

2˛∗

− 0

(0,lt,1m )

2˛∗

2(˛∗ +˛∗ )

− 0

1

1

2

1

+



2˛∗

2

2˛∗

− 0

(0,lt,2m ) 2˛∗

− 0

1





Am [(Y1 k,1 )

− (Y1 0,1 ) 2˛∗

1

2˛∗



1

− (Y1 0,1 )

2˛∗

1]

1

2˛∗ (Y1 0,1 ) 1 ]

2˛∗

2

2˛∗ 2

)

− 0,2

(0,lt,2m )

+



2˛∗

2

2˛∗

− 0,22

(0,lt,2m )

2˛∗

2

,

2˛∗

2

2

2˛∗

2

2˛∗

2˛∗ 2

(0,tt,3m )

1

−

2˛∗

the air side, and pfloat = pT,abr (ı) for the defects induced during the A,ı float production process. If one assumes that the crack shape is the same for both populations, i.e., Y1 = Y2 = Y, the probability of failure for the tin-side strength is given by (4.24). In order to verify the theoretical prediction, we refer again to the results of the experimental campaign by TC129/WG8 of CEN [15]. Glass was tested according to EN 1288-2 [19] (coaxial double ring test with overpressure), recording 741 failure stress values, 371 for the air side and 370 for the tin side. Re-arranging the data to fix some erroneous indications contained in the standard [19] as shown in [23], the air-side data could be very well interpolated with a left-truncated Weibull distribution. To take into account that the state of stress in the loaded specimen is neither homogeneous nor uniform, one has to calculate at each point of the tensile area the principal component of stress  1



−˛∗

ımax2 − ı

+

2˛∗ Am [(Y1 k,1 ) 1

When glass is produced with the float process, the side that has been in contact with the molten tin bath (tin-side) and, successively, with the steel rollers, presents a different statistics in terms of strength from the side that has been exposed to air (air-side). As discussed at length in [1] while elaborating the data from the large experimental campaign of the technical group TC129/WG8 of CEN [15], a left-truncated Weibull distribution of the type considered in Section 3.1.2 can very well reproduce the statistics of strength when specimens are tested with the air-side under tensile stress. This result has been attributed to strict production controls, rejecting glass that does not meet optical requirements, i.e., with visible cracks. For the tin-side, however, the authors have not been able, even re-arranging the data, to find any type of generalized Weibull function that acceptably interprets the experimental findings [11]. Here, we propose to interpret the statistics for the air-side as that for a pristine material, and consider the defects on the tin-side induced during the production phase as a consequence of abrasion, as discussed in Section 4.2. T,pre With the notation of Eq. (4.22), consider then pair = pA,ı (ı) for

−˛1

ı

2˛∗

− (Y1 0,1 )

1

4.3. A practical application to float glass: from air- to tin-side strength

Observe that, here, it has been supposed that the pre-existing cracks and the cracks due to abrasion have the same shape, i.e., the same geometric coefficient Y in the corresponding expression (3.6) of the SIF, but this is not an issue because assuming different shapes, associated respectively with Y1 and Y2 , would lead to slightly more complicated expressions. What should be noticed, in any case, is that the resulting cumulative statistical distribution is much more complex than the simple, left-truncated, Weibull distribution. There are cases in which the abrasive process precedes the material selection through a factory production control, as in the application of the next Section 4.3. This means that both statistics should be interpreted by upper-truncated power law functions of the type



(Yeff )

2˛∗

where 0,lt,2m is again given by (4.20). The resulting statistical distributions becomes rather complex.

(4.21)

˛1 − 1

(Yef )

−A

(4.26)

when  ≥ 0 .

T,pre pA,ı (ı)

= 1 − exp

eqb2

+

1 0,lt,1m



Pf (eqb , A)



2˛∗

2˛∗

2˛∗

0

2˛∗

(

when  ≤ 0 ,

0,2p,2m

( Pf (eqb , A) = 1 − exp

2˛∗2



Pf (eqb , A) = 1 − exp −A

− 0

2

2˛∗

− 0

2



2˛∗

1

2˛∗

− 0

2(˛∗ +˛∗ ) 1

2

1

⎤⎫ ⎬ ⎦ , ⎭

(4.24)

where 0,lt,1m and 0,lt,2m are defined as in (4.20), while 0,tt,3m =

0,lt,1m 0,lt,2m Am





2˛∗

= Am 0

1

and  2 ≤  1 and define the equivalent stress  eq [11] as



2˛∗

− k,11

2˛∗

0

2

 2˛∗

− k,22

1 2(˛∗ +˛∗ ) 1 2

.

