A tensile test until breakage and a creep and relaxation test on a polypropylene fibre are carried out and the resulting creep and stress relaxation curves are fit by a model adopting a fraction-exponential kernel in the viscoelastic operator. The models using fraction-exponential functions are simpler than the complex ones obtained from combination of dashpots and springs and, furthermore, are suitable for fitting experimental data with good approximation allowing, at the same time, obtaining inverse Laplace transform in closed form. Therefore, the viscoelastic response of polypropylene fibres can be modelled straightforwardly through analytical methods. Addition of polypropylene fibres greatly improves the tensile strength of composite materials with concrete matrix. The proposed analytical model can be employed for simulating the mechanical behaviour of composite materials with embedded viscoelastic fibres.

Fibre-reinforced composite materials consist of fibres with high strength and elastic modulus embedded in a matrix to produce a combination of properties that cannot be achieved by single constituents. Usually, fibres are the principal load-carrying members, while the surrounding matrix keeps them in the desired location and orientation. The matrix acts as a load transfer medium between fibres and plays a number of useful functions, for example, protecting the fibres from environmental damage. Fibres may be made up of various materials, such as steel, polymer, glass, or carbon, whereas polymer, metal, or ceramic can be chosen for the matrix material.

Fibre-reinforced polymer composites are probably the most important and widespread fibre-reinforced materials used for commercial and industrial applications. This is due to the combination of their low density, strength-weight ratios, and modulus-weight ratios that make them more attractive than many traditional metallic materials [

A classic example of fibre-reinforced composite used in civil engineering is Fibre-Reinforced Concrete (FRC), widely adopted for industrial pavements, tunnel linings, marine structures, earthquake-resistant structures, and plate and slab foundation [

Recently, the use of macro synthetic fibres made of polymeric materials has been proposed for structural purposes [

Ductility and flexural strength of FRC are defined by energy-dissipation mechanisms during the pullout of the fibres that occurs in the opening propagating cracks [

Pullout of the fibres begins after cracking the concrete matrix (for the stress and strain localizations at the crack tip, see, e.g., [

Another application of synthetic fibres in civil engineering is the fibre-reinforced polymer (FRP) composites for reinforcement and retrofitting of concrete and masonry members, with applications in new buildings as well as for strengthening and/or rehabilitation of existing (prestressed as well as nonprestressed) structural members of both prefabricated and cast-in-place frames. FRP reinforcement consists in strengthening fibres applied to structural elements by a cementitious or polymeric-based layer. The mechanical performances of such systems can be assessed by solving the contact problem between two bounded layers [

This work presents a creep and stress relaxation test performed over a PP synthetic fibre used for FRC. Creep is a time-dependent deformation of a viscoelastic material under the application of a constant stress at a constant temperature. Relaxation is the counterpart of creep, namely, a time-dependent stress of a viscoelastic material under the application of a constant deformation at a constant temperature. Both are complex phenomena for they depend on material properties (e.g., molecular orientation and crystallinity) and external conditions (e.g., applied stress, temperature, and moisture). Moreover, the viscoelastic behaviour of PP fibres embedded in an elastic matrix complicates the modelling of creep and stress relaxation response of the composite material, which depends on many additional parameters such as concentration, aspect ratio, orientation, and, obviously, mechanical properties of PP fibres.

Creep and stress relaxation tests best demonstrate the viscoelastic characteristics of a polymeric solid. In creep test, a constant stress is maintained on a specimen while its deformation is monitored as a function of time, and deformation increases with time. In stress relaxation test, a constant deformation is maintained while the stress on the specimen is monitored as a function of time, and stress decreases with time. Typical creep and stress relaxation diagrams exhibit an instantaneous elastic response followed by a delayed time-dependent response [

Over the experimental creep and stress relaxation tests performed, this paper proposes an analytical model to fit these experimental curves. The model uses fraction-exponential kernel in the viscoelastic operator and was proposed, for the first time, by Scott Blair and Coppen [

