# Size-Independent Flexure Test Technique for the Mechanical Properties of Geocomposites Reinforced by Unidirectional Fibers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{m}*, and its corresponding elasticity modulus, E*. Notably, these values exhibit a pronounced dependency on the size of the testing parameters. Specifically, within a judicious range of support span L relative to specimen height H, spanning a ratio of 10 to 40, these metrics can vary by a factor between 2 and 4. By conducting evaluations across an extensive array of H/L ratios and adhering to the protocols set for comparable composites with a plastic matrix, it becomes feasible to determine the definitive flexural elastic modulus E and shear modulus G, both of which can be viewed as size-neutral material traits. A parallel methodology can be employed to deduce size-agnostic values for flexural strength, σ

_{m}. The established linear relationship between the inverse practical value E* (1/E*) and the squared ratio (H/L)

^{2}is acknowledged. However, a congruent 1/σ

_{m}* relationship has been recently corroborated experimentally, aligning primarily with Tarnopolsky’s theoretical propositions. The parameter T, defined as the inverse gradient of 1/σ

_{m}* about (H/L)

^{2}, is integral to these findings. Furthermore, the significance of the loading displacement rate is underscored, necessitating a tailored consideration for different scenarios.

## 1. Introduction

_{4}and AlO

_{4}tetrahedra. These tetrahedra are cohesively linked by shared oxygen atoms, forming polymeric Si-O-Al bonds [2,3]. Currently, geopolymers are perceived as avant-garde materials, having potential for utilization in coatings, adhesives, fiber-reinforced composites, and viable substitutes for traditional types of cement in concrete mixtures [4]. For over thirty years, composites reinforced with fibers and employing a geopolymer matrix, commonly called geocomposites, have been acknowledged, tracing back to the seminal patent registered by Davidovits [5]. These cutting-edge materials can be fabricated and set either at ambient conditions or via a heat treatment at low temperature. After a brief curing time, these composites exhibit superior attributes, including being lightweight yet possessing remarkable strength. Additionally, they showcase commendable fire-resistant properties, characterized by the emission of benign fumes and minimal smoke, and demonstrate resilience against organic solvents [4,6,7,8,9]. Owing to these distinct characteristics, geopolymer matrix composites are highly beneficial for a spectrum of advanced industries such as aerospace, marine design, terrestrial transportation, and the automotive domain, especially in applications demanding resistance to elevated temperatures [4,6,8,10]. Geopolymer composites offer a cost-effective alternative to lightweight, high-strength composites composed of carbon or glass fibers with ceramic or organic matrices. The high costs associated with the specialized ceramic processing requirements and the limited applicability of most organic matrix composites at temperatures exceeding 200 °C make geopolymer composites a viable option [10,11]. Furthermore, geopolymer matrices can accommodate a wide range of reinforcement fibers, and specific matrices can protect carbon fibers from oxidation [10,12].

## 2. Theory of Size-Independent Flexure-Test Technique

#### 2.1. Modulus of Elasticity

_{f}) is considered a measure of elasticity. It is evaluated from the force-deflection curve for three-point bending in terms of Equation (1), which is entirely valid only for isotropic materials (with high shear modulus).

^{2}, which for κ

^{2}< 4.5 gives

^{2}/4 = 1.1747. In another practically aimed publication, Tarnopolsky rounds up to α = 1.2 [22]. The mutual difference is minimal; we have kept it to 1.175 in all our evaluations.

^{2}. The values of the virtual modulus E are obtained through linear regression, where

^{2}× E*/G should amount to 0.0426. With L/H = 20 it is fulfilled if E*/G = 17, which can be approximately correct for Class III in standard EN ISO 14,125 (plastics reinforced, for example, with unidirectional glass fibers) and for Class IV (plastics reinforced for example with carbon fibers) with recommended L/H = 40 for this class it corresponds to E*/G = 68 [17]. These values seem to be around at the lower end of the actual values with geopolymer unidirectional fiber composites, as will be documented later.

#### 2.2. Strength

_{m}. Theoretical analysis shows a more complicated pattern of stress dislocation across the specimen cross-section, formed from typically non-isotropic material [23]. The inhomogeneity of geopolymer composites is further largely influenced by minor irregularities in the preparation process, which causes considerable scattering of results.

_{m}according to the relationship valid for isotropic materials [15,16,17,19,20]. For rectangular profiles and three-point bending, the basic formula reads

^{2}according to Equation (14) is illustrated in Figure 2.

