Influence of service ageing on polyester-reinforced polyvinyl chloride-coated fabrics reported through mathematical material models

Abstract

In this paper the coupled service (constructional tension) and environmental (sunlight, rainfalls, temperature variations) ageing influence on the polyester-reinforced polyvinyl chloride (PVC)-coated fabric VALMEX is studied. Two cases of the same fabric have been analyzed: one USED for 20 years on the real construction of the Forest Opera in Sopot (Poland), and one kept as a spare material (NOT USED). The following tests have been conducted: uniaxial tensile, biaxial tensile and long-term creep tests. The obtained results have been used for the parameter identification of the piecewise non-linear, Burgers and Bodner–Partom models. Next, the analysis of the influence of environmental conditions on the parameters of these models has been made. It has been concluded that some parameters are more and the others are less sensitive to the exposure to environmental and mechanical conditions. The change of material parameters for fill threads (due to larger deformation) is higher. The obtained results may be useful in the durability evaluation of the textile membranes reinforced with polyester threads and PVC coated. All the constitutive models with the identified parameters may be used for the numerical analysis of structures made of fabrics at the service beginning and after long-term usage.

Keywords

properties strength performance modeling ageing polyester fabrics

Reinforced coated textile membranes are much used as building materials due to their good strength properties, easy construction linked to partial off-site manufacturing, flexibility to obtain sophisticated shapes, low production costs and light weight. In order to fulfill still rising design demands and to guarantee the long-term, fully functional and repair-free lifetime of a roof, the degradation over time of building materials must be taken into account; hence, the issue of the durability estimation of covering materials used for many years and their possible reuse or recycling after a structure is removed. The best way to assess the material durability is an actual field study, including the collection of aged samples, performance of various experiments on these samples and comparison of the identified properties with the producer’s specification for the new manufactured material.

The present paper investigates the influence of the outdoor environment and service stresses on the mechanical properties of the technical fabric VALMEX. This polyvinyl chloride (PVC)-coated material reinforced with polyester threads has been used for almost 20 years as the covering of the open theater in Sopot (Forest Opera, Poland). The impact of the acting stresses on the fabric performance is described through the following constitutive approaches: the piecewise non-linear model (upon the assumption of the elastic behavior of the material in the whole range of strains), the viscoelastic Burgers model and the viscoplastic Bodner–Partom model. This approach, including different types of material modeling, would reveal how ageing influences particular parameters and give the opportunity to describe the long-term durability of the polyester-reinforced PVC-coated roofing materials through the proposed constitutive formulations.

Ageing of roofing materials

Several publications documenting the ageing and weathering of roofing materials have appeared in recent years. Berdahl et al.¹ overviewed several ageing factors, such as: the degradation of plastics induced by ultraviolet (UV) radiation; the effects of moisture on the decay of wood, corrosion of metals and staining of clay; and reduction of solar reflectance due to soiling. Razak et al.² pointed out that a harsher tropical climate (hot and humid) causes an increase in degradation of the material surface. They subjected 10 different types of fabrics to outdoor exposure and demonstrated that cracking and peeling of the coating are greater on the PVC-coated than on the polytetrafluoroethylene (PTFE)-coated fabrics. Moreover, it has been also proven that the addition of TiO₂ films results in better self-cleaning of the PVC-coated fabrics. Xing and Taylor³ tested 13 types of thermoplastic polyolefin (TPO) membranes for UV and thermal breakdown. They observed that TPO membranes designed to work at high temperatures (e.g. 138℃) also have excellent UV stability. They obtained good correlation between laboratory, field and thermal ageing tests. Their work contains the precise presentation of powerful apparatus for a laboratory and natural accelerated weathering.

The methods of the ageing process simulation, both naturally and artificially accelerated, will never take into account the wide spectrum of various factors affecting the performance of materials under natural outdoor exposure. Nothing is more useful and informative than actual field experience.⁴ It provides direct facts on the ageing phenomenon and can serve as a basis for the validation of any simulation program. The in situ research consists of a simple visual inspection and/or the collection of samples for further laboratory testing. An illustration of a field study is the analysis of the 10-year performance of PVC roofing materials that has been accomplished by American military laboratories in years 1982–1993.^5–7 The unreinforced and reinforced types of roofing membranes installed by different techniques were investigated. The basic mechanical properties of textiles have been identified and their evolution in time has been approximated by linear functions, giving satisfying results.

The next example is a group of American, Canadian and Swiss scientists that tested 44 different roof membranes collected from old houses in America and Europe in 2001–2002.^8,9 They performed a series of experiments (e.g. tensile tests, low temperature flexibility, hail resistance, dimensional stability, plasticizer content, seam strength) and compared the results to the requirements of the national standards of ASTM, DIN and SIA.^10–12 The laboratory testing confirmed that, although the products tested lost some of their initial physical properties due to the ageing process, they generally behaved very well in comparison to the normative minimum values for new PVC roofing membranes. It should be highlighted that most of the samples were installed before establishing the first single-ply roofing national standards in particular countries. The DIN and SIA standards were introduced in 1976 and 1977, respectively, while the American standard ASTM was introduced in 1985. Moreover, all the tested on-site PVC roofing systems maintained well (without leakage) and were capable of being welded despite their age. Therefore, the roofing textiles would probably perform satisfactorily for the next decade if they are properly treated.

Cash et al.¹³ presented the comparative analysis of 12 different types of roofing membranes exposed to two and four years of outdoor weathering. They also proposed a rating system to assess their durability. A review and development of other field studies of PVC roof systems has been offered by Koontz.¹⁴

The presented research focuses not only on the influence of the ageing on the physical fabrics characteristics, but it also proceeds to evaluate the ageing effect on the selected constitutive models. These models are willingly used by constructors for the computer analysis and designing purposes of hanging roof structures. In the current research, the difference in the behavior in the fill and warp directions is studied in detail. In addition, different constitutive models for the same material are identified.

