2.1. Optical fiber used

The used monomode fiber has two acrylate coatings (primary and outer coatings).

This fiber was manufactured using the Outside Vapor Deposition (OVD) process which produces a totally synthetic, ultra-pure fiber. It has high strength and low attenuation. Its dual acrylate layer (CPC6) coating provides excellent fiber protection. On the other hand, the operating temperature range was −60 °C to +70 °C. A soft, primary coating has a low module of elasticity, adheres closely to the glass fiber and forms a stable interface. It protects the fragile glass fiber against micro-bending and attenuation. The outer coating protects the primary coating against mechanical damage and acts as a barrier to lateral forces. It has a high glass temperature and Young modulus and a good chemical resistance. The combined coating diameter is 245 μm ± 5 μm, the clad diameter is 125 μm ± 1 μm and the coating thickness is equal to 62.5 μm ± 2 μm.

2.2. Test bench used

Optical fiber was wound onto an alumina mandrel of 2.6, 2.7, 2.8 or 3 mm in diameter (Fig. 1). Electrical engine helps to wound fibers around alumina mandrels. A counter gives the maximum number of circumferences around each mandrel. The winding of each fiber was automated and the same applied stress was obtained for each fiber. The real deformation for each fiber depends on the mandrel diameter; with the use of suspended mass (Fig. 1) each fiber has taken on the exact shape of the mandrel.

Figure 1 Electrical engine for winding of each fiber. 

The extremities were clamped in two oblique cuts simple clamping rings, made of elastomer-rubber, and mounted on the extremities of the alumina mandrels. Once the fiber was wound around the mandrel, it was placed between a transmitter T and a light receiver R (Fig. 2). The light beam cannot reach the receiver and from then on the time of fiber loading is triggered. The mean time to failure is recorded, and this corresponds to the time required for the fiber strength to degrade until it equals the stress applied through winding round the mandrel. The time to failure is measured by optical detection when the ceramic mandrel moves out of the special holder. When fiber breaks, the mandrel rocks from its vertical static position, the light beam can reach the receiver and the time to failure is directly recorded with an accuracy of ±1 s. The testing setup consists of a large number of vats containing 16 holders each.

Figure 2 Test bench with many samples and mandrel sample with wound optical fiber. 

The alumina mandrels have a very low surface roughness. The friction between the fiber polymer coatings and the mandrel is negligible.

For fiber aging in hot water, we developed a water circulation system (Fig. 2). The system comprises a 9 l plastic tank in which is immersed a water heater with a resistance heater and a discharge pump. A water diffusion tube sends water at the desired temperature to the testing bench with mandrels. Thus the fibers (wound around a mandrel) can be aged in water at different temperatures.

2.3. Theoretical relation used

The stress-strain relation of optical fibers was first examined by ^{Mallinder and Proctor (1964)}. They found that the applied stress σ (GPa) for an optical fiber is related nonlinearly to the strain ɛ (where both σ and ɛ are positive in the tensile region), according to:).

(1)

where E ₀ is the Young modulus (=72 GPa for the silica); α' = 0.75α and a value for the non-linearity constant α of 6 can be used (^{Glaesemann, Gulati, & Helfinstine, 1988}

One can note that a wound fiber was subjected to compressive stresses on its internal part (fiber surface close to the curvature center) and a tensile stress on its external part (fiber surface furthest away from the curvature center).

A tensile stress is experienced by the outer surface of the fiber. The bending stress must first be calculated from the mandrel diameter ϕ. This can be performed with (1) and the simple expression for the maximum strain ɛ (^{Griffioen, 1994}):

(2)

ϕ is the mandrel diameter (in μm); d _glace , is the glass fiber diameter (=125 μm); d _fiber is the fiber diameter (=245 μm), including the layer polymer coating. This leads, in the case of a usual fiber, to the corresponding stresses of 3.22 GPa for the calibrated diameter mandrel of 2.8 mm.

