# The Impact of Elongation on Change in Electrical Resistance of Electrically Conductive Yarns Woven into Fabric

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Measuring the Electrical Resistance

_{N}connects to the voltage of the source U in series (Figure 1), or with a resistor of unknown resistance R

_{X}. The value of the unknown resistance is obtained from the value of the known resistance and the ratio of voltage drops—i.e., currents—on both resistors [35].

_{V}. With this connection, the voltage drops across the resistors R

_{X}and R

_{N}are:

_{X}and R

_{N}are significantly less than R

_{V}, the impact of R

_{V}can be neglected, from which follows the expression (2):

#### 2.2. Force–Elongation Diagram for the Fabric

_{0}, a

_{1}, …, and a

_{n}are known coefficients.

_{0}, a

_{1}, …, and a

_{n}. Determining point A between the linear and nonlinear parts (Figure 2) represents the most sensitive part of the problem.

#### 2.3. Mathematical Model of Force and Elongation Dependence

_{i}and F

_{i}are experimentally obtained values of elongation and associated force, respectively, while F is the theoretical value of force.

#### 2.4. Electrical Resistance–Elongation Diagram for the Fabric, and Mathematical Model

_{0}. The curve in the electrical resistance–elongation diagram (Figure 2) consists of a linear part AD and a nonlinear part located on the curve between points D and E. Point E represents the maximum value of electrical resistance R

_{E}at elongation ε

_{E}. In the initial range of the diagram up to point D the electrical resistance is represented as a linear function of extension:

_{0}, b

_{1}, …, and b

_{n}are known coefficients.

_{0}, b

_{1}, …, and b

_{n}is solved by the least squares method. A mathematical model of the dependence of the electrical resistance R and the extension ε is set, which can be written as:

_{i}and R

_{i}are experimentally obtained values of elongation and associated resistance, respectively, while R is the theoretical value of resistance.

## 3. Experimental Part

#### 3.1. Samples of the Conductive Yarns

#### 3.2. Fabrics Samples

_{0}= 20 cm, (Figure 3a); the direction of the warp (0°), where the length of the conductive yarn was equal to the width of the sample, and amounted to c

_{0}= 5 cm (Figure 3b); and at an angle of 45° to the weft, where the length of the conductive yarn d

_{0}= c

_{0}·$\sqrt{2}=7.05\text{}\mathrm{cm}$ (Figure 3c). The direction of action of the tensile force during the performance of the experiment was always the same. For each specified cutting direction of the electrically conductive fabric sample, and for each yarn count, five measurements were performed.

_{1}, and that of yarn 2 is denoted by R

_{2}.

#### 3.3. Method and Manner of Measuring the Electrical Resistance of Electrically Conductive Yarns

_{0}= 200 mm, and copper conductors for measuring voltage change—i.e., resistance—were connected to the ECYs by the process of fixing copper clips (crimping). The samples were subjected to uniaxial tensile loading at a tensile speed of v = 100 mm/min until a break was reached. The tensile properties of all samples were tested according to ISO 13934-1:2008 with a tensile tester.

## 4. Results and Discussion

_{1}denotes the mean value of the measured electrical resistance of the electrically conductive yarn 1; R

_{2}denotes the mean value of the measured electrical resistance of the electrically conductive yarn 2 (Figure 3). The mean value of the electrical resistance of conductive yarns 1 and 2 is denoted by R = (R1 + R2)/2.

_{B}and the corresponding elongation ε

_{B}, along with the coefficients of variation CV, are shown in Table S1.

_{B}has the highest value for sample Z-0 with the highest yarn count cut in the warp direction—amounting to F

_{B}= 1068.8 N—and the lowest value for sample X-90 with the lowest yarn count, cut in the weft direction, amounting to F

_{B}= 809.1 N. The maximum elongation ε

_{B}has the highest value for sample Z-45—amounting to ε

_{B}= 65.98%—and the lowest value for sample X-90, amounting to ε

_{B}= 25.22%. For samples cut in the same direction, the value of maximum force and elongation increases with the increase in the yarn count of the electrically conductive yarn.

