12
Journal of Engineered Fibers and Fabrics 114 http://www.jeffjournal.org Volume 10, Issue 4 – 2015 Woven Fabrics Behavior in Pure Shear Željko Penava 1 , Diana Šimić Penava 2 , Marija Nakić 3 1 University of Zagreb, Faculty of Textile Technology, Zagreb CROATIA 2 University of Zagreb, Department of Engineering, Zagreb CROATIA 3 University of Mostar, Mostar, BOSNIA and HERZEGOVINA Correspondence to: Željko Penava email: [email protected] ABSTRACT Shear behavior is one of the most important mechanical characteristics that contributes to the performance and appearance of woven fabrics. Because of anisotropy, shear properties of woven fabric are tested in various directions. This research is focused on the experimental study of shear properties of plain woven fabric when shear force acts on specimens that are cut at different angles to the direction of the weft. Tests were conducted on woven fabric specimens that were fastened in two parallel clamps of the tensile tester. Five cotton woven fabrics of different weft density and of the same warp density were used. The research results show a very high degree of correlation between shear force and its axial component for all directions of the cutting specimen, and likewise between the relative extension of the diagonal of the specimen and the vertical displacement of the specimen. The initial shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial shear modulus, Pure shear, Shear angle, Shear force, Woven fabric INTRODUCTION Practically used, textile fabrics are subjected to a wide range of complex deformations, so the shear properties of woven fabrics are important in many practical applications. During shear deformation, the woven fabric yarns experience a number of angular variations of warp and weft yarns. Test methods of shear properties of woven fabrics are listed in literature [1-3]. To understand the mechanisms of woven fabric shear behavior, shear apparatuses measuring woven fabric shear properties are described [4-8]. They indicated that the hysteresis produced during shearing is determined entirely by frictional restraints arising in the rotation of the yarns from the intersecting points in woven fabric. In addition, the existing literature proves that shear mechanism is one of the most important properties affecting draping and pliability of woven fabrics [9]. Shear deformation of woven fabrics also affects their bending and tensile properties in various directions other than just in the warp and weft directions [10]. Kilby introduced the classical elasticity theory on the assumption that woven fabric is regarded as an anisotropic material with two planes of symmetry at right angles to each other [11]. He used a simple grid model for woven fabrics and analyzed the relationship between stress and strain in the plane and noted that there is a connection between the Poisson ratio, shear modulus, and modulus of elasticity of woven fabric. He showed that shear rigidity can be calculated from the tensile properties of woven fabric at an angle of 45° in relation to the plane of the plate. Grosberg and Park [12] conducted a mathematical analysis to determine initial shear modulus, frictional shear stress, and shear rigidity. Initial research mainly focused on the shear behavior of woven fabrics in both principal directions because it affects woven fabric behavior to a great extent, and later their attention was directed at woven fabric shear properties in various directions. Consequently, shear properties of woven fabrics should be determined in various directions. Engineering properties were studied by a number of researchers. Anisotropy is a characteristic of woven fabric that affects its physical and mechanical properties [13]. Determination of shear stresses and strains in various directions involves a complex mechanism providing information about shear properties of woven fabrics in various directions where the angles between two sets of yarns change in intersecting points. Due to the inherent nature of woven fabrics the accurate and reliable determination of shear angle and shear stress is a difficult task.

Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Embed Size (px)

Citation preview

Page 1: Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Journal of Engineered Fibers and Fabrics 114 http://www.jeffjournal.org Volume 10, Issue 4 – 2015

Woven Fabrics Behavior in Pure Shear

Željko Penava1, Diana Šimić Penava2, Marija Nakić3

1University of Zagreb, Faculty of Textile Technology, Zagreb CROATIA

2University of Zagreb, Department of Engineering, Zagreb CROATIA

3University of Mostar, Mostar, BOSNIA and HERZEGOVINA

Correspondence to: Željko Penava email: [email protected]

ABSTRACT Shear behavior is one of the most important mechanical characteristics that contributes to the performance and appearance of woven fabrics. Because of anisotropy, shear properties of woven fabric are tested in various directions. This research is focused on the experimental study of shear properties of plain woven fabric when shear force acts on specimens that are cut at different angles to the direction of the weft. Tests were conducted on woven fabric specimens that were fastened in two parallel clamps of the tensile tester. Five cotton woven fabrics of different weft density and of the same warp density were used. The research results show a very high degree of correlation between shear force and its axial component for all directions of the cutting specimen, and likewise between the relative extension of the diagonal of the specimen and the vertical displacement of the specimen. The initial shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial shear modulus, Pure shear, Shear angle, Shear force, Woven fabric INTRODUCTION Practically used, textile fabrics are subjected to a wide range of complex deformations, so the shear properties of woven fabrics are important in many practical applications. During shear deformation, the woven fabric yarns experience a number of angular variations of warp and weft yarns. Test methods of shear properties of woven fabrics are listed in literature [1-3]. To understand the mechanisms of woven fabric shear behavior, shear apparatuses measuring woven fabric shear properties are described [4-8]. They indicated that the hysteresis produced during shearing is determined entirely by frictional restraints arising in the rotation of the yarns

