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Classification of damping properties of fabric-reinforced flat beam-like specimens by a degree of ondulation implying a mesomechanic kinematic

  • Marco Romano and Ingo Ehrlich EMAIL logo
Published/Copyright: August 22, 2024
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Science and Engineering of Composite Materials
From the journal Science and Engineering of Composite Materials Volume 31 Issue 1

Abstract

In order to determine the influence of the ondulations in fabrics on the damping properties of fiber-reinforced plastics, the structural dynamic properties of fabric- and unidirectionally reinforced plastics are investigated. The free decay behavior of flat beam-like specimens is investigated under fixed-free boundary conditions. As the material damping is consistently higher in fabric-reinforced specimens compared to unidirectionally reinforced ones, a contribution of an additionally acting mesomechanic kinematic in fabric weaves is implied. Based on a degree of ondulation, it is possible to classify the enhancement of the material damping and determine the corresponding energy dissipation. The study provides valuable quantitative relations of the additional damping effect due to the mesomechanic kinematic. Compared to the unidirectionally reinforced material, plain weave enhances the material damping by 37…52% at O ˜ PL = 0.0133 , whereas twill weave 2/2 enhances it by 31…40% at O ˜ T2 = 0.0098 . The consideration of the findings contributes to a deeper understanding of the visco-elastic dynamic behavior of fabric-reinforced plastics and allows further applications in research, development, and industry.

Keywords: fiber-reinforced plastics; mesomechanic scale; fabric-reinforced layer; ondulation

1 Introduction

It is presumed that a mesomechanic kinematic in fabric-reinforced single layers influences the structural dynamic properties of fiber-reinforced plastics. The carried-out experimental investigation focuses on the influence of fabric ondulations on damping properties of fiber-reinforced plastics. Therefore, the structural dynamic properties of three comparable sets of fabric- and unidirectionally reinforced plastics are studied. In detail, the free decay behavior of flat, beam-like specimens under fixed-free boundary conditions is investigated.

Fabric-reinforced specimens consistently exhibit higher material damping than unidirectionally reinforced specimens, suggesting additional mesomechanical kinematics in woven fabrics. By introducing a degree of ondulation O ˜ , a valuable quantitative relationship of the enhancement in material damping and corresponding energy dissipation can be stated. Compared to the unidirectionally reinforced material, plain weave enhances the material damping by 37…52% at O ˜ PL = 0.0133 , whereas twill weave 2/2 enhances it by 31…40% at O ˜ T2 = 0.0098 .

The findings support the assumption that the repeated acting of the presumed mesomechanic kinematic enhances the material damping of fabric-reinforced single layers additionally to the purely visco-elastic part of unidirectionally reinforced ones. The dependence of the presumed mesomechanic kinematic on the geometric dimensions is made evident, as indicated by the degree of ondulation O ˜ , which represents the intensity of the ondulated yarn in different fabric weaves.

2 Research environment

The structural mechanic behavior of fabric-reinforced plastics is considered the research environment. Detailed thematic excerpts of relevant literature are given in the review of Romano et al. [1]. There are various approaches and investigations, such as Guan [2] and Guan and Gibson [3] regarding the viscoelastic damping in fabric-reinforced single layers by finite element (FE) calculations and concluding the acting of a mesomechanic mechanism for damping in fabric-reinforced composites. Exemplarily, Matsuda et al. [4] numerically derived a homogenization theory for elastic and viscoplastic material properties for plain weave fabrics, and Nakanishi et al. [5] experimentally investigated the damping properties of glass fabric-reinforced composites by vibration tests of flat beam-like. Ideas of parametrically varying the mesomechanic geometric dimensions are mentioned besides others by Le Page et al. [6] and Ansar et al. [7]. A first thought of basically characterizing fabric-reinforced single layers by its mesomechanic geometric dimensions has been indicated in the study of Kreikmeier et al. [8]. For fabricating 3D-printed polymers with arbitrary variable fabric weave forms and types (i.e., with continuous fiber reinforcement), the fused deposition modeling (FDM) is a promising additive manufacturing technology as presented considering even production parameters in Karimi et al. [9,10]. Analogue aspects regarding the fabrication of adequate reinforced materials in the required material quality for specimens are presented in the study of Cao [11] for the vacuum-assisted material extrusion (MEX). In the studies of Cao et al. [12] and Cao [13] the FDM is presented again focusing on customized 3D-printed core sandwich composites with enhanced interphase properties between the thermoplastic core and glass-fiber-reinforced skin.

Yet, the former mentioned works do not contain an extensive parametric variation and sensitivity analysis regarding the mesomechanic geometric parameters. Thus, this idea has been followed in the study of Ottawa et al. [14] focusing on numerical investigations. The investigations of Valentino et al. [15,16] extended the investigations with basalt fiber-reinforced plastics, as they mechanically characterize different fabric reinforcements via experimental tensile tests and FE calculations. Additionally, more detailed analytical and numerical investigations of plain representative sequences of fabric-reinforced single layers are focused in the studies of Romano [17] and Romano et al. [18,19]. The identified causality and correlation between mesomechanic kinematic caused by geometric constraints lead to the pursued mechanical principle, described in Section 3.

For a distinct characterization of the mesomechanic geometry, a degree of ondulation as a non-dimensional parameter is introduced. It relates the geometric parameters, namely, the amplitude and the length of one complete ondulation, so that it represents the intensity of the ondulation of the respective fabric construction. As a data base of the structural dynamic material properties, the results of the experimental investigations presented in the studies of Romano et al. [20,21] for three sets of comparable flat beam-like specimens. These are all carbon fiber-reinforced epoxy ( 0 ° -unidirectionally and 0 ° plain and twill weave 2/2 fabric-reinforced in the warp direction). The evaluated material damping of fabric-reinforced and unidirectionally reinforced plastics is used for further characterization. The investigations are carried out under constant geometric and constant dynamic conditions. The mesomechanic geometric dimensions are used in order to determine the degree of ondulation. Based on this value as the intensity of the ondulation, the enhancement of the material damping is correlated. It is implied, that the repeated acting of a mesomechanic kinematic dissipates energy and thus additionally contributes to the material damping.

3 Pursued mechanical principle

This section aims to understand the mesomechanic kinematic induced by ondulation in fabric-reinforced composites. Structural dynamic experiments are conducted on flat beam-like specimens to provide a one-dimensional approach. The underlying mechanical principles and simplifying assumptions are briefly described.

Figure 1 illustrates one complete ondulation of a plain weave and twill weave 2/2 fabric, containing the range of dimensions investigated via the three sets of comparable specimens. The geometries are based on a purely analytic sine function y ( x ) = A sin 2 π x L . Plain representative sequences for balanced plain weave and twill weave 2/2 fabrics are derived. The geometric parameters, amplitude A , and length L define the entire cross-section of one complete ondulation. The idealization allows the repetition in series and layerwise as often as required as the sine function is a continuous differentiable function over the whole domain of definition [22,23].

Figure 1 Sequences of one full ondulation of a plain weave (top) and twill weave 2/2 fabric (bottom) with the ranges of the indicated dimensions for the investigated three sets of comparable specimens.
Figure 1

Sequences of one full ondulation of a plain weave (top) and twill weave 2/2 fabric (bottom) with the ranges of the indicated dimensions for the investigated three sets of comparable specimens.

