#### 2.1. Analytical Model

The basic analytical model for the two sensing cables is in line with the traditional strain transfer models developed for bare surface-bonded optical fibers. However, in the present case some additional assumptions are considered and different boundary conditions (BCs) are applied. The model is developed under the following assumptions:

- A1.
All the materials involved in the analysis behave as linear elastic materials and there is perfect bonding at all the layer interfaces.

- A2.
It is assumed that the fiber core and the cladding behave as a unique homogeneus material which is referred to as “optical fiber”.

- A3.
The optical fiber coatings, the corresponding tight tubing, the cable jacket, and the adhesive carry only shear stresses. Indeed, the Young moduli of these cable components are at least one or two orders of magnitude smaller than those of the optical fiber and the specimen.

- A4.
The strain transfer from the structure towards the fiber core depends only on the cable components surrounding the fiber under test. Therefore, referring to

Figure 1a,c only the left half of the two cables, where the strain sensing fiber is embedded, was considered in the development of the model.

- A5.
In the second cable prototype the effect of the reinforced bar is neglected, since, as already said, it is mechanically decoupled from the surrounding cable jacket.

Based on the assumption A4 only one half of the cable is considered, modelling its geometry as outlined in

Figure 2.

The analysis was carried out using cylindrical coordinates. The axial direction, along the axis of the optical fiber, is denoted with x, the radial direction with r, whereas $\vartheta $ represents the azimuth.

The analysis starts considering an infinitesimal fiber segment and imposing the equilibrium condition:

where, referring to

Figure 2,

${r}_{f}$ represents the optical fiber radius,

${\sigma}_{f}$ denotes the normal stress in the optical fiber and

$\tau \left(x,{r}_{f}\right)$ is the shear stress at the interface between the optical fiber and the inner coating.

Then, it is possible to extract the shear stress at the optical fiber boundary as follows:

Recalling the assumption A3, the equilibrium condition in the

x direction of the first layer surrounding the optical fiber, which is the inner fiber coating, leads to Equation (3):

The first integral of Equation (3) is defined within the interval

$\left[\alpha ,\pi -\alpha \right]$, where

$\alpha $ represents the angle between the horizontal direction and the line connecting the center of the optical fiber with the top point of the adhesive layer on the cable surface (see

Figure 2). In the case of an embedded optical fiber the integration interval would be

$\left[0,\text{}2\pi \right]$ as is for the second term of Equation (3). However, for surface bonded optical cables the strain field is not axially symmetric. Hence, the shear stresses in the coating can be expressed as:

Substituting Equation (2) into Equation (4) one obtains:

Assumption A1 allows to use Hooke’s law, relating stresses to strains with the constitutive equations:

where E, G,

$\epsilon $ and

$\gamma $ represent, respectively, the Young’s modulus the shear modulus, the normal strain and the shear strain of a generic layer of the sensing cable. Based on these parameters, Equation (5) can be rewritten as:

where

${G}_{ic}$,

${E}_{f}$ and

${\epsilon}_{f}$ represent the shear modulus of the inner coating, the Young’s modulus and the normal strain of the optical fiber, respectively.

The shear strain can be expressed under the assumption of small displacements:

The radial displacements,

w, are negligible compared to the axial displacements

u. Indeed, the radial displacements are mainly induced by the Poisson contraction occurring in the coating and the displacements along the

x axis are at least one order of magnitude higher than

w. Hence, substituting Equation (8) into Equation (7) leads to:

Then, integrating Equation (9) from the outer optical fiber radius,

${r}_{f}$, to the inner coating boundary,

${r}_{ic}$ one gets:

The result of the integration is given by Equation (11), with

${u}_{ic}$ and

${u}_{f}$ being the axial displacements of the inner coating and the optical fiber, respectively:

Performing the same operation for all the other layers leads to:

where the axial displacement of the structure is denoted with

${u}_{s}$, whereas

${G}_{a}$,

${G}_{j}$,

${G}_{t}$ and

${G}_{oc}$ and

${r}_{j},\text{}{r}_{t}$ and

${r}_{oc}$ are the shear moduli (

G) and the radii (

r) of the adhesive, cable jacket, tight tubing and outer coating, respectively. The thickness of the adhesive,

${t}_{a}$, deserves additional considerations because it is a function of the azimuthal angle