from the intersecting points in woven fabric. In addition, the existing literature proves that shear mechanism is one of the most important properties affecting draping and pliability of woven fabrics [9]. Shear deformation of woven fabrics also affects their bending and tensile properties in various directions other than just in the warp and weft directions [10]. Kilby introduced the classical elasticity theory on the assumption that woven fabric is regarded as an anisotropic material with two planes of symmetry at right angles to each other [11]. He used a simple grid model for woven fabrics and analyzed the relationship between stress and strain in the plane and noted that there is a connection between the Poisson ratio, shear modulus, and modulus of elasticity of woven fabric. He showed that shear rigidity can be calculated from the tensile properties of woven fabric at an angle of 45° in relation to the plane of the plate. Grosberg and Park [12] conducted a mathematical analysis to determine initial shear modulus, frictional shear stress, and shear rigidity. Initial research mainly focused on the shear behavior of woven fabrics in both principal directions because it affects woven fabric behavior to a great extent, and later their attention was directed at woven fabric shear properties in various directions. Consequently, shear properties of woven fabrics should be determined in various directions. Engineering properties were studied by a number of researchers. Anisotropy is a characteristic of woven fabric that affects its physical and mechanical properties [13]. Determination of shear stresses and strains in various directions involves a complex mechanism providing information about shear properties of woven fabrics in various directions where the angles between two sets of yarns change in intersecting points. Due to the inherent nature of woven fabrics the accurate and reliable determination of shear angle and shear stress is a difficult task.

Page 2: Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Journal of Engineered Fibers and Fabrics 115 http://www.jeffjournal.org Volume 10, Issue 4 – 2015

The aim of this study is to determine shear angle, shear forces and shear stresses of woven fabric using clamps that were specially designed in laboratory to measure shear forces. Shear force acts on woven fabric specimens that are cut at different angles in the weft direction. Also, the influence of anisotropy and weft density on the initial shear modulus values of a woven fabric was analyzed. The degree of agreement between the experimental results and calculated values of initial shear modulus was determined. The design of the apparatus and the measurement procedures are described and illustrated in this paper. Woven fabric specimens fastened in two parallel clamps placed in the tensile tester will be used for the experimental determination of the shear properties of woven fabric. THEORETICAL PART Woven fabrics are generally inhomogeneous and discontinuous objects. They are elastic orthotropic materials with a very small deformation defined as orthotropic plates with two mutually perpendicular planes of elastic symmetry [14]. These planes of elastic symmetry are planes of orthotropy. The x-axis runs in the weft direction, and the y-axis runs in the warp direction of woven fabric (Figure 1). In the arbitrary cross section of the woven fabric, specimens under stress generally act with normal stresses σx, σy and shear stresses τxy or σk, σl, τkl.

FIGURE 1. Woven fabric element of an orthotropic plate. For a particular state of plane stress in element we can determine the plane in which only shear stresses act and normal stresses are zero [15]. In the plane whose normal forms an angle φ=45° with the x-axis, normal stress is σk=0, and shear stress is τkl= τ. This kind of stress is called pure shear. In pure shear a relative change in volume is εv=0. Shear stress changes only body shape, but it does not change its volume. The isolated element of woven

fabric ABCD which is subjected to pure shear is shown in Figure 2. During shear deformation, the woven fabric yarns experience a large angular variation between warp and weft yarns.

FIGURE 2. Woven fabric element subjected to pure shear - rectangular shape of element. If the AB side is motionless, then under the action of shear force T acting in the plane of the element side shear stress τ will occur and will come to shearing the DC side parallel with the AB side for an amount of CC'=DD'=δ (mm) which is called absolute shear (Figure 2). ABCD rectangle becomes a ABC'D' parallelogram. The side lengths of woven fabric elements do not change, only the right angles between the element sides change, right angles become sharp or blunt angles. A change in the right angle is denoted with γ (rad) and is called relative shear, shear angle or relative angular (shear) strain. It is a measure of deformation. During elastic deformation shear angle γ is very small and is equal to absolute shear divided by the spacing between the shear plane, Eq. (1):

(1)

The value of absolute shear δ does not depend only on the value of shear stresses, but also on the dimensions of the cut element. Shear stress τ (N/mm2

= MPa) of woven fabric can be directly calculated from shear forces. Shear strain is equal to shear angle γ. It is assumed that the thickness of the woven fabric t (mm) is constant during shearing. Thus, shear stress is calculated from Eq. (2)

(2)

where: T (N) is shear force acting on the DC side of woven fabric element as a result of stress (assuming a