The classical viscoelastic constitutive models represented by a combination of dashpots and springs are usually adopted for simulating creep behaviour of composite materials. However, the simplest ones (Maxwell and Kelvin-Voight) are not sufficiently flexible to match experimental data for real materials. The more complex ones, obtained from combination of different Maxwell and Kelvin-Voight models, require instead many parameters and do not allow obtaining inverse Fourier or Laplace transforms in closed form [

The model developed in the present paper is able to describe the creep curve of a PP fibre carefully and it allows obtaining the creep and stress relaxation test response of a fibre composite material in closed form. Moreover, the adopted fraction-exponential operators can be efficiently employed for the homogenization of synthetic FRC by extending the Maxwell scheme developed for elastic composites to viscoelastic behaviour of the constituents (e.g., [

The fibre consists of PP monofilament with a diameter of 0.78 mm and length of 200 mm. Since the cross section of the fibre is not perfectly circular, the diameter is an average of six measurements: Two measurements in two orthogonal directions in three points of the fibre, namely, at the middle and at both ends. To evaluate elastic Young modulus and tensile strength, tensile tests were performed on four specimens of fibres until their breakage. Each fibre was clamped at its ends and pulled by an electromechanic traction machine under displacement control.

The load cell is a GALDABINI 514262 TYPE TCA, with OUTPUT sensitivity of 2 mV/V. The machine uses a 20-bit A/D converter to acquire the analogical quantities. The resolution of the load cell is 0.002 N over the entire field of use, with a capacity of 250 N. The displacement control was performed by the actuator at a rate of 40 mm/min. The average value of the breakage tensile load is 130.5 N that occurs approximately at a displacement of 10 mm, namely, at 5% of strain.

The elastic Young modulus

Properties of polypropylene fibre.

Diameter (mm) | 0.78 |

Tensile strength (N/mm^{2}) | 273.0 |

Elastic modulus ^{2}) | 5.131 × 10^{3} |

Elastic modulus ^{2}) | 1.959 × 10^{3} |

The experiments were performed at 25°C. Since the glass transition temperature of the PP is approximately −20°C [

(a) Tensile tests until breakage on four specimens of fibre: load versus displacement. (b) Experimental curves: creep versus time (dashed orange curve) and stress relaxation versus time (dashed cyan curve: stress scaled by 1/2000).

Plots of breaking tests are reported in Figure

The creep test was carried out on a sample of length 200 mm subject to a constant tensile force of 60 N by fixing between two clamps and measuring the displacement of the fibre over the time. The load cell is the same used for the breaking test (see Section

It is worth noting that at time longer than 20,000 s the strain exceeds 5% but the fibre does not break (Figure

Similarly, the stress relaxation test was carried out on a sample of length 200 mm subject to a constant displacement of 5 mm and measuring the tensile force over the time. The speed to gain 5 mm was performed at a rate of 40 mm/min. The total duration time was 7 hours, until the force of the fibre was almost stationary. Plots of creep and stress relaxation tests are reported in Figure

Both creep and relaxation tests were performed on a single specimen.

The elastic Young modulus

We wanted to fit experimental creep and stress relaxation curves with an analytical model by using fraction-exponential kernel that on one side fits carefully experimental data and, at the same time, allows analytical expression for inverse Laplace transform. Let us consider the Boltzmann convolution integral that describes the creep strain

By shifting the index

Again, let us consider the Boltzmann convolution integral that describes the relaxation stress

The equations (

Parameters

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(a) Experimental creep curve (dashed orange curve) and analytical creep curve according to (

Polypropylene fibres used for tensile until breakage and creep and stress relaxation tests [

In the present paper we carried out a tensile test until breakage and a creep and stress relaxation test on a PP fibre and fitted these experimental tests by using a viscoelastic model based on fractional-exponential kernel. The curves plotted in Figure

Conversely to complex viscoelastic models based on combinations of Maxwell and Kelvin-Voight schemes, the proposed model requires the calibration of only two parameters (

It has been proved that the model here adopted is able to match carefully experimental data obtained for PP fibres. In a forthcoming work, the present investigation will be extended in order to take into account the effects induced by thermal variations acting on FRC elements (for the thermodynamic aspects of thermoelasticity, see, e.g., [

The author declares that they have no competing interests.