The simplified geometry of the ondulation represents a cross-section obtained by a cut of the warp yarn along its theoretical centerline perpendicular to the fill yarns. Three mechanically distinct regions result: the warp yarn, the fill yarns, and the pure matrix region. The stiffnesses of these regions are predominantly defined via E 1 , E 2 = E 3 , and E m and significantly differ from each other. For high tenacity (HT), carbon fiber-reinfoced epoxy at a fiber volume content of φ f = 60 % homogenization gives approximately E 1 = 150 GPa ≫ E 2 = 11.5 GPa > E m = 3.3 GPa . Qualitatively, it is E 1 ≈ 13 ⋅ E 2 and E 1 ≈ 45 ⋅ E 2 .

Based on the former described structural-mechanical relations with the geometric constraints, a mesomechanic kinematic can be lead back when the ondulated yarn undergoes elastic deformation in the longitudinal direction. It results in additional variations of the amplitude (flattening for elongation or upsetting compression) additionally and directly coupled to the purely elastic deformation. The variation of the amplitude results as superposition of the elastic transversal Poisson effect and the kinematic response due to mesomechanic geometric constraints. This kinematic response is illustrated in Figure 2. Therefore, the centerline of an ondulated yarn is considered, which is presumed to be ideally stiff in the longitudinal direction and at the same time ideally flexible in terms of bending, E l → ∞ and E b → 0 .

Figure 2 Obtained sine-waves for a sinusoid with A = 1 A=1 and L = 2 π L=2\pi : original graph bold solid line; elongated graphs as dashed lines; and shortened graphs as dash-dotted lines.
Figure 2

Obtained sine-waves for a sinusoid with A = 1 and L = 2 π : original graph bold solid line; elongated graphs as dashed lines; and shortened graphs as dash-dotted lines.

This repeated kinematic, induced by geometric constraints, is presumed to enhance the damping properties of fabric-reinforced composites compared to 0 ° -unidirectionally reinforced ones. Evaluation and quantitative investigations are based on the free decay behavior of flat beam-like specimens with fabric-reinforced and unidirectionally reinforced single layers. They are supported with fixed-free boundary conditions and subjected to defined displacement excitations. The vibrating structure undergoes the kinematic in a number of cycles equal to the fundamental frequency. Selected unsupported lengths are defined, so that either constant geometric conditions, i.e., with the same unsupported length, or constant dynamic conditions, i.e., with the same fundamental frequency, can be evaluated.

Validation of this concept has been performed in previous studies (see Ottawa et al. [14] and Valentino et al. [15,16] for one set of comparable specimens of basalt fiber-reinforced epoxy [ 0 ° -unidirectionally and 0 ° twill weave 2/2 fabric-reinforced in the warp direction]). The more detailed structural dynamic investigations in the studies of Romano et al. [20,21] for three sets of comparable specimens of carbon fiber-reinforced epoxy ( 0 ° -unidirectionally and 0 ° plain and twill weave 2/2 fabric-reinforced in warp direction) are considered.

The ranges of the geometric dimensions are indicated in terms for the heights and lengths. It is important to note, that these geometric dimensions are neither variable nor interchangable in the investigated material. As three comparable sets of specimens have been manufactured, prepared, and investigated experimentally, the distinct combinations result based on the kind of fabric and parameter of the curing process is briefly described in Section 5.1. The distinct values for the each set of specimens are listed in Tables 3, 4, and 5.

Table 3

Thicknesses of the fabric-reinforced single layers in dry or uncompacted condition h S,d as indicated in the technical data sheets as theoretical or effective values and in impregnated or compacted in cured condition in the laminate h S,L according to equation (12) as resulting or apparent values

ID Thickness of dry single layer h S , d (mm) Reference sequence Stacking sequence Thickness of beam-like specimen h Number of single layers N (mm) Thickness of single layer in composite h S , L = h ⁄ N (mm)
Ten-Uni — [30,36] [ 0 ] 2 S C 1.95 4 0.488
Ten-Plain 0.64 [31,36] [ ( 0 ⁄ 90 ) ] 2 S C 1.61 4 0.403
Ten-Twill 0.85 [32,36] [ ( 0 ⁄ 90 ) ] 2 S C 2.25 4 0.563
Pyr-Uni — [33,36] [ 0 ] 4 S C 2.05 8 0.256
Pyr-Plain 0.5 [34,36] [ ( 0 ⁄ 90 ) ] 4 S C 2.38 8 0.298
Pyr-Twill 0.5 [35,36] [ ( 0 ⁄ 90 ) ] 4 S C 2.28 8 0.285
Hex-Uni 0.165 [29] [ 0 ] 4 S C 1.42 8 0.178
Hex-Fab 0.227 [29] [ ( 0 ⁄ 90 ) ] 4 S C 1.94 8 0.243
Table 4

Parameter of the ondulation of the sets of comparable specimens based on the thickness of an impregnated, compacted fabric-reinforced single layer in cured condition in the laminate h S,L according to Equation (12) as resulting or apparent value (cf. Table 3)

ID Thickness single layer laminate h S , L = h ⁄ N see Equation (12) (mm) Thickness roving yarn h R = 1 ⁄ 2 ⋅ h S , see Equation (10) (mm) Amplitude ondulation A = 1 ⁄ 4 ⋅ h R , see Equation (10) (mm) Number of roving yarns n R (per cm) Length cross-section roving L R = 1 ⁄ n R , see Equation (11) (mm) Specific factor of fabric λ Length of ondulation L = λ ⋅ L R , see Equation (11) Degree of ondulation O ˜ = A ⁄ L , see Equation (1) Reference
Ten-Uni 0.488 — — — — — — 0 [30,36]
Ten-Plain 0.403 0.202 0.101 2.5 4.0 2 8.0 0.01263 [31,36]
Ten-Twill 0.563 0.282 0.141 3.7 2.7 4 10.8 0.01306 [32,36]
Pyr-Uni 0.256 — — — — — — 0 [33,36]
Pyr-Plain 0.298 0.149 0.075 3.7 2.7 2 5.4 0.01389 [34,36]
Pyr-Twill 0.285 0.143 0.071 3.7 2.7 4 10.8 0.00657 [35,36]
Hex-Uni 0.178 — — — — — — 0 [29]
Hex-Fab 0.243 0.202 0.061 n.d. n.d. — n.d. n.d. [29]
Table 5

Number of representative sequences of every single layer n O , S and in total of the specimen as beam n O , B according to Equation (13) under constant geometric and dynamic conditions

ID Stacking sequence Number of single layers N Length cross-section roving L R = 1 ⁄ n R , see Equation (11) (mm) Specific factor of fabric λ Length of ondulation L = λ ⋅ L R , see Equation (11) (mm) Kind of constant condition Unsupp. length l (mm) Number of rep. seq. per single layer n O , S = l ⁄ L , see Equation (13) Number of rep. seq. per specimen as beam n O , B = l ⁄ L ⋅ N , see Equation (13) Reference
Ten-Uni [ 0 ] 2 S C 4 — — — — 220 0 0 [30,36]
220 0 0
220 0 0
Ten-plain [ ( 0 ⁄ 90 ) ] 2 S C 4 4.0 2 8.0 Geometric 220 27.5 110 [31,36]
220 27.5 110
220 27.5 110
Dynamic 170 21.25 85
168 21 84
168 21 84
Ten-twill [ ( 0 ⁄ 90 ) ] 2 S C 4 2.7 4 10.8 Geometric 220 20.37 81.48 [32,36]
220 20.37 81.48
220 20.37 81.48
Dynamic 210 19.07 76.28
206 19.44 77.76
208 19.26 77.04
Pyr-uni [ 0 ] 4 S C 8 — — — — 220 0 0 [33,36]
220 0 0
220 0 0
Pyr-plain [ ( 0 ⁄ 90 ) ] 4 S C 8 2.7 2 5.4 Geometric 220 40.74 325.92 [34,36]
220 40.74 325.92
220 40.74 325.92
Dynamic 202 37.04 296.32
200 37.41 299.28
202 37.41 299.28
Pyr-twill [ ( 0 ⁄ 90 ) ] 4 S C 8 2.7 4 10.8 Geometric 220 20.37 162.96 [35,36]
220 20.37 162.96
220 20.37 162.96
Dynamic 204 18.89 151.12
202 18.89 151.12
204 18.7 149.6
Hex-uni [ 0 ] 4 S C 8 — — — — 200 0 0 [29]
200 0 0
200 0 0
Hex-fab [ ( 0 ⁄ 90 ) ] 4 S C 8 n.d. n.d. n.d. Geometric 200 n.d. n.d.
200 n.d. n.d.
200 n.d. n.d.
Dynamic 204 n.d. n.d.
196 n.d. n.d.
200 n.d. n.d.