Page 3: Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Journal of Engineered Fibers and Fabrics 116 http://www.jeffjournal.org Volume 10, Issue 4 – 2015

uniform distribution of shear stresses on the surface side), A (mm2) is the surface side DC of the woven fabric element. During shearing, the initial length of

the diagonal of the element 220 baADd +== is

increased by Δd (absolute extension of the diagonal) (Figure 2). The final length of the diagonal AD'= d (mm) is calculated from Eq. (3):

ddd ∆+= 0 or ( )22 δ++= abd (3) If we equalize the above expressions (3), we get the equation for calculating the extension Δd of the diagonal d0=AD:

( ) 022 dabd −++=∆ δ (4)

The relative extension of diagonal εd (%) is:

%1000

⋅∆

=dd

dε (5)

Shear force T is divided into two perpendicular components. Te is the axial component acting in the direction of the diagonal AD' and causes the extension of the diagonal in the element which is subjected to shearing (Figure 2). Values cosθ are obtained from Figure 2, and force values Te are calculated according to Eq. (6)

θcos⋅= TTe , where δ

θ d∆=cos (6)

The connection between shear angle γ and shear stress τ is shown by Eq. (7) which represents Hooke's law of shear and applies for elastic, homogeneous isotropic material in a linear region, i.e., where the relationship between shear stress and shear strain is linear. G (N/mm2 = MPa) is shear modulus.

γτ ⋅= G (7) For an orthotropic and elastic material when the k- and l -axes do not coincide with the main x- and y -axes of orthotropy, anisotropic behavior of materials under loads can be written in matrix form by Hooke's law [14]:

=

kl

l

k

kllk

lll

kl

kk

lk

k

kl

l

k

G

EE

EE

τσσ

αα

αν

αν

γεε

1

1

1

(8)

σk and σl are normal stresses, τkl is shear stress, εk, εl are normal strain (relative extension strain), γkl is shear strain (relative angle strain), Ek, El are modulus of elasticity, Gkl is shear modulus, αk, αl are elasticity coefficients, νkl, νlk are Poisson’s ratio for arbitrary coordinate system k, l. Shear Modulus of Woven Fabrics The functional relationship between stress and strain cannot be determined theoretically, but only by experimental testing of samples made of certain materials. Mechanical properties are mainly investigated within the area of elasticity which means in terms of low load [16]. Diagrams of shearing and diagrams of extension have a similar shape. It is assumed that force - shear angle curve for woven fabric is an approximate straight line before yield point. Therefore, elastic performance equation can apply here. Shear modulus is the initial, linear elastic slope of the stress-strain curve in shear. It is the numerical constant that describes the elastic properties of a woven fabric under the application of shear forces. For orthotropic elastic materials, shear modulus Gkl, in various directions of cutting samples, i.e., in the coordinate system k, l, whose axes do not coincide with the main axes x, y, is obtained by using the expression for the transformation of elastic constants, which states [14]:

( )ϕ

ϕϕ

ϕϕν

GG

EEEG

xy

x

xy

yxkl

1sincos1

sincos42111

222

22

=−⋅+

+⋅⋅

⋅++=

(9)

φ is the cutting angle of the samples due to the weft (Figure 1); Ex, Ey are the modulus of elasticity in two main directions (weft direction φ=0°, warp direction φ=90°); Gxy is the shear modulus between both principal directions; νxy is Poisson's ratio. Shear modulus Gkl changes depending on the angle φ, and will be denoted as Gφ=Gkl.

Page 4: Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Journal of Engineered Fibers and Fabrics 117 http://www.jeffjournal.org Volume 10, Issue 4 – 2015

Shear modulus Gφ in any given direction can be calculated from Eq. (9). The numerical values of Ex, Ey, Gxy, νxy are obtained by experimental testing of woven fabric samples in the laboratory. In standard conditions the value of νxy is very difficult to determine by experimental measurements. The theoretical treatment suggests that measurements of the modulus in two directions (weft and warp direction) are insufficient to define the shear modulus of woven fabric. An investigation of the third direction is therefore necessary, and the most convenient direction is φ=45°. So, measurements were carried out in three directions by considering samples cut along the warp, weft, and 45° directions. Therefore, when considering φ=45° values, Eq. (9) gives:

yxx

xy

EEGE1112

045

−−=⋅ν

(10)

Substituting Eq. (10) into Eq. (9), we get

( )22222

45

sincos1sincos410

ϕϕϕϕϕ

−+

=

GGG(11)

Thus, woven fabric shear modulus in various directions can be predicted from Eq. (11) when its values in the warp or weft directions and under the angle of 45° are measured. Because shear modulus provides a measure of resistance to rotational movements between the warp and weft yarns at the intersecting points when the woven fabric is subjected to small shear deformation, the relationship between both principal directions should be determined. However, if differences in the shear modulus values between the warp and weft directions are large, we take the average value in both principal directions to calculate shear modulus in various directions given in Eq. (11). Eq. (11) indicates the mathematical relation between shear modulus Gφ in any given direction and values of G and G45°. MATERIALS AND METHODS In the experimental part of the paper, the influence of anisotropy of woven fabrics and the influence of yarn density on shear properties were tested. The experimental study of shear properties of woven fabric when shear force acts on the samples that are cut at different angles 0°, 15°, 30°, 45°, 60°, 75°, 90° in the weft direction was conducted. The values of shear force in relation to the shear angle were measured. Based on the experimentally obtained values, initial shear modulus was calculated in various directions.

