Some features of flat fiberglass specimens’ behavior under three-point bending

. The authors carried out the experimental works devoted to the problem of describing the processes occurring under long-term loading of composite materials. They studied a glass-reinforced plastic (GP) flat specimen under three-point bending and assumed the deformations normal to the section are purely elastic, and the transverse shear deformation consists of two parts – elastic, and creep deformations and damages formed as a result of accumulation. The authors considered a variant of the creep ratios according to the hardening theory. In previous research authors determined its parameters based on the analysis of the tubular samples operation. It was found that even for thin beams, in contrast to a purely elastic problem, taking into account creep under transverse shear leads to inelastic deflection components that are comparable with their elastic values. The authors presented the results of numerical calculations. It is concluded that at large time and loads, deformations appear in the sample caused by the accumulation of damage.


Introduction
In this paper, the authors consider one of the special cases of the process of deformation of a flat rod made of fiberglass (the direction of reinforcement is along the sample), namely, the process of three-point kinematic bending. It turned out that under such conditions, some elements of structures made of such composites exhibit effects caused by creep, as well as deformations caused by damage accumulation. In previous research to construct deformation models and to determine the mechanical characteristics included in these models, the authors used a number of well-known and new hypotheses, approaches, and methods described, for example, in [3]. All these allowed to reduce the volume of field studies and to obtain good agreement between the results obtained by calculation and in experiments. The presented approach to the determination of mechanical characteristics is aimed, firstly, at simplifying the problem of their identification, and secondly, at reducing the variety and volume of experiments. It is demonstrated on the problem of determining the mechanical characteristics of these composite samples only for three-point bending; therefore, the problem of constructing physical relationships is considered in a formulation in which the two-dimensional distribution law of the deformed state in the bending plane can be found analytically.

Experimental studies
The authors studied a polymer composite material in the form of flat extended elements reinforced with fiberglass. Environments for research were air environment, distilled water and aqueous solutions of alkali. The first aqueous solution of alkali, simulating the liquid phase of concrete, adopted 8 g NaOH and 22.4 gr. KOH per 1 liter of distilled water (GOST 31938-2012), the pH value of the alkaline solution ranged from 12.6 to 13. To exclude interaction of air with CO 2 and evaporation, the alkaline solution was in a closed container. The concentration of the second aqueous solution of alkali is doubled to that described above.
The studied samples were deformed in the form of a three-hinged fastening. When carrying out intermediate measurements, the universal equipment (see Fig. 1) was disassembled and reassembled after the control measurements. The general view of the studied samples is shown in Fig.2. The test temperature is assumed to be 18±3 degrees. The samples were kept in the selected media (see Fig. 3) with intermediate measurements. The authors studied residual deflection (see Fig. 4) after the testing equipment was disassembled, and determined the load until the deflection returned to its original state of deformation (see Fig. 5). The measurements carried out with cathetometers, dial gauges and reference force meters.

Creep model using hardening theory
Further, only the case of a flat three-point beam bending is considered. To describe the creep process, we introduce the hypothesis of the presence of creep deformations only for transverse shear. This is confirmed by numerous, including the author's, experiments. We accept the Bernoulli hypothesis that the longitudinal strains ε are distributed over the thickness of the sample according to a linear law. Then the relations for linear deformation can be taken as: Here, el  is the linearly elastic part of the deformation, and 0 E is the modulus of elasticity. With a three-point bending, according to the Timoshenko theory, the following relation holds: where P=2 F is the load applied in the center of the specimen, , bh are its width and thickness.
For the transverse shear strain components, we introduce the following assumptions regarding their distribution over the beam height: Here el  is the linearly elastic part of the deformation, G is the shear modulus, k is the coefficient depending on the shape of the section and is equal to 6/5 in our case, creep  is the creep strain. In the case of an elastic problem, relations (2.2), (2.3) are hypotheses of the Timoshenko theory. For the case of a flat three-point bend, the deformation  in absolute value will be constant along the length as well. In the case of creep, it must be considered that this deformation increases with time, and also depends on both stresses and the level of these deformations. If the centrally applied load is permanent (dead load), then the shear stress is constant in absolute value and along the sample (the sign changes only at the central point), but also in time (since a three-point bend is considered). In the experiments proposed for consideration in this work, on the contrary, the deflection of the central section was set constant; therefore, the transverse shear deformation and shear stress change with time.
Next, we accept the following expression for the creep strain rate (one of the simplest versions of the flow theory [1,2] [2], which makes it possible to reduce the number of arguments in the functions used in these relations. Namely, it is accepted that the various components of the deformation do not affect each other and develop depending simply on the stresses and these accumulated deformations [3]. When using relations (2.4), even in the case when P it depends on time, the following expression can be obtained for the creep deformation (for one of the halves of the beam, for which 0 creep   ): For the case of three-point bending, the total elastic deflection at the center of the beam can be found from the formula: 3 3 , 48 4 12 The residual deflection caused by creep deformations is determined by the relation: From the test results, it is possible to obtain total and residual deflections, as well as a function that determines the value of the force P at different points of time.
Identification methods can find all the mechanical characteristics included in the physical relationships that were given above. This is the method of minimizing the quadratic residual of the calculated and experimental values of the total and residual deflections of samples of various lengths under various loads, Next, the authors carried out numerical calculations for three values of specified deflections in the center of the sample under kinematic loading. To verify the obtained models, the calculated and experimental data were compared for these given values of the deflection in the center of the sample. The results of the calculations were carried out at the values of the creep parameters obtained for the GP in earlier works [4 -6], but refined by minimizing the quadratic residuals of the numerical and experimental values of the displacements of the central section of the sample. The reactive force F(t) for each experiment was approximated by a hyperbolic function of the form Below are the results of calculations and experimental data in the problem of three-point bending of fiberglass specimens.
The geometric and mechanical characteristics were as follows. First, the authors considered samples loaded in air. An analysis of the experimental results showed that after 2.4 hours of exposure, a residual deflection of 2.4 mm appears in the sample, which is 9.6 % of the initial elastic one.
Below are the test results for samples placed in distilled water (see Fig.7, Fig.8). An analysis of the experimental data showed that by the time t=1.39 10 6 sec. (third point on the right in the upper graph of Fig. 7) the residual deflection is 2.95 mm, which is already 13 % of the elastic deflection, equal to 22.05 mm. In addition, Fig. 7 shows, that at high loads and large times (the right side of the upper graph), the calculated deflections of the central section, obtained from relations (2.5) -(2.9), are already less than the experimental ones. The residual experimental strains differ from the calculated creep strains by 12 %. This allows us to consider that deformations appear in the sample caused by the accumulation of damage. To find the ratio that describes the process of degradation of the material, it is necessary to conduct experiments at long times with a sufficient amount of experimental data. Since the law of creep is known, then at long times, with a sufficient amount of experimental data, it will be possible to determine the parameters of only those relationships that describe the process of damage accumulation.

Conclusion
The results of the experimental data analysis show that even for thin beams, in contrast to a purely elastic problem, creep accounting in transverse shear leads to deflections comparable to their full values. At the same time, the use of the Timoshenko theory and the assumption that only elastic deformations take place along the axis of the rod makes it possible to obtain the solution of the creep problem for the beam in an analytical form. It also follows from the analysis of experimental data that, at sufficiently large times and loads, deformations appear in the sample caused by the accumulation of damage. Its description requires additional experiments with a longer exposure under load.