Octavio Andrés González-Estrada

Alberto Pertuz
email: bapertuzc@uis.edu.co

Jabid Quiroga

Evaluation of Tensile Properties and Damage of Continuous Fibre Reinforced 3D-printed Parts

Keywords: additive manufacturing, elastic properties, fibre reinforced 3D printing, mechanical properties, composites

printing technology has been traditionally used for the production of prototypes. Recently, developments in 3D printing using Fused Deposition Modelling (FDM) and reinforcement with continuous fibres (fiberglass and carbon fibre), have allowed the manufacture of functional prototypes, considerably improving the mechanical performance of the composite parts. In this work, we characterise the elastic tensile properties of fibre reinforced specimens, considering the variation of several parameters available during the printing process: fibre orientation, volume fraction, fill pattern, reinforcement distribution. Tensile tests were performed according to ASTM D638 to obtain Young's modulus and ultimate strength for different material configurations available during the printing process. We also perform a fractographic analysis using Scanning Electron Microscopy (SEM) to give an insight of the failure mechanisms present in the specimens.

Introduction

Three-dimensional (3D) printing technologies have emerged as a new paradigm for machine design. Functional parts produced by additive manufacturing (AM) are changing the way practitioners tackle design bounds, with models highly optimised using more advanced CAD/CAE tools. Nowadays, the 3D printing industry is growing fast, and moving from the segment of rapid prototyping (RP) and marketing to include fully functional machine parts, leading to a new industrial revolution [START_REF] Bogue | 3D printing: the dawn of a new era in manufacturing?[END_REF], [START_REF] Waterman | Rapid Product Development in the USA, Europe and Japan[END_REF]. Most popular 3D printing technologies include: fused deposition modelling (FDM), selective laser melting (SLM), stereolithography (STL), laminated object manufacturing (LOM) [START_REF] Ian Gibson | Additive Manufacturing Technologies: 3D Printing, Rapid Prototyping, and Direct Digital Manufacturing[END_REF], and are available for printing polymers [START_REF] Dawoud | Mechanical behaviour of ABS: An experimental study using FDM and injection moulding techniques[END_REF], [START_REF] Carneiro | Fused deposition modeling with polypropylene[END_REF] and metals [START_REF] Martínez | Evaluación de las propiedades tribológicas de materiales compuestos de matriz metálica ( MMCs ) procesados por técnicas de fabricación aditiva con haz láser ( SLM )[END_REF], [START_REF] Olakanmi | A review on selective laser sintering/melting (SLS/SLM) of aluminium alloy powders: Processing, microstructure, and properties[END_REF].

Several authors have worked on the evaluation of the dimensional accuracy and the mechanical properties of parts manufactured by FDM, both in industrial and desktop printers [START_REF] Dawoud | Mechanical behaviour of ABS: An experimental study using FDM and injection moulding techniques[END_REF], [START_REF] Durgun | Experimental investigation of FDM process for improvement of mechanical properties and production cost[END_REF]- [START_REF] Domingo-Espin | Mechanical property characterization and simulation of fused deposition modeling Polycarbonate parts[END_REF], desktop printers usually using ABS, polycarbonate (PC) or polylactic acid (PLA) as thermoplastic raw materials [START_REF] Roberson | 3D printer selection: A decision-making evaluation and ranking model[END_REF]. Research on new thermoplastic materials has been done, including thermoplastics with embedded metallic particles [START_REF] Sa'ude | Melt Flow Behavior of Metal Filled in Polymer Matrix for Fused Deposition Modeling (FDM) Filament[END_REF], reinforced with short carbon fibres [START_REF] Gardner | Additive Manufacturing of Multifunctional Components Using High Density Carbon Nanotube Yarn Filaments[END_REF] or natural fibres [START_REF] Dudek | FDM 3D printing technology in manufacturing composite elements[END_REF]- [START_REF] Cardona Uribe | Possibilities of rice husk ash to be used as reinforcing filler in polymer sector -a review[END_REF].

In 2014, a new FDM technology using continuous filament fabrication (CFF™) was presented [START_REF] Inc | Mechanical Properties[END_REF]. The printer allows printing functional parts of composite nylon reinforced with continuous fibreglass, carbon fibre, etc. The addition of carbon fibres considerably increases the stiffness and strength of the pieces, compared to those made in ABS. However, to be able to use these functional parts, and introduce this technology in the design process, it is necessary to investigate the mechanical response of the composite printed parts and the effect of the different printing parameters.

The use of carbon fibre reinforcement was studied by Mori et al. [START_REF] Mori | Dieless forming of carbon fibre reinforced plastic parts using 3D printer[END_REF] using a RepRap, replicating rapid prototype. However, the fibres were manually set during the printing process with a fixed orientation and distribution, and not enough thermal bonding to improve the adherence in the fibrematrix interface. Melenka et al. [START_REF] Melenka | Evaluation and prediction of the tensile properties of continuous fiber-reinforced 3D printed structures[END_REF] studied the effect of the addition of concentric rings of continuous fibres on a nylon matrix. It was verified that the modulus of elasticity and ultimate tensile strength increase with the volume fraction. Additionally, a prediction model of the elastic constants was developed using the average stiffness method.