Material models for ageing analysis

Piecewise linear model

The concept of the piecewise linear material model is based on the observation that even a very sophisticated shape of an experimental stress–strain curve $(σ - ɛ)$ can be approximated by the set of linear functions followed by the ranges of their applicability according to the following equations

\begin{matrix} σ_{1} (ɛ) = f_{1} (ɛ) = a_{1} ɛ + b_{1}, ɛ \in á 0; ɛ_{1} ñ \\ σ_{i} (ɛ) = f_{i} (ɛ) = a_{i} ɛ + b_{i}, ɛ \in á ɛ_{i - 1}; ɛ_{i} ñ, i = 2, 3, \dots \end{matrix}

(1)

where a_i and b_i are the model parameters that have to be identified. The piecewise linear conception is used to conduct the preliminary evaluation of the material behavior on the assumption of the material linear performance in the particular ranges of strains.¹⁵ An example of the model application for the uniaxial tensile test of an architectural fabric is presented in Figure 1, while its application for biaxial tensile tests can be found in Ambroziak.¹⁶ The proposed model is a combination of linear functions, and therefore it cannot describe ideally the stress–strain curved sections, where the curve changes its trajectory in a non-linear way (in the strain region of 0.14–0.18 for the fill direction and 0.08–0.12 for the warp direction). However, the discrepancy between the experimental curve and the model is observed for the greater elongations that are usually beyond the acceptable elongation level practiced in civil engineering structures made of architectural fabrics. In some cases, the particular longitudinal stiffness of a material is equal to its Young’s modulus. However, for technical fabrics their stress–strain response under the tensile test is complex and one should be very careful in estimating the true value of the Young’s modulus, especially in the fill direction of the fabric.¹⁷

Figure 1.

Numerical application of the linear piecewise model for uniaxial tensile testing.

For technical approaches, the units used for the description of technical fabrics stresses and related mechanical parameters are established for the unitary thickness that means ${kN / m}^{2} \cdot m = kN / m$ . Therefore, all mechanical characteristics in this paper will follow the methodology commonly used in this field (e.g. Ambroziak,¹⁵ Beer et al.,⁸ Kłosowski et al.¹⁸).

Viscoelastic Burgers model

The Burgers model is a four-parameter model representing the linear viscoelastic properties of a material using derivative relations. It is the series combination of the Maxwell and Kelvin–Voigt models¹⁹ (Figure 2): the instantaneous deformation $ɛ_{1}$ comes from the Maxwell spring (described by elastic modulus E₁), the viscous deformation $ɛ_{2}$ from the Maxwell dash-pot (described by viscosity coefficient $η_{1}$ ) and the viscoelastic deformation $ɛ_{3}$ from the Kelvin–Voigt unit (described by elastic modulus E₂ and viscosity coefficient $η_{2}$ ).

Figure 2.

Illustration of the Burgers model structure.

The basic mathematical representation of the Burgers model and other transformations can be found, for example, in Findley et al.¹⁹ The final form of the Burgers model for the uniaxial case of creep test obtains the following formula

ɛ (t) = \frac{σ_{0}}{E_{1}} + \frac{σ_{0}}{η_{1}} t + \frac{σ_{0}}{E_{2}} (1 - e^{\frac{- E_{2}}{η_{2}} t})

(2)

where t stands for time. The first two parts of Equation (2) represent elastic strain and viscous flow, respectively, while the last term represents the delayed elastic deformation of the Kelvin–Voigt model.

Among the other simple viscoelastic linear models, only the Burgers model describes precisely the creep behavior with the permanent strain. The Burgers equation is mainly used for the viscoelastic analysis of fluids. The Burgers model is general enough to include the one-dimensional Oldroyd, Maxwell and Navier–Stokes models as subsets.²⁰ Apart from the fluid analysis, the Burgers formulation is willingly applied with satisfactory results into the analysis of metals,¹⁹ polycarbonate panels²¹ and natural fiber/polymer composites.^22,23

Viscoplastic Bodner–Partom model

The Bodner–Partom model describes the elasto-viscoplastic behavior of materials.^24,36 The model is presented in details in the literature,^{18,24–28,37} thus only the elementary equations are presented here in Table 1.

Table 1.

Bodner–Partom model elementary equations^24,36

Inelastic strain rate	${\overset{\cdot}{E}}^{I} = \frac{3}{2} \overset{\cdot}{p} \frac{s'}{J (s')}$
Cumulated strain inelastic rate	$\overset{\cdot}{p} = \frac{2 D_{0}}{\sqrt{3}} exp [- (\frac{Z}{J (s')})^{2 n} \frac{n + 1}{2 n}]$
Additional equations	$\begin{matrix} Z = R + X : \frac{s'}{J (s')} \end{matrix}$
Isotropic hardening	$\begin{matrix} {\overset{\cdot}{W}}^{I} = s' : {\overset{\cdot}{E}}^{I} \\ \overset{\cdot}{R} = m_{1} (R_{1} - R) {\overset{\cdot}{W}}^{I}, R (t = 0) = R_{0} \end{matrix}$
Kinematic hardening	$\overset{\cdot}{X} = m_{2} (\frac{3}{2} D_{1} \frac{s'}{J (s')} - X) {\overset{\cdot}{W}}^{I}$
Material parameters (thermal and recovery effects neglected)	n – strain rate sensitivity parameter [-]
	D₀ – limit (maximum) of strain rate [s^–1]
	D₁ – limit (maximum) of kinematic hardening [s^–1]
	R₀ – initial value of isotropic hardening [kN/m]
	R₁ – limit (maximum) of isotropic hardening [kN/m]
	m₁ – coefficient of isotropic hardening [kN/m]^–1
	m₂ – coefficient of kinematic hardening [kN/m]^–1

The base for the identification of the Bodner–Partom model parameters can be the uniaxial tensile tests with constant strain rates, which according to the literature study seem to be the simplest possible choice. Applied in this study, the complete description of the Bodner–Partom model identification procedure adapted for technical fabrics is given by Kłosowski et al.^28,29

Material types and laboratory tests

VALMEX FR 1000 Hallen type III Universal³¹ is the investigated fabric. It is an architectural fabric built of two orthogonal polyester thread families (the warp and the fill with P2/2 weaving), both sides coated with PVC layers. It has been produced by the Mehler Company in German and since 1990 used for the roof structure of the Forest Opera in Sopot.