Therefore, the term comparable is defined by the material quality of the composite for the sets of specimens. Comparable in this context means that a single layer consists of the same kind of reinforcement fiber, in the fabric-reinforced single layers and 0 ° -unidirectionally ones, with the same polymeric matrix as well as at approximately same fiber volume contents φ f = 60 % and overall thicknesses of the laminate h .

4 Mesomechanic approach

The phenomenon of ondulation in fabric-reinforced composites on the mesomechanic scale is investigated. Within the fabric as textile semi-finished product, the warp yarns are crossed by fill yarns, creating a characteristic patterns or fabric construction. The plain weave and twill weave 2/2 fabric are considered in the study as the two most common fabric types by three comparable sets of specimens.

The aim is to identify the described mesomechanic kinematic correlations and its enhancing effect on the material damping. The beam-like specimens are considered cantilever beams to reduce the analysis to a one-dimensional approach. Plain representative sequences adequately describe the periodic geometry of fabrics, allowing the parametric variation of its geometric dimensions (amplitude A and length of the ondulation in fabric L F or of the cross-section of a roving L R ) to identify the sensitivity of the kinematic with respect to different shapes.

4.1 Degree of ondulation

The degree of ondulation, introduced as O ˜ , quantifies the geometric intensity of the deviation of the predominant fiber direction of the yarns in the fabric. It is defined via the relation of amplitude A to length L F or L R by

(1) O ˜ = A L = A L F = A λ L R ,

representing a non-dimensional measure. Herein, the characteristic factor λ specifies the type of fabric, i.e., λ = 2 for a plain weave and λ = 4 for a twill weave 2/2 fabric

(2) L PL = λ L R = 2 L R = L F and L T2 = λ L R = 4 L R = L F .

The measure O ˜ is already introduced in the studies of Ottawa et al. and Romano et al. [14,17,18,20,21]. The approach is adopted from wave steepness used in nautics (see exemplarily the study of Büsching 2001 [24]).

4.2 Comparison of the structural–mechanical motivations

The structural–mechanical motivations regarding the modification of wave steepness as exemplarily defined in the study of Büsching [24] toward the degree of ondulation O ˜ are explained in detail in the studies of Romano et al. [18,19]. The degree of ondulation O ˜ describes the deviation from the ideally straight orientated unidirectionally yarn caused by the fabric weave ondulation [25]. The amplitude A corresponds to a degree of eccentricity compared to its ideally straight orientation.

The non-dimensional measure O ˜ according to Equation (1) enables comparability between different fabric weaves in general and plain representative sequences in particular. Thus, it is the basis for the validation of the carriedout investigations.

A graphical illustration of O ˜ is shown in Figure 3 over amplitude A and length L of an ondulation. The geometric dimensions are chosen in realistic ranges, as evaluated in the studies of Ottawa et al. [14], Valentino et al. [15,16], as well as Romano et al. [18,19] or reported by Ballhause [26], Matsuda et al. [4], and Kreikmeier et al. [8]. These are for A = 0.05 … 0.25 mm and lengths of the ondulation for plain weave L PL = 5.0 … 15.0 mm and for twill weave 2/2 fabric L T2 = 10.0 … 30.0 mm . The corresponding degree of ondulation O ˜ for a plain weave fabric then lies in the range of 0.00333 … 0.05000 and for a twill weave 2/2 fabric in the range of 0.00166 … and 0.02500, so that O ˜ PL = 2 ⋅ O ˜ T2 .

Figure 3 Graphical illustration of the degree of ondulation O ˜ \tilde{O} over the geometric parameters of a plain weave fabric (top) and a twill weave 2/2 fabric (bottom).
Figure 3

Graphical illustration of the degree of ondulation O ˜ over the geometric parameters of a plain weave fabric (top) and a twill weave 2/2 fabric (bottom).

5 Materials and test procedures

This section describes the used materials, the production process of the test panels, specimen preparation, and experimental procedures for determining fiber volume content. The principle of laser vibrometry for contactless vibrational analysis is briefly described, before the experimental setup is indicated.

5.1 Materials and processing

Test panels with different fabric weaves and unidirectionally reinforced materials have been produced. Comparability via the selection of materials is assured as the roving used in the fabric weave is the same as in the unidirectionally reinforced material. Two sets of comparable materials were investigated: one set of prepregs and two sets of dry textile semi-finished products.

For prepregs, both unidirectional and fabric-reinforced carbon fiber prepregs Hexcel G947 and G939 with a thermoset matrix system HexPly M18/1 [29] have been used. Dry textile semi-finished products were impregnated using filament winding according to DIN 65071-1 and pre-impregnation technique according to DIN 65071-2 [27,28].

The dry textile semi-finished products were composed of the following materials: Tenax HTS40 roving applied in the corresponding plain weave Style 427 and twill weave 2/2 fabric Style 404 [30–32], and Pyrofil TR50S 6K roving applied in the plain weave Sigratex KDL 8051/120 and twill weave 2/2 fabric Sigratex KDL 8052/120 [33–35]. The afore mentioned impregnation techniques were applied using a thermoset matrix system warm-curing, thermoset matrix system Araldite LY 556/Aradur 917/Accelerator DY 070 [36].

The test panels have been cured by autoclave processing under a vacuum bag. The specimens are cut out from the test panels by waterjet cutting to dimensions of l s = 250 mm × b = 25 mm and precisely measured. The density ϱ was calculated using the weighted mass m and measured dimensions and verified experimentally.

After determination of the material density ϱ according to standard DIN EN ISO 1183-1 [37], the fiber volume content was determined by chemical extraction of fibers from the matrix and evaluated according to DIN EN 2564 [38]

(3) φ f = 1 1 + 1 − ψ ψ ⋅ ϱ f ϱ m ,

where ϱ is the density and ψ is the fiber mass content and the subscripts f and m indicate the properties of the fibers and of the matrix, respectively. The densities of the single components are taken out of the data sheets [29–36].

For all test panels at five positions each, a fiber volume content of approx. φ f = 55 % in case of the prepreg material and approx. φ f = 60 % was calculated, ensuring comparability between specimens. Table 1 resumes the properties of the selected reinforcements and test panels, confirming material comparability.