For industrial applications, to ensure that fibre-reinforced 3D-printed components meet the required design specifications, it is necessary to understand their tensile mechanical properties. In this work, the behaviour under tensile stress tests of 3D-printed specimens manufactured in nylon matrix and reinforced with continuous fibres of glass or carbon is characterized. For this study, the 3D printer Mark Two from Markforged [START_REF]Mark Two -Markforged[END_REF] was used. In the next section, we present the parameters of the tensile test according to ASTM D638 [START_REF]Standard test method for tensile properties of plastics[END_REF] and the parameter configurations chosen to manufacture the test specimens. Then, we present the results, including an analysis of the dimensional accuracy of the printer, the results obtained from the tensile tests for the modulus of elasticity, ultimate strength and Poisson's coefficient, for each material configuration. We perform a comparison between different fill patterns, fibre orientation and fibre material. Finally, conclusions are presented.

Materials and methods

Mechanical tests. The 3D-printed specimens were evaluated by performing tensile tests using an MTS Bionix 370.02, with a load cell of 25 KN. Longitudinal deformation and contraction of the cross-section of the test specimens were measured using the LVDT of the machine and an LX500 laser extensometer, with a test speed of 5 mm/min. The tensile test specimens were produce using the printer Mark Two from Markforged, following ASTM D638, geometry type IV [START_REF]Standard test method for tensile properties of plastics[END_REF], as shown in Figure 1. STL files were produced in a CAD software and exported to the software Eiger for preparation. a Figure 1. Geometry of the specimen for tensile test of 3D-printed parts according to ASTM D638 type IV.

Printing parameters and internal structure. The 3D printer Mark Two allows the variation of several printing parameters which could potentially modify the mechanical response of the final produced part. In Error! Reference source not found. we show the chosen values for different printing configurations. In the experimental tests, we studied the influence of the density, fill pattern, fibre material, and fibre orientation on the mechanical response.

Results

Dimensional accuracy. The measurement of the geometry of the specimens was carried out to evaluate the dimensional accuracy of the Mark Two 3D printer, and to verify the cross-sectional area of the specimens. The measurements were compared with the nominal dimensions according to the ASTM D638 TYPE IV standard. A t-Student test with a p-value significance test of <0.05 was applied to statistically determine the difference between the nominal value and the measured values for the section widths Wc, WO and the thickness t, with the highest percentage difference for the thickness, Table 2. The results are consistent with other studies evaluating the dimensional accuracy of 3Dprinted parts [START_REF] Melenka | Evaluation of dimensional accuracy and material properties of the MakerBot 3D desktop printer[END_REF]. Table 1. Printer parameters for nylon and continuous filament fabrication (CFF). Table 2. Measurements of the thickness (t) for reinforced specimens.

Tensile tests results.

The effect on the mechanical properties of the variation of material parameters defined during the printing process is investigated. The modulus of elasticity, the ultimate tensile strength and Poisson's ratio of different configurations were determined from the experimental results. In Table 3 we show the results for two fill densities for the nylon matrix: 20% and 50%, using three different fill patterns: triangular, rectangular, and hexagonal. As expected, the triangular pattern yields higher values for the stiffness and ultimate strength. Increasing the fill density to 50% only increases the elastic modulus by 3.3%, and the strength by 5.5%. Table 3. Experimental mechanical properties at 20% and 50% density for different filling patterns.

In Table 4 we show the results for specimens printed with continuous filament fabrication, using fibreglass at different orientations: 0°, 45° and 90° with respect to the axial direction of the load. We observe the effect of the reinforcement with fibreglass at 0°, which yields the highest elastic modulus (1610 MPa) and ultimate strength (83 MPa). With fibres aligned with the transverse direction, the strength and stiffness are characterized by the nylon matrix. We produced specimens with different types of fibres: fibreglass and carbon fibre, using the parameters in Table 1, see Figure 2. We observe that with carbon fibre reinforcement we reach stiffness values of 4.4 GPa and ultimate strength of 104 MPa, Table 5, for the chosen printing configuration. Table 5. Experimental mechanical properties of CFF fibreglass and carbon fibre.

Microscopy. Three specimens of 3D-printed parts reinforced with fibreglass were prepared for fractographic analysis using scanning electron microscopy (SEM). We show in Figure 3 the images for fibres oriented at (a) 0° and (b) 45°, corresponding to the cross-section in the fracture area and a close-up view of the reinforced areas. The reinforced roof and floor layers, and the infill layer can be clearly identified. Composite models assume perfect bonding between the fibres and the matrix. Note that for the 0° orientation we can see fibre pull-out, which indicates imperfect fibre-matrix bonding.

Summary

Three-dimensional printing using FDM with continuous filament fabrication (CFF) has potential to produce functional parts with high strength and stiffness. To be able to use this technology for industrial applications it is necessary further investigation on the mechanical response of the printed parts. We performed ASTM tensile tests to evaluate the elastic modulus and ultimate strength of FDM-printed parts, modifying printing parameters to obtain different composite material features. Moreover, we obtained SEM images of the fractured cross section to observe the damage mechanisms of the fibre, matrix and interface. Different fibre patterns prove useful to improve mechanical properties of the composite 3D-printed parts.

Figure 2 .

 2 Figure 2. ASTM D638 type IV specimens for tensile tests printed with: fibreglass (top) and carbon fibre (bottom) reinforcement.

Figure 3 .

 3 Scanning electron microscopy (SEM) fractography of 3D-printed parts reinforced with fibreglass, cross section and close-up: (a) fibres oriented at 0°, (b) fibres at 45°.