^{2}over the test data would provide parameters σ

_{m}and E/G. The last term of expansion Equation (14) is, however, relatively small at not too large (H/L)

^{2}values; summary declinations from 5% linearity occur for L/H = 4.5 at E/G = 15 and for L/H = 20 at E/G = 1300. Even those declinations could have been lower if further terms (with alternating signs) had been included. Considering additional natural scatter, the curvature of the regression line could be neglected in the first approximation. The analysis thus shows that the application of the same pattern of straight-line regression as applied for elasticity is elementary justified even for strength. The reciprocal linear regression intercept

_{m}) from

#### 2.3. Displacement Rate

_{m}, within which the specimen should fail at a constant velocity ν of cross-head (displacement rate), or by a strain/deformation rate $\dot{\epsilon}$ = dε/dt. For three-point loading, the interrelation of these paired quantities is

_{m}with plastic composites in the standard [20]; 60 s with ceramic composites [16]. By contrast, ASTM 1341-6 requires only 5 to 10 s for ceramic composites [15]. The higher speed is reasoned for as a “minimization of environmental and force application rate effects when testing in ambient air” [15]. As the product $\dot{\epsilon}\cdot {t}_{m}$ should be approximately constant with the same strain expected at failure, the standards in fact recommend rates $\dot{\epsilon}$ numerically: 167 × 10

^{−6}s

^{−1}in EN 13706-2 [20] and 500 × 10

^{−6}to 5000 × 10

^{−6}s

^{−1}in ASTM 1341-06 [15], resp., (recommended mean 1000 × 10

^{−6}s

^{−1}).

^{−6}to 25,000 × 10

^{−6}s

^{−1}), no matter whether the composite was reinforced with carbon or AR-glass fibers [24]. The increases of modulus E* respective of strength ${\sigma}_{m}^{*}$ due to the rate of acceleration over this scale amounted up to 100 and 50 per cent, resp.

_{0}and dynamic parts. On the basis of dimensional analysis, the dynamic component should have the shape of velocity squared divided by composite density. A reasonable assumption is that such a shear velocity is of a quantity proportional to the displacement rate ν and the ratio (L/H)

^{2}. The important conclusion is that under very high displacement rates there would be a size-independent limit on a curve 1/E* vs. the rate for ν → ∞, which equals 1/E. On the other hand, the limit for ν → 0 should lead to a primary relationship as in Equation (3). As discussed above, the same features for rate dependence as for elasticity can be anticipated for strength based on a similarity between both phenomena. Further elaboration is, of course, essential.

## 3. Experiments

^{2}T700 800 tex (Toho Tenax Europe GmbH) and E-glass 250 g/m

^{2}(Saint-Gobain Vetrotex) and twill (Weave (0/90) carbon 160 g/m

^{2}with HTS 5631 400 tex (Toho Tenax Europe GmbH) and E-glass 163 g/m

^{2}(Saint-Gobain Vetrotex). The slips of fabrics approx. 150 × 150 mm were saturated with matrix gel using a brush, laid into the mold; a roller pushed away the excess, and the composite was left to harden under vacuum at the same conditions as those of the fibers. Also, the final treatment was similar; the strip specimens were cut.

## 4. Results and Discussion

#### 4.1. Compliance with Theory

^{2}values above 0.2. The resulting values ${\sigma}_{m}^{*}$ were deeply under the anticipated dependence on H/L. Evidently, the results were already affected by the clip. Diverse accidental defects in the prepared composite, particularly in shorter specimens, caused sometimes solitary points in the plot, characterized usually by abrupt shifting of the $1/{\sigma}_{m}^{*}$ value upwards. Such an outlying point had to be sensibly eliminated from the regression.

^{2}—requires taking each specimen as an individual with its actual sizes, to counteract the difference in this common trait with homogeneous specimens, which used to be machined so accurately that the results could be grouped, and as such statistically pre-treated. The overall deviations of the extrapolated values are usually in frames of values with the standard testing of brick and concrete materials.

#### 4.2. Displacement Rate

^{2}plots at low rates and low (H/L)

^{2}values. An introductory series taken at 2 mm/min and 9.2 mm/min showed no difference. When applying a larger interval of 2 to 200 mm/min, however, characteristic regression lines appeared, as in Figure 6 and Figure 7.

^{2}→ 0, but the slopes of the lines started to differ. Table 3 shows that the evaluated shear modulus G increases with an increasing deformation rate. It supports, at least formally, the conception of a gross description with G composed of static and dynamic elements, while the limiting value of E stays constant as a static material property. The extrapolated value of flexural strength σ

_{m}behaves similarly, but the quantity of T has unexpectedly reversed course. All these relationships need experimental verification with more significant rate spans concerning the dependence on sizes.

#### 4.3. Unidirectional and Woven Fabrics

^{2}, as did composites prepared from rovings. By contrast, woven fabrics (twill 1/90) express 1/E* vs. (H/L)

^{2}slopes close to zero, and 1/${\sigma}_{m}^{*}$ vs. (H/L)

^{2}slopes even into negatives. Obviously, there is also a qualitative change in the mode of the weave composite failure.

## 5. Conclusions

^{2}, an optimal approach would entail broadening the tests to encompass a variety of specimens with diverse L/H ratios. This methodology, albeit with lesser emphasis, has been previously endorsed in specific standards for evaluating plastic composites. In a refined approach, shear properties, specifically the modulus G and the reciprocal gradient T of the 1/${\sigma}_{m}^{*}\text{}$in relation to (H/L)

^{2}, are concurrently derived as ancillary attributes. These enhance the precise evaluation of a geocomposite and hold potential utility in design computations.