2.4. Boundary test conditions

As has been documented it was known (^{El Abdi, Rujinski, Borda, Severin, & Poulain, 2008}; ^{El Abdi et al., 2010}; ^{El Abdi et al., 2015}; ^{Gougeon, Sangleboeuf, El Abdi, Poulain, & Borda, 2005}), the fiber coating begins to be damaged above 80 °C. It changes color and its odor was released. The chemical environment of the silica surface was modified and the fiber core damage was accelerated.

The results obtained at very high temperatures (higher than 80 °C) cannot be used to predict the behavior of the fiber at lower temperatures. For our tests, a maximum temperature of 70 °C was used so that the polymer coating would not suffer temperature damage.

3.1. Influence of aging environments

The polymer coating was permeable. Thus, when a fiber was put in water, a significant number of water molecules diffused into the interface between the coating and the fiber core and the rate of reaction between the H₂O molecules and the Si-O glass network was determined by the temperature.

Fibers were wound on mandrels and aged in air and distilled water at 20 °C (Fig. 3).

Figure 3 Change of time to failure of optical fiber aged in air and water. 

Thus, upon application of a stress, the fiber damage was much higher in water than in the air (time to failure ratio was about 10). The number of water molecules activated here was considerably higher than those in air.

3.2. Temperature dependence of failure time

The different diameters of the alumina mandrels (2.6, 2.7, 2.8 and 3 mm) produced stress values of 3.47, 3.34, 3.22 and 3.01 GPa respectively. For each temperature T (20, 30, 40, 50, 60 and 70 °C) and for each diameter, 16 mandrels were used each with 1 m wound optical fiber.

The averages of times to failure according to temperature and applied stress are given in Figure 4. For each value, the standard deviation was about 10% (the average value is obtained from 16 measured samples).

Figure 4 Time to failure in inverse proportion to temperature for different mandrel diameters. 

Figure 4 shows that whatever the stress level may be the logarithm of the time to failure tf, increases linearly with respect to the inverse of temperature. Thus, we can write:

(3)

(4)

The fatigue of a silica optical fiber was controlled by the crack growth rate c., which depends on the reaction rate between water and silica. On the other hand, ^{Wiederhorn and Bolz (1970)} developed a chemical kinetic model for the crack growth, based on simple chemical kinetics in which the stress at the crack tip modifies the effective activation energy for the bond rupture via an activation volume (^{Shiue & Matthewson, 2002}):

(5)

where Q is the apparent activation energy and R is the perfect gas constant (R = 8.314 J mol⁻¹ K⁻¹⁾.

^{Wiederhorn and Bolz (1970)} found that Eq. (5) fits the crack velocity data of various studied glasses. From Eq. (5), the time to failure of the material tf, which inherits the temperature dependence of .c , is therefore inverse Arrhenius (^{Matthewson & Kurkjian, 1988}):

(6)

The experimental results have enabled us to determine the values of Q and C for different stress levels. These parameters are summarized in Table 1. According to applied stress, the strain slowly decreases but remains around 4%. The activation energy Q tended to increase when the applied stress decreases. Indeed, for a decrease in stress from 3.47 to 3.01 GPa, the Q parameter increased from 55 to 75 kJ/mol (Fig. 5).

Figure 5 Change of activation energy Q values versus applied stress. 

On the other hand, for a static fatigue test carried out on a silica fiber, ^{Kao (1980)} observed an increase of Q from 45 to 55 kJ/mol. when the applied stress decreases from 3.4 to 2.9 GPa. Similarly, ^{Duncan, France, and Craig (1985)} obtained an activation energy of 50 kJ/mol, with dynamic tests.

The values of Q obtained by Kao and Duncan are slightly different from the ones we measured, because the coating materials in the two studies are different and so the coating/silica interface were be exposed to different ionic environments.