Table 4 .

 4 Experimental mechanical properties of CFF reinforced with fibreglass.


Juan S León

O A González-Estrada

A Pertuz
email: capertuzc@uis.edu.co

Damage in fibreglass composite laminates used for pipes

Keywords: continuum damage mechanics, finite element, composite pipe, fiberglass, damage evolution

In this work, we present a model for the initiation and evolution of damage for a composite fibre-reinforced pipe used in the Oil & Gas industry, based on a commercially available pipe. A continuum damage mechanics model was employed to determine the initiation and evolution of damage. This model was implemented using finite element analysis to investigate the performance of the commercial composite pipe. Initially, the material properties were obtained from experimental data and fitting with static structural simulations. Then, FE simulations with damage were performed, considering three different boundary conditions: open, closed (pressure-vessel type) and fixed ends, the load considered was internal pressure. Results showed differences not only in the stress distribution but on the damage initiation and evolution along the geometry of the pipe. These differences in the damage initiation and propagation can be explained as the result of different axialhoop stress ratio.

Introduction

Composite pipelines have numerous competitive advantages over metal pipelines in Oil & Gas sectors [START_REF] Sas-Jaworsky | Composite Spoolable Pipe Development , Advancements , and Limitations[END_REF]. The costs of installation, operation and maintenance are reduced because they are sold in rolls, requires less personnel and equipment for installation, do not use welding, coatings or X-rays and, most importantly, do not present corrosion. However, the evaluation of structural integrity and the remaining life is more complex than in steel, due to the anisotropic properties of the reinforcing layer.

In this work, we consider the composite pipe as an elastic material coupled with damage, a model which determines the initiation of the damage and the corresponding law of evolution is employed. First, the determination of the stress and strain field has been obtained, then a continuum damage model is used to account for the initiation and progression that describes the nonlinear response of the reinforcement layer for internal pressure. The influence of axial stress in the damage is investigated to determine the relationship between boundary conditions at initiation and propagation of damage.

The stresses generated in the pipe depend on the loads, the geometry, the material properties and the boundary conditions [START_REF] Gómez | Comparative study of the mechanical and vibratory properties of a composite reinforced with fique fibers versus a composite with E-glass fibers[END_REF], [START_REF] Watkins | Structural Mechanics of Buried Pipes[END_REF]. The loads are usually defined in the design stage, the geometry is a circular profile with a specific thickness, the material properties are given by the composite material configuration and the boundary conditions depend on the installation setup. The most typical boundary conditions for internal pressure pipes are: fixed, closed and open. In the closed-end condition, the pressure acts on the lateral faces, which creates longitudinal stresses, in isotropic pipes these are given by 𝜎𝑎=𝑃𝑟/2𝑡 [START_REF] Watkins | Structural Mechanics of Buried Pipes[END_REF]. In the fixed -end conditions, the x-displacement is restricted at ends such that longitudinal stresses are generated, with a value of 𝜎𝑎=𝜈𝑃𝑟/𝑡, related to the Poisson effect [START_REF] Watkins | Structural Mechanics of Buried Pipes[END_REF]. Finally, in the open-end condition ideally there are not longitudinal stresses because very elastic supports exist at the end. Hoop stresses are given by 𝜎=𝑃𝑟/𝑡 [START_REF] Watkins | Structural Mechanics of Buried Pipes[END_REF]. However, in anisotropic cylinders the stress distribution is not as simple as those from the equations aforementioned. Xia et al. [START_REF] Xia | Analysis of multi-layered filament-wound composite pipes under internal pressure[END_REF], based on the anisotropic tridimensional elasticity, give an exact solution for the stresses and strains of an internal pressure pipe. Sun et al [START_REF] Sun | Stress analysis of multi-layered hollow anisotropic composite cylindrical structures using the homogenization method[END_REF] have developed a stress analysis which follows the Lekhnistskii formalism, that uses an homogenization method to describe the response of hollow composite cylindrical structures consisting of multiple layers of different anisotropic materials, subjected to different loads such as pressure, axial force, torsion, shear and bending.

While there have been developed analytical methods to solve exactly the stress distribution on a multilayer composite pipe, numerical methods are still used due to their simplicity and accuracy of the results, mostly because no assumptions or simplifications are made. General 3D stress distributions are obtained, and in problems concerning damage, an analytical analysis would be difficult to perform [START_REF] Kunert | Nonlinear FEM strategies for modeling pipe-soil interaction[END_REF], [START_REF] Anping | FEA-based Comparison of Two Kinds of Steel Wire Reinforced Composite Pipes[END_REF].

If the stresses are known, it is possible to apply a failure criterion to determine if the structure does not sustain irreversible damage. Nevertheless, if the load is increased beyond that point, damage initiates. A damage progression must be done to know the functional failure of a structure. Damage is responsible for the non-linear response of pipes [START_REF] Ferry | Interaction between plasticity and damage in the behaviour of [+φ, -φ]n fibre reinforced composite pipes in biaxial loading (internal pressure and tension)[END_REF], and it can be difficult to treat damage in composites due to different mechanisms, scales and analysis approach used. Ferry et al. [START_REF] Ferry | Interaction between plasticity and damage in the behaviour of [+φ, -φ]n fibre reinforced composite pipes in biaxial loading (internal pressure and tension)[END_REF] show that pipes exhibit varying types of damaged elastoplastic behaviour depending on stress ratio axial /hoop stress, which changes for the different BCs as stated above. The extent of damage and plastic phenomena are responsible for non-linearity on stress-strain curves. They show by micrographic analysis that microcracking is the main damaging process.