Materials To carry out this study, five cotton woven fabrics of different weft density and of the same warp density were available. In this work, only balanced orthogonal plain weave fabric was considered. Structural properties of the tested woven fabrics are shown in Table I.

TABLE I. Description of woven fabric.

Fabric code

Yarn count (tex)

Yarn density (cm-1) Mass

(g/m2) Thickness

(mm) warp weft warp weft S18 30 30 23 18 135 0.38 S20 30 30 23 20 141 0.40 S22 30 30 23 22 149 0.41 S24 30 30 23 24 157 0.42 S26 30 30 23 26 165 0.44

Before testing all samples were conditioned under standard atmospheric conditions (relative humidity 65 ± 2%, temperature of 20 ± 2°C). Five tests were carried out on the tensile tester for each mentioned cutting direction of the specimen (0°, 15°, 30°, 45°, 60°, 75°, 90°) (Figure 3). For each cutting direction, the average values obtained from five measurements will be shown in diagrams and will be used to calculate the initial shear modulus.

FIGURE 3. Schematic view of directions (angles) of cutting specimens. Methods The Uniaxial Bias Extension (UBE) test can be used only to characterize textile-based materials in relation to relatively low shear angles, typically around 40 degrees, before test specimens begin to deform through mechanisms other than trellis shear, such as intra-ply slip. Such mechanisms are a result of the specific conditions imposed by the UBE test and are not particularly representative of actual press or diaphragm forming conditions. In this test range shear angles are limited, and it is difficult to control

Page 5: Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Journal of Engineered Fibers and Fabrics 118 http://www.jeffjournal.org Volume 10, Issue 4 – 2015

shear rate. Picture Frame test induces an approximately homogeneous deformation throughout the test specimen but is known to be susceptible to errors resulting from its boundary conditions. Sample misalignment may produce large forces due to tensile strain along the fiber directions. This test requires a specialized rig. Specimens being tested on the KES shear tester are not subjected to pure shear deformation. Therefore, test results cannot be directly used for the determination of fabric shear modulus and stress-strain relationship. In order to avoid the disadvantages of the above methods of determining fabric shear properties, clamps for shearing woven fabrics were designed, manufactured, and are schematically shown in Figure 4a. They consist of the left (fixed) clamp and the right (movable) clamp. The force acting on the right clamp causes its vertical displacement. The left clamp is placed on the upper plate on which there is a measuring probe, and the right clamp to the lower plate on which the movable clamp is usually placed. The distance between the left and right clamp can be adjusted in the range 0-50 mm. The maximum specimen size that can be fixed within the clamp is 75 mm. For this testing, specimens were cut in dimensions 125 x 75 mm, fastened in the clamps of the instrument at a distance of b=25 mm, and subjected to a force acting in the plane of the fixed side of the specimen till rupture (Figure 4b). Sample dimensions are: a=75 mm and b=25 mm.

a) b)

FIGURE 4. Sample testing in clamps: (a) schematic view, (b) photo.

A tensile tester Statimat M of the German manufacturer "Textechno" was used for testing. This tensile tester is an automatic, microprocessor-controlled instrument operating on the principle of constant rate of extension. Two parallel clamps at a distance of 25 mm are fixed on the tensile tester and pulling speed of the right clamp is 100 mm/min. The action of external force T causes the shearing of the right fixed side of the specimen in relation to the left side by δ. Forces in parallel clamps cause a shear condition in the plane of the woven fabric specimen, whereby woven fabric deformation is uniform. In this study the value of the pulling forces T in the right clamps and its corresponding vertical displacements δ were measured. The angle between the warp and weft yarns change leading to the occurrence of shear strain in the woven fabric specimen. RESULTS AND DISCUSSION Overview of Test Results of Average Shear Forces and Corresponding Displacements Microsoft Excel software was used for the statistical analysis of data at p<0.05 for five measurements. The diagrams (T- δ) of the average values of the test results of action of force T and the corresponding vertical displacements δ until break of the samples that were cut at different angles in the direction of the weft are shown in Figures 5, 6, 7, 8 and 9.

0

50

100

150

200

250

300

0 10 20 30 40 50 60 70 80

δ (mm)

T (N

)

0o

15o 90o

75o

60o

45o

30o

FIGURE 5. Diagram T- δ (force- vertical displacement) for sample S18.

Page 6: Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Journal of Engineered Fibers and Fabrics 119 http://www.jeffjournal.org Volume 10, Issue 4 – 2015

0

50

100

150

200

250

300

350

400

0 10 20 30 40 50 60 70

δ (mm)

T (N

)

0o

15o

90o

75o

60o

45o

30o

FIGURE 6. Diagram T- δ (force- vertical displacement) of sample S20.