Table 1

Selected kinds of comparable reinforcements of unidirectionally and fabric-reinforced test panels, namely prepregs and dry textile semi-finished products, the corresponding polymeric matrix system and relevant properties of the cured composite

ID Reinforcement Type of material Lin.density/areal weight Matrix system Kind of fiber reinforcement Stacking sequence Density ρ of cured composite (g/cm3) Fiber volume content φ f (%) Reference
Ten-Uni Tenax HTS40 Roving 800 tex (12 k) Araldite LY 556/Aradur 917/Acc. DY 070 Unidir. 0 ° Carbon [ 0 ] 2 S C 1.52 60.0 [30,36]
Ten-plain Style 427 Plain weave 400 g⁄m 2 Araldite LY 556/Aradur 917/Acc. DY 070 Plain weave warp 0 ° carbon [ ( 0 ⁄ 90 ) ] 2 S C 1.51 61.0 [31,36]
Ten-twill Style 404 Twill weave 2/2 600 g⁄m 2 Araldite LY 556/Aradur 917/Acc. DY 070 Twill 2/2 warp 0 ° carbon [ ( 0 ⁄ 90 ) ] 2 S C 1.52 60.0 [32,36]
Pyr-uni Pyrofil TR50S 6K Roving 400 tex (6 k) Araldite LY 556/Aradur 917/Acc. DY 070 Unidir. 0 ° carbon [ 0 ] 4 S C 1.54 59.0 [33,36]
Pyr-plain Sigratex KDL 8051/120 Plain weave 300 g⁄m 2 Araldite LY 556/Aradur 917/Acc. DY 070 Plain weave warp 0 ° carbon [ ( 0 ⁄ 90 ) ] 4 S C 1.53 61.0 [34,36]
Pyr-twill Sigratex KDK 8052/120 Twill weave 2/2 300 g⁄m 2 Araldite LY 556/Aradur 917/Acc. DY 070 Twill 2/2 warp 0 ° carbon [ ( 0 ⁄ 90 ) ] 4 S C 1.54 60.00 [35,36]
Hex-Uni Hexcel G947 UD Unidir. prepreg 200 tex (3 k) HexPly M18/1 Unidir. 0 ° carbon [ 0 ] 4 S C 1.52 54.70 [29]
Hex-fab Hexcel G939 fabric Fabric prepreg 220 g⁄m 2 HexPly M18/1 Fabric warp 0 ° carbon [ ( 0 ⁄ 90 ) ] 4 S C 1.51 54.10 [29]

5.2 Laser vibrometry as reference vibration analysis system

Laser scanning vibrometry offers precise and reproducible measurement of structural dynamic vibrations without introducing additional disturbances like the mass of an acceleration sensor and cabling or similar. It provides high sensitivity and contactless measurement across a wide frequency range, meeting the requirement of high reproducibility of experimental investigations. This enables a relatively simple experimental setup. A laser scanning vibrometer of the type PSV 400 from Polytec [39] has been used for the study.

Laser vibrometry targets a low power laser beam onto a moving surface. The reflected light has received a Doppler shift. By interference within the device, the reflected laser beam creates an interference pattern on the detector. Its intensity depends on the frequency shift, and a Bragg cell allows the time-dependent evaluation over the recording time of the velocity of the surface in case of the shift of the frequency of the displacement of the surface in case the shift of phase is considered [40].

5.3 Experimental setup for structural dynamic investigations

In the experimental setup, the specimens are excited into free vibrations through displacement excitations. Each specimen is reproducibly positioned and clamped on one side. Based on the fixed-free boundary condition, the structure can mechanically be treated one-dimensional as a cantilever beam. A defined excitation is applied at the unsupported end, inducing transversal vibrations primarily in the fundamental Eigenmode at the fundamental frequency. The free decay of the vibrating structure is measured by the laser scanning vibrometer. Figure 4 shows a schematic side view and an isometric one of the test setup used for the study.

Figure 4 Test setup of the structural dynamic experimental investigations. Top: schematic side view containing from left to right the clamping, the clamped specimen, the side of excitation, and the laser vibrometer Polytec PSV 400; bottom: isometric view of the clamping with specimen and excitation.
Figure 4

Test setup of the structural dynamic experimental investigations. Top: schematic side view containing from left to right the clamping, the clamped specimen, the side of excitation, and the laser vibrometer Polytec PSV 400; bottom: isometric view of the clamping with specimen and excitation.

As the laser scanning vibrometer measures the velocity of the surface of the vibrating structure, the time signal is evaluated regarding the shift in frequency and shift in phase. The first one corresponds to the velocity over the time signal, and the second one to the displacement over the time signal. As for structural mechanics, the displacement signal is more descriptive, it has been further processed and evaluated.

In the software of the laser scanning vibrometer PSV 400 from Polytec [39], the relevant signal recording settings bandwidth B and the number of lines in the frequency domain N are defined. A bandwidth of B = 1,000 Hz and N = 12,800 lines in the frequency domain yield a sampling rate of f s = 2.56 , B = 2,560 Hz and a total recording time of t tot = N ⁄ B = 12.8 s per measurement. The resolution in the frequency domain is f r = 1 ⁄ t tot = B ⁄ N = 78.125 mHz [39].

5.4 Reproducibility, parameter identification, and sensitivity analysis

Preliminary investigations have been conducted to validate the reproducibility of the test procedure and identify the technically reasonable parameters. Additionally, the sensitivity of the results to variations in experimental boundary conditions was analyzed. Key parameters examined included positioning and clamping procedures, contact pressure, tightening torque, and specimen orientation.

Removable mechanical positioners on the baseplate ensure both the free length and the perpendicular orientation of the specimens, as illustrated in Figure 4 (bottom). The position of the clamping piston is defined by a removable positioner and decoupled from the rotation of the tightening bolt via a ball joint. Clamping accuracy and reproducibility were validated with a defined tightening torque of 12.5 Nm corresponding to a contact pressure of 14.8 N/mm2.

Three specimens of each kind of reinforcement were measured, with each specimen measured five times consecutively to ensure statistically reliable results. This approach enables the statistical consideration of the results via average values and standard deviations for each specimen and reinforcement type.

A relatively constant displacement excitation is necessary to measure and consider the structural dynamic properties within the presumed linear visco-elastic range throughout the whole recording time. Preliminary investigations identified a reasonably defined displacement of h ⁄ 4 as suitable. This displacement was verified to minimize and neglect non-linear effects, such as geometric and material nonlinearities or air friction.

The total recording time t tot has been divided into five isochronous intervals I ( n ) for n = 1 , … , 5 , i.e., with t tot = 12.8 s each isochronous interval is t n = 2.56 s . The material damping was evaluated for each of these intervals, as well as for the entire recording time. At the selected h ⁄ 4 displacement for each specimen, nearly constant material damping values were obtained across the five intervals and the total recording time. This experimental verification confirms the measurement of linear visco-elastic structural dynamic properties only.

6 Results and discussion

The evaluation of the experimental results is conducted in the time and in the frequency domain. The structural dynamic investigations and the presumption of constant geometric and constant dynamic conditions are explained.

The constitutive model employed to describe the free decay behavior of the flat beam-like specimens is the single-mass oscillator with vicious damping [22,23]. The dynamic behavior of the specimens is analytically considered as the one-dimensional analytical model of a vibrating cantilever beam [42,43,46,47]. The constitutive Equations of the single-mass oscillator with viscous damping have been used to mechanically describe the dynamic behavior and methodically evaluate the results [41,43–45].

6.1 Structural dynamic experiments

The unidirectionally reinforced specimens are considered as the basis of the experimental investigations due to the absence of the presumed mesomechanic kinematic.