According to the producer’s specification, cross-sections of the fibers in both directions are the same, while the average counting of the single threads is 110 ± 2/10 cm and 104 ± 2/10 cm for the warp and fill directions, respectively. The analyzed canopy material has been used for about 20 years. During that time it has been impacted by the whole spectrum of environmental conditions usual for that part of the country (north of Poland, temperate climate). It is characterized by average temperature ranges from –3℃ during the winter period to +22℃ in the summer period. The overall year mean temperature is about 8℃, while the average humidity is 82%. The rainfalls are most often during autumn periods (September–November), but sometimes could happen during summer as well. The average year rainfall is about 21 mm. Two types of VALMEX fabric have been tested in this research. The USED material was used for 20 years (1990–2009) as the roof of the Forest Opera. The roof structure was not designed to carry snow loading; therefore, the canopy was a short-term construction and had to be taken down before the winter season and put back up in spring each year. When the roof was tensioned above the scene (summer period from May to September), the conditions directly impacting on the roof surface were sunlight, wind, humidity and rainfalls. During the period of October–April, the roofing material was kept on the scene (in open air conditions), and thus it was still exposed to the moisture and day and night temperature variation impacts. Consequently, over 20 years of usage the USED VALMEX material underwent about 20 cycles of tensioning and releasing, and was subjected to environment impacts. It confidently underwent coupled weathering and mechanical ageing. This has been observed by the change of the color on the fabric side exposed to the sun (Figure 3). The next type of VALMEX fabric is the material that was kept as an additional one to renovate the canopy if necessary, which was deposited at constant temperature (about 10℃) and without light access in the basement of a building, protected from unfavorable environmental conditions (temperature variations, moisture). It underwent only a natural (tension-free) ageing process and therefore it is called the NOT USED type of material.

Figure 3.

VALMEX samples: the USED exposed to the sun (a); the USED not exposed to the sun (b); the NOT USED (c).

In order to find and describe the differences in mechanical properties between the USED and NOT USED fabrics, the samples have been cut separately for the warp and fill directions of the fabric. The investigation started with uniaxial tensile tests performed to obtain the basic tensile properties of both materials. These experiments were realized with three different but constant strain rates of 0.005, 0.001, 0.0001 1/s, which has been necessary to perform the identification of the Bodner–Partom model parameters.²⁴ Dimensions of the specimens were chosen according to the ISO 1421 national standard.³⁰ The width of each specimen was 50 ± 0.5 mm, while the active length (the grip separation) was 200 ± 0.5 mm (Figure 4(a)). The testing machine Zwick Z020 with a video-extensometer (with the gauge length of about 50 mm) was used for the testing (Figure 4(b)). For each strain rate for the USED and NOT USED type of VALMEX fabric, at least five tests were carried out. The tests were conducted for the warp and fill directions separately.

Figure 4.

Sample geometries (a) and strength machines used in the study: Zwick Z0/20 with a video-extensometer (b); biaxial testing machine (c); creep testing machine (d).

Next, biaxial experiments were executed to show how service ageing impacted interactions between the warp and fill threads. Tests with a crosshead speed of 100 mm/minute (the same in both directions) were conducted on the special cruciform samples (Figure 4(a)). The specimen width was 100 mm and the grip separation was 300 mm in each direction. The biaxial strength machine (Figure 4(c)) equipped with a video-extensometer was used for these tests (the extensometer basis was also close to 50 mm). For each type of VALMEX fabric at least three tests were carried out.

Finally, as architectural fabrics are most often characterized by their viscoelastic description, creep tests were realized.³¹ A 10-stands machine for long-term rheological creep experiments with the maximum load of 3 kN per stand was used (Figure 4(d)). It is equipped with induction and laser measurement devices. Due to the machine geometry, the sample width of 50 mm was used and the grip separation was set at 150 mm (Figure 4(a)). The long-term creep experiments had a duration of about 1 month for the fill direction and 2 weeks for the warp direction. An identicial stress level of 30 kN/m for sepecimens cut in the warp and weft directions was introduced. All the tests were conducted for the USED and NOT USED VALMEX samples. At least three tests in the warp and fill directions were carried out. To minimize the recorded errors, the special registration procedure was programmed for the machine. It resulted in data collection with the following resolution: first hour – every 1 s; next 2 hours – every 5 s; next 4 hours – every 10 s; next 17 hours – every 1 minute; then until the end of the test – every 5 min.

Influence of outdoor service ageing

Strength parameters and piecewise linear model

The characteristic typical tensile curves obtained for three different strain rates for the NOT USED and USED VALMEX fabric are depicted in Figures 5(a) and (b), respectively. The analysis of the stress–strain trajectories allows one to distinguish three or four particular ranges for the warp and fill directions of the fabric, respectively. The experimental curve for the fill direction is more complex in its initial stage. Below the stress level of about 20 kN/m for the warp direction there is only one distinct linear phase (identified as the Young’s modulus), while for the fill direction two stages of different slopes can be observed. The difference is caused by different behavior of both thread families during the weaving process. The threads in the warp direction are straight and the threads of the fill are interspersed through them. Besides, the threads in the warp direction are usually pre-stressed during the manufacturing coating process, a technique that infrequently occurred for the fill direction 20 years ago. Therefore, it is set that the first slope in the stress–strain curve in the fill direction is correlated with the stiffness of the PVC coating of the fabric, while the fill threads undergo straightening. Consequently, the second linear phase mirrors the state where fill threads are almost fully straightened and carry subjected force. As a result, according to the observations made previously,¹⁷ in the fill direction of the VALMEX material the second tangent represents the elasticity modulus in a linear range. In addition, Figure 5 shows that the fabric exhibits rate-dependent behavior for both types of samples. Comparing the NOT USED and USED fabrics, it can be noticed that for the warp direction the change of the strain rate causes lower variations of the material response. This is the result of tensioning the USED fabric during service, causing a decrease in the flexibility and increase in the stiffness of the material. For the fill direction, the strain rate has influenced the response of the NOT USED and USED fabrics with a similar scale. The dependence on the strain rate indicates that viscous constitutive models can be implemented for the performance description of this textile.

Figure 5.

Characteristic uniaxial tensile curves for the NOT USED (a) and USED (b) samples of the technical VALMEX fabric for three various strain rates.

In Figure 6(a) the USED and NOT USED types of membrane under uniaxial tension for a strain rate of 0.001 1/s are compared. Comparisons for the other strain rates and for the warp direction have always given very similar results. In contrast, for the fill direction the greatest difference between the USED and NOT USED fabric is observed for the beginning of the stress–strain curve.

Figure 6.

The uniaxial tensile tests: comparison of NOT USED and USED VALMEX fabric for a strain rate of 0.001 1/s (a) and representative sections for the piecewise linear model (b).

In Figure 7(a) the results of the stress–strain plots for the biaxial tests are shown for the warp and fill directions of the VALMEX fabric. As for the uniaxial tests, the analogous trend is noticed here: results for the warp direction are similar comparing the USED and NOT USED samples, whereas for the fill direction the greatest difference arises in the beginning part of the stress–strain curve. In the case of the biaxial tests, the discrepancies for the fill direction are more pronounced. In addition, it can be observed that for the biaxial tests the results of ultimate tensile strength (UTS) are distinctly lower than for the uniaxial tests for both material directions. Only one typical test of the USED and one of the NOT USED samples are demonstrated, as the rest of the outcomes are analogous.