(4.25)

On the other hand, if the two crack populations are correlated with different shape factors for which we suppose Y1 < Y2 = Yef , than one obtains on the safe side

eq =

 2/



/2

(1 cos

2

2

+ 2 sin

)

2˛∗

1

d

,

(4.27)

0

which represents a spatial average of the state of stress, equivalent in statistical terms to the opening stress under the hypothesis that the orientation of the crack planes is due to chance alone, i.e., there

G. Pisano, G. Royer Carfagni / Journal of the European Ceramic Society 37 (2017) 4197–4206

are no preferred directions. Then, introducing the effective area K A as

* KA=

2˛∗  1 dA A eq , 2˛∗ 1 max

(4.28)

 * Pfair (f , A) = 1 − exp −

2˛∗



2˛∗

( 1 − 0,11 ) dA A eq

⎡ = 1 − exp ⎣−A

2˛∗

1 0,lt,1

2˛∗

1

K f

2˛∗

− 0,11

2˛∗

⎤ ⎦,

(4.29)

1 0,lt,1

where  f is the maximum stress in the specimen when failure occurs, and  0,1 is the lower bound for glass strength. The Weibull parameters are determined with a graphical-based regression of the experimental data. The measured failure stress values  i are ranked in ascending order and a failure probability Pi is assigned to each of them according to the probability estimator Pi = i/(N + 1), where N is the total number of data. Then, from (4.29), one has ln A



2˛∗

0,11 2˛∗

+ ln

1 0,lt,1

2˛∗ 1

Weibull distribution [1] of the type

Pf (f , A) = 1 − exp

⎧ ⎨

−Aeff

⎩

⎡ ⎣

2˛∗

f

1

2˛∗

− 0,11

0,lt,1m

2˛∗

+

f

2

2˛∗

− 0,22

0,lt,2m

⎤⎫ ⎬ ⎦ . ⎭ (4.31)

where  max is the maximum tensile stress in the specimen, the cumulative distribution of strengths for the air-side is given by [1]



4205



1 1 − Pfair 2˛∗ 1



=

2˛∗1

ln(f ) + ln



2˛∗ 1





KA 2˛∗

.

1 0,lt,1

(4.30)

Setting G = A 0,1 /0,lt,1 and B = ln K A/0,lt,1 , this is the equation ofthe line y = mx + B inthe Left-Truncated-Weibull (LTW) plane y = ln G + ln 1/(1 − Pfair ) and x = ln f .

For the experimental configuration of EN 1288-2 [19], one has A = (300)2 mm2 and, provided that second order effects are neglected, K = 0.45 (compare [11], Fig. 6). For ˛∗1 = 2.125, ∗ 0,lt,1 = 1462.85 MPa mm1/˛ and  0,1 = 40.2 MPa, one obtains the excellent fitting graphically represented in the LTW plane of Fig. 4. Once the statistics for the air-side is defined, the attempt is to interpolate the tin-side data with a distribution of the type (4.24). Observe that the third term in square brackets is negligible with respect to the other two, since 0,tt,3m = (0,lt,1m 0,lt,2m )/Am is very large for the case considered here. Remarkably, neglecting such a term, the distribution (4.24) becomes a bi-modal generalized

Fig. 4. Linear interpolation in the LTW plane of the air side failure stress measurements in the experimental campaign by CEN-TC129/WG8 [15] (˛∗1 = 2.125; ∗ 0,lt,1 = 1462.85 MPa mm1/˛ ;  0,1 = 40.2 MPa).