In the case of dynamic tests, the deformation can play a major role and the activation energy can change with stress level (^{Inniss, Kurkjian, & Brownlow, 1992}). Kao was the first to demonstrate that the activation energy of lightguide fibers is stress-dependent (^{Kao, 1980}). While ^{Chandan and Kalish (1982)} report temperature dependent fatigue data, they are treated with the power law, as are the temperature-dependent data of ^{Krause and Shute (1988)}. ^{Ritter, Glaesemann, and Jakus (1984)} used an exponential law to fit temperature-dependent dynamic fatigue data, and the predicted curves are used to construct the stress-dependent activation energies.

As indicated in Figure 6, the C logarithm increases linearly according to the applied stress σ . thus, one can write:

(7)

Figure 6 Change of C values versus applied stress. 

^{Shiue and Matthewson (2002)} show that the activation energy Q can be expressed as:

(8)

where Q* is the apparent activation energy under zero stress, and the term (β·σ) represents the stress effect on the apparent activation energy.

If γ is equal to R ·C ₂ (=75 SI), Eq. (6) can be expressed as:

(9)

Eq. (7) can be written as:

(10)

Using Table 1 and a linear regression, one can deduces the C ₁ and C ₂ values (equal to 2.3 × 10⁻²³ and 9.34 respectively), β and ^Q* values (equal to 43.4 kJ/(GPa mol) and 205 (kJ/mol) respectively). So, a complete identification of the time to failure model is obtained.

The term (γ ·T·σ ) has a physical meaning which is a little more complex. When the fiber was subjected to stresses, the Si-O links in the glass network were set in motion to achieve the equilibrium position corresponding to this new state of stress.

This movement is known as the structural relaxation . At room temperature, the relaxation is very slow, but at higher temperatures the atoms and molecule motions were much faster and the relaxation occurs more rapidly.

Thus, (γ ·T·σ ) characterizes the influence of the relaxation on the time to failure for a fiber.

3.3. Change of failure with applied stresses

Figure 7 gives the fatigue curves for different aging temperatures between 20 °C and 70 °C.

Figure 7 Change of time to failure versus applied stress for different temperatures. 

One can note that whatever the temperature, the fatigue curve can be described by a straight line.

Concerning aging in water, many authors have shown (^{Chandan & Kalish, 1982}; ^{Cuellar, Roberts, & Middleman, 1987}; ^{Krause, 1979}; ^{Wang & Zupko, 1978}) that the fatigue curve is not linear over the stress range. From a certain stress level, the fatigue curves present a slope change and therefore a change for the n parameter. This transition in the slope is known as the "knee" phenomenon.

In the case of a standard fiber aged in water at 85 °C, ^{Cuellar, Kennedy, Roberts, and Ritter (1992)} observed a change in the slope for a stress of about 2.7 GPa. The time to failure occurs at 12 days.

This slope change results from coating damage in hot water. The coating damage occurs more rapidly if the water temperature is high.

As we previously stated, all fatigue tests were carried out on a fiber with a new kind of coating. It has been shown (^{Kuyt, Leclercq, & Breuls, 1995}) that this new kind of coating has a good resistance to temperature and to water. It is therefore a very good protection for the fiber when used in harsh environments. All aging tests were conducted at a maximum temperature of 70 °C and the coating does not undergo significant damage after relatively long aging durations.

We can therefore assume that when the water temperature does not exceed 70 °C, the coating does not undergo significant damage during the aging duration. The "knee" phenomenon doesn't occur and the n parameter remains constant during the used stress range.

3.4. Change of the stress corrosion parameter according to temperature

Eq. (9) gives the time to failure versus applied stress, temperature and activation energy. The time to failure can also be obtained as a function of stress corrosion parameter, mechanical and geometrical material parameters and applied stress for a material with the presence of a microcrack.