_{01}. The mean values of the initial electrical resistance of the electrically conductive yarn 2 are denoted by R

_{02}, and the mean value of the initial resistance of the conductive yarns 1 and 2 is R

_{0}= (R

_{01}+ R

_{02})/2. The mean values of the measured maximum electrical resistance at point E of the electrically conductive yarn 1 are denoted by R

_{E1}. The mean values of the measured maximum electrical resistance of the electrically conductive yarn 2 are denoted by R

_{E2}, and the mean value of the maximum electrical resistance of the electrically conductive yarns 1 and 2 is R

_{E}= (R

_{E1}+ R

_{E2})/2.

_{E}has the highest value for sample X-90 with the lowest yarn count—which is cut in the weft direction and amounts to R

_{E}= 177.69 Ω—and has the lowest value for sample Z-0 with the highest yarn count, which is cut in the warp direction and amounts to R

_{E}= 3.09 Ω. The initial electrical resistance has the highest value for sample X-90— amounting to R

_{0}= 53.40 Ω—and the lowest value for sample Z-0, amounting to R

_{0}= 2.99 Ω.

_{A}, F

_{A}) given in Table S3 for different samples of fabrics with ECYs. Point A is common to the experimental curve and the mathematical model; it is obtained from Equations (6) and (7).

_{0}, a

_{1}, a

_{2}, and a

_{3}of the cubic parabola (nonlinear range). The values of these coefficients were calculated using Equations (4), (5) and (8), and are shown in Table S3. The correlation coefficients r between the experimental curve and the mathematical model were calculated, and are shown in Table S3. The correlation coefficients show very high congruence of the experimental curves and the mathematical model in the linear and nonlinear ranges. The equation of lines and of the third-order polynomial describe very well the curve of the ratio between the experimentally obtained values of the forces and the corresponding elongations.

_{D}, F

_{D}) given in Table S4 for different samples of fabric with ECYs. Point D is common to the experimental curve and the mathematical model, and is obtained from Equations (13) and (14).

_{0}, b

_{1}, b

_{2}, and b

_{3}of the cubic parabola (nonlinear range)—were calculated. The values of these coefficients were calculated using Equations (11), (12) and (15), and are shown in Table S4. The associated correlation coefficients r between the experimental curve and the mathematical model were also calculated, and are given in Table S4. The correlation coefficients show good congruence of the experimental curves and the mathematical model in the linear and nonlinear ranges. The equation of lines and the third-order polynomial describe well the curve of the ratio of the experimentally measured values of electrical resistance and the corresponding elongations.

_{0}to point D, where they have the value R

_{D}(Figure 6a–c and Figure 7a–c). For these samples, in the linear part of the force–elongation dependence diagram, the direction coefficient of line k has a positive value (Table S3, Figure 6a–c and Figure 7a–c). Thus, in the linear part, the values of tensile forces and elongation increase, and at the same time the values of electrical resistance decrease (Tables S3 and S4). It can be freely assumed that in the structure of the electrically conductive yarn there is an increase in the number of parallel contact resistances between the filaments, with a simultaneous decrease in series resistances on the surface, which causes an overall decrease in electrical resistance at the tested length of the electrically conductive yarn.

_{0}towards point D, where they have the value R

_{D}(Figure 8a–c). For these samples, in the linear part of the force–elongation diagram, the direction coefficient of line k also has a positive value (Table S3), and the values of the tensile forces and elongation increase, as do the values of electrical resistance (Tables S3 and S4). In these samples, there is a direct action of force and elongation on the conductive yarns, which are consequently elongated, reducing their waviness, which increases their electrical resistance. Due to the action of force and tension, there is an increased number of surface series resistances with simultaneous interruptions of contacts in the structure (between filaments), which eliminates parallel joints; thus, there is a noticeable increase in total electrical resistance on the tested length of woven electrically conductive yarn.

_{D}) are compared (Table S4) for samples of the same yarn count that are cut in different directions (X-0, X-45, and X-90), (Y-0, Y-45, and Y-90), and (Z-0, Z-45, and Z-90), it can be concluded that the values of electrical resistance are the lowest when the samples are cut in the warp direction, and the highest for the samples cut in the weft direction (14.19 Ω, 19.53 Ω, and 54.87 Ω), (10.34 Ω, 14.64 Ω, and 41.20 Ω), and (2.92 Ω, 4.17 Ω, and 11.81 Ω), respectively. Samples cut in the direction of the warp (X-0, Y-0, and Z-0) have a 5-cm length of electrically conductive yarn, which is located at the sample perpendicular to the direction of the tensile force. Under the action of force, the sample is elongated in the direction of the force, and in the transverse direction (perpendicular to the direction of force action) there is a lateral narrowing of the sample and an increase in the corrugation of the electrically conductive yarn. In doing so, its cross-section increases on parts of the yarn due to compression, which has the effect of reducing the electrical resistance. Here we should also assume surface interruptions of series-connected elements, with a simultaneous significant increase in the number of parallel joints in the yarn structure, so it is logical that the total resistance at the measured yarn length decreases.