The rigorous evaluation of the effective area for such a statistical distribution is analytically difficult. As a first order approximation, for the experimental set-up of [19], we may consider Aeff = K A = 0.45  (300)2 mm2 , in analogy with the results of [1]. Moreover, we have here slightly generalized the expression (4.24), by assuming that the lower bound for glass strength  0 may be different for the pre-existing defects ( 0,1 ) and the defects induced through abrasion ( 0,2 ). Setting 0,lt,1m , ˛∗1 and  0,1 equal to the corresponding values for the air-side graphically estimated from the distribution (4.29) and its graphical estimation of Fig. 4, the best fitting of the experimental data for the tin-side is obtained with ˛∗2 = 2, ∗ 0,lt,2m = 1276 MPa mm1/˛ and  0,2 = 46 MPa. Observe that  0,2 is slightly higher than  0,1 . This could mean that the abrasion induced on the tin-side is associated with cracks whose maximum size is smaller than the maximum  size ofpre-existing cracks. The plot in the Weibull plane ln ln 1/(1 − Pf ) vs. ln for the tin side data is shown in Fig. 5. The goodness of fit is evident. It should be observed that the effect of abrasion on the lower fractiles of the distribution is clearly negligible, whereas it becomes important for the higher fractiles. Moreover, the fact that  0,1 <  0,2 implies that the statistics coincides with that of the air side for  0,1 ≤  ≤  0,2 . For a quantitative verification of the goodness of fit, the chisquare test can be used [24]. Such a method leads to the evaluation of the probability that the deviation of the observed from expected data is due to chance alone through a precise parameter, referred to as the p-value. It is customary to accept the 5% rule, according to which a statistical model can be considered reliable for p-value ≥ 0.05 (5%). Since the statistics (4.31) employed here provides p-value 0.06 (6%), it is possible to claim that the proposed statistical model, derived from micromechanical considerations, is acceptable, especially when compared with the best interpolation obtainable with a generalized Weibull distribution, which at most can achieve p-value 0.02 (2%), as discussed in [1]. Of course, the present treatment is based upon some approximations, of which one is certainly the calculation of the effective area. Anyway, the results should be considered completely satisfactory.

Fig. 5. Plot in the Weibull plane of the tin-side failure stress (experiments from [15]). Interpolation with the statistical distribution of Eq. (4.24) (˛∗1 = 2.1; ∗ ∗ 0,lt,2m = 1462.85 MPa mm1/˛ ;  0 = 40.2 MPa; ˛∗2 = 4; 0,lt,2m = 1276 MPa mm1/˛ ; A = 127170 mm2 ).

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G. Pisano, G. Royer Carfagni / Journal of the European Ceramic Society 37 (2017) 4197–4206

5. Conclusions A micromechanically-motivated model, correlating the statistical distributions of surface microcracks in glass with the experimentally-measured macroscopic strengths has been proposed. The defectiveness scenario at the microscopic level is interpreted by a power law distribution of crack lengths, i.e., a highly right-skewed distribution with long right-hand-side tail. This assumption would lead to a 2-parameter Weibull statistics for glass strength, which depends upon the size of the specimens and the state of stress (uniaxial vs. biaxial). The effect of factory production controls is interpreted as a right-hand-side truncation of the population of surface flaws, whose effect is to provide a left-truncated Weibull (LTW) distribution for the macroscopic strengths, already observed in [1]. The effects of corrosion and abrasion due to artificial treatments or natural aging are interpreted by a modification of the population of micro-defects, which modify accordingly the expected macroscopic strength, proposing new forms of generalized Weibull statistics. The process of corrosion modifies the highest fractiles, associated with the minimum crack lengths, which are the most affected by a uniform dissolution of a thin surface layer. The abrasive process can produce new cracks on the surface, which are analyzed by considering the superposition of an additional crack distribution on the one present in the pristine material. This approach is used in particular to justify the difference in the experimentally-measured populations of float-glass strengths when it is either the air- or the tin-side under tensile stress. It is the abrasion due to the contact with the tin bath and rollers that produces a damaging action. The results so obtained are in good agreement with the experimental findings. Of course, the conclusions from this study can only be considered partial. Some important phenomena have not been considered directly, such as the degradation of the borders induced by the cutting process, the possible healing of existing cracks due to thermal processes, the residual stress that accompanies abrasion, and the crack blunting induced by corrosion. Unfortunately, there is a lack of experimental data about these effects, but we believe that our study is of general value and we hope that the proposed micromacro approach will encourage properly designed experimental campaigns. These will allow a step-change improvement of our understanding of the mechanical properties of glass and their variation with aging, promoting a more efficient use of this beautiful material in structural applications. Acknowledgements The authors acknowledge the support of the Italian Dipartimento della Protezione Civile under project ReLUIS-DPC 2014-2018 and the support of the Italian Ministero dell’Istruzione, dell’Università e

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