# Hybrid Geopolymer Composites Based on Fly Ash Reinforced with Glass and Flax Fibers
## 2 Materials and Methods
### Methods
#### 2.2.4 Mechanical Properties, Including Fracture Behavior Analysis  
The resonance method was used to determine the dynamic characteristics of investigated composites before fracture tests were conducted. The prismatic specimens with a nominal dimension of 40 \(\times\) 40 \(\times\) 160 mm were used. The natural frequency of longitudinal and torsional vibrations was assessed using a Handyscope HS4 oscilloscope (TiePie engineering, Sneek, The Netherlands) with an acoustic sensor [39]. The absolute valuesof the dynamic modulus of elasticity and the dynamic Poisson ratio were calculated in compliance with the ASTM C215-19 [40]. Three-point bending (3 PB) tests were performed for the above-mentioned prismatic specimens to determine the selected mechanical fracture characteristics of the investigated geopolymer composites. An initial notch was made one day before fracture tests. A diamond blade saw made the central edge notch with a nominal depth of about 1/3 of the specimen's height. The span length was set to 120 mm. The specimens were tested approximately at the age of 28 days. Fracture tests were conducted using the stiff multipurpose mechanical testing machine LabTest 6.250 (LaborTech, s.r.o., Opava, Czech Republic) with a load range of 0-250 kN. The loading process was governed by a constant increment of displacement of 0.02 mm/min during the whole course of testing. The vertical displacement (midspan deflection) \(d\) was measured using the inductive sensor mounted in a special measurement frame placed on the specimens. The outcome of each test is a vertical force \(F\) vs. \(d\) diagram. The diagrams recorded during the experiment were processed to obtain the correct input values for the consecutive evaluation of mechanical fracture parameters. The processing of diagrams covered the elimination of duplicate points, the shifting of the origin of the coordinate system, the smoothing of the diagram, and the reduction in the number of points. The processing of recorded diagrams was performed in GTDiPS software (v2t53i) [41] which is based on advanced transformation methods used for the processing of extensive point sequences. The \(F-d\) diagrams from fracture tests are introduced in Figure 2. Each curve represents the response of one tested specimen. The difference in fracture response of individual specimens made of the same composite is caused by the heterogeneity of materials. Especially with the addition of fibers, the difference in the post-peak part of the \(F-d\) diagrams can be seen. The addition of fibers mainly influences fracture behavior in this part of the \(F-d\) diagrams and the individual specimen response depends on the distribution of fibers in the cross-section where the crack is propagating. The \(F-d\) diagrams were used to estimate values of the modulus of elasticity, effective fracture toughness, and specific fracture energy. The modulus of elasticity \(E\) was determined from the initial part of the \(F-d\) diagrams according to [PERSON] [42]:  
\[E=\frac{F_{i}}{4 Bd_{i}}\bigg{(}\frac{S}{W}\bigg{)}^{3}\bigg{[}1-0.387\frac{W}{ S}+12.13\bigg{(}\frac{W}{S}\bigg{)}^{2.5}\bigg{]}+\frac{9}{2}\frac{F_{i}}{Bd_{i}} \bigg{(}\frac{S}{W}\bigg{)}^{2}F_{1}(\alpha_{0}) \tag{2}\]  
where \(F_{i}\) and \(d_{i}\) are the vertical force and midspan deflection, respectively, read from the linear part of the \(F-d\) diagram, \(S\) represents span length, and \(B\) and \(W\) are the specimen's width and height, respectively. \(F_{1}(\alpha_{0})\) is the shape function according to [PERSON] [43], and \(\alpha_{0}\) is the relative crack/notch length. The first term of the equation is calculated from the deflection of the whole cross-section (without crack/notch). The second term includes the effect of the notch depth and is based on the Castigliano principles; see [PERSON] [43] for more details. Several adaptations of linear elastic fracture mechanics (LEFMs) have been proposed to consider the nonlinear behavior of quasi-brittle materials in an approximate manner.