Constitutive Modelling of Short-Term Tensile Response of Geotextile Subjected to Mechanical and Abrasion Damages

This paper discusses simple constitutive models to represent the tensile response of a geotextile (GTX) and the influence of two endurance durability factors on that response: mechanical and abrasion damage, acting independently and sequentially. The damage was induced in laboratory under standard conditions. Two types of models were used: polynomial and hyperbolic. The polynomial models (order 4 and 6) approximated the short-term tensile experimental data very well (and better than the hyperbolic model), but its model parameters have no physical meaning. The hyperbolic model parameters did not always have the physical meaning reported in the literature. Correction factors were introduced successfully, to achieve that physical meaning. These correction factors were not materials constants and were affected by the damage induced, and by the type of aggregate used in the mechanical damage tests. Equations to estimate the model parameters of the damaged samples were proposed using the model parameters for the undamaged sample and its properties (tensile strength and corresponding correction factor), and the reduction factors allowing for damage for the initial stiffness (Ji) and tensile strength (Tmax).


Introduction
The tensile properties of geosynthetics are extremely important in geosynthetic design. Particularly for soil reinforcement applications, the relevant design parameters are usually the maximum tensile strength and strain. Numerical models are becoming more and more popular to assist the design of reinforced soil structures. Often, in such models, geosynthetics are represented by simple linear-elastic constitutive models, i.e., a stiffness (sometimes a maxi-mum tensile strength can be included). However, the response of geosynthetics to loading can be represented by non-linear constitutive models. The tensile properties of geosynthetics may be significantly impacted by mechanical and abrasion damages, which may reduce the available tensile strength, strain and stiffness.
Geosynthetics need to be designed to maintain adequate levels of their functional properties throughout their design life, i.e., to have adequate durability. The durability of geosynthetics is affected by endurance factors, related to the resistance of geosynthetics, and degradation factors, resulting from changes of the polymer at molecular level [1,2]. Examples of factors affecting the durability of geosynthetics include: mechanical damage, creep, stress relaxation, abrasion, high temperatures, oxidation, weathering factors (particularly ultraviolet, UV, radiation), and chemical factors. The relevance of the different factors depends on the project, the particular site conditions, and the construction processes and timings.
Transportation and handling of the materials and their installation on site often lead to mechanical damage. The damage mechanisms and degree of mechanical damage depend on the geosynthetic (structure, constituent polymer and some physical and mechanical properties), the backfill material (particle size and angularity), the foundation soil (stiffness and strength), the installation method (procedures and equipment used, height of soil layers) and the weather conditions [3][4][5][6]. For some applications, mechanical stresses associated with installation are the highest stresses geosynthetics are submitted to during their design life [5]. The consequences of installation damage may include: superficial abrasion; localised thickness reductions; fibre cutting; puncturing; complete disintegration within a given area of geosynthetic, which likely will lead to reductions in tensile strength [7]. Although usually installation damage is associated with loss of resistance, a residual stiffness modulus measures the resistance to site installation damage more adequately for woven polyester geogrids and polyethylene geogrids [6,8]. The assessment of installation damage and its consequences to the functional properties of geosynthetics should be done using field tests, under conditions similar to the project, and following standardised procedures [9]. Examples of such procedures include: Annex D of BS 8600-1 [10], ASTM D5818 [11] and NorGeoSpec 2012 [12]. Nevertheless, many authors have used laboratory tests to simulate such damage [5,6,[13][14][15]. Some authors carried out mechanical damage laboratory tests by adapting a standard procedure [16,17] using a natural aggregate and/ or changing the intensity of the cyclic load applied [5,15,18,19].
Mechanical damage associated with installation is common to all applications of geosynthetics, as they are transported, handled, and placed on site. Additionally, for particular applications of geosynthetics, abrasion damage is also key (e.g., railways, temporary roads, canal revetments, seashores with sediments and sliding masses washing up and down). Abrasion damage is time dependent as it occurs mostly during service, particularly when cyclic relative motion (friction) between a geosynthetic and contact soil occurs. The abrasion damage leads to reduction of tensile strength and is related to the mass per unit area of the geosynthetic and, for railway applications, to the volume of traffic [35]. In hydraulic applications, abrasion can also be caused by turbid flow [36]. The existing standardised procedure to induce abrasion damage in laboratory [37] can be linked to in-field abrasion using equations proposed by ISO [38].
The design of geosynthetics is often carried out using reduction factors, each representing a single factor or mechanism affecting the durability of geosynthetics, that are multiplied [1,39,40]. Nevertheless, positive or negative synergy between factors can occur, as reported by Rosete et al. [20] and Pinho-Lopes et al. [23]. Rosete et al. [20] studied the influence of mechanical and abrasion damage acting isolated and combined on the short-term tensile response and hydraulic properties of geosynthetics. These authors concluded that there can be interaction, positive or negative synergy between mechanical and abrasion damage, represented by a combined reduction factor different than the product between the corresponding independent reduction factors.
Numerical models, particularly using the finite element method, are becoming more and more popular for example to assist the design of structures with geosynthetics and/or analyse their performance. Often, in such models geosynthetics are represented by a stiffness and a tensile strength. As durability is key for the design and performance of geosynthetics, representing their response using non-linear constitutive models allowing for durability is essential. Thus, it is important to understand how damage may influence the load-strain response of geosynthetics and how such changes can influence the parameters used in design. Paula and Pinho-Lopes [41,42] have used simple constitutive models allowing for durability aspects and presented initial work on this topic. This paper focuses on two factors affecting the durability of geosynthetics: mechanical damage, often associated with installation, and abrasion damage, acting independently and sequentially. The influence of the type of aggregate used to induce mechanical damage is also analysed. The short-term tensile response of a geotextile submitted to mechanical damage and abrasion damage is represented using constitutive models. The model parameters are analysed and their ability to represent the damage induced is discussed. Equations to estimate the model parameters of the damaged samples are proposed using the model parameters for the undamaged sample and its properties, and the reduction factors for the damaged sample. Table 1 describes the different types of samples analysed to derive constitutive models and their parameters to represent the short-term tensile response of the geotextile (GTX) studied.