Figure 7.

The biaxial tensile tests: comparison of NOT USED and USED VALMEX fabric for the crosshead speed of 100 mm/minute (a) and representative sections for the piecewise linear model (b).

The estimation of the tensile properties of the material was made. For the warp direction two ( $E_{Wi}$ , where i = 1,2) and for the fill direction three ( $E_{Fj}$ , where j = 0, 1, 2) longitudinal stiffnesses were determined according to the methodology of the piecewise linear model presented in Figures 6(b) and 7(b) for the uniaxial and biaxial tests, respectively. The boundary points ( $ɛ_{0 / 1}; ɛ_{1 / 2}; ɛ_{f}$ ) corresponding to the strains ranges, where the identified longitudinal stiffness can be used, were calculated. The values of the longitudinal stiffness and the intersection points were obtained as the mean values of all the tests from all three strain rates of the uniaxial tests, separately for the warp and fill directions. As the biaxial tests were performed with the same crosshead rate for both directions, the results of the piecewise model parameters were calculated as the mean value from all the tests, regarding the warp and fill directions. The uniaxial and biaxial study of fabric behavior was limited to the strain level of 0.05 in the warp direction ( $ɛ_{f} = 0.05$ ) and to the strain level of 0.1 ( $ɛ_{f} = 0.1$ ) in the fill direction, which was proposed previously by Kłosowski et al.²⁸

The obtained results (± their standard deviation) are collected in Tables 2 and 3 for the uniaxial and biaxial tests, respectively. The relative difference between the NOT USED and USED samples has been calculated according to the formula

δ = [(NOT USED - USED) / NOT USED] \times 100 %

(3)

Table 2.

The uniaxial experiment results

Direction		WARP			FILL
Fabric type		NOT USED	USED	δ	NOT USED	USED	δ
Parameter	Unit	NOT USED	USED	[%]	NOT USED	USED	[%]
$E_{F 0}$	[kN/m]	-	-	-	140 ± 20	190 ± 10	–36
$ɛ_{0 / 1}$	[-]	-	-	-	0.0356 ± 0.0005	0.0345 ± 0.0002	3
$E_{F 1}$ ; $E_{W 1}$	[kN/m]	1830 ± 30	1800 ± 70	2	380 ± 30	390 ± 10	–3
$ɛ_{1 / 2}$	[-]	0.0091 ± 0.0003	0.0088 ± 0.0005	4	0.0576 ± 0.0004	0.056 ± 0.002	2
$E_{F 2}$ ; $E_{W 2}$	[kN/m]	420 ± 10	400 ± 10	5	240 ± 10	250 ± 10	–4
$ɛ_{ult}$	[-]	120 ± 10	100 ± 10	16	60 ± 10	60 ± 10	0
UTS	[kN/m]	0.150 ± 0.001	0.136 ± 0.006	9	0.181 ± 0.016	0.173 ± 0.004	5
$σ_{02}$	[kN/m]	16.7 ± 1.0	17.1 ± 1.1	–2	14.6 ± 1.2	16.7 ± 0.8	–15

Table 3.

The biaxial experiments results

Direction		WARP			FILL
Fabric type		NOT USED	USED	δ	NOT USED	USED	δ
Parameter	Unit	NOT USED	USED	[%]	NOT USED	USED	[%]
$E_{F 0}$	[kN/m]	-	-	-	400 ± 10	490 ± 10	–23
$ɛ_{0 / 1}$	[-]	-	-	-	0.048 ± 0.002	0.045 ± 0.002	6
$E_{F 1}$ ; $E_{W 1}$	[kN/m]	2200 ± 20	1950 ± 10	11	280 ± 10	280 ± 10	0
$ɛ_{1 / 2}$	[-]	0.012 ± 0.0001	0.012 ± 0.0005	0	0.094 ± 0.008	0.080 ± 0.003	14
$E_{F 2}$ ; $E_{W 2}$	[kN/m]	480 ± 10	450 ± 10	6	390 ± 10	370 ± 10	5
$ɛ_{ult}$	[-]	100 ± 10	80 ± 10	20	40 ± 10	40 ± 10	–1
UTS	[kN/m]	0.127 ± 0.004	0.109 ± 0.001	14	0.121 ± 0.005	0.100 ± 0.001	18
$σ_{02}$	[kN/m]	18.2 ± 0.7	16.8 ± 0.6	8	17.8 ± 0.7	20.0 ± 0.8	–12

UTS: ultimate tensile strength.

The main change can be noticed in the fill direction for the slope $E_{F 0}$ , which is related to the covering material – PVC. For the USED fabric the modulus $E_{F 0}$ is about 36% (uniaxial) and 23% (biaxial) higher than for the NOT USED material. This is because the PVC becomes more rigid due to the environmental ageing. Next, the inclinations $E_{W 1}$ and $E_{F 1}$ were analyzed. They are related to the polyester threads of fabric reinforcement and can be assumed as the stiffness moduli of the material in the elastic range for the warp and fill directions, respectively. The values of $E_{W 1}$ and $E_{F 1}$ stiffness for both types of VALMEX fabric do not change, except for the case of biaxial testing for the results in the warp direction (difference of about 11%). This outcome indicates that the polyester threads have kept their initial elastic properties for 20 years. It suggests that polyester threads have been sufficiently protected by PVC coating from unfavorable external conditions or that the polyester threads are not sensitive to external environment influence.

The UTS declines for the USED material by about 9–18%, except for the uniaxial test in the fill direction. The values of the elongation at break ( $ɛ_{ult}$ ) decrease for the warp direction, achieving for the biaxial tests even an 20% reduction. These observations confirm that the UTS and elongation at break ( $ɛ_{ult}$ ) were influenced by service and environmental conditions, as well as passing time. In the fill direction for the uniaxial tests the drops in parameter values are not high. The threads in this direction were crimped during a weaving and coating process, which resulted in their higher elongation limits when they are simply tensioned. However, during the biaxial test, when interrelation between the warp and fill threads appears, the material loses its high UTS. This proves that for comprehensive analysis of the technical fabric behavior, biaxial tensile tests are recommended. The outcomes of the yield limit $σ_{02}$ (found for 0.02% of the total elongation) show that it grows for the USED fabric, especially in the fill direction (15% of rise for uniaxial tests). This is in accordance with the reasoning described before. The USED material was tensioned during service, and therefore the yield limit $σ_{02}$ increased. The threads in the fill direction, initially crimped, were firstly straightened and then tensioned under working conditions and therefore the greatest growth of the yield limit $σ_{02}$ is observed in this direction. The other obtained parameters do not change their values.