6.1.1 Definition of constant geometric and constant dynamic conditions

To obtain constant geometric conditions, both unidirectionally and fabric-reinforced specimens were tested at the same unsupported length. Fabric-reinforced specimens exhibited lower fundamental frequencies f due to their nominally lower material stiffness. The overall elastic modulus E of the specimen material was derived from the unsupported length l and the fundamental frequency f by [41–45]

(4) E = 2 π λ 2 ⋅ f ⋅ l 2 2 ⋅ ϱ ⋅ A I = 12 ⋅ 2 π λ 2 ⋅ f ⋅ l 2 2 ⋅ ϱ h ,

where l is the unsupported length, f is the fundamental frequency, I is the geometric moment of inertia, ϱ is the density, A is the area of the cross-section, h is the thickness of the specimen, and λ = 1.875 is the wave number for the fundamental frequency of the shear-stiff beam according to Euler–Bernoulli.

Additional structural dynamic investigations have been conducted on fabric-reinforced specimens with varied unsupported lengths l to match the fundamental frequency of unidirectionally reinforced specimens, ensuring identical dynamic conditions. Solving the relation (4) for the varied unsupported length l by [41–45]

(5) l = λ 2 2 π f ⋅ E I ϱ A = λ 2 h 2 π f ⋅ E 12 ϱ ,

where in this context f equals the fundamental frequency of the unidirectionally reinforced specimens. The basic unsupported length of the unidirectionally reinforced specimens is l = 220 mm for the dry fabric weave-based material and l = 200 mm for the prepreg-based one.

6.1.2 Evaluation of experimental results

Material damping was assessed using the logarithmic decrement Λ based on the displacement–time signal in the time domain. Subsequently, the signal was transferred to the frequency domain using the fast Fourier transform (FFT) algorithm to determine the fundamental frequency f . These steps facilitated the calculation of the dynamic modulus in terms of storage and loss modulus.

The material damping is evaluated via the logarithmic decrement [41–45]:

(6) Λ = 1 n ⋅ ln w ( t i ) w ( t i + n ) , for n = 1 , 2 , … , n ,

where n is the counter for the maximum amplitudes in the displacement–time signal and t i and t i + n are the instants of time of their appearance. The dynamic or complex modulus

(7) E * = E ′ + i E ″ = ( 1 + i η ) E ′ ,

is evaluated. A detailed processing is given in the studies of Gibson, the guideline VDI 3830 and Schmidt [41–45]. Thereby, E ′ is the storage modulus, E ″ is the loss modulus, and η is the loss factor. It is the relation between the energy dissipated in one load cycle and the maximum total energy stored in the structure.

Presuming weak damping D 2 ≪ 1 , where D is the dimensionless Lehr damping factor, the Eigen angular frequency of the damped system ω d is approximately the Eigen angular frequency of the undamped system ω 0 , ω d ≈ ω 0 . Equation (7) then gives the approximation [41–45]

(8) E ′ ≈ E ,

representing the real part of the dynamic modulus E * .

The loss modulus E ″ as the complex part of the dynamic modulus E * is approximated via D = Λ ⁄ 4 π 2 + Λ 2 ≈ Λ ⁄ 2 π for a damped harmonic oscillator. It can be expressed in terms of the Lehr damping factor D and the storage modulus E ′ and yields [41–45]

(9) E ″ = 2 D E ′ = 2 Λ 4 π 2 + Λ 2 E ′ ≈ Λ π E ′ .

In detail, the aforementioned evaluation of the experimental results is carried out in MATLAB. Therefore, the logarithmic decrement Λ is evaluated via Equation (6) based on the signal in the time domain, and the exact term is taken into account.

6.2 Mesomechanic geometric relations and degree of ondulation

The evaluated results of the structural dynamic experiments are lead back to geometric relations, i.e., mesomechanic dimensions and degree of ondulation O ˜ (1).

The thickness of the fabric weave h S yields the thickness of one roving yarn by the division by 2 and the amplitude A by the division by 4

(10) h R = 1 2 h S and A = 1 4 h S ,

as illustrated in Figure 1, where the subscripts R and S indicate the roving and single layer, respectively.

The number of roving yarns in warp or fill direction n R per unit length yields the length of the cross-section of a roving in the warp and fill direction by the reciprocal value, and the length of one complete ondulation L follows via consideration of the specific factor of the fabric weave λ :

(11) L R = 1 n R and L = λ L R = λ 1 n R ,

with λ = 2 for a plain and λ = 4 for a twill weave 2/2, cf. Equation (2).

In the technical data sheets of the fabrics (first set Tenax [31,32], second set Pyrofil [34,35], third set Hexcel [29]), the thicknesses of the dry or uncompacted single layers h S,d are indicated, where the subscript d indicates the dry condition. For the two sets of comparable specimens based on the dry fabrics, the number of roving yarns n R per unit length (here: per cm) in the warp and fill direction is indicated. Because in the three sets, the fabrics are balanced, the consideration of warp and fill direction nominally yields the same properties and specific values.

The thickness of an impregnated, compacted fabric weave single layer in cured condition in the laminate h S,L follows by division of the measured thickness of the laminate h by the number of single layers N :

(12) h S , L = h N ,

with subscripts S and L for single layer and laminate.

With the length of one complete ondulation L according to Equations (2) and (11) and the selected unsupported length of the flat beam-like specimen l , the number of representative sequences or ondulations of each single layer n O , S follows by division, and the total number of ondulations of one specimen as a beam n O , B by a subsequent multiplication by the number of single layers in the layup N by

(13) n O , S = l L and n O , B = l L N ,

with subscripts O and B for ondulations and specimen as a beam.

6.3 Results

The evaluated results obtained under constant geometric and constant dynamic conditions are presented.

6.3.1 Structural dynamic experiments

The results of structural dynamic experiments are provided for each specimen, averaging five consecutive measurements. Figure 5 illustrates Λ (top) and the fundamental frequency f (bottom). Figure 6 illustrates the storage modulus E ′ (top) and loss modulus E ″ (bottom). In both figures, the left side represents constant geometric conditions, whereas the right side represents constant dynamic conditions. Each of the single specimens of a set (cf. Table 1) is considered individually.

Figure 5 Logarithmic decrement Λ \Lambda (upper row) and fundamental frequency f f (lower row) for every specimen under the presumption of constant geometric conditions (left) and constant dynamic conditions (right).
Figure 5

Logarithmic decrement Λ (upper row) and fundamental frequency f (lower row) for every specimen under the presumption of constant geometric conditions (left) and constant dynamic conditions (right).

Figure 6 Storage modulus E ′ E^{\prime} (upper row) and loss modulus E ″ {E}^{^{\prime\prime} } (lower row) for every specimen under the presumption of constant geometric conditions (left) and constant dynamic conditions (right).
Figure 6

Storage modulus E ′ (upper row) and loss modulus E ″ (lower row) for every specimen under the presumption of constant geometric conditions (left) and constant dynamic conditions (right).

Table 2 summarizes the results of the structural dynamic investigations for each specimen. For every set, the results of the unidirectionally reinforced specimens are indicated first. Then, the fabric-reinforced specimens are considered: first the results obtained at constant geometric conditions (e.g., unsupported length l = 220 mm and l = 200 mm ) and second the ones obtained constant dynamic conditions. For the latter, varied unsupported lengths l result according to Equation (5) resulting in the same fundamental frequency f .