0

50

100

150

200

250

300

350

0 10 20 30 40 50 60 70 80

δ (mm)

T (N

)

0o

15o

90o

75o

60o

45o

30o

FIGURE 7. Diagram T- δ (force- vertical displacement) of sample S22.

0

50

100

150

200

250

300

350

400

450

0 10 20 30 40 50 60 70

δ (mm)

T (N

)

0o

15o 90o

75o

60o

45o

30o

FIGURE 8. Diagram T- δ (force- vertical displacement) of sample S24.

0

50

100

150

200

250

300

350

400

0 10 20 30 40 50 60 70 80

δ (mm)

T (N

)

0o

15o

90o 75o

60o

45o

30o

FIGURE 9. Diagram T- δ (force- vertical displacement) of sample S26. Based on the diagrams of the measured values of force T and corresponding vertical displacement δ in Figures 5, 6, 7, 8 and 9, the average values of shear angle γ and corresponding shear stress τ are calculated from Eq. (1) and Eq. (2). These values are shown in the diagrams in Figures 10, 11, 12, 13 and 14, up to the value of the shear angle γ = 8°. The surface on which τ acts is A= t·a (mm2). The values of shear forces at the complementary angles and in the warp and weft direction vary with shear strain. When vertical displacements or shear strains are very small, the values of shear forces in the warp and weft direction are very close. This also applies to the values of shear force for complementary angles. However, at a later stage, the difference between the above mentioned values increases when strain increases. This may be due to the fact that the orthotropical assumption is inapplicable during this later stage, while the orthotropical structure is true when shear angle (shear strain) is very small at the initial stage. During woven fabric shearing, buckling in the woven fabric specimen occurs at a certain shear angle. According to Kawabata [17], woven fabric specimens which are subjected to shearing after exceeding the value of the shear angle γ = 8° tend to buckle. A shear angle of γ = 8° corresponds to a vertical displacement of δ=6.98 mm. This value specifies the limit of in plane deformation after which the sample begins its out of plane deformation and is called the buckling zone. The occurrence of buckling causes errors in the measurement results. The shear stress and shear angle curve up to the value of γ = 8° (until the beginning of buckling in shearing) are shown in Figures 10, 11, 12, 13 and 14.

Page 7: Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Journal of Engineered Fibers and Fabrics 120 http://www.jeffjournal.org Volume 10, Issue 4 – 2015

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

0 2 4 6 8

γ (o)

τ (M

Pa)

0153045607590

FIGURE 10. Diagram τ-γ (shear stress–shear angle) of sample S18

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

0 2 4 6 8

γ (o)

τ (M

Pa)

0153045607590

FIGURE 11. Diagram τ-γ (shear stress–shear angle) of sample S20.

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

0 2 4 6 8

γ (o)

τ (M

Pa)

0153045607590

FIGURE 12. Diagram τ-γ (shear stress–shear angle) of sample S22.

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

0 2 4 6 8

γ (o)

τ (M

Pa)

0153045607590

FIGURE 13. Diagram τ-γ (shear stress–shear angle) of sample S24.

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

0 2 4 6 8

γ (o)

τ (M

Pa)

0153045607590

FIGURE 14. Diagram τ-γ (shear stress–shear angle) of sample S26. In Figures 10, 11, 12, 13 and 14, at the same shear angle the highest value of shear stress τ occured in the samples which were cut at an angle of 45°. The values of shear stresses τ are very similar for complementary angles of the cutting samples. This is because balanced orthogonal plain weave fabric was considered. Shear stress τ assumes the minimum values in the warp and weft directions. In the samples that were cut in the weft and warp direction with a very small increment of shear stress the values of shear angle remarkably increased. Shear stress in relation to the shear angle has the highest increase at an angle of 45°. This applies only to the area in which there is no buckling of the woven fabric. When the fabric sample is subjected to the deformation imposed by pure shear test, in the initial stage, warp

Page 8: Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Journal of Engineered Fibers and Fabrics 121 http://www.jeffjournal.org Volume 10, Issue 4 – 2015

and weft yarns are perpendicular to each other, and the rotation relative to each other is limited by internal friction between the warp and weft. By contrast, at a later stage, the fabric structure changes from orthotropic to skewed structure, where the warp and weft yarns are no longer perpendicular to each other. The difference between warp and weft increases more and more with increasing shear angle (shear strain). In the course of shearing, the sample changes its shape, a rectangle becomes a parallelogram. The external sides of the specimen do not change their length, but an extension of the diagonal d0=AD=79.06 mm for value Δd (Figure 2) occurs. The values of absolute extension Δd of the diagonal d0 are calculated using the measured values δ and Eq. (4). The values of relative extension εd of the diagonal d0 are calculated from Eq. (5). The axial component Te of the measured force T is calculated by Eq. (6). Relation of axial force component Te and its relative extension εd of the diagonal for sample S22 is shown in Figure 15. Their relation is shown up to εd = 4.186 %, which corresponds to γ = 8° when woven fabric specimen starts to buckle during shear.