_{0}.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Characteristic diagrams of the tensile force–elongation (F-ε) curve of the woven fabric and the electrical resistance–elongation (R-ε) curve of the conductive fabric yarn.

**Figure 3.**Fabric samples with woven ECYs: (

**a**) sample cut in the weft direction (0°); (

**b**) sample cut in the warp direction (90°); (

**c**) sample cut at a 45° angle.

**Figure 6.**Force–elongation (F-ε) and electrical resistance–elongation (R-ε) diagrams for fabric samples cut in the warp direction: (

**a**) for sample X-0; (

**b**) for sample Y-0; and (

**c**) for sample Z-0. (

**d**) Experimental and mathematical models of F-ε and R-ε curves for sample Y-0.

**Figure 7.**Force–elongation (F-ε) and electrical resistance–elongation (R-ε) diagrams for fabric samples cut at a 45° angle: (

**a**) for sample X-45; (

**b**) for sample Y-45; and (

**c**) for sample Z-45. (

**d**) Experimental and mathematical models of F-ε and R-ε curves for sample Y-45.

**Figure 8.**Force–elongation (F-ε) and electrical resistance–elongation (R-ε) diagrams for fabric samples cut in the weft direction: (

**a**) for sample X-90; (

**b**) for sample Y-90; and (

**c**) for sample Z-90. (

**d**) Experimental and mathematical models of F-ε and R-ε curves for sample Y-90.

**Figure 9.**Correlation between the slope of the line k of force and the slope of the line p of electric resistance for samples: (

**a**) X-0, Y-0, and Z-0; (

**b**) X-45, Y-45, and Z-45; and (

**c**) X-90, Y-90, and Z-90.

Code Name | X | Y | Z |
---|---|---|---|

Material | Polyamide 6.6 filament | Polyamide 6.6 filament | Polyamide 6.6 filament |

Metal-plated | 99% Pure silver | 99% Pure silver | 99% Pure silver |

Coating | Yes | Yes | Yes |

Filaments | 17 | 17 | 36 |

Ply | 1 | 2 | 2 |

Yarn count, raw (dtex) | 117f17 | 117f17 | 235f36 |

Yarn count, silverized (dtex) | 142 | 295 | 604 |

Resistivity | <500 Ω/m | <300 Ω/m | 80 Ω/m |

Warp Direction | Weft Direction | |||||||
---|---|---|---|---|---|---|---|---|

Fabric Structure | Yarn Fibers | Yarn Count (tex) | Density (cm^{−1}) | Yarn Fibers | Yarn Count (tex) | Density (cm^{−1}) | Weight (g/m^{2}) | Fabric Thickness (mm) |

Plain weave | 50%Cotton/ 50%PA | 33 × 2 | 33 | 50%Cotton/ 50%PA | 50 | 25 | 208.2 | 0.463 |

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**MDPI and ACS Style**

Knezić, Ž.; Penava, Ž.; Penava, D.Š.; Rogale, D.
The Impact of Elongation on Change in Electrical Resistance of Electrically Conductive Yarns Woven into Fabric. *Materials* **2021**, *14*, 3390.
https://doi.org/10.3390/ma14123390

**AMA Style**

Knezić Ž, Penava Ž, Penava DŠ, Rogale D.
The Impact of Elongation on Change in Electrical Resistance of Electrically Conductive Yarns Woven into Fabric. *Materials*. 2021; 14(12):3390.
https://doi.org/10.3390/ma14123390

**Chicago/Turabian Style**

Knezić, Željko, Željko Penava, Diana Šimić Penava, and Dubravko Rogale.
2021. "The Impact of Elongation on Change in Electrical Resistance of Electrically Conductive Yarns Woven into Fabric" *Materials* 14, no. 12: 3390.
https://doi.org/10.3390/ma14123390