In [START_REF] Liu | Failure analysis of carbon fiber/epoxy composite cylindrical laminates using explicit finite element method[END_REF], the authors show that an important parameter for the failure evolution is stiffness degradation. A progressive failure analysis is made using continuum damage mechanics (CDM) for a composite pressure vessel with an aluminium layer. An explicit finite element method is used, which proved to be more convenient due to the convergence issues of the implicit method. Almeida et al. [START_REF] Almeida | Damage and failure in carbon/epoxy filament wound composite tubes under external pressure: Experimental and numerical approaches[END_REF] proposed a damage model for a carbon-fibre reinforced composite pipes under external pressure. A non-linear geometric analysis was performed based on the arc-length method to predict the buckling load. Also, a damage model using CDM was developed to evaluate de gradual stiffness degradation, showing good agreement between experimental and numerical results.

Finite element and damage model

Typically, a composite pipe for oil transport is formed by an internal thermoplastic pressure barrier, an adhesion layer, a laminate which may be glass fibre reinforced with an epoxy matrix and on the outside, is covered by a wear resistance thermoplastic layer. The study was made on a Fiberspar© pipe with an internal diameter of the HDPE layer equal to 2 inches (50.8 mm), the internal diameter of the epoxy laminate is 55.78 mm, the internal diameter of the exterior HDPE is 59.79 and the outer diameter is 64.77 mm.

For the first ply failure (FPF), the Hashin's criterion [START_REF] Hashin | Failure Criteria for Unidirectional Fiber Composites[END_REF] was used. It has two types of failure mechanisms that are considered: in the fibre and in the matrix. Two expressions are used for identifying each failure by considering separately the traction and compression. Above the FPF load, the stiffness of the laminate will change and a more complex analysis must be performed which is called progressive damage or failure. Due to the ease of implementation and relatively good results [START_REF] Barbero | Finite Element Analysis of Composite Materials Using ANSYS, Second[END_REF], a CDM method was chosen for the modelling of damage in the multilayer composite pipe. The CDM method chosen is presented in Matzenmiller et al. [START_REF] Matzenmiller | A constitutive model for anisotropic damage in fiber-composites[END_REF]. Damage change the material stiffness 𝐶 in function of an internal variable 𝐶(𝐷) by considering the evolving and irreversible conditions of the damage. The response of a damaged solid is a nonlinear function of the applied load and depends on the history load.

Experimental procedure

For the evaluation of the mechanical properties of the composite laminate, a tensile test was performed on a UMT Bionix tensile machine, using a strip of a longitudinal cut of the composite pipe and, then, compared to a numerical model, Figure 1

Validation and results

We define a numerical model for the multilayer composite pipe using the finite element method. Material properties of the pipe are shown in Table 1. [START_REF] González-Estrada | Análisis de integridad estructural de tuberías de material compuesto para el transporte de hidrocarburos por elementos finitos[END_REF]. The axial stress distribution was plotted with respect to a dimensional radius defined with respect to the outer and inner radius as R=(r-rint)/(rout-rint).

Figure 2 shows the axial stress distribution with respect to the boundary condition. Note that axial stress varies with the thickness, with higher values at the stiffer reinforcing layer, as expected. The highest axial stress is for the closed-end condition and the lowest value for the open-end. For the three boundary conditions tested, the hoop stress distribution shows the same value [START_REF] González-Estrada | Análisis de integridad estructural de tuberías de material compuesto para el transporte de hidrocarburos por elementos finitos[END_REF]. These results are in agreement with those presented by Xia et al. [START_REF] Xia | Analysis of multi-layered filament-wound composite pipes under internal pressure[END_REF].

Figure 2. Axial stress distribution along the thickness

Once the stress and strain fields are obtained, it is possible to apply a failure criterion to determine the first ply failure. Thus, the load was increased until the FPF pressure was obtained according to three different failure criteria, shown in Table 2. From that point, the progressive damage analysis initiates, using load steps, and the solution for each incremental step is obtained and the stresses, strains and damage status of each layer is found. The values of the damage parameters used were: energy dissipated per unit area for tensile damage 490 [N/mm], for compressive fibre damage 10 [N/mm], and for matrix 11 [N/mm]. All the viscous regularization variables were set to 0.01.

Figure 3 shows the stress distribution for the first layer of a closed end condition, for two different values of pressure: 8 MPa and 12 MPa. The stress distribution tends to concentrate at the edges, which is in agreement with the work of Baranger et al. [15]. This non-uniform stress distribution at the edges propagates to other layers and, for higher loads, we even find more non-uniform stress distributions, as shown in Figure 4 for the most external HDPE layer at a load of 15 MPa. The burst strength was found numerically to be 22 [MPa], which is the point when a single thickness point fails in all the layers, whereas the experimental burst strength is 27 [MPa].

Conclusions

This model is a useful tool in the assessment of the remaining life of composite pipes. Under real conditions, damage can be locally initiated due to different events such as impacts, earthquakes, thermal distortion or interaction with the environment, this damage can be measured and evaluated using the above-mentioned method.