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

0 1 2 3 4 5

εd (%)

Te (N

)

0153045607590

FIGURE 15. Diagram Te - εd (axial component of force – relative extension of diagonal) of sample S22. At the same relative extension of the diagonal the biggest axial component Te appears in a sample which has been cut at an angle of 45°. The values of Te are very similar for complementary angles because the balanced orthogonal plain weave fabric was considered. Te assumes minimum values in the warp and weft directions. The values of relative extension of the diagonal considerably increased with a very small increment of axial force component in the samples that are cut in the weft and warp directions.

The woven fabric area is directly related to the number of crossovers in the material. A larger weft density of the sample would have more yarns resulting in more crossovers between the yarns. With an increased number of yarns and crossovers, a larger shear force is required to shear the sample. With increasing the measured shear forces T, the calculated values of axial force component Te also increase. Determining the Initial Shear Modulus Based on the experimentally obtained force-shear angles of curves, the values of initial shear modulus were obtained and compared with the corresponding calculated values. Percent deviations among experimental and calculated values of initial shear modulus will also be calculated. Experimental Values of the Initial Shear Modulus From the presented diagrams in Figures 5, 6, 7, 8 and 9, the values of shear force in the elastic range are used. We determined shear modulus Gφ from a particular region on force – shear angle curve that is determined by monitoring the experimental data obtained from an experimental set-up with regression control chart [18]. In this area of the curve, the relationship between shear stress and shear strain is linear. The shear modulus Gφ is defined as a slope of its shear stress- shear strain curve (τ- γ) in the elastic deformation region. Hooke’s law for shearing can be applied:

( )MPata

TtgG⋅⋅

===γγ

ταϕ (12)

where A=a·t is the area of the sample in which shear force acts. Using values T and γ in elastic range and using Eq. (12), the average values of initial shear modulus Gφ in relation to an arbitrary direction of cutting of the woven fabric samples are calculated. Linear regression equations are placed on the shear stress- shear strain curves in the elastic range. In Figure 16, the slope of the curve, i.e. the coefficient of line direction represents the shear modulus Gφ for sample S22 at angles of 0°, 45° and 90°. In Figure 16, the curve τ- γ is shown up to the value of the shear angle γ= 8° (0.14 rad).

Page 9: Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Journal of Engineered Fibers and Fabrics 122 http://www.jeffjournal.org Volume 10, Issue 4 – 2015

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16

γ (rad)

τ (M

Pa)

τ=0.82γ90o

45o

0o τ=0.76γ

τ=5.18γ

FIGURE 16. Diagram of the shear stress-shear angle (τ- γ) and regression lines of woven fabric sample S22. TABLE II. Experimentally obtained values of shear modulus Gφ (MPa).

Sample G0° G15° G30° G45° G60° G75° G90° S18 0.48 0.56 1.71 3.98 1.40 0.58 0.52

S20 0.64 0.72 2.01 4.55 1.78 0.85 0.72

S22 0.76 1.14 2.23 5.18 2.40 1.03 0.82

S24 0.89 1.39 2.73 5.36 2.78 1.53 0.99

S26 1.04 1.60 3.20 5.85 3.63 1.96 1.30

The obtained experimental values of shear modulus Gφ in dependence on the change of the cutting angle of the samples are given in Table II. The diagram of experimental values Gφ for each 15° is shown in Figure 17.

0

1

2

3

4

5

6

7

0 15 30 45 60 75 90

φ (o)

Gφ (M

Pa)

S18S20S22S24S26

FIGURE 17. Diagram of experimental values of shear modulus Gφ (MPa) for each 15°. The diagram is almost a symmetrical curve in relation to an angle of 45°. At that angle Gφ assumes the highest value for all woven fabric samples

because the initial slope of the stress-strain curve in shear is the biggest at that angle, Figure 10-14. When the samples are cut in the warp direction (φ=90°) and weft direction (φ=0°) shear modulus Gφ have the lowest value for all woven fabric samples because the initial slope of the stress-strain curve in shear is the smallest one at that angle, Figure 10-14. The values Gφ in the warp and weft direction are almost equal for each woven fabric sample or it is observable that shear modulus Gφ is almost equal for complementary angles. This is because balanced orthogonal plain weave fabric was considered. Because the warp densities of these fabrics are kept constant, any changes in the trends of diagrams of shear modulus can be considered from the different weft densities of these fabrics. The highest shear modulus is observed in woven fabric S26, while the lowest is in S18 fabric, Figure 17. In case of different weft densities, the results in Figure 17 show that the values of a shear modulus increase with the increasing weft density of woven fabrics. This is because a loose structure (lower weft density) has lower interyarn friction and allows a relative movement of warp and weft yarns. As a result, loose fabrics (with lowest weft density) have the lowest shear modulus. On the other hand, a tight structure hinders yarn movement, thus increasing shear modulus. Likewise, a higher sample weft density would require more yarns resulting in more crossovers among the yarns. Calculation of Initial Shear Modulus in Relation to an Arbitrarily Selected Coordinate System According to Eq. (11) and based on the experimental data G0° or G90° and G45° from Table II, the values of initial shear modulus Gφ were calculated, depending on a change in the cutting angle of the samples. The calculated values of Gφ for each 15° are shown in Table III. TABLE III. Theoretically calculated values of initial shear modulus Gφ (MPa).