Geosynthetic
GTX is a nonwoven geotextile formed by continuous polypropylene (PP) filaments mechanically bonded (Fig. 1). GTX has the following nominal properties: tensile strength T nom = 50 kN/m; strain for the tensile strength ε nom = 65%; mass per unit area μ nom = 800 g/m 2 ; thickness d nom = 6 mm. The tensile response of GTX was characterized using wide-width tensile tests, EN ISO 10319 [43], for different types of samples: reference material (undamaged) and after damage. GTX was submitted to index tests ( Fig. 2) for inducing: (i) mechanical damage; (ii) abrasion damage; (iii) mechanical damage followed by abrasion damage.

Laboratory Mechanical Damage Test
A standardised procedure [16] was used to induce mechanical damage on GTX. During the test, a specimen of geosynthetic is placed between two layers of a synthetic aggregate (sintered aluminium oxide, particle sizes between 5 and 10 mm) and submitted to cyclic loading.
The mechanical damage test is carried out using a container formed by two half boxes (Fig. 2a). Their dimensions are: 0.30 × 0.30 m 2 in plan section and 0.075 m high, each; the total height of the container is 0.15 m. The load is applied using a rigid load plate with plan area of 0.10 × 0.20 m 2 .
During the assembly of the test, two layers of aggregate are placed and compacted in the lower half-box, up to a total height of 0.075 m. For the compaction, a static pressure of 200 kPa is applied during 1 min using a compaction plate with 0.30 × 0.30 m 2 in plan section. Then, a specimen of geosynthetic is placed on top of the lower half-box. The upper half-box is assembled and filled with loose aggregate (0.075 m high). The loading plate is positioned, centred and  (Fig. 2a). The test ends after a cyclic loading ranging between 5 and 900 kPa, with frequency of 1 Hz for 200 cycles, is applied. The dimensions of the specimens submitted to mechanical damage (0.20 m wide and 1.0 m long) are adequate to perform wide-width tensile tests for characterising the mechanical damage induced. The area of the load plate coincides with the area of geosynthetic to be placed between clamps for a tensile test. A minimum of five specimens were tested.
The test method used [16] and later (updated) versions (such as [44]) define ways of ensuring that the aggregate maintains its characteristics during the tests. Thus, if necessary, after every three uses the aggregate should be wet sieved (using a 5-mm aperture sieve), and particles smaller than 5 mm should be discarded. All aggregate must be completely discarded after 20 uses.
Besides the tests performed with the synthetic aggregate prescribed in the test standard, two additional aggregates were used. These were natural aggregates, with minimum particle size analogue to that of the synthetic aggregate, and with different petrography: granite; limestone (Table 2). Thus, three sets of mechanical damage tests were carried out following the procedures described and with the synthetic aggregate (SYN), the granite aggregate (GRA), and the limestone aggregate (LIM).