Summing up the uniaxial and biaxial tensile tests, it can be concluded that the environmental conditions and the service tension have mainly influenced primarily the stiffness properties of the PVC coating. Properties of the polyester threads have been unaffected by ageing, but the ultimate strength properties (UTS, $ɛ_{ult}$ ) have decreased. All in all, the technical VALMEX fabric has maintained its initial elastic tensile properties over 20 years of usage and could probably perform further as the roof structure without breakdown for forthcoming years. However, in this particular case, the roof had to be replaced due to the bad condition of the sewn links of the fabric and some holes made accidentally by theater workers.

Viscoelastic Burgers model

The results of the long-term creep tests and the representative model of creep behavior with particular values necessary for the identification of the Burgers model are presented in Figures 8(a) and (b), respectively. The responses of the NOT USED material always give greater values of the strain level. This is because the USED material has been tensioned several times during service, resulting in permanent elongations, so its initial strain due to the creep test would be always lower than for the NOT USED fabric.

Figure 8.

The creep tests results of the USED and NOT USED VALMEX fabric (a) and the parameter relations of the Burgers model³³ (b).

The identification of the Burgers model parameters was conducted according the following algorithm. It is presumed that for the selected stress level $σ_{0} = constant$ , the immediate strain $ɛ_{0}$ is established at time t₀. Consequently, the instantaneous elastic modulus E₁ can be obtained as $E_{1} = σ_{0} / ɛ_{0}$ . For the strain versus time curve of the typical creep test (Figure 8(b)), two distinct phases can be selected: the non-linear (in the time range $t_{0} - t_{1}$ ) and the proportional (in the time range $t_{1} - t_{2}$ ). From the latter one the viscous coefficient $η_{1}$ can be easily written as $η_{1} = σ_{0} (t_{2} - t_{1}) / (ɛ_{2} - ɛ_{1})$ . Due to the proportional part of Equation (2) it is possible to find the value of the auxiliary strain $ɛ_{k} = ɛ_{1} - (σ_{0} / η_{1}) t$ . The difference between the immediate strain $ɛ_{0}$ and the auxiliary strain $ɛ_{k}$ is due to the non-linear part, which results in the relation $E_{2} = σ_{0} / (ɛ_{k} - ɛ_{0})$ . The coefficient $η_{2}$ is obtained from the non-linear part of the curve using the previously obtained parameters E₁, E₂, ɛ_k, η₁ and the least square method for approximation of Equation (2).

For the identification methodology, most important are the time limits

t_{0}, t_{1}, t_{2}

that divide the experimental stress–strain curve into the non-linear and the linear parts (Figure 8(b)). The boundary value t₁ between these ranges has to be subjectively set. Therefore, it has been decided to perform the identification procedure independently for two different values of

t_{1} = 2 e + 5 s

and

t_{1} = 4 e + 5 s

, and for both thread directions of the fabric. It has been proven before³² that the level of immediate elongation

ɛ_{0}

also influences the quality of performed identification; therefore, it was decided to take for calculations two different values of

ɛ_{0}

for each fabric direction (Table 4).

Table 4.

The values of the immediate elongation $ɛ_{0}$ taken for calculations

Fabric type	Fabric direction
Fabric type	WARP		FILL
NOT USED	0.055	0.065	0.113	0.143
USED	0.049	0.059	0.115	0.125

The selected values of immediate elongation

ɛ_{0}

must have been dissimilar for the warp and fill directions, as can be observed in Figure 8(a). It is believed that taking the various values of the boundary time t₁ and immediate elongation

ɛ_{0}

would make the comparative analysis more precise and show if the difference between the USED and NOT USED VALMEX material depends on these constraints. The identified parameters of the Burgers material model (the arithmetic mean value ± its standard deviation) for the NOT USED and USED VALMEX fabric are compiled in Tables 5 and 6 for the warp and fill directions, respectively.

Table 5.

Identification results of the Burgers model for the warp direction of the VALMEX technical fabric

WARP NOT USED
Initial values	t ₀	[s]	0
	t ₁	[s]	4.00E+05		2.00E+05
	t ₂	[s]	12E+05
	$ɛ_{0}$	[-]	0.0485	0.0585	0.0485	0.0585
Identification results	E ₁	[kN/m]	560 ± 90	560 ± 80	570 ± 90	550 ± 80
	E ₂	[kN/m]	3040 ± 240	4110 ± 250	3040 ± 280	4300 ± 270
	$η_{1}$	[-]	6.79 +09 ± 1.26E+09	6.79E+09 ± 1.26E+09	6.16E+09 ± 1.24E+09	6.16E+09 ± 1.24E+09
	$η_{2}$	[-]	1.38E+08 ± 1.76E+05	1.40E+08 ± 2.82E+06	1.03E+08 ± 2.92E+05	1.05E+08 ± 1.92E+06
WARP USED
Initial values	t ₀	[s]	0
	t ₁	[s]	4.00E+05		2.00E+05
	t ₂	[s]	12E+05
	$ɛ_{0}$	[-]	0.055	0.065	0.055	0.065
Identification results	E ₁	[kN/m]	510 ± 10	490 ± 10	510 ± 10	500 ± 10
	E ₂	[kN/m]	3160 ± 80	4220 ± 90	3200 ± 60	4460 ± 50
	$η_{1}$	[-]	7.47E+09 ± 3.04E+08	7.47E+09 ± 3.04E+08	6.82E+09 ± 3.11E+08	6.82E+09 ± 3.11E+08
	$η_{2}$	[-]	1.39E+08 ± 9.27E+05	1.40E+08 ± 3.21E+06	1.04E+08 ± 9.30E+05	1.05E+08 ± 2.17E+06

Table 6.