^{2}is not solely confined to the flexural modulus, as one might anticipate based on insights from plastic composites, but also extends to flexural strength. The theoretical foundation for this latter relationship is rooted in the earlier work of Tarnopolsky and his colleagues. However, for elevated E/G values, the theoretical trajectory tends to deviate from the empirically observed linearity.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Theoretical size dependences of ${\sigma}_{m}/{\sigma}_{m}^{*}$ ratios at different E/G (in the legend).

**Figure 5.**Reciprocal effective flexural properties ((

**a**,

**c**,

**e**) for flexural modulus and (

**b**,

**d**,

**f**) for flexural strength) of composites carbon HTS 5631 1600tex 24k-M1 matrix at different temperatures of curing in legends.

**Figure 6.**Correlation of flexural modulus for geopolymer M2-basalt; rate of deformation in mm/min in the legend.

**Figure 7.**Correlation of flexural strength for geopolymer M2-basalt; rate of deformation in mm/min in the legend.

**Figure 8.**Comparison of E in carbon-woven (twill) and unidirectional fabrics, last numbers in the legend: days of cooled matrix storage.

**Figure 9.**Comparison of σ

_{m}in carbon-woven (twill) and unidirectional fabrics, last numbers in the legend: days of cooled matrix storage.

**Figure 10.**Comparison of E in E-glass-woven (twill) and unidirectional fabrics, last numbers in the legend: days of cooled matrix storage.

**Figure 11.**Comparison of σ

_{m}in E-glass-woven (twill) and unidirectional fabrics, last numbers in the legend: days of cooled matrix storage.

**Table 1.**Results of the comparative tests on the composites with matrices M2 and M1, and carbon fiber H1600 in two laboratories; ± sign marks standard deviation and variation.

Matrix | Laboratory | Flexural Modulus | Shear Modulus | Flexural Strength | |||
---|---|---|---|---|---|---|---|

E [GPa] | G [GPa] | E/G | σ_{m}[MPa] | ||||

M2 | VUANCH | 109 ± 32 | ±29% | 0.76 | 143 | 631 ± 75 | ±12% |

M2 | TUL | 111 ± 14 | ±13% | 0.73 | 152 | 571 ± 75 | ±13% |

M1 | VUANCH | 96 ± 26 | ±27% | 0.35 | 278 | 316 ± 18 | ±6% |

M1 | TUL | 128 ± 21 | ±17% | 0.33 | 392 | 323 ± 18 | ±5% |

**Table 2.**Compiled results of the estimation of basic mechanical properties E and σ

_{m}of geocomposites cured at 85 °C.

Matrix | Fiber | Flexural Modulus | Shear Modulus | Flexural Strength | Slope | Product | ||
---|---|---|---|---|---|---|---|---|

E [GPa] | G [GPa] | E/G | σ_{m} [MPa] | T [MPa] | P | ∏ | ||

M1 | basalt | 58.1 ± 3.1 | 0.45 | 129 | 400 ± 29 | 3.27 | 0.18 | 1.06 |

carbon | 179.5 ± 24.9 | 0.47 | 380 | 672 ± 68 | 2.95 | 0.28 | 1.66 | |

E-glass | 61.1 ± 11.6 | 0.16 | 388 | 145 ± 14 | 0.94 | 0.42 | 2.53 | |

M2 | basalt | 61.0 ± 4.0 | 0.50 | 122 | 306 ± 28 | 1.16 | 0.08 | 0.46 |

carbon | 138.5 ± 10.7 | 0.81 | 171 | 649 ± 75 | 1.58 | 0.08 | 0.42 | |

E-glass | 65.0 ± 5.2 | 0.46 | 142 | 276 ± 50 | 1.19 | 0.12 | 0.61 |

Speed | Flexural Modulus | Shear Module | Flexural Strength | Reciprocal Slope |
---|---|---|---|---|

[mm/min] | E [GPa] | G [GPa] | σ_{m} [MPa] | T [MPa] |

2 | 69.7 ± 6.5 | 0.7 | 505.2 ± 129.7 | 0.7 |

20 | 73.3 ± 10.2 | 0.5 | 546.0 ± 145.2 | 1.1 |

200 | 83.4 ± 6.3 | 0.2 | 518.1 ± 78.6 | 1.8 |

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**MDPI and ACS Style**

Tran Doan, H.; Kroisova, D.; Bortnovsky, O.
Size-Independent Flexure Test Technique for the Mechanical Properties of Geocomposites Reinforced by Unidirectional Fibers. *Ceramics* **2023**, *6*, 2053-2069.
https://doi.org/10.3390/ceramics6040126

**AMA Style**

Tran Doan H, Kroisova D, Bortnovsky O.
Size-Independent Flexure Test Technique for the Mechanical Properties of Geocomposites Reinforced by Unidirectional Fibers. *Ceramics*. 2023; 6(4):2053-2069.
https://doi.org/10.3390/ceramics6040126

**Chicago/Turabian Style**

Tran Doan, Hung, Dora Kroisova, and Oleg Bortnovsky.
2023. "Size-Independent Flexure Test Technique for the Mechanical Properties of Geocomposites Reinforced by Unidirectional Fibers" *Ceramics* 6, no. 4: 2053-2069.
https://doi.org/10.3390/ceramics6040126