Laboratory Abrasion Test
The abrasion damage was simulated using the procedures in [45]. In this test, a geosynthetic specimen is placed on an upper plate of a stationary platform and rubbed by a P100 abrasive, placed on the lower plate and moved along a horizontal axis under controlled pressure (Fig. 2b). Some adjustments were made to the test procedure [20]: a P24 abrasive film was placed between the specimen and the upper plate, to ensure that during the test the specimen did not adhere to the abrasive film placed on the lower plate (no additional damage was observed).

Short-Term Tensile Response
The short-term tensile response of the geosynthetics was characterised following the procedures described in [45]. For each type of sample studied, five valid specimens were tested.
The tensile tests results are summarised in Table 3: tensile strength, T max (herein defined as the maximum tensile force mobilised during the test); strain for maximum load, ε max ; secant stiffness for 2% and 5% strain, J s2% and J s5% , respectively. The results refer to five valid specimens per sample and include the corresponding coefficient of variation (CV). The secant stiffness for x% strain, J sx% , was determined using Eq. 1, where T x% is the force for x% strain measured during the tests. Table 2 Properties of the aggregates used in the mechanical damage tests: minimum and maximum particle size and Los Angeles (LA) coefficient D min minimum particle sizes, D max maximum particle sizes, LA coefficient-Los Angeles abrasion coefficient  The tensile strength of GTX was reduced after damage, with retained tensile strength ranging between 61 and 88%. Samples of GTX submitted sequentially to mechanical damage (with the synthetic aggregate) and abrasion damage (MEC SYN + ABR) exhibited the largest reduction in tensile strength. The secant stiffness of GTX was also reduced by damage, with retained values ranging between 61 and 70% for 2% strain and between 51 and 61% for 5% strain.
The tensile response after mechanical damage was affected by the aggregate used in damage tests. The reductions in tensile strength were higher with the synthetic aggregate (33%) than with the natural aggregates (granite, 22% and limestone, 12%). The natural aggregates, with particles larger than those of the synthetic aggregate, were less aggressive to GTX. Particles of these aggregates responded differently to loading: the synthetic aggregate particles fragmented into smaller elements with sharp and angular edges and faces, more likely to induce damage; the natural aggregates, and in particular the granite aggregate, exhibited less fragmentation. Figure 3 illustrates the load-strain responses obtained from the wide-width tensile tests for representative specimens of each sample. The reductions of tensile strength and corresponding strain and the reductions of stiffness are evident.
Simple constitutive modes, such as polynomial models and hyperbolic-based models, are be used to represent the short-time tensile response of geosynthetics. Curve fitting experimental data, namely using the least-squares method, allows defining the model parameters [46]. Equation 2 (1) represents a generic polynomial model, where T is the load per unit width, ε is the axial tensile strain, a i is the polynomial coefficient of order i and n is the order of the polynomial. Equation 3 represents the tangent stiffness (J tε% ) of the geosynthetic for a strain ε (in %), obtained by derivation of Eq. 2. Equation 4 represents a hyperbolic model. According to Liu and Ling [47], a and b are model parameters representing the inverse of the initial stiffness, J i , and the inverse of the tensile strength of the geosynthetic, T max (Eqs. 6 and 7, respectively).