Identification results of the Burgers model for the fill direction of the VALMEX technical fabric

FILL NOT USED
Initial values	t ₀	[s]	0
	t ₁	[s]	4.00E+05		2.00E+05
	t ₂	[s]	27.0E+05
	$ɛ_{0}$	[-]	0.1325	0.1425	0.1325	0.1425
Identification results	E ₁	[kN/m]	230 ± 40	230 ± 40	230 ± 40	230 ± 40
	E ₂	[kN/m]	1960 ± 550	2600 ± 610	1970 ± 640	2810 ± 770
	$η_{1}$	[-]	1.37 +10 ± 5.02E+09	1.37E+10 ± 5.02E+09	1.07E+10 ± 3.63E+09	1.07E+10 ± 3.63E+09
	$η_{2}$	[-]	1.35E+08 ± 2.74E+06	1.39E+08 ± 1.77E+06	1.02E+08 ± 1.64E+05	1.05E+08 ± 1.54E+06
FILL USED
Initial values	t ₀	[s]	0
	t ₁	[s]	4.00E+05		2.00E+05
	t ₂	[s]	27.0E+05
	$ɛ_{0}$	[-]	0.115	0.125	0.1325	0.1425
Identification results	E ₁	[kN/m]	230 ± 10	260 ± 10	260 ± 10	260 ± 10
	E ₂	[kN/m]	2270 ± 110	2920 ± 160	2190 ± 320	3200 ± 220
	$η_{1}$	[-]	1.63E+10 ± 1.27E+09	1.63E+10 ± 1.27E+09	1.29E+10 ± 5.66E+08	1.29E+10 ± 5.66E+08
	$η_{2}$	[-]	1.36E+08 ± 1.22E+06	1.40E+08 ± 2.87E+05	1.03E+08 ± 4.25E+05	1.06E+08 ± 2.16E+05

The identification results were verified by the numerical simulations of the uniaxial creep tests performed by the own code in the commercial software SigmaPlot. The simulation curves have been obtained using the identified parameters and Equation (2). The laboratory and simulated curves are compared in Figure 9, confirming the correctness of the performed identification for the NOT USED and USED types of fabric, for the warp and fill directions and for all proposed values of the immediate elongation $ɛ_{0}$ .

Figure 9.

Verification of the Burgers model parameter identification by numerical simulation of the creep test for the warp (a) and fill (b) directions of the fabric.

The relative difference (calculated according to Equation (3)) between the Burgers model parameters obtained for the USED and NOT USED samples for the different values of the time limits (

t_{1} = 2 e + 5 s

and

t_{1} = 4 e + 5 s

) and the immediate elongations

ɛ_{0}

are collected in Table 7, for both material directions separately.

Table 7.

Relative errors between Burgers constitutive parameters of both VALMEX fabric types (NOT USED versus USED) and for both the warp and fill directions

WARP
t ₁	[s]	4.00E+05		2.00E+05
$ɛ_{0} (NOTUSED / USED)$	[-]	0.055/0.0485	0.065/0.0585	0.055/0.0485	0.065/0.0585
Parameters		Relative difference δ [%]
E ₁	[kN/m]	9	13	11	9
E ₂	[kN/m]	–4	–3	–5	–4
$η_{1}$	[-]	–10	–10	–11	–11
$η_{2}$	[-]	–1	0	0	0
FILL
t ₁	[s]	4.00E+05		2.00E+05
$ɛ_{0} (NOTUSED / USED)$	[-]	0.1325/0.115	0.1425/0.125	0.1325/0.115	0.1425/0.125
Parameters		Relative difference δ [%]
E ₁	[kN/m]	–13	–13	–13	–13
E ₂	[kN/m]	–16	–12	–11	–14
$η_{1}$	[-]	–19	–19	–20	–20
$η_{2}$	[-]	–1	–1	–1	–1

Despite the various values of the immediate strain $ɛ_{0}$ used for identification purposes, the dispersion of the identification results for the NOT USED and USED fabrics is always the same. It can be concluded for both fabric directions that the identification results are not influenced by the level of the immediate strain $ɛ_{0}$ taken for calculations. Coming to the study of the model parameters, it can be seen that the alteration of the stiffness modulus E₁ is similar for the warp and fill directions of the material. The change of the modulus E₂ is two, three or even four times greater for the fill direction than for the warp one (depending on the other analysis parameters). For the viscous coefficient $η_{1}$ , the change for the fill direction in comparison with the warp direction is two times larger. The last parameter (the viscous coefficient $η_{2}$ ) stays unchanged throughout all the performed identification processes, including changes of the material directions.

It is clear from this study that the influence of the ageing is more pronounced when observing results for the fill direction of the fabric. The natural environmental ageing has affected the parameters $E_{1}, E_{2}, η_{1}$ identified from the linear part of the creep experiment. On the contrary, the coefficient $η_{2}$ (related to the non-linear part of the creep curve) can be omitted in the analysis of the VALMEX fabric, including ageing effects, as it stays at a constant level throughout different analyses. Consequently, it is demonstrated that ageing has not influenced the early phase of creep. It is suggested that during the cyclic usage of the polyester-reinforced PVC-coated VALMEX textile for about 20 years, each time the material would have adapted to the preferred geometry, without significant ageing influence on its initial creep performance. Lastly, for the ageing analysis of the VALMEX behavior through the Burgers model, the parameters $E_{1}, E_{2}, η_{1}$ seem to be of supreme importance, as they change significantly due to ageing and therefore contribute to the material behavior description considerably.

Following this, the parameters $E_{2}, η_{1}$ are identified from the linear part of the strain–time curve and are related to the long-term behavior of the material. As a result, for the numerical calculations of the long-term durability of the fabric, these parameters are the most important ones. The underestimation of the elongations caused by the long-term loadings may cause fabric “pods”, where snow or water can collect, finally resulting in the failure of the structure. To avoid such situations, the level of elongation should be predicted correctly to provide the desirable tensioning of the whole structure all the time the hanging canopy is used. Some practical examples of the importance of that observation have been presented in another paper.³³

Viscoplastic Bodner–Partom model

The identification of the Bodner–Partom model parameters was accomplished in accordance with description presented previously by Kłosowski et al.^18,37 Some necessary improvements in the identification procedure concerning complex behavior of the fabric in the fill direction was introduced in accordance with Kłosowski et al.²⁷ and Żerdzicki.¹⁷ Table 8 collects the final results of the identified parameters. The identification results of the Bodner–Partom model parameters have been confirmed by the computer simulations of the uniaxial tensile tests. The description of appropriate software can be found in Woźnica and Kłosowski.³⁴ Figures 10(a) and (b) present the experimental results compared to the numerical simulations of the uniaxial tensile test for the strain rate of 0.001 1/s for the NOT USED and USED samples, respectively. For the modeling of the fill direction, the results are introduced by two different protocols. Firstly, the longitudinal stiffness modulus

E_{F 1}

has been taken into the calculations (black dash line in Figure 10). In the second approach, the values of

E_{F 0}

and

E_{F 1}

have been applied to the numerical simulations (gray line in Figure 10). For more details concerning these concepts, refer to Kłosowski et al.²⁸ The stress–strain simulation curves for the warp direction are presented in Figure 11. The high values of determination coefficient R² of all the numerical simulations confirmed the correctness of the performed parameter identification.