The model for the multilayer composite pipe was numerically defined. Interest was given in the definition of the fibre reinforced layer, and the model was validated through experimental tests. It was shown that the FPF pressure for the open-end or pure internal pressure condition was the highest.

The damage progression initiates near the boundary conditions on the cross sections, and propagates towards the centre of the pipe. This damage can lead to delamination which is not considered in this study. The stiffness degradation causes an irregular stress distribution that propagates to different layers. Matrix cracking is the most critical mechanism of damage in composite pipes.

The functional failure predicted by the open-end condition is near 22 MPa (the point at which an element in all layers has failed). The experimental burst strength pressure was about 27 MPa. Differences can be accounted to the fact that, for the burst strength, an entire region must fail.

Figure 1

 1 Figure 1. a) Tensile test on a commercial pipe, b) tension on hand laid-up specimen, c) compression on hand laid-up specimen.

Figure 3 .

 3 Hoop stress distribution of the first layer for: a) 8MPa, and b) 12 MPa.

Figure 4 .

 4 Figure 4. Hoop stress distribution of HDPE outer layer.

Table 1 .

 1 Material properties for the layers of the composite pipe.

		Young´s Modulus	Poisson Ratio	Shear Modulus
	Layer	E1 [MPa]	E2 [MPa]	E3 [MPa]	ν12 ν23 ν13	G12 [MPa]	G23 [MPa]	G13 [MPa]
	Interior HDPE	1,340	/	/	0.4	/	/	478.5	/	/
	Epoxy-glass	35,000	9,000	9,000 0.28 0.4 0.28 4,700	3,500	4,700
	fiber laminate
	Exterior HDPE	1,340	/	/	0.4	/	/	478.5	/	/

For the model, we use the following stress limits: tensile in X, 780

[MPa]

, tensile in Y, 31

[MPa]

, compressive in X, 480

[MPa]

, compressive in Y, 100

[MPa]

, shear in YZ, 35

[MPa]

and shear in other directions, 60

[MPa]

. A nominal internal pressure load of 5 MPa was applied, considering the three different boundary condition. The stresses across the thickness were obtained, stresses do not change with the theta or Z coordinate

Table 2 .

 2 FPF for different BC using different failure criteria.

		Open-end	Fixed-end	Closed-end
		[MPa]	[MPa]	[MPa]
	Max. stress	13.73	11.15	9.25
	Tsai-Hill	10.94	9.96	8.62
	Hashin	11.06	10.04	8.68


Jabid Quiroga

Octavio Andres Gonzalez-Estrada

Yesid Rueda Ordonez

Giema

Stress Sensitivity of the T(0,1) mode velocity for cylindrical waveguides

Keywords: Torsional guided waves, Stressed cylinders, phase velocity, stress monitoring

In this paper, the stress influence in the guided wave velocity of the fundamental torsional mode is presented. Two analytical models, based on the Acoustoelasticity effect, to compute the fundamental torsional mode velocity propagating in a specimen subject to an axial stress are studied. These models are obtained due to the relation between the T (0, 1) guided wave velocity and the bulk shear velocity. The analytical models to calculate the guided wave velocity are functions of the stress, second and third order elastic constants. A series of axial stress levels applied to a cylindrical waveguide is investigated with numerical simulations (Finite Elements) to estimate variations of the T (0, 1) guided wave velocity. This analysis provides a criterion to evaluate the practical implementation of a stress monitoring scheme based on velocity variations of the fundamental torsional mode.

Introduction

Lately, guided waves are gaining the attention of the NDT and SHM community. Guided waves are suitable to explore large material volume for long distances from one probe position. all material discontinuity (notches, cracks, erosion, etc) is revealed on the wave field once the ultrasonic pulse interacts with the material disturbance. The ultrasonic guided wave analysis allow the inspection of buried and underwater structures, coated structures, and structures encapsulated in insulation and concrete. Inspection of tubes and pipelines by guided waves is probable the most popular application of this technique. However, in order to have success in pipe inspection, a deep understanding of the technique is required. Several factors may affect the guided wave propagation such as environmental conditions, material structure and the presence of stress in the waveguide. Stressed media as predicted by the Acoustoelasticity effect produces velocity variations in the bulk wave, longitudinal and transversal, and consequently in the guided waves generated by them.

The stress influence in guided waves have been tackled in several studies, the change in the guided wave velocity has been used to stress monitoring in plates with biaxial loads [START_REF] Shi | In situ estimation of applied biaxial loads with Lamb waves[END_REF], in bolts to verify tightening [START_REF] Chaki | Combination of Longitudinal and Transverse Ultrasonic Waves for In Situ Control of the Tightening of Bolts[END_REF], in rails [START_REF] Bartoli | Stress dependence of ultrasonic guided waves in rails[END_REF], [START_REF] Chen | The effect of load on guided wave propagation[END_REF], [START_REF] Loveday | Semi-Analytical Finite Element Analysis of the Influence of Axial Loads on Elastic Waveguides[END_REF], steel strands [START_REF] Chaki | Guided ultrasonic waves for non-destructive monitoring of the stress levels in prestressed steel strands[END_REF], [START_REF] Chaki | Stress level measurement in prestressed steel strands using acoustoelastic effect[END_REF], [START_REF] Nucera | Monitoring load levels in multi-wire strands by nonlinear ultrasonic waves[END_REF], in grouted tendons [START_REF] Beard | Ultrasonic Guided Waves for Inspection of Grouted Tendons and Bolts[END_REF] and finally in pipes [START_REF] Jabid E Quiroga | Guided Ultrasonic Wave for Monitoring Stress Levels in Pipelines[END_REF]. Among guided waves families of modes that propagate in cylindrical structures i.e. longitudinal, flexural and torsional, the latter is preferred in some inspections. Torsional guided waves barely are affected by fluids inside or outside of the cylinder, so the energy leakage from the waveguide to the fluid is minimal because of the shearing nature of this type of guided wave. In addition, the fundamental torsional mode T (0, 1) is non-dispersive i.e. phase and group velocities are frequency independent and equal. Finally, due to the axisymmetric character of T (0, 1) the wavefield energy is distributed uniformly over the cross-section avoiding blind zones to the ultrasonic exploration.

