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Data Processing 239

Figure 7.17 Typical stress–strain (σ –) curve of a ﬂax ﬁber (sample DII.04)

obtained with the microcycling tensile machine. We added, for a graphic

comparison, a smaller image [8] with the qualitative shape of the σ – curve

of a ﬂax ﬁber, as it can be found in the literature.

on the gauge length, on the ﬁber’s diameter, and on the moisture

absorbed.

57

57

From the mathematical point of view we made an aﬃne transformation. The real

stress (σ *) was standardized (σ ) according to the equation

σ = K

L

K

d

K

H

σ

∗

,

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240 The Mechanical Multiparametric Dating Method

From the σ – standardized diagrams of the tested samples we

evaluated the following mechanical parameters:

58

• Breaking strength σ

R

• Elastoplastic strain at break

R

• Final elastic modulus E

f

• Elastic strain at break

E

• Decreasing elastic modulus E

c

• Direct loss factor η

D

• Inverse cycle loss factor η

I

In addition to the preliminary selection criteria (Section 7.3),

to minimize the data scattering resulting from the presence of

ﬁbers’ microdamage and/or contamination not visible in stereomi-

croscopic analysis, we introduced two exclusion criteria once the

test had been concluded:

(a) Fibers with an evident anomalous stress–strain curve must be

excluded.

(b) Each standardized breaking strength must be compared with

the average one within each sample series. Whenever the

breaking strength of a sample is outside the allowed variation

range (between 2.5 times larger and 2.5 times smaller than

the average breaking stress of the series), the sample must be

excluded.

Table 7.3 summarizes the number of tensile-tested samples for

each series and the ﬁnal number of those used for identifying

particular bijective dependencies with time.

where K

L

, K

d

, and K

H

are the coeﬃcients in the above-mentioned order and

deﬁned as follows:

K

L

=

1479

1500 − 20.93L

K

d

=

67.62

89.41 − 1.452d

K

H

=

743.0

580.4 + 3.250H

,

in which L (mm) and d (μm) are, respectively, the measured ﬁber gauge length and

diameter and H)(%) is the laboratory humidity.

58

For details on the deﬁnitions the reader can consult Section A.13 of the appendix.

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Results 241

Table 7.3 Number of samples subjected to a tensile test. The table

shows only the ﬂax ﬁbers that passed the preliminary selection,

rules (a), (b), and (c), discussed in Section 7.3. The causes of the

tensile test failure were accidental breakage of the ﬁber during

its gluing on the tab (tabbing technique), its positioning on the

microcycling tensile machine, and cutting of the lateral bridges

(Section A.12)

Samples Samples Samples

Tabbed Not tested Tested excluded excluded used for

Name ﬁbers samples samples by d) by e) dating

A 21 7 14 2 0 12

B 19 4 15 2 0 13

DII 4 0 4 0 0 4

D21 15 6 1 0 5

FII 19 9 10 0 0 10

NI 5 2 3 0 0 3

E5 1 4 0 0 4

HII 5 2 3 1 0 2

K8 5 3 0 0 3

LII 10 7 3 0 0 3

7.8 Results

Following the preliminary selection criteria of the samples (Sec-

tion 7.3) and applying the two exclusion criteria to the data obtained

from the tensile tests (Section 7.7), we evaluated for each series the

average values of the breaking strength σ

R

, the elastoplastic strain

at break

R

, the ﬁnal elastic modulus E

f

, the elastic strain at break

E

, the decreasing elastic modulus E

c

, the (direct) loss factor η

D

,and

the inverse cycle η

I

loss factor.

Then, for each of the above-mentioned mechanical properties we

placed the representative value of each of the nine tested series in a

diagram with the years on the horizontal axis

59

in order to identify

bijective correlations between the mechanical characteristics and

the date in agreement with condition 2 (Section 6.1).

59

What we did is qualitatively shown in Fig. 6.3 on the left. It is suﬃcient to imagine

those diagrams with only nine points because nine are the tested series for each

measured mechanical property Y.

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242 The Mechanical Multiparametric Dating Method

The trend analysis allowed us to identify the following bijective

property–time relationships:

• Breaking strength σ

R

vs. time

• Final elastic modulus E

f

vs. time

• Decreasing elastic modulus E

c

vs. time

• Loss factor η

D

vs. time

• Inverse cycle loss factor η

I

vs. time

Supposing that natural cellulose degradation obeys [88] ﬁrst-

order reaction kinetics, namely an exponential decay law over time

such as the one characterizing the radiocarbon decay (Chapter 4),

we demonstrated that both the breaking strength σ

R

and the two

Young’s moduli E

f

and E

c

follow an exponential decay law over

time, too, determining the corresponding equation (condition 5)

by means of the least squares ﬁtting method.

60

In other words,

these three mechanical properties exponentially decrease with the

fabric’s age. We of course considered other kinds of least squares

interpolation curves, ﬁrst of all the linear law and the polynomial

one, but the exponential law was the curve that best interpolated

the data according to the Pearson correlation coeﬃcient.

61

Taking, as we actually did, semilogarithmic coordinates in

plotting the diagrams of the mechanical properties σ

R

, E

f

,andE

c

,

the above-mentioned exponential relationships with time turn out

to be a straight line. We used this form in the diagrams for dating the

Shroud. We point out that this transformation has no eﬀect on the

results, because it is simply a diﬀerent way of presenting the data

in a graph. For example, Fig. 7.18a shows the dating diagram of the

Young’s modulus E

f

.

60

We refer the reader to Section 6.1 for more details on this method.

61

The interpolation curves with Pearson’s correlation coeﬃcient R are the following:

σ

R

= 139.14e

0.0009678x

± 336 years R = 0.943

E

f

= 6.2219e

0.000709386x

± 418 years R = 0.915

E

c

= 8.1758e

0.00060588x

± 537 years R = 0.910

See footnote 12 on p. 193 for the meaning of the coeﬃcient R.

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Results 243

Figure 7.18 Two of the ﬁve dating diagrams [61] obtained by the authors:

(a) dating diagram of the Young’s modulus E

f

; (b) dating diagram of the

direct loss factor η

D

.

Both direct loss factor η

D

and inverse cycle loss factor η

I

show

a diﬀerent bijective correlation with time, which is linear

62

instead

of the exponential type. The loss factor (direct and inverse) grows

with the antiquity of the sample. This would suggest that this

mechanical property is not correlated to the resistance of the

cellulose chains but rather to their degree of crystallinity. Since

the degree of crystallinity is supposed to decrease with aging, a

greater micromobility of the polymer chains of the ﬁbers is allowed,

with the macroscopic consequence of an increase of the internal

dissipation energy when the ﬁber is subjected to cyclic loading–

unloading conditions. Figure 7.18b shows the dating diagram of the

direct (percentage) loss factor η

D

.

We utilized the ﬁve dating diagrams with the relative uncertainty

evaluation

63

(condition 7) for dating the Shroud as follows.

62

The interpolation curves with Pearson’s correlation coeﬃcient R are the following:

η

D%

= 7.5376 − 0.0013721x ± 193 years R = 0.955

η

I %

= 4.2604 − 0.0011458x ± 385 years R = 0.900

63

The lines above and below the central one shown in Fig. 7.18 delimit the uncertainty

band.