Table 2

Evaluated results of the structural dynamic investigations in terms of the logarithmic decrement Λ and the fundamental frequency f as well as in terms of the storage modulus E ′ and loss modulus E ″ as real and imaginary part of the dynamic modulus E *

ID Reinf. Type of material Stacking sequence Thickness h (Init. displ. h ⁄ 4 ) Unsupp. length l (mm) Logarithmic decrement Λ Fundamental frequency f (Hz) Storage modulus E ′ (GPa) Loss modulus E ″ (MPa)
Ten-uni Tenax HTS40 Roving [ 0 ] 2 S C 1.96 mm (0.49 mm) 220 3.56 × 1 0 ‒ 3 49.92 99.44 36.50
220 3.58 × 1 0 ‒ 3 51.72 101.48 36.78
220 3.54 × 1 0 ‒ 3 50.63 99.75 35.76
Ten-plain Style 427 Plain weave [ ( 0 ⁄ 90 ) ] 2 S C 1.61 mm (0.40 mm) 220 5.40 × 1 0 ‒ 3 29.77 48.12 26.35
220 5.37 × 1 0 ‒ 3 30.16 48.84 26.57
220 5.39 × 1 0 ‒ 3 29.77 48.40 26.43
170 5.23 × 1 0 ‒ 3 49.69 61.87 32.78
168 5.11 × 1 0 ‒ 3 50.94 62.05 32.13
168 5.14 × 1 0 ‒ 3 51.02 63.31 32.96
Ten-twill Style 404 Twill weave 2/2 [ ( 0 ⁄ 90 ) ] 2 S C 2.25 mm (0.56 mm) 220 4.86 × 1 0 ‒ 3 45.08 54.56 26.85
220 4.93 × 1 0 ‒ 3 45.63 55.94 27.93
220 5.03 × 1 0 ‒ 3 45.55 55.70 28.36
210 4.99 × 1 0 ‒ 3 49.38 54.35 26.12
206 4.96 × 1 0 ‒ 3 51.88 55.59 27.93
208 4.96 × 1 0 ‒ 3 50.63 54.98 27.61
Pyr-uni Pyrofil TR 505 6k Roving [ 0 ] 4 S C 2.07 mm (0.52 mm) 220 3.71 × 1 0 ‒ 3 55.47 106.25 39.95
220 3.69 × 1 0 ‒ 3 56.17 103.03 38.49
220 3.66 × 1 0 ‒ 3 55.00 105.62 39.48
Pyr-plain Sigratex KDL 8051/120 Plain weave [ ( 0 ⁄ 90 ) ] 4 S C 2.38 mm (0.60 mm) 220 5.07 × 1 0 ‒ 3 46.95 56.32 27.22
220 5.03 × 1 0 ‒ 3 46.88 56.32 28.70
220 5.08 × 1 0 ‒ 3 46.64 55.99 28.80
202 5.32 × 1 0 ‒ 3 55.23 64.44 32.16
200 5.34 × 1 0 ‒ 3 56.09 62.78 33.96
202 5.38 × 1 0 ‒ 3 54.92 61.20 33.36
Pyr-twill Sigratex KDK 8052/120 Twill weave 2/2 [ ( 0 ⁄ 90 ) ] 2 S C 2.30 mm (0.58 mm) 220 4.92 × 1 0 ‒ 3 47.73 58.61 29.22
220 4.80 × 1 0 ‒ 3 47.11 58.50 27.25
220 4.80 × 1 0 ‒ 3 47.27 57.26 27.85
204 5.18 × 1 0 ‒ 3 55.08 57.69 30.25
202 4.90 × 1 0 ‒ 3 55.39 57.48 28.57
204 5.10 × 1 0 ‒ 3 54.38 56.03 28.12
Hex-uni Hexcel G947 UD Unidir. prepreg [ 0 ] 4 S C 1.42 mm (0.34 mm) 200 5.15 × 1 0 ‒ 3 42.97 104.82 57.14
200 5.39 × 1 0 ‒ 3 46.80 106.06 57.97
200 5.23 × 1 0 ‒ 3 44.77 106.07 56.18
Hex-fab Hexcel G939 Fabric Fabric prepreg [ 0 / 90 ] 4 S C 1.93 mm (0.48 mm) 200 6.78 × 1 0 ‒ 3 44.38 57.58 39.58
200 6.63 × 1 0 ‒ 3 44.53 57.82 38.87
200 6.79 × 1 0 ‒ 3 44.84 57.29 39.42
204 6.74 × 1 0 ‒ 3 42.66 56.47 38.55
196 6.61 × 1 0 ‒ 3 46.41 59.10 39.60
200 6.79 × 1 0 ‒ 3 44.84 57.29 39.42
Hun-PM — Pure matrix — 1.98 mm (0.50 mm) 220 3.52 × 1 0 ‒ 2 10.98 3.25 36.41
220 3.54 × 1 0 ‒ 2 10.71 3.09 34.81
220 3.55 × 1 0 ‒ 2 11.42 3.34 38.37

6.3.2 Mesomechanic geometric dimensions and relations

Table 3 confronts the values of the thickness of the dry or uncompacted and preimpregnated fabric-reinforced single layers h S,d indicated in the technical data sheets [31,32,34,35] to the resulting values of thickness of the fabric-reinforced single layers in the laminate h S,L according to Equation (12).

In contrast to the indications h S,d in the technical data sheets, the evaluation of the thickness of the single layers in the laminate h S,L according to Equation (12) always yields lower values. Due to the preimpregnation process of the dry fabrics, in combination with the applied pressure during the autoclave processing, and a more probable “out-of-phase” arrangement of the adjacent fabrics in the laminate, the test panels or the specimens, based on the dry fabrics, are thinner than the theoretical value, according to a so-called “in-phase” arrangement.

Thereby, the thicknesses of a dry or uncompacted fabric-reinforced single layer h S,d indicated in the data sheets correspond to a theoretical or effective value, whereas the thicknesses of an impregnated, compacted fabric in cured condition h S,L according to Equation (12) correspond to a resulting or apparent value. In detail, the resulting values of the thickness h S,L are approx. 66–57% of the theoretical or effective thicknesses h S,d . In contrast, this correlation is not valid in case of the carbon fiber prepregs [29]. Due to the automatic impregnation process and the higher viscous matrix system, the prepregs do not exhibit any sensitivity regarding the applied pressure during the autoclave processing and the kind of arrangement of the single layers in the laminate.

In the following, the thicknesses of an impregnated, compacted fabric-reinforced single layer in cured condition in the laminate h S,L according to Equation (12) are used further. The values yield the resulting or apparent values, as they are directly based on the thickness of the specimen h .

These values correspond to the number of single layers in the layup N of impregnated, compacted fabric-reinforced single layers in cured condition in the laminate. In this condition, the structural dynamic experiments are carried out, so that the validation of the presumed mesomechanic kinematic due to geometric constraints is based just on the described condition.

Table 4 contains the evaluated results as parameters of the ondulation of the fabric-reinforced single layers of the three sets of comparable specimens according to Equations (10) and (11). As previously explained, the calculation of the degree of ondulation O ˜ according to Equation (1) is based on the thicknesses of the fabric-reinforced single layers in the laminate h S,L (see Equation (12)).

Table 5 contains the number of representative sequences or ondulations of each fabric-reinforced single layer n O , S and the total number of ondulations of one specimen as a beam n O , B according to Equation (13). The corresponding number of ondulations is indicated regarding the unsupported lengths l under constant geometric and constant dynamic conditions. The evaluation of the number of representative sequences or ondulations according to Equation (13) reflects the different numbers of relevant ondulations in the different kinds of fabrics, namely, plain and twill weave 2/2, whereas the intensity of the geometric deviation of the single roving yarns is the same.

7 Conclusions and outlook

Specimens reinforced with both unidirectional fibers and fabric weave have been mechanically characterized by structural dynamic investigations. The free decay behavior of flat beam-like specimens has been measured and analyzed using a contactless method, via a laser scanning vibrometer PSV 400 from Polytec [39]. Analysis of the recorded signals, in both time and frequency domain, enabled distinct conclusions regarding the structural dynamic properties of each type of reinforcement.