two analytical models to compute the phase velocity of the T (0, 1) mode propagating in a specimen subject to an axial stress are studied based on the acoustoelasticity effect. A series of FEM simulations are performed to study the propagation of torsional guided waves in presence of stress in the waveguide.

Fundamental torsional guided waves in cylindrical waveguides

A simplified analytical model of torsional guided waves assumes that the hollow cylindrical system is geometrically axisymmetric, infinitely long, and stress-free in the boundaries. The material is elastic, homogeneous and isotropic. The waves will be assumed to be continuous, the frequency real (steady state), and the energy is finite and constant. The solutions to motion equation will only be sought explored for guided waves, which are propagated axially.

The torsional modes are characterized mainly by a displacement primarily in the θ-direction. The axisymmetric torsional mode corresponds to an uniform azimuthal displacement in θ-direction (angular displacement) of the entire cylindrical waveguide.

The generalized guided wave propagation model is based on the combination of Euler's equation of motion and the generalized Hooke's law. Both relations yield the Navier's displacement equation of motion given by

(λ + 2µ) ∇ (∇ • u) + µ∇ × (∇ × u) = ρ ∂ 2 u ∂t 2 , ( 1
)
where u is the displacement vector, ρ is the mass density per unit volume and λ, µ are the Lame's constants. The resulting Navier's equations, where (λ + 2µ) ∇ (∇ • u) considers the dilatation (compressional) portion and, µ∇ × (∇ × u) the rotational (equivoluminal) portion of the model. These two terms are decoupled by the Helmholtz decomposition and individually belong to a wave equation of the scalar field Φ and the H vector field are written as:

C 1 ∇ 2 Φ = ∂ 2 Φ ∂t 2 , (2)
C 2 ∇ 2 H = ∂ 2 H ∂t 2 , (3)
where C 1 and C 2 are the longitudinal and shear bulk velocities respectively. Since Equation ( 1) is separable in cylindrical coordinates, the solution may be divided into the product of functions of each one of the spatial dimensions in cylindrical coordinates. Assuming no propagation in the radial direction (r) and the displacement field without variation in θ-direction or z-direction except for the harmonic oscillation described by the wavenumber, the wave fields can be described by:

Φ, H = Γ Φ,H (r) e ipθ e i(ξz-ωt) , (4)
where Γ Φ,H (r) describe the wave field variation in r coordinate, ξ is the component of the complex vector wavenumber in the z-direction, since only propagation in the cylinder axis-direction is considered, p is referred to as the circumferential order which must be a whole number.

Recalling from Equation (1) and Hooke's law, the field variables such as displacements and stresses can be expressed in terms of potential functions, which can be numerically solved (see [START_REF] Rose | Ultrasonic Waves in Solid Media[END_REF][START_REF] Viola | Exact Analysis of Wave Motions in Rods and Hollow Cylinders[END_REF][START_REF] Auld | Acoustic Fields and Waves in Solids[END_REF][START_REF] Graff | Wave Motion in Elastic Solids[END_REF] for more details on this subject).

On the other hand, the family of torsional modes results when only the u θ displacement is assumed to exist (u r , u z = 0). Such a displacement field is obtained only if rotational potential function in z, h z = 0. Then, for sake of brevity, only the expressions for u θ , h z , and the stress (σ rθ as boundary condition), are used in forward to study the axisymmetric torsional modes as follows [START_REF] Graff | Wave Motion in Elastic Solids[END_REF].

h z (r) = K 1 J p (βr), β 2 = ω 2 C 2 2 -ξ 2 (5)
u θ = h z (r) cos (pθ) e i(ξz-ωt) , (6)
σ rθ = µ -2h z -β 2 h z , (7)
where J p is the Bessel function of the first kind an order p, r is the cylinder radius, and K 1 is a constant. For the case of the axisymmetric modes, p=0, and by using the property of Bessel functions J 0 ' (x) = -J 1 (x), the corresponding u θ can be expressed as:

u θ = - ∂h z ∂r = (K 1 β)J 1 (βr) e i(ξz-ωt) . (8)
The frequency equation for the torsional modes may be obtained by using the stress-free boundary condition Eq.( 7) as follows:

β 2 a 2 J 0 (βa) -2βaJ 1 (βa) -β 2 b 2 J 0 (βb) -2βbJ 1 (βb) = 0. ( 9
)
where a and b are the inner and outer radius of the waveguide respectively . The fundamental torsional mode, the first root of Equation ( 9), is β = 0. But, this mode is not adequately described by the Bessel equations [START_REF] Armenakas | Free Vibrations of Circular Cylindrical Shells[END_REF]. So this particular mode is described by the RHS relation expressed in Eq. ( 5) which will be used to obtain the phase velocity of T (0, 1).