Table 8.

The Bodner–Partom constitutive parameters comparison between the USED and NOT USED fabrics

Direction		WARP			FILL
Fabric type		NOT USED	USED	δ	NOT USED	USED	δ
Parameter	Unit	NOT USED	USED	[%]	NOT USED	USED	[%]
m ₁	[kN/m]^–1	0.83 ± 0.06	0.60 ± 0.03	28	1.69 ± 0.23	1.24 ± 0.23	27
m ₂	[kN/m]^–1	34 ± 1	44 ± 6	–31	27 ± 7	18 ± 2	34
D ₀	[kN/m]	1	1	-	1	1	-
D ₁	[kN/m]	3.0 ± 0.2	2.4 ± 0.6	21	2.8 ± 0.8	1.3 ± 0.3	54
n	[-]	1.89; R²= 1	1.79; R²= 0.99	5	1.42; R²= 0.97	2.36; R²= 0.97	–67
R ₀	[kN/m]	30; R²= 1	32; R²= 0.99	–5	31; R²= 0.97	27; R²= 0.97	12
R ₁	[kN/m]	90 ± 7	102 ± 5	–13	72 ± 8	74 ± 5	–2

Figure 10.

Stress–strain numerical curves of the VALMEX fabric for the fill direction, for the strain rate of 0.001 1/s: (a) the NOT USED; (b) the USED.

Figure 11.

Stress–strain numerical curves of the VALMEX fabric for the warp direction, for the strain rate of 0.001 1/s: (a) the NOT USED; (b) the USED.

Comparing the USED and the NOT USED material it is clearly seen that the main differences in both directions appear for the parameters $m_{1}, m_{2}$ (related to the isotropic and kinematic hardening coefficients, respectively) and for the parameter D₁, which represents the limiting (maximum) value for the kinematic hardening. Other observations indicate that all parameters for the fill direction (except for the limiting value of the isotropic hardening R₁) change significantly.

Conclusions

In this investigation, the VALMEX fabric has been described by three different constitutive models, separately for the warp and fill directions of the material called the USED and NOT USED fabrics. Parameters of the piecewise, Burgers and Bodner–Partom models have been successfully obtained and compared, leading to some conclusions presented below.

The comparative analysis of the piecewise model parameters between the NOT USED and USED materials has revealed that the environmental, service ageing of the VALMEX fabric has caused a decrease in the UTS for the warp and fill directions and increase of the yield limit for the fill direction. The greatest change, however, has been observed in the growth of PVC coating stiffness (of about 36%). In the case of the biaxial tests, almost all of the strength properties have decreased when comparing the NOT USED and USED materials, but the stiffness of PVC has increased again. It seems to be obvious that outdoor exposure has influenced mostly the coating PVC layer. In the same time the polyester threads responsible for the mechanical strength of the canopy have been unaffected by the ageing or sufficiently protected by the PVC coating from unfavorable environmental conditions.

Coming to the Burgers model, it has been observed that ageing has influenced only the parameters related to the linear part of the creep curve ( $E_{1}, E_{2}, η_{1}$ ). The viscous coefficient $η_{2}$ has remained constant throughout the whole identification process. This suggests that it is not susceptible to ageing and can be treated as constant in further ageing influence analyses. However, for different materials with non-linear behavior, the experimental and identification procedures proposed here should be repeated to find out if the observation concerning the viscous coefficient $η_{2}$ stays correct. Finally, the main differences between the NOT USED and USED fabrics have been noticed for the values of E₂ and $η_{1}$ , which have been at least two times greater for the fill direction than for the warp direction. As long as the analyzed fabrics are treated as anisotropic material, the constitutive parameters in the warp and fill directions are not the same. The observation that in the fill direction some parameters are two times greater can simplify the calculations. Furthermore, while the parameters $E_{2}, η_{1}$ are related to the long-term behavior of the material, it can be concluded that the threads in the fill direction undergo damage faster due to ageing and will affect the lifetime of the whole roofing structure more than the threads in the warp direction. The analysis of the Bodner–Partom model has revealed that ageing has affected the parameters $m_{1}, m_{2}$ (related to the isotropic and kinematic hardening, respectively) for the warp and fill directions, causing a change of about 30%. The greatest increase has been reported for the parameter n, which for the fill direction increased by almost 67%. The fill direction is generally more susceptible to outdoor exposure, as the change of other parameters is more significant in comparison to the ones for the warp direction. Summing up, the findings have proven that the polyester-reinforced PVC-coated VALMEX membrane has kept its initial strength parameters and is in good overall condition after two decades of usage. Another overall conclusion states that the fabrics are more sensitive to the coupled environmental and mechanical ageing in the fill direction than in the warp direction. This can be explained by the fact that the threads in the fill direction underwent greater elongations during the exploitation of the fabric. The threads in the fill direction are interspersed through the threads in the warp direction, and therefore their elongations are greater. Despite the fact that the material from the Forest Opera was tensioned in both directions at least 20 times during its use (due to taking down the roof for the winter time), the threads in the fill direction underwent, apart from the simple tensioning like for the warp threads, additional damages coming from straightening and waving back the fill threads about 20 times.

All the outcomes and drawn conclusions presented here may be used to describe the long-term durability of the polyester-reinforced PVC-coated roofing materials through the proposed constitutive models.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Faculty of Civil and Environmental Engineering (Grant GRAM) at Gdansk University of Technology, the Polish Ministry of Science and Higher Education with a scholarship for young Polish researches and PhD students aimed at their scientific and developmental work and the French Government (through Bourse du Gouvernement Français).

References

Berdahl

Akbari

Levinson

et al.

Weathering of roofing materials – an overview. Constr Build Mater 2008; 22: 423–433.

Razak

Chua

Toyoda

. Weatherability of coated fabrics as roofing material in tropical environment. Build Environ 2004; 39: 87–92.

Xing

Taylor

. Correlating accelerated laboratory, field, and thermal aging TPO membranes. J ASTM Int 2011; 8: 50–70.