$\vartheta $ (see

Figure 2). In Equation (12),

${t}_{a}$ is assumed equal to the average adhesive thickness and is calculated as outlined in the following expression:

where

t is the minimum adhesive thickness (see

Figure 2). Substituting Equation (13) into Equation (12), and introducing the shear lag parameter

k, one gets:

where

k is defined by the following equation:

Since the axial strain is defined as the derivative of the longitudinal displacement with respect to the

x variable, the differentiation of Equation (14) with respect to

x leads to:

with

${\epsilon}_{s}$ being the axial strain of the structure. Equation (16) is a second order linear non-homogeneous differential equation with constant coefficients. Adding up the homogeneous and the particular solutions, one obtains:

where

C_{1} and

C_{2} represent the integration constants whose value can be computed imposing the corresponding BCs. Normally, the strain values at the optical fiber extremities are assumed equal to zero. However, this is not the case in real applications, where the strain does not suddenly reduce to zero, although the cable is not subjected to external loads.

Figure 3 represents the actual situation.

Assuming a null strain level in the optical fiber at the two extremities of the bonding length generates a discontinuity in the first derivative of the strain profile which is unlikely to occur. In addition, since the fiber core stiffness is higher than that of the other cable components, the related deformation at the fiber boundaries is expected to be significantly lower with respect to the outer layers. Consequently, the fiber core prevents the cable jacket from stretching whereas the cable jacket tends to stretch the fiber core. This results in a self-equilibrating configuration where the fiber core experiences a tensile load whereas the other cable components undergo a compressive load. Such effect vanishes after few cable diameters (

Figure 3) based on the De Saint Venant principle (stresses are free to redistribute along the structure).

In addition, in a surface-bonded cable the two ends tend to bend upwards as a result of the shear strains acting in those sections. If the optical fiber core is not perfectly centered in the cable structure, the misalignment with the neutral axis produces an additional axial load.

Based on these considerations, the BCs applied to Equation (17) are not null and assumed equal to:

where

L is half of the bonded length and the

p parameter symbolizes the percentage of residual strain in the optical fiber core, thus

$p\in \left[0,1\right]$.

Imposing the BCs defined in Equation (18), the integration constant

C_{1} and

C_{2} can be found to be:

Then, the substitution of

C_{1} and

C_{2} into Equation (17) leads to an expression for the strain profile of the fiber core as a function of

x:

Equation (20) holds when $x\in \left[-L,L\right]$.

For

$x>L$ and

$x<-L$, it is assumed that, in accordance with the De Saint Venant principle, the axial strain shows an exponential decay as follows:

Equation (21) represents an even function in line with the fact that the strain profile should be symmetric with respect to the sensing cable midpoint. The a, b, and p parameters can be determined by fitting the experimental data. However, the authors propose the following methodology to assess their value without any prior test. The b parameter represents the exponential strain decay in the optical cable beyond the bonding length (i.e.,

$x>L\text{}\vee x-L$). Hence, an estimate of b can be carried out using the same approach used to determine the shear lag parameter k. However, in this case the adhesive layer is not present and the first term of Equation (3) should be integrated from 0 to

$2\pi $ since the strain propagates with no preferential direction as in the case of a fully embedded optical fiber. These considerations lead to the following expression for

b:

The other two parameters,

a and

p, can be evaluated by imposing the continuity of the strain profile and its derivative at the two extremities, where

$x=\pm L$. The derivative of the strain profile is estimated differentiating Equation (20) along the

x axis as follows:

Considering for example the interval

$x\in \left[L,+\infty \right]$ it is possible to write the following system of equations:

The system solved for

a and

b returns:

Once every parameter of the model is determined and the corresponding strain profile is computed, it is convenient to introduce the so-called effective bonding length

${\mathrm{L}}_{\mathrm{eff}}$, which has been defined in the literature by several authors using various expressions [

11,

25,

26]. In this study,

${\mathrm{L}}_{\mathrm{eff}}$ is defined as the minimum half fiber length to be bonded such that at the midpoint (i.e.,

$x=0$) of the fiber core the strain level reaches 95% of the strain present in the structure. Assuming

${\epsilon}_{f}\left(0\right)=0.95{\epsilon}_{s}$,

${\mathrm{L}}_{\mathrm{eff}}$ can be obtained from Equation (20) as follows:

Hence, the higher the shear lag parameter, the lower the corresponding effective bonding length. Moreover, since p represents the percentage of strain in the fiber core at $x=\pm L$ with respect to the strain present in the structure, it can be stated that high values of p entail lower values of ${\mathrm{L}}_{\mathrm{eff}}$. An alternative conservative approach would be to apply Equation (25) with $p=0$.