V p = ω ξ = C 2 = D 66 ρ (10)
where D 66 belongs to the lineal elasticity matrix of an isotropic specimen. Eq. [START_REF] Jabid E Quiroga | Guided Ultrasonic Wave for Monitoring Stress Levels in Pipelines[END_REF] shows that T (0, 1) propagates at a constant phase velocity equivalent to the bulk shear velocity of the material.

2 The Acoustoelasticity Effect in T(0,1) mode

The acoustoelasticity effect in mechanics establishes the mathematical relationship between ultrasonic bulk velocities and mechanical stresses in the studied material. In this sense, the acoustoelasticity establishes five elastic constants to describe the relation between stress and bulk wave velocity for isotropic materials subject to uniaxial stress; the second order elastic constant, Lamé constants (λ, µ) and the third order elastic constants (TOEC), Murnaghans constants (m, n, l), in our case. The TOEC highly depend on the material processing, such as casting, rolling, or drawing. Now, acoustoelasticity in [START_REF] Duquennoy | Effective elastic constants in acoustoelasticity[END_REF], it is tackled by replacing the elastic constants (second order) in the stiffness tensor with effective elastic constants (EECs) to produce a modified elasticity matrix which terms consider the influence of stress. In particular, the term D 66 is re-expressed in EEC as:

D 66 = µ + σ E 3µ + n 4 + (1 -2ν) λ + m - n 4 ( 11
)
where E is the Youngs modulus, ν is the Poisson ratio and σ is the stress (positive for tension and negative for compression stresses). On the other hand, in [START_REF] Chaki | Guided ultrasonic waves for non-destructive monitoring of the stress levels in prestressed steel strands[END_REF] a first order approximation, considering finite deformation theory and third-order terms, for the acoustoelastic shear velocity propagating in the same direction as the applied stress is stated as follows:

C 2 = µ ρ 1 + σ 2µ (3λ + 2µ) 4λ + 4µ + m + λn 4µ . (12)
So, based on the result expressed by Eq.( 10) and the analytical models of the acoustoelastic shear velocity expressed by Eqs.( 11)- [START_REF] Viola | Exact Analysis of Wave Motions in Rods and Hollow Cylinders[END_REF] two approximate expressions for calculating the stressed phase velocity of the T (0, 1), are given by:

V σ T (0,1) = µ ρ 1 + σ 2µ (3λ + 2µ) 4λ + 4µ + m + λn 4µ . ( 13
) V σ T (0,1) = µ + σ E 3µ + n 4 + (1 -2ν) λ + m -n 4 ρ (14)
where V σ T (0,1) is the velocity of the T (0, 1) mode when the torsional wave propagates under a stress (σ) along the cylinder axis, ρ is the mass density. Eq. ( 13) represents the approach named in figures as finite deformation, meanwhile Eq. ( 14) is the EEC approach. Expression inside curly brackets in Eq. ( 13) represents the stress influence in phase velocity of T (0, 1).

Clearly, the new expression for the phase velocity of the T (0, 1) depends on the second and third order elastic constants and the applied stress. In addition, in absence of dispersion, phase and group velocities are equal.

Although, T (0, 1) presents notable advantages as mentioned above (no dispersion and no leakage of energy through the fluid), is few sensible in term of velocity change to the acoustoelasticity effect. Experimental tests showed in [START_REF] Egle | Measurement of Acoustoelatic and Third-Order Elastic Constants for Rail Steel[END_REF], the smallest relatives changes associated with shear waves propagating perpendicular to the load and polarized perpendicular to the axial loads.

Sensitivity analysis of the stressed wave velocity of T(0,1) mode

In order to analytically estimate the variations of T (0, 1) velocity produced by uniform axial stress applied to the cylinder, the Murnaghan's and Lame's constants are required. In this work, it is used the reported values in [START_REF] Egle | Measurement of Acoustoelatic and Third-Order Elastic Constants for Rail Steel[END_REF]. Although the steel composition may be different among specimens, the close agreement of the reported TOEC values for different steels suggest that TOEC for steel

Finite deformation approach EEC approach

Figure 1: Velocity of T(0,1) for different stress levels (σ) may be relatively independent of the specific composition. Here, ρ = 7800kg/m 3 , µ = 79.9 GPa, λ = 115.8 GPa, l = -248 GPa, m = -623 GPa, and n = -714 GPa are considered. Fig. 1 shows the velocity of the fundamental torsional mode propagating in a specimen under different stress levels (-40 to 40 MPa). This plot is obtained by using Eqs.( 13)-( 14), where small velocity variations can be noted with a difference between the extreme cases of only 1 m/s. Comparing both models, the slope difference suggests an increasing gap in calculated velocity for greater stress values. For this reason, it is performed a FEM simulation to verify which approach provides a velocity value closer to the numerical model. Finally, as presented in Fig. 1 velocity magnitude increases for compression stresses and the opposite for tension stresses.