Lacasse MA and Vanier DJ (eds). Durability of Building Materials and Components 8. Institute for Research in Construction, Ottawa, Ontario, Canada, 1999, pp. 1083–1092.

Bailey DM, Foltz SD, Rossiter WJ, et al. Performance of polyvinyl chloride (PVC) roofing: results of ten-year field study. In: proceedings of the fourth international symposium on roofing technology, Gaithersburg, USA, 17-19 September 1997, pp. 253–264.

Rosenfiled MJ and Wilcoski J. Experimental polyvinyl chloride (PVC) roofing: field tests results. TR M-87/04/ADA178193, USACERL, 1987.

Foltz S and Bailey DM. Polyvinyl chloride (PVC) roofing: preliminary field test results. In: Wallace TJ and Rossiter WJ (eds) Roofing research and standards development: second volume ASTM STP 1088, New York, ASTM, 1990; pp.63-81.

Beer HR, Delgado AH, Paroli RM, et al. Durability of PVC roofing membranes - proof by testing after long term field exposure. In: 10DBMC international conference on durability of building materials and components, Lyon, France, 17-20 April 2005, paper no. TT2-157. Centre scientifique et technique du bâtiment (CSTB).

Whelan BJ, Graveline S, Delgado AH, et al. Field investigation and laboratory testing of exposed poly(vinyl chloride) roof systems. In: CIB world building congress, “building for the future”, Toronto, Canada, 1-7 May 2004, paper no. 835, pp. 1–14. Ottawa: National Research Council of Canada.

10.

ASTM D-4434:1985. Standard specification for polyvinyl chloride sheet roofing.

11.

DIN16726:1983-05. Plastic sheets – Testing.

12.

SIA V280:1986-12. Kunstoff-Dichtungsbahnen (Polymer-Dichtungsbahnen) – Anforderungswerte und Materialprüfung.

13.

Cash G, Bailey DM, Davies AG, et al. Predictive service life tests for roofing membranes. In: proceedings of 10DBMC international conference on durability of building materials and components, Lyon, France, 17–20 April 2005, paper number TT4-213. Centre scientifique et technique du bâtiment (CSTB).

14.

Koontz JD. Field evaluation and laboratory testing of PVC roof systems. In: proceedings of fourth international symposium on roofing technology (NRCA), Gaithersburg, USA, 17-19 September 1997, pp.22–27. NRCA.

15.

Ambroziak

. Mechanical properties of PVDF-coated fabric under tensile tests. Journal of Polymer Engineering 2015; 35: 377–390.

16.

Ambroziak

. Mechanical properties of polyester coated fabric subjected to biaxial loading. Journal of Materials in Civil Engineering 2015; 27: 1–8.

17.

Żerdzicki K. Durability evaluation of textile hanging roofs materials. PhD Thesis, Gdańsk University of Technology, Orleans University, 2005.

18.

Kłosowski

Zagubień

Woźnica

. Investigation on rheological properties of technical fabric “Panama”. Arch Appl Mech 2004; 73: 661–681.

19.

Findley

Lai

Onaran

. Creep and relaxation of nonlinear viscoelastic materials, Amsterdam-New York-Oxford: North-Holland, 1976.

20.

Siginer

. Stability of non-linear constitutive formulations for viscoelastic fluids, New York: Springer, 2014.

21.

Żerdzicki K. Właściwości mechaniczne poliwęglanu w różnych temperaturach. In: Buchacz A (ed.) Badania i analizy wybranych zagadnień z budownictwa. Gliwice: Wydawnictwo Politechniki Śląskiej, 2011, pp.423–430.

22.

Lei

et al.

Creep behavior of bagasse fiber reinforced polymer composites. Bioresour Technol 2010; 101: 3280–3286.

23.

Chatree

Thanate

Wiriya

. Time–temperature and stress dependent behaviors of composites made from recycled polypropylene and rubberwood flour. Constr Build Mater 2014; 66: 98–104.

24.

Bodner

Partom

. Constitutive equations for elastic–viscoplastic strain-hardening materials. J Appl Mech 1985; 42: 385–389.

25.

Andersson

. An implicit formulation of the Bodner–Partom constitutive equations. Comput Struct 2003; 81: 1405–1414.

26.

Zaïri

Naït-Abdelaziz

Woźnica

et al.

Constitutive equations for the viscoplastic-damage behaviour of a rubber-modified polymer. Eur J Mech A-Solids 2005; 24: 169–182.

27.

Rubin

Bodner

. A three-dimensional nonlinear model for dissipative response of soft tissue. Int J Solids Struct 2002; 39: 5081–5099.

28.

Kłosowski

Żerdzicki

Woźnica

. Identification of Bodner-Partom model parameters for technical fabrics. Comput Struct 2017; 187: 114–121.

29.

Zerdzicki K, Klosowski P and Woźnica K. Application of the Bodner-Partom constitutive equations for modelling the technical fabric Valmex used for the hanging roof of the Forest Opera in Sopot. In: Pietraszkiewicz W and Gorski J (eds) Shell Structures: Theory and Applications. Taylor & Francis, London, 2014, pp.579–582.

30.

POLNAM Fabrics Manufacture. Report on strength properties of VALMEX FR 1000 Hallentype III Universal fabric, Częstochowa, 1989.

31.

ISO 1421:2001. Fabrics coated with rubber or plastics – determination of breaking strength and elongation at break.

32.

Żerdzicki

Woźnica

Kłosowski

. Experimental research on technical fabric of the Forest Opera roof in Sopot. Mach Dyn Res 2012; 36: 136–145.

33.

Żerdzicki K, Kłosowski P and Woźnica K. Modelling of the viscoelastic properties of the technical fabric VALMEX. In: Kleiber W, Burczyński T, Wilde K, et al. (eds) Advances in mechanics: theoretical, computational and interdisciplinary issues. Taylor & Francis Group, London, 2016, pp.611–614.

34.

Ambroziak

Klosowski

. Mechanical testing of technical woven fabrics. J Reinf Plast Compos 2013; 32: 726–739.

35.

Woźnica

Kłosowski

. Evaluation of viscoplastic parameters and its application for dynamic behaviour of plates. Arch Appl Mech 2000; 70: 561–570.

36.

Bodner SR. Unified plasticity – an engineering approach. Final Report. Faculty of Mechanical Engineering. Technion – Israel Institute of Technology Haifa, 2000.

37.

Kłosowski

Woźnica

. Numerical treatment of elasto viscoplastic shells in the range of moderate and large rotations. Comput Mech 2004; 34: 194–212.