#### 2.1.1. Cable-Specimen Interaction

It is worth to consider in the analysis the mutual interaction between the sensing cable and the structure if the former is particularly stiff with respect to the latter. Referring to

Figure 3, it is possible to relate the theoretical strain in the structure with the actual strain, i.e., the result of the reciprocal interaction between the sensing cable and the structure. The equilibrium condition for the system is given by:

where

${\sigma}_{t}$ is the true stress applied to the structure,

${\sigma}_{s}$ is the corresponding actual stress,

${\sigma}_{f}$ is the stress acting in the optical fiber, whereas

${A}_{s}$ and

${A}_{f}$ are the cross section of the structure and the optical fiber, respectively. Exploiting the Hooke’s law and substituting the values of the relative cross sections one has:

where

h and

w are the two cross section dimensions of the structure (

Figure 2), and

${E}_{s}$ is its modulus of elasticity. Solving for the actual longitudinal strain in the substrate structure,

${\epsilon}_{s}$, leads to:

Hence, when the cable stiffness is not negligible with respect the host structure, the mutual interaction must be considered.

#### 2.1.2. Interrogator Resolution

The interrogator resolution has an impact on the measured strain profile,

${\epsilon}_{m}$. In [

27] J.M. Henault et al., estimated the interrogator effect on the strain transfer mechanism by convolving the strain profile in the fiber core,

${\epsilon}_{f}$, with a rectangular function

${\mathsf{\Pi}}_{\mathrm{i}}\left(x\right)$. The interval width of

${\mathsf{\Pi}}_{\mathrm{i}}\left(x\right)$ corresponds to the resolution of the measuring system. Hence, the filtering operation due to the interrogator can be expressed by:

where the symbol

$\otimes $ denotes the convolution operator. For a consistent comparison between the analytical and the experimental data it is necessary to filter the analytical model results according to Equation (29).

#### 2.3. Numerical Model

The numerical model was developed to validate the proposed analytical model with additional data and provide verified modelling strategies for a complex fiber cable. The Abaqus/CAE

^{TM} was used to this aim. In particular, since strain transfer analysis can be considered as a static problem, Abaqus/Standard was chosen as solver. All the model parts were meshed using the C3D8R element type. A preliminary analysis using a microscope was carried out to analyze the cross section of the two sensing cables and thus estimate their effective shape. The material properties and dimensions of the two cables and the specimen were defined according to

Table 1,

Table 2 and

Table 3, respectively.

Figure 7 shows the cable cross section of the two meshed models and the reference frame (

x axis pointing inward).

The analysis was performed applying two symmetry BCs to the two cables in order to simulate one quarter of the model, thus minimizing the computational cost. The first symmetry BC was applied to the cable cross section at the midpoint (

$x=0$). This condition is obtained by posing the displacement along the

x direction equal to zero and fixing the rotation with respect to the other directions. The other symmetry BC was applied on the x-z plane in correspondence of the dashed red lines in

Figure 7, assuming a zero displacement along the y direction and no rotation with respect the

x and

z axes.

In both numerical models, the adhesive layer was connected to the structure and cable jacket using a tie constrain between the respective surfaces. In the second numerical model, shown in

Figure 7b, the reinforcing bar was coupled to the outer layer with a frictionless connection.

Following the testing procedure outlined in

Section 2.2.3, six numerical models were generated. Due to the first symmetry BC, for each cable the simulations were carried out for L equal to 135 mm, 120 mm and 105 mm. In order to take into account the strain variation beyond the bonding length, in the numerical model the two cables were extended by 50 mm, which is more than 10 cable diameters in both cases. This choice is the result of a tradeoff between minimizing the computational cost of the simulation and avoiding any alteration of the strain transfer in the extended region, i.e., for

$x>L$.

The different strain levels were imposed applying a fixed displacement along the x direction, $\u2206u$, in the specimen cross section at the end of the bonding length ($x=L)$. The value of $\u2206u$ for the different load cases was obtained from the average strain values measured by the strain gauges, $S{G}_{avg}$, and the corresponding bonding lengths.

Finally, in order to compare the results with the experiments, the computed strain profile along the fiber was convolved with the interrogator resolution according to Equation (29). The latter corresponds to the shift resolution, $\u2206x$, selected in the data processing area of the OBR 4413 system.