Results and Discussion
The experimental data were fitted with three simple constitutive models: two polynomials, order 4 and order 6, and a hyperbolic model. The model parameters were obtained via curve fitting of the experimental data for each specimen; then, average model parameters were calculated. Table 4 summarises the average model parameters for different samples of GTX: undamaged (UND) and submitted to laboratory damage: mechanical (MEC), with three different aggregates (SYN, GRA, LIM); abrasion (ABR); and mechanical damage (with the synthetic aggregate) followed by abrasion damage (MEC SYN + ABR). Table 4 includes values for the coefficient of determination (R 2 ) of some of model parameters. Figure 4 includes load-strain curves for selected specimens of each sample obtained experimentally and with the constitutive models analysed.
The polynomial models fit the experimental data very well (Table 4; Fig. 4), with coefficient of determination, R 2 , of 1.0 and coefficient of variation lower than 0.005%. The order 4 and order 6 polynomials provide a good approximation of the experimental data. The polynomial coefficients of order 0-4 are different for the two polynomial models studied; however, the order 6 polynomial model had very small coefficients of order 5 and order 6 (a 5 : 1.69 × 10 -7 to 3.64 × 10 -8 ; a 6 : − 9.44 × 10 -11 to − 6.21 × 10 -10 ). Thus, order 4 polynomials were considered adequate approximations of the tensile response of GTX before and after damage. Nevertheless, for the GTX and the type of samples analysed, the short-term tensile response is defined mostly by the coefficients of order 0-2. It must be noted that the polynomial models are of little practical use as they are simple mathematical functions fitted to the experimental data. The model parameters have no physical meaning and, thus, no relation to the tensile response of GTX. The hyperbolic model (Eq. 3) fitted well the experimental data (Table 4; Fig. 4), with R 2 of 0.997-0.999. Hyperbolicbased models are often associated with tensile parameters of the material, as represented in Eqs. 5 and 6. Herein the initial stiffness was estimated as the tangent stiffness for 0% strain, obtained from the order 6 polynomial models. Figure 5 illustrates how parameters a and b for the hyperbolic models related to the tensile parameters of GTX. The data (Fig. 5a, c) show that a and b do not follow Eqs. 6 and 7. However, that changes if correction factors are included for J i and for T max , herein designated as C J and C T , for a and b, respectively (Eqs. 8 and 9).
These correction factors were derived for each type of sample studied, to ensure the corresponding parameters a or b, obtained from curve fitting, were related to J i or T max for the same type of sample. For the undamaged sample (GTX UND) C J = 107.64 and C T = 0.584. These correction factors changed after damage ( Table 5), showing that they are not material constants and that they are influenced by aspects relative to the durability of GTX, including the type of damage induced and the aggregate used for the mechanical damage test.
One of the goals of this work was to estimate model parameters for the damaged samples using tensile properties of the undamaged sample and the corresponding model parameters. Some trends were noticed and analysed further to then proposed relations for estimating model parameters after damage. The reduction factor for the tensile strength of each sample, RF dam (Eq. 10), can be used for estimating the correction factor C T obtained for each type of sample studied ( Fig. 6a and Eq. 11). Therefore, the parameter b for each type of sample can be estimated reasonably using Eq. 12, which does not depend on the damage induced. This is consistent with the values for parameter b presented in Table 4: after damage b was the practically the same for all damaged samples, although 8% and 17% smaller than for the undamaged samples of GTX.
For the damaged samples, parameter a can be reasonably estimated from Eq. 13 (Fig. 6b), combining the model parameter a for the undamaged sample (a und ) and (10) RF dam,T = T max, und T max,dam .

Conclusions
In this paper, the short-term tensile response of a geotextile was approximated using simple constitutive models (polynomial and hyperbolic). The influence of two endurance durability factors on that response was analysed, namely mechanical and abrasion damage, acting independently and sequentially; the influence of the aggregate used to induce mechanical damage was also analysed. From the results the main conclusions can be summarised: • The polynomial models (order 4 and order 6) approximated the short-term tensile experimental data very well and better than the hyperbolic model. However, the polynomial model parameters have no physical meaning. • The hyperbolic-model parameters (a and b) obtained from the experimental data did not always have the physical meaning referred in the literature. To ensure these model parameters had relevant physical meaning, correction factors for the initial stiffness (C J ) and for the tensile strength (C T ) were introduced. The correction factors were not material characteristics, as they were affected by mechanical and abrasion damage, isolated or sequential, and by the type of aggregate in contact with GTX. • The hyperbolic-model parameters of the damaged samples were estimated from those of the undamaged samples using properties of the undamaged sample of GTX, namely its tensile strength and the corresponding correction factor (C T ), and the reduction factors allowing for damage applicable to the properties represented by the model parameters a and b, respectively the initial stiffness (J i ) and the tensile strength (T max ). • The hyperbolic-model parameter b was approximately constant for all samples of GTX and, thus, little affected by the durability factors studied.
The model developed and the equations proposed to estimate constitutive-model parameters after damage will have a significant positive impact on the design of geosynthetics. These proposals enable a simple way of estimating the tensile response of GTX after mechanical and abrasion damage. The base information used are: (13) a dam ∼ a und × RF dam,Ji .
• undamaged material-tensile properties (tensile strength and initial stiffness) and correction factors for the tensile strength and initial stiffness; • damaged samples-the corresponding reduction factors allowing for damage and applicable to two key properties in the design of geosynthetics: tensile strength and stiffness.
These trends need to be confirmed for other materials, but they are good indicators of links between model parameters representing the tensile response of the undamaged material and the material submitted to mechanical and abrasion damage, isolated or sequentially.
Thus, it will be possible to considerer the effects of the durability of geosynthetics in their design by implementing these constitutive models, particularly when using the finite element method or the finite differences method. Therefore, more realistic designs and analysis will be possible.