Sample G0° G15° G30° G45° G60° G75° G90° S18 0.48 0.61 1.37 3.98 1.50 0.67 0.52

S20 0.64 0.81 1.73 4.55 1.97 0.92 0.72

S22 0.76 0.96 2.07 5.18 2.20 1.04 0.82

S24 0.89 1.12 2.31 5.36 2.54 1.25 0.99

S26 1.04 1.29 2.55 5.85 3.27 1.64 1.30

The diagram of the calculated values Gφ (MPa) for each 5° is shown in Figure 18.

Page 10: Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Journal of Engineered Fibers and Fabrics 123 http://www.jeffjournal.org Volume 10, Issue 4 – 2015

0

1

2

3

4

5

6

7

0 15 30 45 60 75 90

φ (o)

Gφ (M

Pa)

S18S20S22S24S26

FIGURE 18. Diagram of calculated values of shear modulus Gφ (MPa) for each 5°. Table IV shows the percent deviations among the experimental values of Gφ from Table II and the calculated values of Gφ in Table III. Deviations D (%) were calculated using Eq. (13):

( )%100exp,

,exp, ⋅−

ϕϕ

GGG

D calc (13)

TABLE IV. Deviations D in (%) among experimental values and calculated values Gφ.

Sample 0° 15° 30° 45° 60° 75° 90°

S18 0 -9.4 19.9 0 -6.8 -16.3 0

S20 0 -1.3 13.8 0 -10.5 -8.1 0

S22 0 15.5 7.3 0 8.5 -1.1 0

S24 0 19.2 1.3 0 8.6 18.5 0

S26 0 19.4 20.5 0 9.8 16.1 0

In the warp (90°) and weft (0°) directions and at an angle of 45°, the percent deviations among experimental and calculated values of the initial shear modulus Gφ are 0%. It follows from Eq. (11) due to the periodicity of the sine and cosine functions for these values. The negative deviation values show that the obtained calculated values Gφ are higher than the experimental values of Gφ. The absolute values of the deviations among the experimental and calculated values of the shear modulus are in a range from 0% to 20.46%. For sample S26 the deviations are maximal. For sample S22 the deviations are minimal.

CONCLUSION Woven fabrics can be defined as orthotropic materials to which, in the linear elastic range of the curve shear stress-shear angle, Hooke's law can be applied for anisotropic material behavior in calculating shear modulus when samples of woven fabrics are cut in an arbitrary direction. Shear properties of woven fabric are determined when shear force acts on woven fabric specimens which are cut at different angles in the weft direction. Because of woven fabric anisotropy its shear properties change in various directions. Shear modulus varies depending on angle φ (cutting direction of the sample). Fundamental differences in shear properties in various directions among different fabrics occur due to some inherent differences in their physical behavior, their finishes, and perhaps yarn or fiber stiffness, contact area at intersecting points of two sets of yarns, or fiber packing density in yarns. Any combination of these factors can impart different shear characteristics and shear modulus in woven fabrics, even though they are made from the same material. Shear modulus Gφ is almost symmetrical to the angle of 45° and the maximum value is reached exactly at that angle. Shear modulus values increase with an increasing weft density for any cutting direction of the sample. In the woven fabric with the lowest weft density (18 yarns/cm), shear modulus values range from 0.48 MPa to 3.98 MPa, and in the woven fabric with the highest weft density (26 yarns/cm) shear modulus values range from 4.1 MPa to 5.85 MPa. An increase in density in the weft direction, from 18 to 26 yarns/cm, significantly affects the increase of shear modulus. The minimum value of shear modulus increased by 116%, and the maximum value increased by 47%. In the woven fabric with the lowest weft density (18 yarns/cm), at a value of shear angle γ=8°, the values of shear stresses range from 0.06 MPa to 1.2 MPa, and in the woven fabric with the highest weft density (26 yarns/cm), the values of shear stresses range from 0.16 MPa to 1.52 MPa. With increasing weft density the value of shear stress increased by 130%, and the maximum value increased by 26%. A good agreement of experimental results and the calculated values of the initial shear modulus is shown, so the theoretical equations can be used with high accuracy to calculate the initial shear modulus of woven fabric in various directions. Thus,

Page 11: Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Journal of Engineered Fibers and Fabrics 124 http://www.jeffjournal.org Volume 10, Issue 4 – 2015

measurements are necessary when shear force acts only in the specimens cut in the warp and weft direction and at an angle of 45°. The woven fabric was subjected to shear up to γ = 8° due to the fact that after this point the woven fabric specimen tended to buckle. The appearance of buckling causes faults in measurement results requiring attention. The woven fabric samples that were cut at an angle of 45° have the highest shear resistance. The samples cut in the warp and weft direction showed the lowest shear resistance. The same values of shear forces in woven fabrics produced maximum shear deformation when the samples were cut in the warp and weft direction and minimum shear deformation when the samples were cut at an angle of 45°. This applied only to the area in which there was no buckling of woven fabric. REFERENCES [1] Go, Y.; Shitrohara, A.; Matsuhashi, F.,

"Viscoelastic Studies of Textile Fabrics, Part 3: On the Shearing Buckling of Textile Fabrics", Sen-i Gakkaishi, 13(7), 1957, pp. 460-465.