Under the presumption of constant geometric and dynamic conditions, low standard deviations indicate high reproducibility of experimental procedures and mechanical quality of the material. The consistent displacement excitation by h ⁄ 4 applied to each specimen is found suitable for further experimental investigations. The former indicated conditions allow the measurement of only linear viscoelastic properties, with non-linear effects either avoided or reduced to a minimum, thus negligible.

Three sets of comparable specimens have been investigated. The first and second set consist of unidirectionally and plain and twill weave 2/2-reinforced specimens based on the roving Tenax HTS40 and Pyrofil TR50S 6K [30,33]. The third set consists of unidirectionally and fabric-reinforced Hexcel prepreg material G947 and G939 [29]. Across all sets, the logarithmic decrement Λ of the fabric-reinforced specimens generally exceeds that of the unidirectionally ones.

7.1 Conclusions

The results show that in either case, the material damping in terms of the logarithmic decrement Λ of the fabric-reinforced specimens is generally higher than the material damping in the unidirectionally reinforced material. Additionally, when the fabric-reinforced specimens are addressed, in each case, the plain weave-reinforced specimens exhibited higher values of the material damping as the twill weave 2/2-reinforced ones.

The described observations justify the initially stated assumption that the repeated acting of the presumed mesomechanic kinematic due to geometric constraints enhances the damping properties of fabric-reinforced single layers compared to unidirectionally reinforced ones. The observations further allow the conclusion that the presumed kinematic depends on mesomechanic geometric dimensions. These purely geometric conditions can directly be indicated in terms of a degree of ondulation, which represents the intensity of the single ondulated yarns in different kinds of fabrics (e.g.., plain weave, twill weave, satin).

7.1.1 Influence of the degree of ondulation O ˜

Table 6 lists the relatively higher material damping of the fabric-reinforced specimens compared to the 0 ° -unidirectionally reinforced specimens in % against the degree of ondulation O ˜ for geometric and dynamic constant conditions. Figure 7 graphically illustrates the values plotted against the degree of ondulation O ˜ . Under both kinds of constant conditions, the material damping of the plain weave-reinforced specimens is higher than that of the twill weave 2/2-reinforced ones.

Table 6

Relatively higher material damping of the fabric-reinforced specimens compared to the 0 ° -unidirectionally reinforced specimens in % listed against the degree of ondulation O ˜ for geometric and dynamic constant conditions

ID Degree of ondulation O ˜ = A ⁄ L see Equation (1) Kind of constant condition Number of rep. sequence per specimen as beam n O , b = l ⁄ L ⋅ N see Equation (13) Relatively higher damping compared to 0 ° -UD (%) Reference Matrix system
Ten-uni 0 — 0 — [30] Huntsman [36]
Ten-plain 0.01263 Geometric 110 51.4 [31]
Dynamic 84.3 45.0
Ten-twill 0.01306 Geometric 81.5 38.7 [32]
Dynamic 77 39.60
Pyr-uni 0 — 0 — [33]
Pyr-plain 0.01389 Geometric 325.9 37.2 [34]
Dynamic 298.3 45.1
Pyr-twill 0.00657 Geometric 163 31.3 [35]
Dynamic 150.6 37.7
Hex-uni 0 — 0 — [29] HexPly M18/1 [29]
Hex-fab n.d. Geometric n.d. 28.1
Dynamic n.d. 27.7
Figure 7 Relatively higher material damping of the fabric-reinforced specimens compared to the 0 ° {0}^{^\circ } -unidirectionally reinforced specimens in % plotted against the degree of ondulation O ˜ \tilde{O} under the presumption of constant geometric conditions at comparable unsupported lengths, l = l UD = l F l={l}_{{\rm{UD}}}={l}_{{\rm{F}}} (top), and constant dynamic conditions at comparable fundamental frequencies, f = f UD = f F f={f}_{{\rm{UD}}}={f}_{{\rm{F}}} (bottom).
Figure 7

Relatively higher material damping of the fabric-reinforced specimens compared to the 0 ° -unidirectionally reinforced specimens in % plotted against the degree of ondulation O ˜ under the presumption of constant geometric conditions at comparable unsupported lengths, l = l UD = l F (top), and constant dynamic conditions at comparable fundamental frequencies, f = f UD = f F (bottom).

For the first set of specimens, based on Tenax HTS40 [30], the plain and twill weave 2/2-reinforced specimens exhibit a degree of ondulation O ˜ PL = 0.01263 and O ˜ T2 = 0.01306 . The relatively similar values follow from the slightly different geometric parameters of the ondulation, as indicated in Table 4. In contrast, the plain and twill weave 2/2-reinforced specimens of the second set of specimens based on Pyrofil TR50S 6K [33] exhibit a degree of ondulation O ˜ PL = 0.01389 and O ˜ T2 = 0.00657 . Due to the relatively similar geometric parameters of the ondulation, as indicated in Table 4, in this case, the values approximately follow the relation O ˜ PL = 2 ⋅ O ˜ T2 . The afore described relations regarding the total number of representative sequences or ondulations of one specimen as a beam n O , B , as indicated in Table 5, behave analogous to the geometric parameters of the ondulation.

7.1.2 Constant geometric and constant dynamic conditions

When analyzing the results under constant geometric conditions, as illustrated in Figure 7 (top), it becomes evident that fabric-reinforced specimens exhibit higher material damping, quantified by the logarithmic decrement Λ , compared to the unidirectionally reinforced ones. In the first set of specimens based on Tenax HTS40 [30] rovings, plain and twill weave 2/2-reinforced specimens yield higher damping values of 51.4 and 38.7%. In the second set of specimens based on Pyrofil TR50S 6K [33] rovings, plain and twill weave 2/2-reinforced specimens yield higher damping values of 37.2 and 31.3%. For the Hexcel prepreg material G947 and G939 [29], the fabric-reinforced specimens yield a higher damping of 28.1%.

A similar tendency is observed when examining the results under constant dynamic conditions, as illustrated in Figure 7 (bottom). In this case, too, across all three sets of specimens, the fabric-reinforced with varied unsupported length l generally exhibits higher damping values Λ compared to unidirectionally reinforced ones. In the first set of specimens based on Tenax HTS40 [30] rovings, plain and twill weave 2/2-reinforced specimens yield higher damping values of 45.0 and 39.6%. For the second set of specimens based on Pyrofil TR50S 6K [33] rovings, plain and twill weave 2/2-reinforced specimens yield higher damping values of 45.1 and 37.7%. For the Hexcel prepreg material G947 and G939 [29] the fabric-reinforced specimens yield a higher material damping of 27.7%.

7.1.3 Influence of the two different kinds of constant conditions

The storage modulus E ′ and loss modulus E ″ of both the plain weave-reinforced specimens, based on Tenax HTS40 [30] and Pyrofil TR50S 6K [33] rovings, show a slight dependence on the fundamental frequency f . Higher fundamental frequencies f result in higher values for both moduli, E ′ and E ″ . In contrast, the real and imaginary components of the dynamic modulus E * do not exhibit a dependence on the fundamental frequency f in this extent when both twill weave 2/2-reinforced specimens, based on Tenax HTS40 [30] and Pyrofil TR50S 6K [33] rovings, or on the fabric-reinforced Hexcel prepreg material G939 [29], are considered.