The subcritical crack growth model which assumes the presence of well-defined sharp surface crack which locally amplify the applied stress at the crack tip is related to the stress intensity factor, K _I , for a crack of length c subjected to a remotely applied stress σ by the following relation:

(11)

where Y is a parameter of order unity which describe the crack shape shape (=1.24 for a halfpenny shaped crack existing on the fiber surface), K _I is a measure of the intensity of the stress field at the crack tip and when it exceeds a critical value, K _IC , the intrinsic strength of the material is exceeded and catastrophic failure occurs. The combined influence of stress at the crack tip and reactive species in the environment (particularly water) leads the strain in the bonds at the crack tip to reduce the activation energy for the chemical reaction between water and the silica so that silicon-oxygen bonds are slowly broken, progressively advancing the crack (^{Wiederhorn, 1972}). A power law relationship between the crack growth rate c. and the applied stress intensity at the crack tip is assumed:

(12)

A is a parameter which is environment dependent and n is the stress corrosion parameter which refers to the crack growth parameter.

Therefore, the applied stress causes the crack to extend, which itself increases K _I , leading to an increases in the growth rate. In most reliability models, the applied stress is assumed to be static (σ constant) so that the tie to failure, t _f , is given by (^{Evans & Wiederhorn, 1974}):

(13)

where S _i is the middle value of the initial intrinsic strength of the material (GPa).

^{Tiffany and Masters (1964)}, ^{Wiederhorn (1973)} and ^{Tetelman (1972)} give the time to failure according to initial intrinsic fiber strength and applied stress as follow:

(14)

where B is equal to

(15)

Eq. (14) can be expressed as:

(16)

The use of Eq. (16) gives the stress corrosion parameter n versus t _f and thus versus temperature.

The n values were determined through experimental results given in Figure 7 and the use of Eq. (16). The n parameter decreased with temperature (Fig. 8). This parameter was proportional to the inverse temperature (1/T ) and a straight line passes through all the experimental values.

Figure 8 Measured values of stress corrosion parameter n in inverse proportion to temperature. 

The n parameter can be expressed as: where a ₀ = 15,180 K and b ₀ = −28.

(17)

Eq. (17) gives a good approximation for n according to temperature. A high stress corrosion parameter n involves a significant slope for the curve of the time to failure and therefore great fiber sensitivity to applied stresses.

This equation was also observed by ^{Sakaguchi and Hibino (1984)} during a dynamic test carried out on a sample of solid silica glass for a temperature range from 20 °C to 80 °C.

By examination of previous work on silica glass both with and without damage, one may postulate four distinct regimes which, without any other information, would be expected a priori to have four distinct applicable micromechanics models (^{Matthewson, 1993}). One can note that four types of defects expected to be encountered in silica:

Eq. (17) forms the basis of the most reliability models since it relates the expected life of the material to the applied stress and starting strength (^{Matthewson, 1993}); or conversely it specifies the maximum permitted service stress for a given service life. This equation is sensitive to the value of the stress corrosion parameter, n , which it is typically around 20 or more for silica fibers.

Figure 8 shows that the n parameter was between 16 and 26 according to temperature and leads to conclude that the crack initiation was dominated by residual stresses.

3.5. Fatigue curve for different temperatures

If one makes an extrapolation to high stress values of the experimental results given in Figure 7, one obtains the schematic representation shown in Figure 9.

Figure 9 Fatigue curve versus applied stress for different temperatures. 

Figure 9 shows that for a very particular stress value, denoted σ _E , the different static fatigue curves meet each other. When the applied stress becomes greater than σ _E , the temperature has an opposite effect and a low temperature led to a low time to failure.

The stress σ _E has a much higher value than the stresses commonly used. Thus, under σ _E , the time to failure t _fE was well below the times to failure measured during experimental tests.

In our tests, for an aging temperature of 70 °C and for a stress equal to 3.47 GPa, the time to failure remained at less than half an hour. This implies that for a stress higher or equal to σ _E , the time to failure would be extremely small. ^{Kurkjian (1989)} has pointed out that this would correspond to time to failure on the order of a few microseconds. From a practical point of view, the use of this high stress would be unrealistic because it would lead to an instantaneous failure of the fiber. For industrial uses, the minimum value of the applied stress must be optimized. The fiber must not support a low stress (so that the mechanical properties of the fiber can be used) and not a high stress to avoid premature failure of the fiber.