[2] Behre, B., "Mechanical Properties of Textile Fabrics Part I: Shearing", Textile Research Journal, 31(2), 1961, pp. 87-93.

[3] Nguyen, M.; Herszberg, I.; Paton, R., "The shear properties of woven carbon fabric", Composite Structures, 47(1-4), 1999, pp. 767–779.

[4] Pan N.; Yoon, M-Y, "Structural Anisotropy, Failure Criterion, and Shear Strength of Woven Fabrics", Textile Research Journal, 66(4), 1996, pp. 238-244.

[5] Lindnberg, J.; Behre, B.; Dahlberg, B., "Mechanical Properties of Textile Fabrics Part III: Shearing and Buckling of Various Commercial Fabrics", Textile Research Journal, 31(2), 1961, pp. 99-122.

[6] Mahar, T. J.;Wheelwright, P.; Dhingre, R. C.; Postle, R., "Measuring and Interpreting Fabric Low Stress Mechanical and Surface Properties, Part V: Fabric Handle Attributes and Quality Descriptors", Textile Research Journal, 60(1), 1990, pp. 7-17.

[7] Morner, B.; Eeg-Olofsson, T., "Measurement of the Shearing Properties of Fabrics", Textile Research Journal, 27(8), 1957, pp. 611-615.

[8] Kawabata, S.; Niwa, M.; Ito, K.; Nitta, M., "Application of objective measurements to clothing manufacture", International Journal of Clothing Science and Technology, 2(3), 1972, pp. 18–33.

[9] Leaf, G.A.V., "Analytical Plain Weave Fabric Mechanics and the Estimation of Initial Shear Modulus", The Journal of the Textile Institute, 92(3), 2001, pp. 70-79.

[10] Spivak, S. M.; Treloar, L. R. G., "The behavior of fabrics in shear Part III: The relation between bias extension and simple shear", Textile Research Journal, 38(9), 1968, pp. 963-971.

[11] Kilby, W. F., "Shear Properties in Relation to Fabric Hand", Textile Research Journal, 31(1), 1961, pp. 72-73.

[12] Grosberg, P.; Leaf, G. A. V.; Park, B. J., "The Mechanical Properties of Woven Fabrics, Part VI: The Elastic Shear Modulus of Woven Fabrics", Textile Research Journal, 38(11), 1968, pp. 1085-1100.

[13] Alamdar-Yazdi, A., "Shearing properties of the skewed woven fabrics", International Journal of Engineering, 18(2), 2005, pp. 177–185.

[14] Penava, Ž.; Šimić Penava, D.; Knezić, Ž., "Determination of the Elastic Constants of a Plain Woven Fabrics by Tensile Test in Various Directions", Fibres and Textiles in Eastern Europe, 22(2), 2014, pp. 57-63.

[15] Lekhnitskii, S. G., "Theory of Elasticity of an Anisotropic Elastic Body", Moscow: Mir Publishers, 1981.

[16] Bassett, R.J.; Postle, R.; Pan, N., "Experiment Methods for Measuring Fabric Mechanical Properties: a Review and Analysis", Textile Research Journal, 69(11), 1999, pp. 866–875.

[17] Kawabata, S.; Niwa, M.; Ito, K.; Nitta, M., "Application of Objective Measurement to Clothing Manufacture", International Journal of Clothing Science and Technology, 2(3), 1990, pp. 18-33.

[18] Ozkul, B.; Karaoglan, D. "Regression control chart for determination of Young’s modulus: A case study", Scientific Research and Essays, 6(30), 2011, pp. 6393-6403.

Page 12: Woven Fabrics Behavior in Pure Shear - jeffjournal.org13) Ž. Penava.pdf · shear modulus of woven fabrics was determined experimentally and theoretically. Keywords: Anisotropy, Initial

Journal of Engineered Fibers and Fabrics 125 http://www.jeffjournal.org Volume 10, Issue 4 – 2015

AUTHORS’ ADDRESESS Željko Penava University of Zagreb Faculty of Textile Technology Prilaz B. Filipovića 28A 10000 Zagreb CROATIA Diana Šimić Penava Department of Engineering University of Zagreb Kačićeva 26 10000 Zagreb CROATIA Marija Nakić University of Mostar Trg Hrvatskih Velikana 1 8800 Mostar BOSNIA AND HERZEGOVINA