The practical implementation of a stress monitoring scheme based on variations of the phase velocity has some difficulties due to its small effect. Nevertheless, in order to determine the wave velocity change, a Time of Flight (TOF) estimation of the waves at different load conditions are performed. Now, the TOF value in practice also consider the material elongation by applied load. Thus, the effect of the material elongation in the variation of the time of flight with respect to the stressfree condition has to be addressed. The variation of time of flight ∆ (T OF ) comparing with the stress-free condition is given by:

∆ (T OF ) = T OF σ -T OF 0 , (15)
where T OF σ is the time of propagation of T (0, 1) mode subject to an axial load, T OF 0 for the stress-free wave. T OF σ is affected by the acoustoelasticity effect and the material elongation caused by the applied load as follows.

T OF 0 = l 0 V 0 , (16)
T OF σ = l 0 (1 + ) V σ , (17)
where l 0 is the original length of the material, is the strain after the load is applied, V σ and V 0 are the phase velocity of T (0, 1) affected by the acoustoelasticity and stress-free respectively. Replacing Eqs ( 16)- [START_REF] Egle | Measurement of Acoustoelatic and Third-Order Elastic Constants for Rail Steel[END_REF] in Eq.( 15) and using the constitutive relation yields.

∆ (T OF ) = l 0 V 0 V 0 (1 + (σ/E)) -V σ V σ , ( 18
)
where E is the elasticity modulus. Based on the results presented in Fig. 1 and Eq. ( 18), it can be concluded that both effects (material elongation and acoustoelasticity) contribute to increase ∆ (T OF ) independently the kind of stress (tension or compression). Therefore, Eq. ( 18) suggests the use of the ∆ (T OF ) as a stress indicator in a stress guided wave monitoring scheme based on the propagation of T (0, 1). Two analytical expressions of fundamental torsional mode velocity propagating in a medium under mechanical stress are studied based on the acoustoelasticity effect. The expression depends on the material properties including the Lamé and the Murnaghans constants and the applied stress. 3 Finite Element Method approach

In order to investigate the Acoustoelastic effect in propagation of T (0, 1), FEM simulations are performed. A 3D FEM model was built representing a cylindrical specimen with a limited length to reduce the size of the FEM model and consequently the computational cost. A steel pipe of 1 inch schedule 40 (outer diameter: 33.4 mm and wall thickness: 3.38 mm) is modeled as a hollow cylinder with an axial length of 0.42 m. Changes in stress are configured varying the displacement of two constrains located at the left and right ends of the cylinder. The material properties used for steel were assumed as follows: Density ρ = 7830 kg/m3, Young's modulus (E) = 210 GPa and Poisson's ratio ν = 0.3. To ensure an adequate mesh refinement level, the minimum allowed inter-nodal length L min is calculated. The lowest phase velocity C T (i.e., transverse or shear wave speed), and consequently the shortest wavelength establishes the minimum permissible mesh size so spatial aliasing due to the finite element discretization does not occur [START_REF] Han | Finite Element Analysis of Lamb Waves[END_REF]. Considering the frequency and the steel shear wave velocity, L min is calculated as follows:

L min = C T n min f max = λ min n min , (19)
where, n min is the number of elements across the smallest wavelength of interest (assumed in this case as n min ≥ 10) [START_REF] Premesh Shehan Lowe | Inspection of Cylindrical Structures Using the First Longitudinal Guided Wave Mode in Isolation for Higher Flaw Sensitivity[END_REF], and f max the maximum frequency of interest. Considering n min = 15, f max = 50000 Hz and C T = 3200 m/s, the minimum element length results, approximately 6.66 mm. Therefore, seeds size of 3 mm can be considered as a sufficient mesh refinement. In addition, the simulation is executed in two stages: first, uniaxial stress is applied as boundary displacement in both end sides of the cylinder. The stress and strain fields resulting are used as a predefined field in the next stage. The first stage is executed using a standard step and the next stage an explicit scheme. Explicit schemes are preferred as time marching process to simulate guided waves. So, an adequate integration time step ∆t assures a more accurate solution. In general, simulation accuracy can be increased with increasingly smaller integration time steps but punished by a higher computational cost. So, the time step ∆t has to be smaller than the critical time step ∆t cr which is the transit time of a dilatational wave through the smallest element in the model which can be calculated by [START_REF] Brice | Efficient Finite Element Modelling of Ultrasound Waves in Elastic Media[END_REF].

∆t ≤ ∆t cr = L min C L , (20)
where C L is the velocity of the dilatational wave. A ∆t of 5 nsec meets these criterion (Considering L min = 3 mm and C L = 5944 m/s) and it is used to solve the model. A total of 9980 linear eight node brick element (C3D8) has been used with 52 elements around the circumferential section of the pipe. The torsional wave is produced by a shear load at the left end face of the cylinder by a 5 cycles Hanning-window tone burst of 50 kHz. The model is configured such as the torsional wave freely propagates along the z-axis.

As presented in Fig. 2 the estimation of the T (0, 1) velocity by EEC has a better agreement with the result of the FEM simulation. It is also observed a slope of approximately 6.1 m/s per GPa, which result in a low sensitivity expressed in time i.e. 90 nsec of ∆(T OF ) per GPa if actuator and sensor are separated by 1 meter. Although some improvements can be done in sensitivity increasing the distance between T(0,1) generation and its capture. Clearly, the exploitation of the velocity change of T(0,1) with the purpose of waveguide stress monitoring has limitations for its low sensitivity.

Figure 2 :

 2 Figure 2: Phase velocity of T(0,1) for different stress levels (σ)