7.1.4 Remark on the mesomechanic effect in the specimen as a beam

During the free decay of the transversal vibrations, the specimens as cantilever beams undergo a repeated and characteristic distribution of the transversal displacements w ( x ) . Due to the defined excitation w = h ⁄ 4 at the free end the specimen almost solely, the first eigenmode occurs at the fundamental frequency. Thereby, the absolute values of the displacements ∣ w ( x ) ∣ reach its maximum at the free end x = l , ∣ w ( x = l ) ∣ = w max , and vanish at the clamping x = 0 , ∣ w ( x = 0 ) ∣ = w min = 0 . In detail, the eigenmodes do not differ significantly, when the shear-stiff (or technical) beam theory and the first-order shear-deformation theory (FOSD-theory) are considered [46,47]. Yet, both theoretical considerations presume linearly distributed displacements u ( z ) in the transversal z -direction, i.e., over the thickness of the structure h . In the case of symmetric and balanced stacking sequences of the specimens, this causes linearly and steadily distributed normal stresses in longitudinal x -direction. Thereby, both the strains and the stresses in the longitudinal x -direction at the clamping x = 0 reach its maximum at its top or bottom side z = ± h ⁄ 2 , and vanish at the free end x = l . Under dynamic transversal loading, such as free decay of transversal vibrations, the signs of strains, and resulting stresses alternate with the respective frequency [46,47].

Depending on the mesomechanic geometric dimensions, the kind of fabric, and the unsupported length l , the specimens exhibit a different number of representative sequences of every single layer n O , S and in total as a beam n O , B , as indicated in Table 5. Thereby, every representative sequence contributes to the material damping in different amounts, depending on its position in the longitudinal x -direction and in the transversal z -direction. For further investigations, it is suggested to weight the contribution depending on the position. In the longitudinal x -direction, a distribution approximately linear or contrarious to the first Eigenmode W ( x ) , i.e., with the maximum value at x = 0 and the minimum one at x = l , is suggested. Regarding the transversal z -direction, it is advantageous to suggest a linear contribution over the thickness of the structure h , analogous to the distribution of the stains and resulting stresses in case of symmetric and balanced stacking sequences. Figure 8 illustrates the suggested simplified distribution of the effect of the mesomechanic kinematic in the longitudinal x -direction and transversal z -direction, which, at the same time, correspond to weighting functions of the effect of enhanced material damping in fabrics.

Figure 8 Simplified presumed distribution of the mesomechanic kinematic during the free decay of transversal vibrations in longitudinal x x -direction and transversal z z -direction causing additional energy dissipation as material damping in fabric-reinforced single layers.
Figure 8

Simplified presumed distribution of the mesomechanic kinematic during the free decay of transversal vibrations in longitudinal x -direction and transversal z -direction causing additional energy dissipation as material damping in fabric-reinforced single layers.

7.2 Outlook

The findings clearly prove that the repeated acting of the mesomechanic kinematic enhances the material damping of fabric-reinforced single layers. This occurs additionally to the purely visco-elastic part of the material. The dependence on the geometric dimensions is quantitatively described via the degree of ondulation O ˜ . Yet, the distribution over the length and thickness, and thus over the continuum of the flat beam-like specimens, is not completely revealed. Further, an extent to a two- or three-dimensional structure (thin or tick plate theory with corresponding specimens) is necessary in the future.

For an identification of parameters and an analysis of the sensitivity to them, further investigations of kinematic correlations due to geometric constraints acting in the mesomechanic scale are necessary. Therefore, more sets of comparable specimens are required. If it is possible to involve the weave process and eventually the impregnation process for obtaining prepreg material quality, a more detailed experimental material characterization is possible. Additionally, further numerical investigations using the finite element analysis, focusing parametrical variations of geometric dimensions with the aim of identifying a kinematic coupling between longitudinal and transversal deformation, analogous to Ottawa et al. [14], Valentino et al. [15,16] or Romano et al. [18,19], could extend and complete the scientific considerations.

# Dedicated to Prof. Dr.-Ing. habil. Exzellenter Emeritus Norbert Gebbeken for his 70th birthday.

Acknowledgements

The publication of this article was funded by the Open Access Publishing Fund of OTH Regensburg. The carried out experimental investigations have been enabled by the founded project “Lebensdauerüberwachung von faserverstärkten Kunststoffen auf Basis der strukturdynamischen Werkstoffdämpfung – DampSIM,” financially supported by the Bayerische Forschungsstiftung (BFS), Project No. AZ-1089-13, in cooperation with an industrial partner. In this context, it was possible to carry out the experimental investigations with a laser scanning vibrometer of the type PSV 400 from Polytec, Waldbronn. The authors would like to thank the industrial partner and the found for the excellent collaboration and their financial support. Further thanks go to Mr Bastian Jungbauer, B.Eng., Ms Carolin Renner, B.Eng., and Mr Simon Walbrun, M.Sc.

  1. Funding information: The publication of this article was funded by the Open Access Publishing Fund of OTH Regensburg. The carried out experimental investigations have been enabled by the founded project “Lebensdauerüberwachung von faserverstärkten Kunststoffen auf Basis der strukturdynamischen Werkstoffdämpfung – DampSIM,” financially supported by the Bayerische Forschungsstiftung (BFS), Project No. AZ-1089-13, in cooperation with an industrial partner. In this context, it was possible to carry out the experimental investigations with a laser scanning vibrometer of the type PSV 400 from Polytec, Waldbronn. The authors would like to thank the industrial partner and the found for the excellent collaboration and their financial support. Further thanks go to Mr Bastian Jungbauer, B.Eng., Ms Carolin Renner, B.Eng., and Mr Simon Walbrun, M.Sc.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. The design of experiments was carried out collaboratively by IE and MR. MR carefully conducted the experiments considering the scientific standards described in detail in the manuscript. The subsequent validation of the experiments and evaluation of the results was done by MR, with careful and intense review and discussions with IE, ensuring the robustness of the methods, results, and conclusions. MR prepared and submitted the manuscript, incorporating the contributions and feedback of co-author IE at each stage.

  3. Conflicts of interest: Authors state no conflict of interest.

  4. Data availability statement: The datasets generated and/or analyzed during the current study are available from both the authors upon reasonable request. Yet, the most relevant evaluated results during the carried out experimental investigations of this study are included in the published article – see detailed tables and figures.

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Received: 2024-03-13
Revised: 2024-05-05
Accepted: 2024-05-12
Published Online: 2024-08-22

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/secm-2024-0019
Keywords for this article
fiber-reinforced plastics; mesomechanic scale; fabric-reinforced layer; ondulation
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Research on damage evolution mechanisms under compressive and tensile tests of plain weave SiCf/SiC composites using in situ X-ray CT
  3. Structural optimization of trays in bolt support systems
  4. Continuum percolation of the realistic nonuniform ITZs in 3D polyphase concrete systems involving the aggregate shape and size differentiation
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  10. Influence of Ti2SnC content on arc erosion resistance in Ag–Ti2SnC composites
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  15. Roles of corn starch and gellan gum in changing of unconfined compressive strength of Shanghai alluvial clay
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  23. Beneficial effect of 4% Ta addition on the corrosion mitigation of Ti–12% Zr alloy after different immersion times in 3.5% NaCl solutions
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  26. Research on uniaxial compression performance and constitutive relationship of RBP-UHPC after high temperature
  27. Experimental analysis of frost resistance and failure models in engineered cementitious composites with the integration of Yellow River sand
  28. Influence of tin additions on the corrosion passivation of TiZrTa alloy in sodium chloride solutions
  29. Microstructure and finite element analysis of Mo2C-diamond/Cu composites by spark plasma sintering
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